Daily Papers

We propose Speech Enhancement based on Drifting Models (DriftSE), a novel generative framework that formulates denoising as an equilibrium problem. Rather than relying on iterative sampling, DriftSE natively achieves one-step inference by evolving the pushforward distribution of a mapping function to directly match the clean speech distribution. This evolution is driven by a Drifting Field, a learned correction vector that guides samples toward the high-density regions of the clean distribution, which naturally facilitates training on unpaired data by matching distributions rather than paired samples. We investigate the framework under two formulations: a direct mapping from the noisy observation, and a stochastic conditional generative model from a Gaussian prior. Experiments on the VoiceBank-DEMAND benchmark demonstrate that DriftSE achieves high-fidelity enhancement in a single step, outperforming multi-step diffusion baselines and establishing a new paradigm for speech enhancement.

Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which n component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter L. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is mathcal{O}( n + nLvarepsilon^{-1}), which improves upon existing methods by a factor up to n. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

Two-player, constant-sum games are well studied in the literature, but there has been limited progress outside of this setting. We propose Joint Policy-Space Response Oracles (JPSRO), an algorithm for training agents in n-player, general-sum extensive form games, which provably converges to an equilibrium. We further suggest correlated equilibria (CE) as promising meta-solvers, and propose a novel solution concept Maximum Gini Correlated Equilibrium (MGCE), a principled and computationally efficient family of solutions for solving the correlated equilibrium selection problem. We conduct several experiments using CE meta-solvers for JPSRO and demonstrate convergence on n-player, general-sum games.

We investigate the mechanism design problem faced by a principal who hires multiple agents to gather and report costly information. Then, the principal exploits the information to make an informed decision. We model this problem as a game, where the principal announces a mechanism consisting in action recommendations and a payment function, a.k.a. scoring rule. Then, each agent chooses an effort level and receives partial information about an underlying state of nature based on the effort. Finally, the agents report the information (possibly non-truthfully), the principal takes a decision based on this information, and the agents are paid according to the scoring rule. While previous work focuses on single-agent problems, we consider multi-agents settings. This poses the challenge of coordinating the agents' efforts and aggregating correlated information. Indeed, we show that optimal mechanisms must correlate agents' efforts, which introduces externalities among the agents, and hence complex incentive compatibility constraints and equilibrium selection problems. First, we design a polynomial-time algorithm to find an optimal incentive compatible mechanism. Then, we study an online problem, where the principal repeatedly interacts with a group of unknown agents. We design a no-regret algorithm that provides mathcal{O}(T^{2/3}) regret with respect to an optimal mechanism, matching the state-of-the-art bound for single-agent settings.

In this paper, we develop an approximation scheme for solving bilevel programs with equilibrium constraints, which are generally difficult to solve. Among other things, calculating the first-order derivative in such a problem requires differentiation across the hierarchy, which is computationally intensive, if not prohibitive. To bypass the hierarchy, we propose to bound such bilevel programs, equivalent to multiple-followers Stackelberg games, with two new hierarchy-free problems: a T-step Cournot game and a T-step monopoly model. Since they are standard equilibrium or optimization problems, both can be efficiently solved via first-order methods. Importantly, we show that the bounds provided by these problems -- the upper bound by the T-step Cournot game and the lower bound by the T-step monopoly model -- can be made arbitrarily tight by increasing the step parameter T for a wide range of problems. We prove that a small T usually suffices under appropriate conditions to reach an approximation acceptable for most practical purposes. Eventually, the analytical insights are highlighted through numerical examples.

We apply the tensor cross interpolation (TCI) algorithm to solve equilibrium quantum impurity problems with high precision based on the weak-coupling expansion. The TCI algorithm, a kind of active learning method, factorizes high-dimensional integrals that appear in the perturbative expansion into a product of low-dimensional ones, enabling us to evaluate higher-order terms efficiently. This method is free from the sign problem which quantum Monte Carlo methods sometimes suffer from, and allows one to directly calculate the free energy. We benchmark the TCI impurity solver on an exactly solvable impurity model, and find good agreement with the exact solutions. We also incorporate the TCI impurity solver into the dynamical mean-field theory to solve the Hubbard model, and show that the metal-to-Mott insulator transition is correctly described with comparable accuracy to the Monte Carlo methods. Behind the effectiveness of the TCI approach for quantum impurity problems lies the fact that the integrands in the weak-coupling expansion naturally have a low-rank structure in the tensor-train representation.

Deep Equilibrium Models (DEQs) and Neural Ordinary Differential Equations (Neural ODEs) are two branches of implicit models that have achieved remarkable success owing to their superior performance and low memory consumption. While both are implicit models, DEQs and Neural ODEs are derived from different mathematical formulations. Inspired by homotopy continuation, we establish a connection between these two models and illustrate that they are actually two sides of the same coin. Homotopy continuation is a classical method of solving nonlinear equations based on a corresponding ODE. Given this connection, we proposed a new implicit model called HomoODE that inherits the property of high accuracy from DEQs and the property of stability from Neural ODEs. Unlike DEQs, which explicitly solve an equilibrium-point-finding problem via Newton's methods in the forward pass, HomoODE solves the equilibrium-point-finding problem implicitly using a modified Neural ODE via homotopy continuation. Further, we developed an acceleration method for HomoODE with a shared learnable initial point. It is worth noting that our model also provides a better understanding of why Augmented Neural ODEs work as long as the augmented part is regarded as the equilibrium point to find. Comprehensive experiments with several image classification tasks demonstrate that HomoODE surpasses existing implicit models in terms of both accuracy and memory consumption.

Offline learning of strategies takes data efficiency to its extreme by restricting algorithms to a fixed dataset of state-action trajectories. We consider the problem in a mixed-motive multiagent setting, where the goal is to solve a game under the offline learning constraint. We first frame this problem in terms of selecting among candidate equilibria. Since datasets may inform only a small fraction of game dynamics, it is generally infeasible in offline game-solving to even verify a proposed solution is a true equilibrium. Therefore, we consider the relative probability of low regret (i.e., closeness to equilibrium) across candidates based on the information available. Specifically, we extend Policy Space Response Oracles (PSRO), an online game-solving approach, by quantifying game dynamics uncertainty and modifying the RL objective to skew towards solutions more likely to have low regret in the true game. We further propose a novel meta-strategy solver, tailored for the offline setting, to guide strategy exploration in PSRO. Our incorporation of Conservatism principles from Offline reinforcement learning approaches for strategy Exploration gives our approach its name: COffeE-PSRO. Experiments demonstrate COffeE-PSRO's ability to extract lower-regret solutions than state-of-the-art offline approaches and reveal relationships between algorithmic components empirical game fidelity, and overall performance.

Scalable sampling of molecular states in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann generators tackle this problem by pairing normalizing flows with importance sampling to obtain uncorrelated samples under the target distribution. In this paper, we extend the Boltzmann generator framework with two key contributions, denoting our framework Sequential Boltzmann Generators (SBG). The first is a highly efficient Transformer-based normalizing flow operating directly on all-atom Cartesian coordinates. In contrast to the equivariant continuous flows of prior methods, we leverage exactly invertible non-equivariant architectures which are highly efficient during both sample generation and likelihood evaluation. This efficiency unlocks more sophisticated inference strategies beyond standard importance sampling. In particular, we perform inference-time scaling of flow samples using a continuous-time variant of sequential Monte Carlo, in which flow samples are transported towards the target distribution with annealed Langevin dynamics. SBG achieves state-of-the-art performance w.r.t. all metrics on peptide systems, demonstrating the first equilibrium sampling in Cartesian coordinates of tri-, tetra- and hexa-peptides that were thus far intractable for prior Boltzmann generators.

We consider the problem of decentralized multi-agent reinforcement learning in Markov games. A fundamental question is whether there exist algorithms that, when adopted by all agents and run independently in a decentralized fashion, lead to no-regret for each player, analogous to celebrated convergence results in normal-form games. While recent work has shown that such algorithms exist for restricted settings (notably, when regret is defined with respect to deviations to Markovian policies), the question of whether independent no-regret learning can be achieved in the standard Markov game framework was open. We provide a decisive negative resolution this problem, both from a computational and statistical perspective. We show that: - Under the widely-believed assumption that PPAD-hard problems cannot be solved in polynomial time, there is no polynomial-time algorithm that attains no-regret in general-sum Markov games when executed independently by all players, even when the game is known to the algorithm designer and the number of players is a small constant. - When the game is unknown, no algorithm, regardless of computational efficiency, can achieve no-regret without observing a number of episodes that is exponential in the number of players. Perhaps surprisingly, our lower bounds hold even for seemingly easier setting in which all agents are controlled by a a centralized algorithm. They are proven via lower bounds for a simpler problem we refer to as SparseCCE, in which the goal is to compute a coarse correlated equilibrium that is sparse in the sense that it can be represented as a mixture of a small number of product policies. The crux of our approach is a novel application of aggregation techniques from online learning, whereby we show that any algorithm for the SparseCCE problem can be used to compute approximate Nash equilibria for non-zero sum normal-form games.

Equilibrium propagation (EP) is a compelling alternative to the backpropagation of error algorithm (BP) for computing gradients of neural networks on biological or analog neuromorphic substrates. Still, the algorithm requires weight symmetry and infinitesimal equilibrium perturbations, i.e., nudges, to estimate unbiased gradients efficiently. Both requirements are challenging to implement in physical systems. Yet, whether and how weight asymmetry affects its applicability is unknown because, in practice, it may be masked by biases introduced through the finite nudge. To address this question, we study generalized EP, which can be formulated without weight symmetry, and analytically isolate the two sources of bias. For complex-differentiable non-symmetric networks, we show that the finite nudge does not pose a problem, as exact derivatives can still be estimated via a Cauchy integral. In contrast, weight asymmetry introduces bias resulting in low task performance due to poor alignment of EP's neuronal error vectors compared to BP. To mitigate this issue, we present a new homeostatic objective that directly penalizes functional asymmetries of the Jacobian at the network's fixed point. This homeostatic objective dramatically improves the network's ability to solve complex tasks such as ImageNet 32x32. Our results lay the theoretical groundwork for studying and mitigating the adverse effects of imperfections of physical networks on learning algorithms that rely on the substrate's relaxation dynamics.

In this paper we study the out-of-equilibrium phase diagram of the quantum version of Derrida's Random Energy Model, which is the simplest model of mean-field spin glasses. We interpret its corresponding quantum dynamics in Fock space as a one-particle problem in very high dimension to which we apply different theoretical methods tailored for high-dimensional lattices: the Forward-Scattering Approximation, a mapping to the Rosenzweig-Porter model, and the cavity method. Our results indicate the existence of two transition lines and three distinct dynamical phases: a completely many-body localized phase at low energy, a fully ergodic phase at high energy, and a multifractal "bad metal" phase at intermediate energy. In the latter, eigenfunctions occupy a diverging volume, yet an exponentially vanishing fraction of the total Hilbert space. We discuss the limitations of our approximations and the relationship with previous studies.

Query-based object detectors directly decode image features into object instances with a set of learnable queries. These query vectors are progressively refined to stable meaningful representations through a sequence of decoder layers, and then used to directly predict object locations and categories with simple FFN heads. In this paper, we present a new query-based object detector (DEQDet) by designing a deep equilibrium decoder. Our DEQ decoder models the query vector refinement as the fixed point solving of an {implicit} layer and is equivalent to applying {infinite} steps of refinement. To be more specific to object decoding, we use a two-step unrolled equilibrium equation to explicitly capture the query vector refinement. Accordingly, we are able to incorporate refinement awareness into the DEQ training with the inexact gradient back-propagation (RAG). In addition, to stabilize the training of our DEQDet and improve its generalization ability, we devise the deep supervision scheme on the optimization path of DEQ with refinement-aware perturbation~(RAP). Our experiments demonstrate DEQDet converges faster, consumes less memory, and achieves better results than the baseline counterpart (AdaMixer). In particular, our DEQDet with ResNet50 backbone and 300 queries achieves the 49.5 mAP and 33.0 AP_s on the MS COCO benchmark under 2times training scheme (24 epochs).

Brain-like intelligent systems need brain-like learning methods. Equilibrium Propagation (EP) is a biologically plausible learning framework with strong potential for brain-inspired computing hardware. However, existing im-plementations of EP suffer from instability and prohibi-tively high computational costs. Inspired by the structure and dynamics of the brain, we propose a biologically plau-sible Feedback-regulated REsidual recurrent neural network (FRE-RNN) and study its learning performance in EP framework. Feedback regulation enables rapid convergence by reducing the spectral radius. The improvement in con-vergence property reduces the computational cost and train-ing time of EP by orders of magnitude, delivering perfor-mance on par with backpropagation (BP) in benchmark tasks. Meanwhile, residual connections with brain-inspired topologies help alleviate the vanishing gradient problem that arises when feedback pathways are weak in deep RNNs. Our approach substantially enhances the applicabil-ity and practicality of EP in large-scale networks that un-derpin artificial intelligence. The techniques developed here also offer guidance to implementing in-situ learning in physical neural networks.

We introduce an Implicit Game-Theoretic MPC (IGT-MPC), a decentralized algorithm for two-agent motion planning that uses a learned value function that predicts the game-theoretic interaction outcomes as the terminal cost-to-go function in a model predictive control (MPC) framework, guiding agents to implicitly account for interactions with other agents and maximize their reward. This approach applies to competitive and cooperative multi-agent motion planning problems which we formulate as constrained dynamic games. Given a constrained dynamic game, we randomly sample initial conditions and solve for the generalized Nash equilibrium (GNE) to generate a dataset of GNE solutions, computing the reward outcome of each game-theoretic interaction from the GNE. The data is used to train a simple neural network to predict the reward outcome, which we use as the terminal cost-to-go function in an MPC scheme. We showcase emerging competitive and coordinated behaviors using IGT-MPC in scenarios such as two-vehicle head-to-head racing and un-signalized intersection navigation. IGT-MPC offers a novel method integrating machine learning and game-theoretic reasoning into model-based decentralized multi-agent motion planning.

Observed dwarf galaxies tend to have linearly rising rotation curves, which indicate flat density cores in their centers. Furthermore, disk galaxies show a wide range of rotation curves shapes. High resolution simulations of cold collisionless dark matter do not reproduce flat central profiles, or the observed diversity of rotation curve shapes; even hydrodynamic simulations incorporating baryonic feedback cannot do that robustly. However, numerical simulations are not the only way to make predictions about density profiles of equilibrium dark matter halos. A theoretical model based on statistical mechanics shows that maximum entropy solutions for cold collisionless self-gravitating dark matter halos can have a range of inner density profiles, including flat density cores. These theoretical profiles, called DARKexp, have only one shape parameter, and are able to fit the observed rotation curves of galaxies with last measured velocities in the range ~20-200 km/s. Here we present fits to 96 SPARC catalog galaxies, and the Milky Way. DARKexp also provides good fits to the projected stellar density distributions of ultrafaint dwarfs that show cores, suggesting that the dark matter halo hosts could have flat density cores. Thus, DARKexp appears to be able to address the core-cusp problem and the diversity of rotation curves with cold collisionless dark matter alone, without baryonic feedback.

This study proposes a hybrid deep-learning-metaheuristic framework with a bi-level architecture for road network design problems (NDPs). We train a graph neural network (GNN) to approximate the solution of the user equilibrium (UE) traffic assignment problem and use inferences made by the trained model to calculate fitness function evaluations of a genetic algorithm (GA) to approximate solutions for NDPs. Using three test networks, two NDP variants and an exact solver as benchmark, we show that on average, our proposed framework can provide solutions within 1.5% gap of the best results in less than 0.5% of the time used by the exact solution procedure. Our framework can be utilized within an expert system for infrastructure planning to determine the best infrastructure planning and management decisions under different scenarios. Given the flexibility of the framework, it can easily be adapted to many other decision problems that can be modeled as bi-level problems on graphs. Moreover, we foreseen interesting future research directions, thus we also put forward a brief research agenda for this topic. The key observation from our research that can shape future research is that the fitness function evaluation time using the inferences made by the GNN model was in the order of milliseconds, which points to an opportunity and a need for novel heuristics that 1) can cope well with noisy fitness function values provided by deep learning models, and 2) can use the significantly enlarged efficiency of the evaluation step to explore the search space effectively (rather than efficiently). This opens a new avenue for a modern class of metaheuristics that are crafted for use with AI-powered predictors.

We introduce the use of generative adversarial learning to compute equilibria in general game-theoretic settings, specifically the generalized Nash equilibrium (GNE) in pseudo-games, and its specific instantiation as the competitive equilibrium (CE) in Arrow-Debreu competitive economies. Pseudo-games are a generalization of games in which players' actions affect not only the payoffs of other players but also their feasible action spaces. Although the computation of GNE and CE is intractable in the worst-case, i.e., PPAD-hard, in practice, many applications only require solutions with high accuracy in expectation over a distribution of problem instances. We introduce Generative Adversarial Equilibrium Solvers (GAES): a family of generative adversarial neural networks that can learn GNE and CE from only a sample of problem instances. We provide computational and sample complexity bounds, and apply the framework to finding Nash equilibria in normal-form games, CE in Arrow-Debreu competitive economies, and GNE in an environmental economic model of the Kyoto mechanism.

Binarization of neural networks is a dominant paradigm in neural networks compression. The pioneering work BinaryConnect uses Straight Through Estimator (STE) to mimic the gradients of the sign function, but it also causes the crucial inconsistency problem. Most of the previous methods design different estimators instead of STE to mitigate it. However, they ignore the fact that when reducing the estimating error, the gradient stability will decrease concomitantly. These highly divergent gradients will harm the model training and increase the risk of gradient vanishing and gradient exploding. To fully take the gradient stability into consideration, we present a new perspective to the BNNs training, regarding it as the equilibrium between the estimating error and the gradient stability. In this view, we firstly design two indicators to quantitatively demonstrate the equilibrium phenomenon. In addition, in order to balance the estimating error and the gradient stability well, we revise the original straight through estimator and propose a power function based estimator, Rectified Straight Through Estimator (ReSTE for short). Comparing to other estimators, ReSTE is rational and capable of flexibly balancing the estimating error with the gradient stability. Extensive experiments on CIFAR-10 and ImageNet datasets show that ReSTE has excellent performance and surpasses the state-of-the-art methods without any auxiliary modules or losses.

When applied to question answering and other text generation tasks, language models (LMs) may be queried generatively (by sampling answers from their output distribution) or discriminatively (by using them to score or rank a set of candidate outputs). These procedures sometimes yield very different predictions. How do we reconcile mutually incompatible scoring procedures to obtain coherent LM predictions? We introduce a new, a training-free, game-theoretic procedure for language model decoding. Our approach casts language model decoding as a regularized imperfect-information sequential signaling game - which we term the CONSENSUS GAME - in which a GENERATOR seeks to communicate an abstract correctness parameter using natural language sentences to a DISCRIMINATOR. We develop computational procedures for finding approximate equilibria of this game, resulting in a decoding algorithm we call EQUILIBRIUM-RANKING. Applied to a large number of tasks (including reading comprehension, commonsense reasoning, mathematical problem-solving, and dialog), EQUILIBRIUM-RANKING consistently, and sometimes substantially, improves performance over existing LM decoding procedures - on multiple benchmarks, we observe that applying EQUILIBRIUM-RANKING to LLaMA-7B outperforms the much larger LLaMA-65B and PaLM-540B models. These results highlight the promise of game-theoretic tools for addressing fundamental challenges of truthfulness and consistency in LMs.

Mathematical reasoning has proven to be a critical yet challenging task for large language models (LLMs), as they often struggle with complex multi-step problems. To address these limitations, we introduce the Monte Carlo Nash Equilibrium Self-Refine Tree (MC-NEST) algorithm, an enhancement of the Monte Carlo Tree Self-Refine (MCTSr) approach. By integrating Nash Equilibrium strategies with LLM-based self-refinement and self-evaluation processes, MC-NEST aims to improve decision-making for complex mathematical reasoning tasks. This method ensures balanced exploration and exploitation of potential solutions, leveraging Upper Confidence Bound (UCT) scores and various selection policies. Through iterative critique and refinement, MC-NEST enhances the reasoning capabilities of LLMs, particularly for problems requiring strategic decision-making. Comparative analysis reveals that GPT-4o, equipped with MC-NEST using an Importance Sampling Policy, achieved superior accuracy in domains such as Number Theory and Geometry. These results suggest that both LLMs GPT-4o and Phi-3-mini can benefit from MC-NEST, with iterative self-refinement proving especially effective in expanding the reasoning capacity and problem-solving performance of LLMs. We evaluate the effectiveness of MC-NEST on challenging Olympiad-level benchmarks, demonstrating its potential to significantly boost complex mathematical reasoning performance in LLMs.

We study the problem of learning a Nash equilibrium (NE) in an imperfect information game (IIG) through self-play. Precisely, we focus on two-player, zero-sum, episodic, tabular IIG under the perfect-recall assumption where the only feedback is realizations of the game (bandit feedback). In particular, the dynamic of the IIG is not known -- we can only access it by sampling or interacting with a game simulator. For this learning setting, we provide the Implicit Exploration Online Mirror Descent (IXOMD) algorithm. It is a model-free algorithm with a high-probability bound on the convergence rate to the NE of order 1/T where T is the number of played games. Moreover, IXOMD is computationally efficient as it needs to perform the updates only along the sampled trajectory.

Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many-query scenarios, e.g., uncertainty quantification, remeshing applications, topology optimization, and so forth. This limitation has motivated the application of data-driven surrogate models, where the microscale computations are substituted with a surrogate, usually acting as a black-box mapping between macroscale quantities. These models offer significant speedups but struggle with incorporating microscale physical constraints, such as the balance of linear momentum and constitutive models. In this contribution, we propose Equilibrium Neural Operator (EquiNO) as a complementary physics-informed PDE surrogate for predicting microscale physics and compare it with variational physics-informed neural and operator networks. Our framework, applicable to the so-called multiscale FE^{,2}, computations, introduces the FE-OL approach by integrating the finite element (FE) method with operator learning (OL). We apply the proposed FE-OL approach to quasi-static problems of solid mechanics. The results demonstrate that FE-OL can yield accurate solutions even when confronted with a restricted dataset during model development. Our results show that EquiNO achieves speedup factors exceeding 8000-fold compared to traditional methods and offers an optimal balance between data-driven and physics-based strategies.

Robustness against adversarial attacks and distribution shifts is a long-standing goal of Reinforcement Learning (RL). To this end, Robust Adversarial Reinforcement Learning (RARL) trains a protagonist against destabilizing forces exercised by an adversary in a competitive zero-sum Markov game, whose optimal solution, i.e., rational strategy, corresponds to a Nash equilibrium. However, finding Nash equilibria requires facing complex saddle point optimization problems, which can be prohibitive to solve, especially for high-dimensional control. In this paper, we propose a novel approach for adversarial RL based on entropy regularization to ease the complexity of the saddle point optimization problem. We show that the solution of this entropy-regularized problem corresponds to a Quantal Response Equilibrium (QRE), a generalization of Nash equilibria that accounts for bounded rationality, i.e., agents sometimes play random actions instead of optimal ones. Crucially, the connection between the entropy-regularized objective and QRE enables free modulation of the rationality of the agents by simply tuning the temperature coefficient. We leverage this insight to propose our novel algorithm, Quantal Adversarial RL (QARL), which gradually increases the rationality of the adversary in a curriculum fashion until it is fully rational, easing the complexity of the optimization problem while retaining robustness. We provide extensive evidence of QARL outperforming RARL and recent baselines across several MuJoCo locomotion and navigation problems in overall performance and robustness.

Randomized mechanisms can have good normative properties compared to their deterministic counterparts. However, randomized mechanisms are problematic in several ways such as in their verifiability. We propose here to derandomize such mechanisms by having agents play a game instead of tossing a coin. The game is designed so an agent's best action is to play randomly, and this play then injects ``randomness'' into the mechanism. This derandomization retains many of the good normative properties of the original randomized mechanism but gives a mechanism that is deterministic and easy, for instance, to audit. We consider three related methods to derandomize randomized mechanism in six different domains: voting, facility location, task allocation, school choice, peer selection, and resource allocation. We propose a number of novel derandomized mechanisms for these six domains with good normative properties. Each mechanism has a mixed Nash equilibrium in which agents play a modular arithmetic game with an uniform mixed strategy. In all but one mixed Nash equilibrium, agents report their preferences over the original problem sincerely. The derandomized methods are thus ``quasi-strategy proof''. In one domain, we additionally show that a new and desirable normative property emerges as a result of derandomization.

The statistical mechanics of Gibbs is a juxtaposition of subjective, probabilistic ideas on the one hand and objective, mechanical ideas on the other. In this paper, we follow the path set out by Jaynes, including elements added subsequently to that original work, to explore the consequences of the purely statistical point of view. We show how standard methods in the equilibrium theory could have been derived simply from a description of the available problem information. In addition, our presentation leads to novel insights into questions associated with symmetry and non-equilibrium statistical mechanics. Two surprising consequences to be explored in further work are that (in)distinguishability factors are automatically predicted from the problem formulation and that a quantity related to the thermodynamic entropy production is found by considering information loss in non-equilibrium processes. Using the problem of ion channel thermodynamics as an example, we illustrate the idea of building up complexity by successively adding information to create progressively more complex descriptions of a physical system. Our result is that such statistical mechanical descriptions can be used to create transparent, computable, experimentally-relevant models that may be informed by more detailed atomistic simulations. We also derive a theory for the kinetic behavior of this system, identifying the nonequilibrium `process' free energy functional. The Gibbs relation for this functional is a fluctuation-dissipation theorem applicable arbitrarily far from equilibrium, that captures the effect of non-local and time-dependent behavior from transient driving forces. Based on this work, it is clear that statistical mechanics is a general tool for constructing the relationships between constraints on system information.

We initiate the study of Multi-Agent Reinforcement Learning from Human Feedback (MARLHF), exploring both theoretical foundations and empirical validations. We define the task as identifying Nash equilibrium from a preference-only offline dataset in general-sum games, a problem marked by the challenge of sparse feedback signals. Our theory establishes the upper complexity bounds for Nash Equilibrium in effective MARLHF, demonstrating that single-policy coverage is inadequate and highlighting the importance of unilateral dataset coverage. These theoretical insights are verified through comprehensive experiments. To enhance the practical performance, we further introduce two algorithmic techniques. (1) We propose a Mean Squared Error (MSE) regularization along the time axis to achieve a more uniform reward distribution and improve reward learning outcomes. (2) We utilize imitation learning to approximate the reference policy, ensuring stability and effectiveness in training. Our findings underscore the multifaceted approach required for MARLHF, paving the way for effective preference-based multi-agent systems.

Recent unsupervised domain adaptation methods have utilized vicinal space between the source and target domains. However, the equilibrium collapse of labels, a problem where the source labels are dominant over the target labels in the predictions of vicinal instances, has never been addressed. In this paper, we propose an instance-wise minimax strategy that minimizes the entropy of high uncertainty instances in the vicinal space to tackle the stated problem. We divide the vicinal space into two subspaces through the solution of the minimax problem: contrastive space and consensus space. In the contrastive space, inter-domain discrepancy is mitigated by constraining instances to have contrastive views and labels, and the consensus space reduces the confusion between intra-domain categories. The effectiveness of our method is demonstrated on public benchmarks, including Office-31, Office-Home, and VisDA-C, achieving state-of-the-art performances. We further show that our method outperforms the current state-of-the-art methods on PACS, which indicates that our instance-wise approach works well for multi-source domain adaptation as well. Code is available at https://github.com/NaJaeMin92/CoVi.

Traditional Reinforcement Learning from Human Feedback (RLHF) often relies on reward models, frequently assuming preference structures like the Bradley-Terry model, which may not accurately capture the complexities of real human preferences (e.g., intransitivity). Nash Learning from Human Feedback (NLHF) offers a more direct alternative by framing the problem as finding a Nash equilibrium of a game defined by these preferences. In this work, we introduce Nash Mirror Prox (Nash-MP), an online NLHF algorithm that leverages the Mirror Prox optimization scheme to achieve fast and stable convergence to the Nash equilibrium. Our theoretical analysis establishes that Nash-MP exhibits last-iterate linear convergence towards the beta-regularized Nash equilibrium. Specifically, we prove that the KL-divergence to the optimal policy decreases at a rate of order (1+2beta)^{-N/2}, where N is a number of preference queries. We further demonstrate last-iterate linear convergence for the exploitability gap and uniformly for the span semi-norm of log-probabilities, with all these rates being independent of the size of the action space. Furthermore, we propose and analyze an approximate version of Nash-MP where proximal steps are estimated using stochastic policy gradients, making the algorithm closer to applications. Finally, we detail a practical implementation strategy for fine-tuning large language models and present experiments that demonstrate its competitive performance and compatibility with existing methods.

Optimization is a ubiquitous modeling tool and is often deployed in settings which repeatedly solve similar instances of the same problem. Amortized optimization methods use learning to predict the solutions to problems in these settings, exploiting the shared structure between similar problem instances. These methods have been crucial in variational inference and reinforcement learning and are capable of solving optimization problems many orders of magnitudes times faster than traditional optimization methods that do not use amortization. This tutorial presents an introduction to the amortized optimization foundations behind these advancements and overviews their applications in variational inference, sparse coding, gradient-based meta-learning, control, reinforcement learning, convex optimization, optimal transport, and deep equilibrium networks. The source code for this tutorial is available at https://github.com/facebookresearch/amortized-optimization-tutorial.

We study multi-agent reinforcement learning (MARL) for the general-sum Markov Games (MGs) under the general function approximation. In order to find the minimum assumption for sample-efficient learning, we introduce a novel complexity measure called the Multi-Agent Decoupling Coefficient (MADC) for general-sum MGs. Using this measure, we propose the first unified algorithmic framework that ensures sample efficiency in learning Nash Equilibrium, Coarse Correlated Equilibrium, and Correlated Equilibrium for both model-based and model-free MARL problems with low MADC. We also show that our algorithm provides comparable sublinear regret to the existing works. Moreover, our algorithm combines an equilibrium-solving oracle with a single objective optimization subprocedure that solves for the regularized payoff of each deterministic joint policy, which avoids solving constrained optimization problems within data-dependent constraints (Jin et al. 2020; Wang et al. 2023) or executing sampling procedures with complex multi-objective optimization problems (Foster et al. 2023), thus being more amenable to empirical implementation.

Traditional reinforcement learning from human feedback (RLHF) approaches relying on parametric models like the Bradley-Terry model fall short in capturing the intransitivity and irrationality in human preferences. Recent advancements suggest that directly working with preference probabilities can yield a more accurate reflection of human preferences, enabling more flexible and accurate language model alignment. In this paper, we propose a self-play-based method for language model alignment, which treats the problem as a constant-sum two-player game aimed at identifying the Nash equilibrium policy. Our approach, dubbed Self-Play Preference Optimization (SPPO), approximates the Nash equilibrium through iterative policy updates and enjoys theoretical convergence guarantee. Our method can effectively increase the log-likelihood of the chosen response and decrease that of the rejected response, which cannot be trivially achieved by symmetric pairwise loss such as Direct Preference Optimization (DPO) and Identity Preference Optimization (IPO). In our experiments, using only 60k prompts (without responses) from the UltraFeedback dataset and without any prompt augmentation, by leveraging a pre-trained preference model PairRM with only 0.4B parameters, SPPO can obtain a model from fine-tuning Mistral-7B-Instruct-v0.2 that achieves the state-of-the-art length-controlled win-rate of 28.53% against GPT-4-Turbo on AlpacaEval 2.0. It also outperforms the (iterative) DPO and IPO on MT-Bench and the Open LLM Leaderboard. Notably, the strong performance of SPPO is achieved without additional external supervision (e.g., responses, preferences, etc.) from GPT-4 or other stronger language models.

Reinforcement learning from human feedback (RLHF) has become essential for improving language model capabilities, but traditional approaches rely on the assumption that human preferences follow a transitive Bradley-Terry model. This assumption fails to capture the non-transitive nature of populational human preferences. Nash learning from human feedback (NLHF), targeting non-transitive preferences, is a problem of computing the Nash equilibrium (NE) of the two-player constant-sum game defined by the human preference. We introduce Extragradient preference optimization (EGPO), a novel algorithm for NLHF achieving last-iterate linear convergence to the NE of KL-regularized games and polynomial convergence to the NE of original games, while being robust to noise. Unlike previous approaches that rely on nested optimization, we derive an equivalent implementation using gradients of an online variant of the identity preference optimization (IPO) loss, enabling more faithful implementation for neural networks. Our empirical evaluations demonstrate EGPO's superior performance over baseline methods when training for the same number of epochs, as measured by pairwise win-rates using the ground truth preference. These results validate both the theoretical strengths and practical advantages of EGPO for language model alignment with non-transitive human preferences.

This work builds on the theoretical frameworks presented in "Liquidity pools as mean field games: A new framework" and "Liquidity pools as mean field games with transaction costs" by the same author, where the strategic interactions among traders in a constant-product market-making protocol were modelled using mean field games (MFG), first without transaction costs and then incorporating them. Here we present the formulation of a more complete model that integrates three types of agents: traders, liquidity providers (LPs), and arbitrageurs. While we do not establish existence results for this general model, the formulation identifies the main technical difficulties and lays the groundwork for future work. The LP acts as a dominating player in the sense of 'Mean field games with a dominating player' by Bensoussan, Chau, and Yam: its strategy influences the mean field distribution of the traders, and the equilibrium is sought as a solution to the coupled system of three problems that constitute a Major-Minor game. The arbitrageurs operate by solving the optimization problem presented in "An analysis of uniswap markets" by Angeris et al., and their impact on the LP is captured through the loss-versus-rebalancing of "Automated market making and loss-versus-rebalancing" by Milionis et al. The material in this article should be read as an open research proposal rather than a collection of closed results.

We introduce Equilibrium Matching (EqM), a generative modeling framework built from an equilibrium dynamics perspective. EqM discards the non-equilibrium, time-conditional dynamics in traditional diffusion and flow-based generative models and instead learns the equilibrium gradient of an implicit energy landscape. Through this approach, we can adopt an optimization-based sampling process at inference time, where samples are obtained by gradient descent on the learned landscape with adjustable step sizes, adaptive optimizers, and adaptive compute. EqM surpasses the generation performance of diffusion/flow models empirically, achieving an FID of 1.90 on ImageNet 256times256. EqM is also theoretically justified to learn and sample from the data manifold. Beyond generation, EqM is a flexible framework that naturally handles tasks including partially noised image denoising, OOD detection, and image composition. By replacing time-conditional velocities with a unified equilibrium landscape, EqM offers a tighter bridge between flow and energy-based models and a simple route to optimization-driven inference.

The computational study of equilibria involving constraints on players' strategies has been largely neglected. However, in real-world applications, players are usually subject to constraints ruling out the feasibility of some of their strategies, such as, e.g., safety requirements and budget caps. Computational studies on constrained versions of the Nash equilibrium have lead to some results under very stringent assumptions, while finding constrained versions of the correlated equilibrium (CE) is still unexplored. In this paper, we introduce and computationally characterize constrained Phi-equilibria -- a more general notion than constrained CEs -- in normal-form games. We show that computing such equilibria is in general computationally intractable, and also that the set of the equilibria may not be convex, providing a sharp divide with unconstrained CEs. Nevertheless, we provide a polynomial-time algorithm for computing a constrained (approximate) Phi-equilibrium maximizing a given linear function, when either the number of constraints or that of players' actions is fixed. Moreover, in the special case in which a player's constraints do not depend on other players' strategies, we show that an exact, function-maximizing equilibrium can be computed in polynomial time, while one (approximate) equilibrium can be found with an efficient decentralized no-regret learning algorithm.

We introduce Equilibrium Propagation, a learning framework for energy-based models. It involves only one kind of neural computation, performed in both the first phase (when the prediction is made) and the second phase of training (after the target or prediction error is revealed). Although this algorithm computes the gradient of an objective function just like Backpropagation, it does not need a special computation or circuit for the second phase, where errors are implicitly propagated. Equilibrium Propagation shares similarities with Contrastive Hebbian Learning and Contrastive Divergence while solving the theoretical issues of both algorithms: our algorithm computes the gradient of a well defined objective function. Because the objective function is defined in terms of local perturbations, the second phase of Equilibrium Propagation corresponds to only nudging the prediction (fixed point, or stationary distribution) towards a configuration that reduces prediction error. In the case of a recurrent multi-layer supervised network, the output units are slightly nudged towards their target in the second phase, and the perturbation introduced at the output layer propagates backward in the hidden layers. We show that the signal 'back-propagated' during this second phase corresponds to the propagation of error derivatives and encodes the gradient of the objective function, when the synaptic update corresponds to a standard form of spike-timing dependent plasticity. This work makes it more plausible that a mechanism similar to Backpropagation could be implemented by brains, since leaky integrator neural computation performs both inference and error back-propagation in our model. The only local difference between the two phases is whether synaptic changes are allowed or not.

A quantum Cournot game of which classical form game has multiple Nash equilibria is examined. Although the classical equilibria fail to be Pareto optimal, the quantum equilibrium exhibits the following two properties, (i) if the measurement of entanglement between strategic variables chosen by the competing firms is sufficiently large, the multiplicity of equilibria vanishes, and, (ii) the more strongly the strategic variables are entangled, the more closely the unique equilibrium approaches to the optimal one.

Both classical and quantum version of two models of price competition in duopoly market, the one is realistic and the other is idealized, are investigated. The pure strategy Nash equilibria of the realistic model exists under stricter condition than that of the idealized one in the classical form game. This is the problem known as Edgeworth paradox in economics. In the quantum form game, however, the former converges to the latter as the measure of entanglement goes to infinity.

The paper sets forth comprehensive basics of Discrete Thermodynamics of Chemical Equilibria (DTD), developed by the author during the last decade and spread over series of publications. Based on the linear equations of irreversible thermodynamics, De Donder's definition of the thermodynamic force, and the Le Chatelier principle, DTD brings forward a notion of chemical equilibrium as a balance of internal and external thermodynamic forces, acting against a chemical system. The basic expression of DTD is a logistic map that ties together energetic characteristics of the chemical transformation in the system, its deviation from true thermodynamic equilibrium, and the sum of thermodynamic forces, causing that deviation. System deviation from thermodynamic equilibrium is the major variable of the theory. Solutions to the basic map define the chemical system domain of states comprising bifurcation diagrams with four areas, from true thermodynamic equilibrium to chaos, having specific distinctive meaning for chemical systems. The theory is derived from the currently recognized ideas of chemical thermodynamics and binds classical and contemporary thermodynamics of chemical equilibria into a unique concept. DTD opens new opportunities in understanding and analysis of equilibria in chemical systems. Some new results, included in the paper, have never been published before.

Recent work by Baratin et al. (2021) sheds light on an intriguing pattern that occurs during the training of deep neural networks: some layers align much more with data compared to other layers (where the alignment is defined as the euclidean product of the tangent features matrix and the data labels matrix). The curve of the alignment as a function of layer index (generally) exhibits an ascent-descent pattern where the maximum is reached for some hidden layer. In this work, we provide the first explanation for this phenomenon. We introduce the Equilibrium Hypothesis which connects this alignment pattern to signal propagation in deep neural networks. Our experiments demonstrate an excellent match with the theoretical predictions.

Recently, remarkable progress has been made by approximating Nash equilibrium (NE), correlated equilibrium (CE), and coarse correlated equilibrium (CCE) through function approximation that trains a neural network to predict equilibria from game representations. Furthermore, equivariant architectures are widely adopted in designing such equilibrium approximators in normal-form games. In this paper, we theoretically characterize benefits and limitations of equivariant equilibrium approximators. For the benefits, we show that they enjoy better generalizability than general ones and can achieve better approximations when the payoff distribution is permutation-invariant. For the limitations, we discuss their drawbacks in terms of equilibrium selection and social welfare. Together, our results help to understand the role of equivariance in equilibrium approximators.

The principle of detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process. For many real physico-chemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and irreversible reactions. In this case, the principle of detailed balance cannot be applied directly. We represent irreversible reactions as limits of reversible steps and obtain the principle of detailed balance for complex mechanisms with some irreversible elementary processes. We proved two consequences of the detailed balance for these mechanisms: the structural condition and the algebraic condition that form together the extended form of detailed balance. The algebraic condition is the principle of detailed balance for the reversible part. The structural condition is: the convex hull of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span of the stoichiometric vectors of the reversible reaction. Physically, this means that the irreversible reactions cannot be included in oriented pathways. The systems with the extended form of detailed balance are also the limits of the reversible systems with detailed balance when some of the equilibrium concentrations (or activities) tend to zero. Surprisingly, the structure of the limit reaction mechanism crucially depends on the relative speeds of this tendency to zero.

We investigate learning the equilibria in non-stationary multi-agent systems and address the challenges that differentiate multi-agent learning from single-agent learning. Specifically, we focus on games with bandit feedback, where testing an equilibrium can result in substantial regret even when the gap to be tested is small, and the existence of multiple optimal solutions (equilibria) in stationary games poses extra challenges. To overcome these obstacles, we propose a versatile black-box approach applicable to a broad spectrum of problems, such as general-sum games, potential games, and Markov games, when equipped with appropriate learning and testing oracles for stationary environments. Our algorithms can achieve Oleft(Delta^{1/4}T^{3/4}right) regret when the degree of nonstationarity, as measured by total variation Delta, is known, and Oleft(Delta^{1/5}T^{4/5}right) regret when Delta is unknown, where T is the number of rounds. Meanwhile, our algorithm inherits the favorable dependence on number of agents from the oracles. As a side contribution that may be independent of interest, we show how to test for various types of equilibria by a black-box reduction to single-agent learning, which includes Nash equilibria, correlated equilibria, and coarse correlated equilibria.

This script offers an implementation-oriented introduction to deep learning methods for solving and estimating high-dimensional dynamic stochastic models in economics and finance. Its starting point is the curse of dimensionality: heterogeneous-agent economies, overlapping-generations models with aggregate risk, continuous-time models with occasionally binding constraints, climate-economy models, and macro-finance environments with many assets and frictions generate state and parameter spaces that strain classical tensor-product grid methods. The exposition is organized around four complementary methodologies. Deep Equilibrium Nets embed discrete-time equilibrium conditions into neural-network loss functions. Physics-Informed Neural Networks approximate continuous-time Hamilton--Jacobi--Bellman, Kolmogorov forward, and related partial differential equations. Deep surrogate models provide fast, differentiable approximations to expensive structural models, while Gaussian processes add a probabilistic layer that quantifies approximation uncertainty; together they support estimation, sensitivity analysis, and constrained policy design. Gaussian-process-based dynamic programming, combined with active learning and dimension reduction, extends value-function iteration to very large continuous state spaces. Applications span representative-agent and international real business cycle models, overlapping-generations and heterogeneous-agent economies, continuous-time macro-finance, structural estimation by simulated method of moments, and climate economics under uncertainty. Companion notebooks in TensorFlow and PyTorch invite hands-on experimentation. These notes are a deliberately subjective and inevitably incomplete snapshot of a rapidly evolving field, aimed at equipping PhD students and researchers to engage with this frontier hands-on.

We continue study of equilibrium of two species of 2d coulomb charges (or point vortices in 2d ideal fluid) started in Lv. Although for two species of vortices with circulation ratio -1 the relationship between the equilibria and the factorization/Darboux transformation of the Schrodinger operator was established a long ago, the question about similar relationship for the ratio -2 remained unanswered. Here we present the answer.

We prove stability for a class of heterogeneous catalysis models in the L_p-setting. We consider a setting in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. Under a reasonable condition on the involved parameters, we show that given equilibria are normally stable, i.e. solutions are attracted at an exponential rate. The potential incidence of instability is discussed as well.

In order for agents in multi-agent systems (MAS) to be safe, they need to take into account the risks posed by the actions of other agents. However, the dominant paradigm in game theory (GT) assumes that agents are not affected by risk from other agents and only strive to maximise their expected utility. For example, in hybrid human-AI driving systems, it is necessary to limit large deviations in reward resulting from car crashes. Although there are equilibrium concepts in game theory that take into account risk aversion, they either assume that agents are risk-neutral with respect to the uncertainty caused by the actions of other agents, or they are not guaranteed to exist. We introduce a new GT-based Risk-Averse Equilibrium (RAE) that always produces a solution that minimises the potential variance in reward accounting for the strategy of other agents. Theoretically and empirically, we show RAE shares many properties with a Nash Equilibrium (NE), establishing convergence properties and generalising to risk-dominant NE in certain cases. To tackle large-scale problems, we extend RAE to the PSRO multi-agent reinforcement learning (MARL) framework. We empirically demonstrate the minimum reward variance benefits of RAE in matrix games with high-risk outcomes. Results on MARL experiments show RAE generalises to risk-dominant NE in a trust dilemma game and that it reduces instances of crashing by 7x in an autonomous driving setting versus the best performing baseline.

We consider the problem of locating a facility to serve a set of agents located along a line. The Nash welfare objective function, defined as the product of the agents' utilities, is known to provide a compromise between fairness and efficiency in resource allocation problems. We apply this welfare notion to the facility location problem, converting individual costs to utilities and analyzing the facility placement that maximizes the Nash welfare. We give a polynomial-time approximation algorithm to compute this facility location, and prove results suggesting that it achieves a good balance of fairness and efficiency. Finally, we take a mechanism design perspective and propose a strategy-proof mechanism with a bounded approximation ratio for Nash welfare.

We establish an existence of equilibrium result for a class of non-Markovian mean-field games with unbounded control space in weak formulation. Our result is based on new existence and stability results for quadratic-growth generalized McKean-Vlasov BSDEs. Unlike earlier approaches, our approach does not require boundedness assumptions on the model parameters or time horizons and allows for running costs that are quadratic in the control variable.

We consider the obnoxious facility location problem (in which agents prefer the facility location to be far from them) and propose a hierarchy of distance-based proportional fairness concepts for the problem. These fairness axioms ensure that groups of agents at the same location are guaranteed to be a distance from the facility proportional to their group size. We consider deterministic and randomized mechanisms, and compute tight bounds on the price of proportional fairness. In the deterministic setting, we show that our proportional fairness axioms are incompatible with strategyproofness, and prove asymptotically tight epsilon-price of anarchy and stability bounds for proportionally fair welfare-optimal mechanisms. In the randomized setting, we identify proportionally fair and strategyproof mechanisms that give an expected welfare within a constant factor of the optimal welfare. Finally, we prove existence results for two extensions to our model.

In their seminal work, Nayyar et al. (2013) showed that imperfect information can be abstracted away from common-payoff games by having players publicly announce their policies as they play. This insight underpins sound solvers and decision-time planning algorithms for common-payoff games. Unfortunately, a naive application of the same insight to two-player zero-sum games fails because Nash equilibria of the game with public policy announcements may not correspond to Nash equilibria of the original game. As a consequence, existing sound decision-time planning algorithms require complicated additional mechanisms that have unappealing properties. The main contribution of this work is showing that certain regularized equilibria do not possess the aforementioned non-correspondence problem -- thus, computing them can be treated as perfect-information problems. Because these regularized equilibria can be made arbitrarily close to Nash equilibria, our result opens the door to a new perspective to solving two-player zero-sum games and yields a simplified framework for decision-time planning in two-player zero-sum games, void of the unappealing properties that plague existing decision-time planning approaches.

Performative Reinforcement Learning (PRL) refers to a scenario in which the deployed policy changes the reward and transition dynamics of the underlying environment. In this work, we study multi-agent PRL by incorporating performative effects into Markov Potential Games (MPGs). We introduce the notion of a performatively stable equilibrium (PSE) and show that it always exists under a reasonable sensitivity assumption. We then provide convergence results for state-of-the-art algorithms used to solve MPGs. Specifically, we show that independent policy gradient ascent (IPGA) and independent natural policy gradient (INPG) converge to an approximate PSE in the best-iterate sense, with an additional term that accounts for the performative effects. Furthermore, we show that INPG asymptotically converges to a PSE in the last-iterate sense. As the performative effects vanish, we recover the convergence rates from prior work. For a special case of our game, we provide finite-time last-iterate convergence results for a repeated retraining approach, in which agents independently optimize a surrogate objective. We conduct extensive experiments to validate our theoretical findings.

We study offline multi-agent reinforcement learning (RL) in Markov games, where the goal is to learn an approximate equilibrium -- such as Nash equilibrium and (Coarse) Correlated Equilibrium -- from an offline dataset pre-collected from the game. Existing works consider relatively restricted tabular or linear models and handle each equilibria separately. In this work, we provide the first framework for sample-efficient offline learning in Markov games under general function approximation, handling all 3 equilibria in a unified manner. By using Bellman-consistent pessimism, we obtain interval estimation for policies' returns, and use both the upper and the lower bounds to obtain a relaxation on the gap of a candidate policy, which becomes our optimization objective. Our results generalize prior works and provide several additional insights. Importantly, we require a data coverage condition that improves over the recently proposed "unilateral concentrability". Our condition allows selective coverage of deviation policies that optimally trade-off between their greediness (as approximate best responses) and coverage, and we show scenarios where this leads to significantly better guarantees. As a new connection, we also show how our algorithmic framework can subsume seemingly different solution concepts designed for the special case of two-player zero-sum games.

We study a fair allocation problem of indivisible items under additive externalities in which each agent also receives values from items that are assigned to other agents. We propose several new fairness concepts. We extend the well-studied envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) to this setting, and we propose a new fairness concept called general fair share (GFS). We undertake a detailed study and present algorithms for finding fair allocations.

During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

Understanding and analyzing markets is crucial, yet analytical equilibrium solutions remain largely infeasible. Recent breakthroughs in equilibrium computation rely on zeroth-order policy gradient estimation. These approaches commonly suffer from high variance and are computationally expensive. The use of fully differentiable simulators would enable more efficient gradient estimation. However, the discrete allocation of goods in economic simulations is a non-differentiable operation. This renders the first-order Monte Carlo gradient estimator inapplicable and the learning feedback systematically misleading. We propose a novel smoothing technique that creates a surrogate market game, in which first-order methods can be applied. We provide theoretical bounds on the resulting bias which justifies solving the smoothed game instead. These bounds also allow choosing the smoothing strength a priori such that the resulting estimate has low variance. Furthermore, we validate our approach via numerous empirical experiments. Our method theoretically and empirically outperforms zeroth-order methods in approximation quality and computational efficiency.

In the endeavor to make autonomous robots take actions, task planning is a major challenge that requires translating high-level task descriptions into long-horizon action sequences. Despite recent advances in language model agents, they remain prone to planning errors and limited in their ability to plan ahead. To address these limitations in robotic planning, we advocate a self-refining scheme that iteratively refines a draft plan until an equilibrium is reached. Remarkably, this process can be optimized end-to-end from an analytical perspective without the need to curate additional verifiers or reward models, allowing us to train self-refining planners in a simple supervised learning fashion. Meanwhile, a nested equilibrium sequence modeling procedure is devised for efficient closed-loop planning that incorporates useful feedback from the environment (or an internal world model). Our method is evaluated on the VirtualHome-Env benchmark, showing advanced performance with better scaling for inference computation. Code is available at https://github.com/Singularity0104/equilibrium-planner.

Boltzmann generators approach the sampling problem in many-body physics by combining a normalizing flow and a statistical reweighting method to generate samples in thermodynamic equilibrium. The equilibrium distribution is usually defined by an energy function and a thermodynamic state. Here we propose temperature-steerable flows (TSF) which are able to generate a family of probability densities parametrized by a choosable temperature parameter. TSFs can be embedded in generalized ensemble sampling frameworks to sample a physical system across multiple thermodynamic states.

We study a discrete model for generalized exchange-driven growth in which the particle exchanged between two clusters is not limited to be of size one. This set of models include as special cases the usual exchange-driven growth system and the coagulation-fragmentation system with binary fragmentation. Under reasonable general condition on the rate coefficients we establish the existence of admissible solutions, meaning solutions that are obtained as appropriate limit of solutions to a finite-dimensional truncation of the infinite-dimensional ODE. For these solutions we prove that, in the class of models we call isolated both the total number of particles and the total mass are conserved, whereas in those models we can non-isolated only the mass is conserved. Additionally, under more restrictive growth conditions for the rate equations we obtain uniqueness of solutions to the initial value problems.

We consider a fair division model in which agents have positive, zero and negative utilities for items. For this model, we analyse one existing fairness property - EFX - and three new and related properties - EFX_0, EFX^3 and EF1^3 - in combination with Pareto-optimality. With general utilities, we give a modified version of an existing algorithm for computing an EF1^3 allocation. With -alpha/0/alpha utilities, this algorithm returns an EFX^3 and PO allocation. With absolute identical utilities, we give a new algorithm for an EFX and PO allocation. With -alpha/0/beta utilities, this algorithm also returns such an allocation. We report some new impossibility results as well.

For a two-player imperfect-information extensive-form game (IIEFG) with K time steps and a player action space of size U, the game tree complexity is U^{2K}, causing existing IIEFG solvers to struggle with large or infinite (U,K), e.g., differential games with continuous action spaces. To partially address this scalability challenge, we focus on an important class of 2p0s games where the informed player (P1) knows the payoff while the uninformed player (P2) only has a belief over the set of I possible payoffs. Such games encompass a wide range of scenarios in sports, defense, cybersecurity, and finance. We prove that under mild conditions, P1's (resp. P2's) equilibrium strategy at any infostate concentrates on at most I (resp. I+1) action prototypes. When Ill U, this equilibrium structure causes the game tree complexity to collapse to I^K for P1 when P2 plays pure best responses, and (I+1)^K for P2 in a dual game where P1 plays pure best responses. We then show that exploiting this structure in standard learning modes, i.e., model-free multiagent reinforcement learning and model predictive control, is straightforward, leading to significant improvements in learning accuracy and efficiency from SOTA IIEFG solvers. Our demonstration solves a 22-player football game (K=10, U=infty) where the attacking team has to strategically conceal their intention until a critical moment in order to exploit information advantage. Code is available at https://github.com/ghimiremukesh/cams/tree/iclr

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under Hölder metric subregularity of the function defining the equation and Hölder continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the Łojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the Łojasiewicz gradient inequality assumption.

This paper generalises the treatment of compositional game theory as introduced by the second and third authors with Ghani and Winschel, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the existing notion of an open game is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issue all at once. To achieve this we make significant use of category theory, especially the 'coend optics' of Riley.

A theoretical approach to describing transport of an entire ensemble of clusters with different sizes as a single species in gas has been developed. The major assumption is an existence of local partial chemical equilibrium between the clusters. It is shown that thermal diffusion emerges in the collective description as a significant factor even if it is negligible when transport of the original molecular species is considered. Analytical expressions for the effective diffusion and thermal diffusion coefficients at temperature, pressure, and chemical composition gradients have been derived. The theory has been applied to a technology of H2S conversion in a centrifugal plasma-chemical reactor and has made it possible to account for sulfur clusters in numerical process modeling.

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

We improve the framework of open games with agency by showing how the players' counterfactual analysis giving rise to Nash equilibria can be described in the dynamics of the game itself (hence diegetically), getting rid of devices such as equilibrium predicates. This new approach overlaps almost completely with the way gradient-based learners are specified and trained. Indeed, we show feedback propagation in games can be seen as a form of backpropagation, with a crucial difference explaining the distinctive character of the phenomenology of non-cooperative games. We outline a functorial construction of arena of games, show players form a subsystem over it, and prove that their 'fixpoint behaviours' are Nash equilibria.

This paper extends the creep tide theory to exoplanetary systems with significant obliquities. The extended theory allows us to obtain the stellar and planetary hydrodynamic equilibrium tides and the evolution of the rotational state of the bodies. The dynamic ellipsoidal figure of equilibrium of the body is calculated taking into account that its reaction to external forces is delayed by its viscosity. The derived equations are used to determine the motion of the tidal bulge of the planetary companion CoRoT-3b (a brown dwarf) and its host star. We show how the tides deform the figure of the companion and how its tidal bulge moves close to the substellar meridian from one hemisphere to another. The stellar lag is mostly positive and is braking the star's rotation.

In this work, we explore two classes of density dependent relativistic mean-field models, their predictions of proton fractions at high densities and neutron star structure. We have used a metamodelling approach to these relativistic density functionals. We have generated a large ensemble of models with these classes and then applied constraints from theoretical and experimental nuclear physics and astrophysical observations. We find that both models produce similar equations of state and neutron star mass-radius sequences. But, their underlying compositions, denoted by the proton fraction in this case, are vastly different. This reinstates previous findings that information on composition gets masqueraded in beta-equilibrium. Additional observations of non-equilibrium phenomena are necessary to pin it down.

Common-pool resource governance requires users to cooperate and avoid overexploitation, but defection and free-riding often undermine cooperation. We model a human-environmental system that integrates dynamics of resource and users' strategies. The resource follows a logistic function that depends on natural growth rate, carrying capacity, and extraction rates of cooperators and defectors. The users' strategies evolve according to different processes that capture effects of payoff, resource, and noise. We analyze the feedback between resource availability and strategic adaptation, and explores the conditions for the emergence and maintenance of cooperation. We find different processes lead to different regimes of equilibrium solutions and resource levels depending on the parameter configuration and initial conditions. We also show that some processes can enhance the sustainability of the resource by making the users more responsive to the resource scarcity. The paper advances the understanding of human-environmental system and offers insights for resource governance policies and interventions.

In this paper, a mathematical model is developed to describe the evolution of the concentration of compounds through a gas chromatography column. The model couples mass balances and kinetic equations for all components. Both single and multiple-component cases are considered with constant or variable velocity. Non-dimensionalisation indicates the small effect of diffusion. The system where diffusion is neglected is analysed using Laplace transforms. In the multiple-component case, it is demonstrated that the competition between the compounds is negligible and the equations may be decoupled. This reduces the problem to solving a single integral equation to determine the concentration profile for all components (since they are scaled versions of each other). For a given analyte, we then only two parameters need to be fitted to the data. To verify this approach, the full governing equations are also solved numerically using the finite difference method and a global adaptive quadrature method to integrate the Laplace transformation. Comparison with the Laplace solution verifies the high degree of accuracy of the simpler Laplace form. The Laplace solution is then verified against experimental data from BTEX chromatography. This novel method, which involves solving a single equation and fitting parameters in pairs for individual components, is highly efficient. It is significantly faster and simpler than the full numerical solution and avoids the computationally expensive methods that would normally be used to fit all curves at the same time.

Inference-time scaling has emerged as a critical avenue for enhancing Large Language Models' performance, yet real-world deployment is constrained by strict computational budgets. In this work, we formulate inference budget allocation as a global constrained optimization problem governed by economic principles. By modeling per-query reasoning utility with a shifted-surge function, we derive an optimal allocation policy based on a global shadow price that equilibrates marginal utility under resource scarcity. Based on this theory, we propose Constrained Latent-utility Equilibrium Allocation for Reasoning (CLEAR). It performs rational abandonment and reallocates resources from insolvent queries to solvable queries near their emergence thresholds. Extensive experiments on several reasoning tasks with different traffic streams demonstrate that CLEAR significantly improves the Pareto frontier of total token cost versus mean accuracy. In resource-scarce regimes, CLEAR achieves up to a 3x improvement in global accuracy compared to uniform allocation.

We study the perfect information Nash equilibrium between a broker and her clients -- an informed trader and an uniformed trader. In our model, the broker trades in the lit exchange where trades have instantaneous and transient price impact with exponential resilience, while both clients trade with the broker. The informed trader and the broker maximise expected wealth subject to inventory penalties, while the uninformed trader is not strategic and sends the broker random buy and sell orders. We characterise the Nash equilibrium of the trading strategies with the solution to a coupled system of forward-backward stochastic differential equations (FBSDEs). We solve this system explicitly and study the effect of information, profitability, and inventory control in the trading strategies of the broker and the informed trader.

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of alpha-coherent function for which we provide convergence analysis. We show that for strictly alpha-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in alpha-coherent class of functions.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

We provide a concise framework for generalized ensemble theory through a matrix-based approach. By introducing an observation matrix, any discrete probability distribution, including those for non-equilibrium steady states, can be expressed as a generalized Boltzmann distribution, with observables and conjugate variables as the basis and coordinates in a linear space. In this framework, we identify the minimal sufficient statistics required for inferring the Boltzmann distribution. Furthermore, we show that the Hadamard and Vandermonde matrices are suitable observation matrices for spin systems and random walks. In master equation systems, the probability flux observation matrix facilitates the identification of detailed balance violations. Our findings provide a new approach to developing generalized ensemble theory for non-equilibrium steady-state systems.

We propose a novel independent and payoff-based learning framework for stochastic games that is model-free, game-agnostic, and gradient-free. The learning dynamics follow a best-response-type actor-critic architecture, where agents update their strategies (actors) using feedback from two distinct critics: a fast critic that intuitively responds to observed payoffs under limited information, and a slow critic that deliberatively approximates the solution to the underlying dynamic programming problem. Crucially, the learning process relies on non-equilibrium adaptation through smoothed best responses to observed payoffs. We establish convergence to (approximate) equilibria in two-agent zero-sum and multi-agent identical-interest stochastic games over an infinite horizon. This provides one of the first payoff-based and fully decentralized learning algorithms with theoretical guarantees in both settings. Empirical results further validate the robustness and effectiveness of the proposed approach across both classes of games.

Many examples of cooperation exist in biology. In chemical systems however, which can sometimes be quite complex, we do not appear to observe intricate cooperative interactions. A key question for the origin of life, is then how can molecular cooperation first arise in an abiotic system prior to the emergence of biological replication. We postulate that selection at the molecular level is a driving force behind the complexification of chemical systems, particularly during the origins of life. In the theory of multilevel selection the two selective forces are: within-group and between-group, where the former tends to favor "selfish" replication of individuals and the latter favor cooperation between individuals enhancing the replication of the group as a whole. These forces can be quantified using the Price equation, which is a standard tool used in evolutionary biology to quantify evolutionary change. Our central claim is that replication and heredity in chemical systems are subject to selection, and quantifiable using the multilevel Price equation. We demonstrate this using the Graded Autocatalysis Replication Domain computer model, describing simple protocell composed out of molecules and its replication, which respectively analogue to the group and the individuals. In contrast to previous treatments of this model, we treat the lipid molecules themselves as replicating individuals and the protocells they form as groups of individuals. Our goal is to demonstrate how evolutionary biology tools and concepts can be applied in chemistry and we suggest that molecular cooperation may arise as a result of group selection. Further, the biological relation of parent-progeny is proposed to be analogue to the reactant-product relation in chemistry, thus allowing for tools from evolutionary biology to be applied to chemistry and would deepen the connection between chemistry and biology.

In online ad markets, a rising number of advertisers are employing bidding agencies to participate in ad auctions. These agencies are specialized in designing online algorithms and bidding on behalf of their clients. Typically, an agency usually has information on multiple advertisers, so she can potentially coordinate bids to help her clients achieve higher utilities than those under independent bidding. In this paper, we study coordinated online bidding algorithms in repeated second-price auctions with budgets. We propose algorithms that guarantee every client a higher utility than the best she can get under independent bidding. We show that these algorithms achieve maximal coalition welfare and discuss bidders' incentives to misreport their budgets, in symmetric cases. Our proofs combine the techniques of online learning and equilibrium analysis, overcoming the difficulty of competing with a multi-dimensional benchmark. The performance of our algorithms is further evaluated by experiments on both synthetic and real data. To the best of our knowledge, we are the first to consider bidder coordination in online repeated auctions with constraints.

We present a vector-based method to balance chemical reactions. The algorithm builds candidates in a deterministic way, removes duplicates, and always prints coefficients in the lowest whole-number form. For redox cases, electrons and protons/hydroxide are treated explicitly, so both mass and charge are balanced. We also outline the basic principles of the vector formulation of stoichiometry, interpreting reactions as integer vectors in composition space, this geometric view supports compact visualizations of reagent-product interactions and helps surface distinct reaction families. The method enumerates valid balances for arbitrary user-specified species lists without special-case balancing rules or symbolic tricks, and it provides a clean foundation for developing new algorithmic variants (e.g., alternative objectives or constraints). On representative examples (neutralization, double displacement, decomposition, classical redox, small multicomponent sets) and a negative control, the method produced correct integer balances. When multiple balances exist, we report a canonical one - minimizing the total coefficient sum with a simple tie-breaker - without claiming global optimality beyond the solutions the search enumerates. The procedure applies per reaction and extends to reaction networks via consistent per-reaction application. We do not report runtimes, broader benchmarking and code/data release are planned.

We study mean field portfolio games under Epstein-Zin preferences, which naturally encompass the classical time-additive power utility as a special case. In a general non-Markovian framework, we establish a uniqueness result by proving a one-to-one correspondence between Nash equilibria and the solutions to a class of BSDEs. A key ingredient in our approach is a necessary stochastic maximum principle tailored to Epstein-Zin utility and a nonlinear transformation. In the deterministic setting, we further derive an explicit closed-form solution for the equilibrium investment and consumption policies.

We consider the problem of fairly allocating a sequence of indivisible items that arrive online in an arbitrary order to a group of n agents with additive normalized valuation functions. We consider both the allocation of goods and chores and propose algorithms for approximating maximin share (MMS) allocations. When agents have identical valuation functions the problem coincides with the semi-online machine covering problem (when items are goods) and load balancing problem (when items are chores), for both of which optimal competitive ratios have been achieved. In this paper, we consider the case when agents have general additive valuation functions. For the allocation of goods, we show that no competitive algorithm exists even when there are only three agents and propose an optimal 0.5-competitive algorithm for the case of two agents. For the allocation of chores, we propose a (2-1/n)-competitive algorithm for n>=3 agents and a square root of 2 (approximately 1.414)-competitive algorithm for two agents. Additionally, we show that no algorithm can do better than 15/11 (approximately 1.364)-competitive for two agents.

Bundle adjustment is the common way to solve localization and mapping. It is an iterative process in which a system of non-linear equations is solved using two optimization methods, weighted by a damping factor. In the classic approach, the latter is chosen heuristically by the Levenberg-Marquardt algorithm on each iteration. This might take many iterations, making the process computationally expensive, which might be harmful to real-time applications. We propose to replace this heuristic by viewing the problem in a holistic manner, as a game, and formulating it as a reinforcement-learning task. We set an environment which solves the non-linear equations and train an agent to choose the damping factor in a learned manner. We demonstrate that our approach considerably reduces the number of iterations required to reach the bundle adjustment's convergence, on both synthetic and real-life scenarios. We show that this reduction benefits the classic approach and can be integrated with other bundle adjustment acceleration methods.

Accurate free-energy estimation is essential in molecular simulation, yet the periodic boundary conditions (PBC) commonly used in computer simulations have rarely been explicitly exploited. Equilibrium methods such as umbrella sampling, metadynamics, and adaptive biasing force require extensive sampling, while non-equilibrium pulling with Jarzynski's equality suffers from poor convergence due to exponential averaging. Here, we introduce a physics-informed, score-based diffusion framework: by mapping PBC simulations onto a Brownian particle in a periodic potential, we derive the Fokker-Planck steady-state score that directly encodes free-energy gradients. A neural network is trained on non-equilibrium trajectories to learn this score, providing a principled scheme to efficiently reconstruct the potential of mean force (PMF). On benchmark periodic potentials and small-molecule membrane permeation, our method is up to one order of magnitude more efficient than umbrella sampling.

Zero-shot coordination in cooperative artificial intelligence (AI) remains a significant challenge, which means effectively coordinating with a wide range of unseen partners. Previous algorithms have attempted to address this challenge by optimizing fixed objectives within a population to improve strategy or behaviour diversity. However, these approaches can result in a loss of learning and an inability to cooperate with certain strategies within the population, known as cooperative incompatibility. To address this issue, we propose the Cooperative Open-ended LEarning (COLE) framework, which constructs open-ended objectives in cooperative games with two players from the perspective of graph theory to assess and identify the cooperative ability of each strategy. We further specify the framework and propose a practical algorithm that leverages knowledge from game theory and graph theory. Furthermore, an analysis of the learning process of the algorithm shows that it can efficiently overcome cooperative incompatibility. The experimental results in the Overcooked game environment demonstrate that our method outperforms current state-of-the-art methods when coordinating with different-level partners. Our demo is available at https://sites.google.com/view/cole-2023.

We present a new approach to formulating and solving heterogeneous agent models with aggregate risk. We replace the cross-sectional distribution with low-dimensional prices as state variables and let agents learn equilibrium price dynamics directly from simulated paths. To do so, we introduce a structural reinforcement learning (SRL) method which treats prices via simulation while exploiting agents' structural knowledge of their own individual dynamics. Our SRL method yields a general and highly efficient global solution method for heterogeneous agent models that sidesteps the Master equation and handles problems traditional methods struggle with, in particular nontrivial market-clearing conditions. We illustrate the approach in the Krusell-Smith model, the Huggett model with aggregate shocks, and a HANK model with a forward-looking Phillips curve, all of which we solve globally within minutes.

The online knapsack problem is a classic problem in the field of online algorithms. Its canonical version asks how to pack items of different values and weights arriving online into a capacity-limited knapsack so as to maximize the total value of the admitted items. Although optimal competitive algorithms are known for this problem, they may be fundamentally unfair, i.e., individual items may be treated inequitably in different ways. We formalize a practically-relevant notion of time fairness which effectively models a trade off between static and dynamic pricing in a motivating application such as cloud resource allocation, and show that existing algorithms perform poorly under this metric. We propose a parameterized deterministic algorithm where the parameter precisely captures the Pareto-optimal trade-off between fairness (static pricing) and competitiveness (dynamic pricing). We show that randomization is theoretically powerful enough to be simultaneously competitive and fair; however, it does not work well in experiments. To further improve the trade-off between fairness and competitiveness, we develop a nearly-optimal learning-augmented algorithm which is fair, consistent, and robust (competitive), showing substantial performance improvements in numerical experiments.

Competitions for shareable and limited resources have long been studied with strategic agents. In reality, agents often have to learn and maximize the rewards of the resources at the same time. To design an individualized competing policy, we model the competition between agents in a novel multi-player multi-armed bandit (MPMAB) setting where players are selfish and aim to maximize their own rewards. In addition, when several players pull the same arm, we assume that these players averagely share the arms' rewards by expectation. Under this setting, we first analyze the Nash equilibrium when arms' rewards are known. Subsequently, we propose a novel SelfishMPMAB with Averaging Allocation (SMAA) approach based on the equilibrium. We theoretically demonstrate that SMAA could achieve a good regret guarantee for each player when all players follow the algorithm. Additionally, we establish that no single selfish player can significantly increase their rewards through deviation, nor can they detrimentally affect other players' rewards without incurring substantial losses for themselves. We finally validate the effectiveness of the method in extensive synthetic experiments.

Congestion is a common failure mode of markets, where consumers compete inefficiently on the same subset of goods (e.g., chasing the same small set of properties on a vacation rental platform). The typical economic story is that prices decongest by balancing supply and demand. But in modern online marketplaces, prices are typically set in a decentralized way by sellers, and the information about items is inevitably partial. The power of a platform is limited to controlling representations -- the subset of information about items presented by default to users. This motivates the present study of decongestion by representation, where a platform seeks to learn representations that reduce congestion and thus improve social welfare. The technical challenge is twofold: relying only on revealed preferences from the choices of consumers, rather than true preferences; and the combinatorial problem associated with representations that determine the features to reveal in the default view. We tackle both challenges by proposing a differentiable proxy of welfare that can be trained end-to-end on consumer choice data. We develop sufficient conditions for when decongestion promotes welfare, and present the results of extensive experiments on both synthetic and real data that demonstrate the utility of our approach.

We study a game between liquidity provider and liquidity taker agents interacting in an over-the-counter market, for which the typical example is foreign exchange. We show how a suitable design of parameterized families of reward functions coupled with shared policy learning constitutes an efficient solution to this problem. By playing against each other, our deep-reinforcement-learning-driven agents learn emergent behaviors relative to a wide spectrum of objectives encompassing profit-and-loss, optimal execution and market share. In particular, we find that liquidity providers naturally learn to balance hedging and skewing, where skewing refers to setting their buy and sell prices asymmetrically as a function of their inventory. We further introduce a novel RL-based calibration algorithm which we found performed well at imposing constraints on the game equilibrium. On the theoretical side, we are able to show convergence rates for our multi-agent policy gradient algorithm under a transitivity assumption, closely related to generalized ordinal potential games.

The study demonstrates the capabilities of a vector-based approach for calculating stoichiometric coefficients in chemical equations, using black powder as an illustrative example. A method is proposed for selecting and constraining intermediate interactions between reactants, as well as for identifying final products. It is shown that even a small number of components can lead to a large number of final and intermediate products. Through concrete calculations, a correlation is established between the number of possible chemical equations and the number of reactants. A methodology is proposed for computing all possible chemical equations within a reaction system for arbitrary component ratios, enabling the derivation of all feasible chemical reactions. Additionally, a method is developed for calculating the chemical composition for a fixed set of reactants, allowing for the evaluation of the set of products resulting from all possible chemical interactions given a specified initial composition.

In this paper, we present a conceptual model game to examine the dynamics of asymmetric interactions in games with imperfect information. The game involves two agents with starkly contrasting capabilities: one agent can take actions but has no information of the state of the game, whereas the other agent has perfect information of the state but cannot act or observe the other agent's actions. This duality manifests an extreme form of asymmetry, and how differing abilities influence the possibility of attaining common knowledge. Using Kripke structures and epistemic logic we demonstrate that, under these conditions, common knowledge of the current game state becomes unattainable. Our findings advance the discussion on the strategic limitations of knowledge in environments where information and action are unevenly distributed.

We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

Classical theories of chemical kinetics assume independent reactions in dilute solutions, whose rates are determined by mean concentrations. In condensed matter, strong interactions alter chemical activities and create inhomogeneities that can dramatically affect the reaction rate. The extreme case is that of a reaction coupled to a phase transformation, whose kinetics must depend on the order parameter -- and its gradients, at phase boundaries. This Account presents a general theory of chemical kinetics based on nonequilibrium thermodynamics. The reaction rate is a nonlinear function of the thermodynamic driving force (free energy of reaction) expressed in terms of variational chemical potentials. The Cahn-Hilliard and Allen-Cahn equations are unified and extended via a master equation for non-equilibrium chemical thermodynamics. For electrochemistry, both Marcus and Butler-Volmer kinetics are generalized for concentrated solutions and ionic solids. The theory is applied to intercalation dynamics in the phase separating Li-ion battery material Li_xFePO_4.

The relationship between balance laws and the Principle of Virtual Work as well as the structure of contact interactions in continua remain foundational issues in Mechanics. In this work, we revisit these issues within the distributional framework emphasized by Paul Germain. We show that while the Principle of Virtual Work implies balance of forces and moments for nth-gradient continua, balance laws alone do not suffice to characterize equilibrium for n geq 2. We then reexamine Noll's classical theorem asserting that surface contact forces depend solely on the unit normal to the surface and identify the precise role of his additional assumptions, namely the absence of edge and wedge contact interactions and the boundedness of the surface contact density on the space of oriented surfaces. We demonstrate that these hypotheses fail for general higher-gradient continua. Consequently, the presence of curvature-dependent surface contact forces in such materials does not conflict with Noll's theorem and refutes his claim of their nonexistence.

Cooperation in multi-agent learning (MAL) is a topic at the intersection of numerous disciplines, including game theory, economics, social sciences, and evolutionary biology. Research in this area aims to understand both how agents can coordinate effectively when goals are aligned and how they may cooperate in settings where gains from working together are possible but possibilities for conflict abound. In this paper we provide an overview of the fundamental concepts, problem settings and algorithms of multi-agent learning. This encompasses reinforcement learning, multi-agent sequential decision-making, challenges associated with multi-agent cooperation, and a comprehensive review of recent progress, along with an evaluation of relevant metrics. Finally we discuss open challenges in the field with the aim of inspiring new avenues for research.

We initiate the study of statistical inference and A/B testing for first-price pacing equilibria (FPPE). The FPPE model captures the dynamics resulting from large-scale first-price auction markets where buyers use pacing-based budget management. Such markets arise in the context of internet advertising, where budgets are prevalent. We propose a statistical framework for the FPPE model, in which a limit FPPE with a continuum of items models the long-run steady-state behavior of the auction platform, and an observable FPPE consisting of a finite number of items provides the data to estimate primitives of the limit FPPE, such as revenue, Nash social welfare (a fair metric of efficiency), and other parameters of interest. We develop central limit theorems and asymptotically valid confidence intervals. Furthermore, we establish the asymptotic local minimax optimality of our estimators. We then show that the theory can be used for conducting statistically valid A/B testing on auction platforms. Numerical simulations verify our central limit theorems, and empirical coverage rates for our confidence intervals agree with our theory.

In this work, we extend the Equilibrium Propagation framework to skew-gradient systems and show an equivalence between deep Energy-Based Models and Hamiltonian neural networks. We focus on networks of diffusively coupled Fitzhugh-Nagumo neurons as a prototypical example. We show that since stationary solutions of the Fitzhugh-Nagumo model are described by self-adjoint operators, the methods of equilibrium propagation for performing credit assignment can be applied. Furthermore, for Fitzhugh-Nagumo networks with the topology of a deep residual network, we show that the steady state solutions admit a (spatial) Hamiltonian, and thus the methods of Hamiltonian Echo Backpropagation can be applied. We end by deriving an explicit layer-wise Hamiltonian recurrence relation governing inference for stationary solutions of both deep Fitzhugh-Nagumo networks and deep Energy-Based Models.

We focus on a simple, one-dimensional collective decision problem (often referred to as the facility location problem) and explore issues of strategyproofness and proportionality-based fairness. We introduce and analyze a hierarchy of proportionality-based fairness axioms of varying strength: Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we characterize the family of mechanisms that satisfy the axiom and strategyproofness. We show that imposing strategyproofness renders many of the axioms to be equivalent: the family of mechanisms that satisfy proportionality, unanimity, and strategyproofness is equivalent to the family of mechanisms that satisfy UFS and strategyproofness, which, in turn, is equivalent to the family of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a unique such mechanism: the Uniform Phantom mechanism, which is studied in Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom mechanism as the unique (pure) equilibrium outcome for any mechanism that satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the approximation guarantees, in terms of optimal social welfare and minimum total cost, obtained by mechanisms that are strategyproof and satisfy each proportionality-based fairness axiom. We show that the Uniform Phantom mechanism provides the best approximation of the optimal social welfare (and also minimum total cost) among all mechanisms that satisfy UFS.

We consider strategy proof mechanisms for facility location which maximize equitability between agents. As is common in the literature, we measure equitability with the Gini index. We first prove a simple but fundamental impossibility result that no strategy proof mechanism can bound the approximation ratio of the optimal Gini index of utilities for one or more facilities. We propose instead computing approximation ratios of the complemented Gini index of utilities, and consider how well both deterministic and randomized mechanisms approximate this. In addition, as Nash welfare is often put forwards as an equitable compromise between egalitarian and utilitarian outcomes, we consider how well mechanisms approximate the Nash welfare.

Achieving robust coordination and cooperation is a central challenge in multi-agent reinforcement learning (MARL). Uncovering the mechanisms underlying such emergent behaviors calls for a dynamical understanding of learn processes. In this work, we investigate the dynamics of actor-critic agents in stochastic games, focusing on the impact of entropy regularization. By leveraging time-scale separation, we derive the system's evolution equations, which are then formally analyzed using dynamical systems theory. We find that in the constant-sum game of Matching Pennies, the system exhibits chaotic behavior. Entropy regularization mitigates this chaos and drives the dynamics toward convergence to fair cooperation. In contrast, in the general-sum game of the Prisoner's Dilemma, the system displays multistability. Interestingly, the three stable equilibria of the system correspond to the well-known ALLC (Always Cooperate), ALLD (Always Defect), and GRIM (Grim Trigger) strategies from evolutionary game theory (EGT). Entropy regularization strengthens system resilience by enlarging the basin of attraction of the cooperative equilibrium. Our findings reveal a close link between the mechanism of direct reciprocity in EGT and how cooperation emerges in MARL, offering insights for designing more robust and collaborative multi-agent systems.

We give a quantum algorithm for computing an epsilon-approximate Nash equilibrium of a zero-sum game in a m times n payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time O(m + ncdot epsilon^{-2.5} + epsilon^{-3}) and outputs a classical representation of the epsilon-approximate Nash equilibrium. This improves upon the best prior quantum runtime of O(m + n cdot epsilon^{-3}) obtained by [vAG19] and the classic O((m + n) cdot epsilon^{-2}) runtime due to [GK95] whenever epsilon = Omega((m +n)^{-1}). We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.

Finding the mixed Nash equilibria (MNE) of a two-player zero sum continuous game is an important and challenging problem in machine learning. A canonical algorithm to finding the MNE is the noisy gradient descent ascent method which in the infinite particle limit gives rise to the {\em Mean-Field Gradient Descent Ascent} (GDA) dynamics on the space of probability measures. In this paper, we first study the convergence of a two-scale Mean-Field GDA dynamics for finding the MNE of the entropy-regularized objective. More precisely we show that for each finite temperature (or regularization parameter), the two-scale Mean-Field GDA with a suitable {\em finite} scale ratio converges exponentially to the unique MNE without assuming the convexity or concavity of the interaction potential. The key ingredient of our proof lies in the construction of new Lyapunov functions that dissipate exponentially along the Mean-Field GDA. We further study the simulated annealing of the Mean-Field GDA dynamics. We show that with a temperature schedule that decays logarithmically in time the annealed Mean-Field GDA converges to the MNE of the original unregularized objective.

We present a new approach to modeling sequential data: the deep equilibrium model (DEQ). Motivated by an observation that the hidden layers of many existing deep sequence models converge towards some fixed point, we propose the DEQ approach that directly finds these equilibrium points via root-finding. Such a method is equivalent to running an infinite depth (weight-tied) feedforward network, but has the notable advantage that we can analytically backpropagate through the equilibrium point using implicit differentiation. Using this approach, training and prediction in these networks require only constant memory, regardless of the effective "depth" of the network. We demonstrate how DEQs can be applied to two state-of-the-art deep sequence models: self-attention transformers and trellis networks. On large-scale language modeling tasks, such as the WikiText-103 benchmark, we show that DEQs 1) often improve performance over these state-of-the-art models (for similar parameter counts); 2) have similar computational requirements to existing models; and 3) vastly reduce memory consumption (often the bottleneck for training large sequence models), demonstrating an up-to 88% memory reduction in our experiments. The code is available at https://github.com/locuslab/deq .

We study the Levine hat problem, a classic combinatorial puzzle introduced by Lionel Levine in 2010. This problem involves a game in which n geq 2 players, each seeing an infinite stack of hats on each of their teammates' heads but not on their own, must simultaneously guess the index of a black hat on their own stack. If one of the players fails to do so, the team loses collectively. The players must therefore come up with a good strategy before the game starts. While the optimal winning probability V_{n} remains unknown even for n=2, we make three key advances. First, we develop a novel geometric framework for representing strategies through measurable functions, providing a new expression of V_{n} and a unified treatment of the game for finite and for infinite stacks via integral formulations. Secondly, we construct a new strategy K_{5} that reaches the conjectured optimal probability of victory : 0.35. We also show that K_{5} is part of a larger class of strategies that allow us to improve current bounds and resolve conjectured inequalities. Finally, we introduce and entirely solve a continuous generalization of the problem, demonstrating that extending to uncountable hat stacks increases the optimal winning probability to exactly 1/2. This generalization naturally leads to a broader and smoother strategic framework, within which we also describe how to compute optimal responses to a range of strategies.

We formulate nonequilibrium thermodynamics in which intensive variables acquire operational meaning through measurement protocols consistent with local reciprocity. Using physical equilibrium as a reference, conjugate observables are constructed by continuously adjusting devices along the local tangent space of the statistical manifold. In this relativity of observation, Onsager reciprocity holds locally, allowing inference-based Lagrange multipliers to be directly measured. This provides a systematic method to extend operational definitions of intensive variables to nonequilibrium states, highlighting their context-dependent nature and offering a concrete experimental strategy.

We present a numerical framework to treat infinite time horizon spatially distributed optimal control problems via the associated canonical system derived by Pontryagin's Maximum Principle. The basic idea is to consider the canonical system in two steps. First we perform a bifurcation analysis of canonical steady states using the continuation and bifurcation package pde2path, yielding a number of so called flat and patterned canonical steady states. In a second step we link pde2path to the two point boundary value problem solver TOM to study time dependent canonical paths to steady states having the so called saddle point property. As an example we consider a shallow lake model with diffusion.

We address the problem of model-free distributed stabilization of heterogeneous multi-agent systems using reinforcement learning (RL). Two algorithms are developed. The first algorithm solves a centralized linear quadratic regulator (LQR) problem without knowing any initial stabilizing gain in advance. The second algorithm builds upon the results of the first algorithm, and extends it to distributed stabilization of multi-agent systems with predefined interaction graphs. Rigorous proofs are provided to show that the proposed algorithms achieve guaranteed convergence if specific conditions hold. A simulation example is presented to demonstrate the theoretical results.

We propose a reinforcement learning (RL) framework under a broad class of risk objectives, characterized by convex scoring functions. This class covers many common risk measures, such as variance, Expected Shortfall, entropic Value-at-Risk, and mean-risk utility. To resolve the time-inconsistency issue, we consider an augmented state space and an auxiliary variable and recast the problem as a two-state optimization problem. We propose a customized Actor-Critic algorithm and establish some theoretical approximation guarantees. A key theoretical contribution is that our results do not require the Markov decision process to be continuous. Additionally, we propose an auxiliary variable sampling method inspired by the alternating minimization algorithm, which is convergent under certain conditions. We validate our approach in simulation experiments with a financial application in statistical arbitrage trading, demonstrating the effectiveness of the algorithm.

Fair division considers the allocation of scarce resources among agents in such a way that every agent gets a fair share. It is a fundamental problem in society and has received significant attention and rapid developments from the game theory and artificial intelligence communities in recent years. The majority of the fair division literature can be divided along at least two orthogonal directions: goods versus chores, and divisible versus indivisible resources. In this survey, besides describing the state of the art, we outline a number of interesting open questions and future directions in three mixed fair division settings: (i) indivisible goods and chores, (ii) divisible and indivisible goods (mixed goods), and (iii) indivisible goods with subsidy which can be viewed like a divisible good.

We study the problem of fairly allocating indivisible goods among a set of agents. Our focus is on the existence of allocations that give each agent their maximin fair share--the value they are guaranteed if they divide the goods into as many bundles as there are agents, and receive their lowest valued bundle. An MMS allocation is one where every agent receives at least their maximin fair share. We examine the existence of such allocations when agents have cost utilities. In this setting, each item has an associated cost, and an agent's valuation for an item is the cost of the item if it is useful to them, and zero otherwise. Our main results indicate that cost utilities are a promising restriction for achieving MMS. We show that for the case of three agents with cost utilities, an MMS allocation always exists. We also show that when preferences are restricted slightly further--to what we call laminar set approvals--we can guarantee MMS allocations for any number of agents. Finally, we explore if it is possible to guarantee each agent their maximin fair share while using a strategyproof mechanism.

Modeling the interaction between traffic agents is a key issue in designing safe and non-conservative maneuvers in autonomous driving. This problem can be challenging when multi-modality and behavioral uncertainties are engaged. Existing methods either fail to plan interactively or consider unimodal behaviors that could lead to catastrophic results. In this paper, we introduce an integrated decision-making and trajectory planning framework based on Bayesian game (i.e., game of incomplete information). Human decisions inherently exhibit discrete characteristics and therefore are modeled as types of players in the game. A general solver based on no-regret learning is introduced to obtain a corresponding Bayesian Coarse Correlated Equilibrium, which captures the interaction between traffic agents in the multimodal context. With the attained equilibrium, decision-making and trajectory planning are performed simultaneously, and the resulting interactive strategy is shown to be optimal over the expectation of rivals' driving intentions. Closed-loop simulations on different traffic scenarios are performed to illustrate the generalizability and the effectiveness of the proposed framework.

Large Language Models (LLMs) have achieved remarkable progress in reasoning, yet sometimes produce responses that are suboptimal for users in tasks such as writing, information seeking, or providing practical guidance. Conventional alignment practices typically assume that maximizing model reward also maximizes user welfare, but this assumption frequently fails in practice: models may over-clarify or generate overly verbose reasoning when users prefer concise answers. Such behaviors resemble the prisoner's dilemma, where individually rational choices lead to socially suboptimal outcomes. The fundamental challenge is the lack of a principled decision making mechanism that mutually benefits both the LLM and the user. We propose Game-Theoretic Alignment (GTAlign), an alignment framework that integrates game-theoretic decision making into both reasoning and training. During reasoning, the model explicitly treats user-LLM interaction as a strategic game: it constructs payoff matrices within its reasoning chain to estimate welfare for both itself and the user, and then selects actions that are mutually beneficial. During training, we introduce a mutual welfare reward that reinforces cooperative responses, aligning model behavior with socially efficient outcomes. In addition, we introduce an inference technique that leverages game-theoretic reasoning to dynamically adapt LLM's response when pricing policies of LLM service change. Extensive experiments demonstrate that GTAlign substantially improves reasoning efficiency, answer quality, and mutual welfare compared to baselines across diverse tasks. The code is available at https://github.com/ulab-uiuc/GTAlign .

The one-dimensional J_1-J_2 q-state Potts model is solved exactly for arbitrary q, based on using OpenAI's latest reasoning model o3-mini-high to exactly solve the q=3 case. The exact results provide insights to outstanding physical problems such as the stacking of atomic or electronic orders in layered materials and the formation of a T_c-dome-shaped phase often seen in unconventional superconductors. The work is anticipated to fuel both the research in one-dimensional frustrated magnets for recently discovered finite-temperature application potentials and the fast moving topic area of AI for sciences.

The entropic fictitious play (EFP) is a recently proposed algorithm that minimizes the sum of a convex functional and entropy in the space of measures -- such an objective naturally arises in the optimization of a two-layer neural network in the mean-field regime. In this work, we provide a concise primal-dual analysis of EFP in the setting where the learning problem exhibits a finite-sum structure. We establish quantitative global convergence guarantees for both the continuous-time and discrete-time dynamics based on properties of a proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, our primal-dual framework entails a memory-efficient particle-based implementation of the EFP update, and also suggests a connection to gradient boosting methods. We illustrate the efficiency of our novel implementation in experiments including neural network optimization and image synthesis.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

Graphon games have been introduced to study games with many players who interact through a weighted graph of interaction. By passing to the limit, a game with a continuum of players is obtained, in which the interactions are through a graphon. In this paper, we focus on a graphon game for optimal investment under relative performance criteria, and we propose a deep learning method. The method builds upon two key ingredients: first, a characterization of Nash equilibria by forward-backward stochastic differential equations and, second, recent advances of machine learning algorithms for stochastic differential games. We provide numerical experiments on two different financial models. In each model, we compare the effect of several graphons, which correspond to different structures of interactions.

In this paper, we develop and implement an efficient asymptotic-preserving (AP) scheme to solve the gas mixture of Boltzmann equations under the disparate mass scaling relevant to the so-called "epochal relaxation" phenomenon. The disparity in molecular masses, ranging across several orders of magnitude, leads to significant challenges in both the evaluation of collision operators and the designing of time-stepping schemes to capture the multi-scale nature of the dynamics. A direct implementation of the spectral method faces prohibitive computational costs as the mass ratio increases due to the need to resolve vastly different thermal velocities. Unlike [I. M. Gamba, S. Jin, and L. Liu, Commun. Math. Sci., 17 (2019), pp. 1257-1289], we propose an alternative approach based on proper truncation of asymptotic expansions of the collision operators, which significantly reduces the computational complexity and works well for small varepsilon. By incorporating the separation of three time scales in the model's relaxation process [P. Degond and B. Lucquin-Desreux, Math. Models Methods Appl. Sci., 6 (1996), pp. 405-436], we design an AP scheme that captures the specific dynamics of the disparate mass model while maintaining computational efficiency. Numerical experiments demonstrate the effectiveness of the proposed scheme in handling large mass ratios of heavy and light species, as well as capturing the epochal relaxation phenomenon.

Macroscopic entropy production σ^{(tot)} in the general nonlinear isothermal chemical reaction system with mass action kinetics is decomposed into a free energy dissipation and a house-keeping heat: σ^{(tot)}=σ^{(fd)}+σ^{(hk)}; σ^{(fd)}=-rd A/rd t, where A is a generalized free energy function. This yields a novel nonequilibrium free energy balance equation rd A/rd t=-σ^{(tot)}+σ^{(hk)}, which is on a par with celebrated entropy balance equation rd S/rd t=σ^{(tot)}+η^{(ex)} where η^{(ex)} is the rate of entropy exchange with the environment.For kinetic systems with complex balance, σ^{(fd)} and σ^{(hk)} are the macroscopic limits of stochastic free energy dissipation and house-keeping heat, which are both nonnegative, in the Delbrück-Gillespie description of the stochastic chemical kinetics.Therefore, we show that a full kinetic and thermodynamic theory of chemical reaction systems that transcends mesoscopic and macroscopic levels emerges.

In this paper, we consider a varying terminal time structure for the stochastic optimal control problem under state constraints, in which the terminal time varies with the mean value of the state. In this new stochastic optimal control system, the control domain does not need to be convex and the diffusion coefficient contains the control variable. To overcome the difficulty in the proof of the related Pontryagin's stochastic maximum principle, we develop asymptotic first- and second-order adjoint equations for the varying terminal time, and then establish its variational equation. In the end, two examples are given to verify the main results of this study.