Asuman Ozdaglar

Matthew O. Jackson, Qiaozhu Me, Stephanie W. Wang et al.

We outline a foundation for a new field of ``AI Behavioral Science,'' covering three perspectives. First, as AI becomes ubiquitous and is increasingly proprietary and opaque, it becomes vital to develop techniques for assessing AI behavior. We outline how tools developed to assess people's behaviors by social scientists can be used to assess and infer AI's behaviors biases, tendencies, and heuristics. Second, we also discuss how AI can change the ways in which we learn about human behavior. Beyond its computational power, AI offers new techniques for simulating, inferring, and predicting human behaviors that we outline and discuss. Third, as humans and AI are interacting in increasingly complex and intertwined systems, we need to understand the implications for the resulting economic and political outcomes. We outline issues that are increasingly pressing concerning the future of human-AI interactions and potential changes and disruptions that can ensue.

Gagik Magakyan, Pablo Parrilo, Asuman Ozdaglar

Orthonormalized update rules have rapidly become a leading choice of optimizer for training large language models, with recent open-source state-of-the-art models adopting Muon. To keep these updates tractable, Muon performs the orthonormalization with the Newton--Schulz (NS) iteration. Since NS is only approximate, directions with small singular values fail to be orthonormalized. In Muon, NS is applied to the momentum matrix at every step, yet little is known about how the singular value spectrum of these momentum matrices behaves during training, or how that behavior changes with model size. We present the first systematic study of this question. Tracking singular value quantiles of the momentum buffer across layers in models ranging from 77M to 2.8B parameters, we observe a consistent picture: after a short burn-in, the quantiles stabilize at a value determined by the layer type and model size. These stabilization values follow remarkably clean power laws in model size, with layer-dependent exponents. Layers up to mid-late depth scale very mildly with model size $M$ (around $M^{-0.25}$), so the standard 5-step NS configuration used at academic scale will continue to orthonormalize them at much larger scales. Some of the late layers, however, scale much more aggressively (up to $M^{-0.96}$) and will fall into the NS failure regime at frontier scale unless one uses more NS iterations or better-tuned coefficients. NS iterations are computationally expensive at scale; our laws give practitioners a principled, layer-aware recipe for choosing the minimum NS configuration that still orthonormalizes the directions that matter -- avoiding unnecessary computation without sacrificing update quality.

Daron Acemoglu, Giacomo Como, Fabio Fagnani et al.

We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society fail to converge almost surely, the belief profile keeps on fluctuating in an ergodic fashion, and it converges in law to a non-degenerate random vector. The structure of the network and the location of the stubborn agents within it shape the opinion dynamics. The expected belief vector evolves according to an ordinary differential equation coinciding with the Kolmogorov backward equation of a continuous-time Markov chain with absorbing states corresponding to the stubborn agents and converges to a harmonic vector, with every regular agent's value being the weighted average of its neighbors' values, and boundary conditions corresponding to the stubborn agents'. Expected cross-products of the agents' beliefs allow for a similar characterization in terms of coupled Markov chains on the network. We prove that, in large-scale societies which are highly fluid, meaning that the product of the mixing time of the Markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large, a condition of \emph{homogeneous influence} emerges, whereby the stationary beliefs' marginal distributions of most of the regular agents have approximately equal first and second moments.

Mingyang Liu, Asuman Ozdaglar, Tiancheng Yu et al.

In this paper, we study regret minimization in repeated games with \emph{adaptive} opponents who can respond based on histories of play. The standard metric of \emph{external regret} in online learning is known to fail to capture such adaptivity. To account for players' counterfactual reasoning, we introduce {\tt Repeated Policy Regret (RP-Regret)}, a game-theoretic metric that measures the difference between the \emph{realized} and the \emph{best-in-hindsight} accumulated utility when all players can \emph{respond} to the history of play. Compared to existing regret notions in this setting, ours is native to repeated game playing, enabling stronger comparators and opponents with fewer constraints, while maintaining the possibility of finding better equilibria when all players minimize it. We first identify necessary conditions for obtaining {\tt RP-Regret} sublinear in time, on the variation of the player's comparator strategies in the regret definition and on the memories of both the comparator and opponents' strategies. We then study additional conditions and provable algorithms to minimize {\tt RP-Regret}, which is by definition \emph{non-convex} in the strategy space. To address this challenge, we propose three algorithms: (i) one based on an optimization oracle, as assumed in some prior work in online non-convex learning; (ii) one that minimizes a convex and \emph{linearized} surrogate of {\tt RP-Regret} at each iteration; (iii) one that directly minimizes {\tt RP-Regret} when opponents change strategies slowly. Furthermore, when all players can run algorithms to minimize the {\tt RP-Regret} (or its linearized variant), certain subgame perfect equilibria of the repeated game can be learned. We also provide experiments showing that minimizing our regret notions can lead to more cooperative solutions with higher utility in games such as Stag-Hunt.

Ermin Wei, Asuman Ozdaglar, Ali Jadbabaie

Most existing work uses dual decomposition and subgradient methods to solve Network Utility Maximization (NUM) problems in a distributed manner, which suffer from slow rate of convergence properties. This work develops an alternative distributed Newton-type fast converging algorithm for solving network utility maximization problems with self-concordant utility functions. By using novel matrix splitting techniques, both primal and dual updates for the Newton step can be computed using iterative schemes in a decentralized manner with limited information exchange. Similarly, the stepsize can be obtained via an iterative consensus-based averaging scheme. We show that even when the Newton direction and the stepsize in our method are computed within some error (due to finite truncation of the iterative schemes), the resulting objective function value still converges superlinearly to an explicitly characterized error neighborhood. Simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing subgradient methods based on dual decomposition.

Francesca Parise, Asuman Ozdaglar

We provide a unified variational inequality framework for the study of fundamental properties of the Nash equilibrium in network games. We identify several conditions on the underlying network (in terms of spectral norm, infinity norm and minimum eigenvalue of its adjacency matrix) that guarantee existence, uniqueness, convergence and continuity of equilibrium in general network games with multidimensional and possibly constrained strategy sets. We delineate the relations between these conditions and characterize classes of networks that satisfy each of these conditions.

Mingyang Liu, Asuman Ozdaglar, Tiancheng Yu et al.

In this paper, we investigate the power of {\it regularization}, a common technique in reinforcement learning and optimization, in solving extensive-form games (EFGs). We propose a series of new algorithms based on regularizing the payoff functions of the game, and establish a set of convergence results that strictly improve over the existing ones, with either weaker assumptions or stronger convergence guarantees. In particular, we first show that dilated optimistic mirror descent (DOMD), an efficient variant of OMD for solving EFGs, with adaptive regularization can achieve a fast $\tilde O(1/T)$ {last-iterate convergence rate for the output of the algorithm} in terms of duality gap and distance to the set of Nash equilibrium (NE) without uniqueness assumption of the NE. Second, we show that regularized counterfactual regret minimization (\texttt{Reg-CFR}), with a variant of optimistic mirror descent algorithm as regret-minimizer, can achieve $O(1/T^{1/4})$ best-iterate, and $O(1/T^{3/4})$ average-iterate convergence rate for finding NE in EFGs. Finally, we show that \texttt{Reg-CFR} can achieve asymptotic last-iterate convergence, and optimal $O(1/T)$ average-iterate convergence rate, for finding the NE of perturbed EFGs, which is useful for finding approximate extensive-form perfect equilibria (EFPE). To the best of our knowledge, they constitute the first last-iterate convergence results for CFR-type algorithms, while matching the state-of-the-art average-iterate convergence rate in finding NE for non-perturbed EFGs. We also provide numerical results to corroborate the advantages of our algorithms.

Asuman Ozdaglar, Sarath Pattathil, Jiawei Zhang et al.

Minimax optimization has served as the backbone of many machine learning (ML) problems. Although the convergence behavior of optimization algorithms has been extensively studied in the minimax settings, their generalization guarantees in stochastic minimax optimization problems, i.e., how the solution trained on empirical data performs on unseen testing data, have been relatively underexplored. A fundamental question remains elusive: What is a good metric to study generalization of minimax learners? In this paper, we aim to answer this question by first showing that primal risk, a universal metric to study generalization in minimization problems, which has also been adopted recently to study generalization in minimax ones, fails in simple examples. We thus propose a new metric to study generalization of minimax learners: the primal gap, defined as the difference between the primal risk and its minimum over all models, to circumvent the issues. Next, we derive generalization error bounds for the primal gap in nonconvex-concave settings. As byproducts of our analysis, we also solve two open questions: establishing generalization error bounds for primal risk and primal-dual risk, another existing metric that is only well-defined when the global saddle-point exists, in the strong sense, i.e., without strong concavity or assuming that the maximization and expectation can be interchanged, while either of these assumptions was needed in the literature. Finally, we leverage this new metric to compare the generalization behavior of two popular algorithms -- gradient descent-ascent (GDA) and gradient descent-max (GDMax) in stochastic minimax optimization.

Zaiwei Chen, Kaiqing Zhang, Eric Mazumdar et al.

We study two-player zero-sum stochastic games, and propose a form of independent learning dynamics called Doubly Smoothed Best-Response dynamics, which integrates a discrete and doubly smoothed variant of the best-response dynamics into temporal-difference (TD)-learning and minimax value iteration. The resulting dynamics are payoff-based, convergent, rational, and symmetric among players. Our main results provide finite-sample guarantees. In particular, we prove the first-known $\tilde{\mathcal{O}}(1/ε^2)$ sample complexity bound for payoff-based independent learning dynamics, up to a smoothing bias. In the special case where the stochastic game has only one state (i.e., matrix games), we provide a sharper $\tilde{\mathcal{O}}(1/ε)$ sample complexity. Our analysis uses a novel coupled Lyapunov drift approach to capture the evolution of multiple sets of coupled and stochastic iterates, which might be of independent interest.

Sarath Pattathil, Kaiqing Zhang, Asuman Ozdaglar

Multi-agent interactions are increasingly important in the context of reinforcement learning, and the theoretical foundations of policy gradient methods have attracted surging research interest. We investigate the global convergence of natural policy gradient (NPG) algorithms in multi-agent learning. We first show that vanilla NPG may not have parameter convergence, i.e., the convergence of the vector that parameterizes the policy, even when the costs are regularized (which enabled strong convergence guarantees in the policy space in the literature). This non-convergence of parameters leads to stability issues in learning, which becomes especially relevant in the function approximation setting, where we can only operate on low-dimensional parameters, instead of the high-dimensional policy. We then propose variants of the NPG algorithm, for several standard multi-agent learning scenarios: two-player zero-sum matrix and Markov games, and multi-player monotone games, with global last-iterate parameter convergence guarantees. We also generalize the results to certain function approximation settings. Note that in our algorithms, the agents take symmetric roles. Our results might also be of independent interest for solving nonconvex-nonconcave minimax optimization problems with certain structures. Simulations are also provided to corroborate our theoretical findings.

Asuman Ozdaglar, Sarath Pattathil, Jiawei Zhang et al.

Offline reinforcement learning (RL) aims to find an optimal policy for Markov decision processes (MDPs) using a pre-collected dataset. In this work, we revisit the linear programming (LP) reformulation of Markov decision processes for offline RL, with the goal of developing algorithms with optimal $O(1/\sqrt{n})$ sample complexity, where $n$ is the sample size, under partial data coverage and general function approximation, and with favorable computational tractability. To this end, we derive new \emph{error bounds} for both the dual and primal-dual formulations of the LP, and incorporate them properly as \emph{constraints} in the LP reformulation. We then show that under a completeness-type assumption, $O(1/\sqrt{n})$ sample complexity can be achieved under standard single-policy coverage assumption, when one properly \emph{relaxes} the occupancy validity constraint in the LP. This framework can readily handle both infinite-horizon discounted and average-reward MDPs, in both general function approximation and tabular cases. The instantiation to the tabular case achieves either state-of-the-art or the first sample complexities of offline RL in these settings. To further remove any completeness-type assumption, we then introduce a proper \emph{lower-bound constraint} in the LP, and a variant of the standard single-policy coverage assumption. Such an algorithm leads to a $O(1/\sqrt{n})$ sample complexity with dependence on the \emph{value-function gap}, with only realizability assumptions. Our properly constrained LP framework advances the existing results in several aspects, in relaxing certain assumptions and achieving the optimal $O(1/\sqrt{n})$ sample complexity, with simple analyses. We hope our results bring new insights into the use of LP formulations and the equivalent primal-dual minimax optimization for offline RL, through the error-bound induced constraints.

Chanwoo Park, Kaiqing Zhang, Asuman Ozdaglar

We study a new class of Markov games, \emph(multi-player) zero-sum Markov Games} with \emph{Networked separable interactions} (zero-sum NMGs), to model the local interaction structure in non-cooperative multi-agent sequential decision-making. We define a zero-sum NMG as a model where {the payoffs of the auxiliary games associated with each state are zero-sum and} have some separable (i.e., polymatrix) structure across the neighbors over some interaction network. We first identify the necessary and sufficient conditions under which an MG can be presented as a zero-sum NMG, and show that the set of Markov coarse correlated equilibrium (CCE) collapses to the set of Markov Nash equilibrium (NE) in these games, in that the product of per-state marginalization of the former for all players yields the latter. Furthermore, we show that finding approximate Markov \emph{stationary} CCE in infinite-horizon discounted zero-sum NMGs is \texttt{PPAD}-hard, unless the underlying network has a ``star topology''. Then, we propose fictitious-play-type dynamics, the classical learning dynamics in normal-form games, for zero-sum NMGs, and establish convergence guarantees to Markov stationary NE under a star-shaped network structure. Finally, in light of the hardness result, we focus on computing a Markov \emph{non-stationary} NE and provide finite-iteration guarantees for a series of value-iteration-based algorithms. We also provide numerical experiments to corroborate our theoretical results.

Mingyang Liu, Gabriele Farina, Asuman Ozdaglar

LiteEFG is an efficient library with easy-to-use Python bindings, which can solve multiplayer extensive-form games (EFGs). LiteEFG enables the user to express computation graphs in Python to define updates on the game tree structure. The graph is then executed by the C++ backend, leading to significant speedups compared to running the algorithm in Python. Moreover, in LiteEFG, the user needs to only specify the computation graph of the update rule in a decision node of the game, and LiteEFG will automatically distribute the update rule to each decision node and handle the structure of the imperfect-information game.

Zaiwei Chen, Kaiqing Zhang, Eric Mazumdar et al.

In this paper, we consider two-player zero-sum matrix and stochastic games and develop learning dynamics that are payoff-based, convergent, rational, and symmetric between the two players. Specifically, the learning dynamics for matrix games are based on the smoothed best-response dynamics, while the learning dynamics for stochastic games build upon those for matrix games, with additional incorporation of the minimax value iteration. To our knowledge, our theoretical results present the first finite-sample analysis of such learning dynamics with last-iterate guarantees. In the matrix game setting, the results imply a sample complexity of $O(ε^{-1})$ to find the Nash distribution and a sample complexity of $O(ε^{-8})$ to find a Nash equilibrium. In the stochastic game setting, the results also imply a sample complexity of $O(ε^{-8})$ to find a Nash equilibrium. To establish these results, the main challenge is to handle stochastic approximation algorithms with multiple sets of coupled and stochastic iterates that evolve on (possibly) different time scales. To overcome this challenge, we developed a coupled Lyapunov-based approach, which may be of independent interest to the broader community studying the convergence behavior of stochastic approximation algorithms.

Mingyang Liu, Gabriele Farina, Asuman Ozdaglar

Policy gradient methods have become a staple of any single-agent reinforcement learning toolbox, due to their combination of desirable properties: iterate convergence, efficient use of stochastic trajectory feedback, and theoretically-sound avoidance of importance sampling corrections. In multi-agent imperfect-information settings (extensive-form games), however, it is still unknown whether the same desiderata can be guaranteed while retaining theoretical guarantees. Instead, sound methods for extensive-form games rely on approximating \emph{counterfactual} values (as opposed to Q values), which are incompatible with policy gradient methodologies. In this paper, we investigate whether policy gradient can be safely used in two-player zero-sum imperfect-information extensive-form games (EFGs). We establish positive results, showing for the first time that a policy gradient method leads to provable best-iterate convergence to a regularized Nash equilibrium in self-play.

Ali ParandehGheibi, Mardavij Roozbehani, Asuman Ozdaglar et al.

This paper examines the value of storage in securing reliability of a system with uncertain supply and demand, and supply friction. The storage is frictionless as a supply source, but once used, it cannot be filled up instantaneously. The focus application is a power supply network in which the base supply and demand are assumed to match perfectly, while deviations from the base are modeled as random shocks with stochastic arrivals. Due to friction, the random surge shocks cannot be tracked by the main supply sources. Storage, when available, can be used to compensate, fully or partially, for the surge in demand or loss of supply. The problem of optimal utilization of storage with the objective of maximizing system reliability is formulated as minimization of the expected discounted cost of blackouts over an infinite horizon. It is shown that when the stage cost is linear in the size of the blackout, the optimal policy is myopic in the sense that all shocks are compensated by storage up to the available level of storage. However, when the stage cost is strictly convex, it may be optimal to curtail some of the demand and allow a small current blackout in the interest of maintaining a higher level of reserve to avoid a large blackout in the future. The value of storage capacity in improving system's reliability, as well as the effects of the associated optimal policies under different stage costs on the probability distribution of blackouts are examined.

Mingyang Liu, Yongshan Chen, Zhiyuan Fan et al.

Online learning in arbitrary, and possibly adversarial, environments has been extensively studied in sequential decision-making, and it is closely connected to equilibrium computation in game theory. Most existing online learning algorithms rely on \emph{numeric} utility feedback from the environment, which may be unavailable in human-in-the-loop applications and/or may be restricted by privacy concerns. In this paper, we study an online learning model in which the learner only observes a \emph{ranking} over a set of proposed actions at each timestep. We consider two ranking mechanisms: rankings induced by the \emph{instantaneous} utility at the current timestep, and rankings induced by the \emph{time-average} utility up to the current timestep, under both \emph{full-information} and \emph{bandit} feedback settings. Using the standard external-regret metric, we show that sublinear regret is impossible with instantaneous-utility ranking feedback in general. Moreover, when the ranking model is relatively deterministic, \emph{i.e.}, under the Plackett-Luce model with a temperature that is sufficiently small, sublinear regret is also impossible with time-average utility ranking feedback. We then develop new algorithms that achieve sublinear regret under the additional assumption that the utility sequence has sublinear total variation. Notably, for full-information time-average utility ranking feedback, this additional assumption can be removed. As a consequence, when all players in a normal-form game follow our algorithms, repeated play yields an approximate coarse correlated equilibrium. We also demonstrate the effectiveness of our algorithms in an online large-language-model routing task.

Amirhossein Reisizadeh, Khashayar Gatmiry, Asuman Ozdaglar

Modern data-driven and distributed learning frameworks deal with diverse massive data generated by clients spread across heterogeneous environments. Indeed, data heterogeneity is a major bottleneck in scaling up many distributed learning paradigms. In many settings however, heterogeneous data may be generated in clusters with shared structures, as is the case in several applications such as federated learning where a common latent variable governs the distribution of all the samples generated by a client. It is therefore natural to ask how the underlying clustered structures in distributed data can be exploited to improve learning schemes. In this paper, we tackle this question in the special case of estimating $d$-dimensional parameters of a two-component mixture of linear regressions problem where each of $m$ nodes generates $n$ samples with a shared latent variable. We employ the well-known Expectation-Maximization (EM) method to estimate the maximum likelihood parameters from $m$ batches of dependent samples each containing $n$ measurements. Discarding the clustered structure in the mixture model, EM is known to require $O(\log(mn/d))$ iterations to reach the statistical accuracy of $O(\sqrt{d/(mn)})$. In contrast, we show that if initialized properly, EM on the structured data requires only $O(1)$ iterations to reach the same statistical accuracy, as long as $m$ grows up as $e^{o(n)}$. Our analysis establishes and combines novel asymptotic optimization and generalization guarantees for population and empirical EM with dependent samples, which may be of independent interest.

Gagik Magakyan, Amirhossein Reisizadeh, Chanwoo Park et al.

Adaptability has been regarded as a central feature in the foundation models, enabling them to effectively acclimate to unseen downstream tasks. Parameter-efficient fine-tuning methods such as celebrated LoRA facilitate efficient adaptation of large foundation models using labeled, high-quality and generally scarce task data. To mitigate data scarcity in fine-tuning of foundation models, we propose to leverage task similarity across multiple downstream users. Intuitively, users with similar tasks must be able to assist each other in boosting the effective fine-tuning data size. We propose Collaborative Low-Rank Adaptation, or CoLoRA, which exploits task similarity to collaboratively and efficiently fine-tune personalized foundation models. The main idea in CoLoRA is to train one shared adapter capturing underlying task similarities across all tasks, and personalized adapters tailored to user-specific tasks. We theoretically study CoLoRA on heterogeneous linear regression and provide provable guarantees for ground truth recovery. We also conduct several natural language experiments with varying task similarity, which further demonstrate that when trained together with similar tasks, individual performances are significantly boosted.

Chanwoo Park, Ziyang Chen, Asuman Ozdaglar et al.

Large language models (LLMs) are increasingly deployed as "agents" for decision-making (DM) in interactive and dynamic environments. Yet, since they were not originally designed for DM, recent studies show that LLMs can struggle even in basic online DM problems, failing to achieve low regret or an effective exploration-exploitation tradeoff. To address this, we introduce Iterative Regret-Minimization Fine-Tuning (Iterative RMFT), a post-training procedure that repeatedly distills low-regret decision trajectories back into the base model. At each iteration, the model rolls out multiple decision trajectories, selects the k-lowest regret ones, and fine-tunes itself on them. Unlike prior methods that (a) distill action sequences from known DM algorithms or (b) rely on manually crafted chain-of-thought templates, our approach leverages the regret metric to elicit the model's own DM ability and reasoning rationales. This reliance on model-generated reasoning avoids rigid output engineering and provides more flexible, natural-language training signals. Empirical results show that Iterative RMFT improves LLMs' DM performance across diverse models - from Transformers with numerical input/output, to open-weight LLMs, and advanced closed-weight models like GPT-4o mini. Its flexibility in output and reasoning formats enables generalization across tasks with varying horizons, action spaces, reward processes, and natural-language contexts. Finally, we provide theoretical insight showing that a single-layer Transformer under this paradigm can act as a no-regret learner in a simplified setting. Overall, Iterative RMFT offers a principled and general post-training framework for enhancing LLMs' decision-making capabilities.

Chanwoo Park, Mingyang Liu, Dingwen Kong et al.

Reinforcement learning from human feedback (RLHF) has been an effective technique for aligning AI systems with human values, with remarkable successes in fine-tuning large-language models recently. Most existing RLHF paradigms make the underlying assumption that human preferences are relatively homogeneous, and can be encoded by a single reward model. In this paper, we focus on addressing the issues due to the inherent heterogeneity in human preferences, as well as their potential strategic behavior in providing feedback. Specifically, we propose two frameworks to address heterogeneous human feedback in principled ways: personalization-based one and aggregation-based one. For the former, we propose two approaches based on representation learning and clustering, respectively, for learning multiple reward models that trades off the bias (due to preference heterogeneity) and variance (due to the use of fewer data for learning each model by personalization). We then establish sample complexity guarantees for both approaches. For the latter, we aim to adhere to the single-model framework, as already deployed in the current RLHF paradigm, by carefully aggregating diverse and truthful preferences from humans. We propose two approaches based on reward and preference aggregation, respectively: the former utilizes both utilitarianism and Leximin approaches to aggregate individual reward models, with sample complexity guarantees; the latter directly aggregates the human feedback in the form of probabilistic opinions. Under the probabilistic-opinion-feedback model, we also develop an approach to handle strategic human labelers who may bias and manipulate the aggregated preferences with untruthful feedback. Based on the ideas in mechanism design, our approach ensures truthful preference reporting, with the induced aggregation rule maximizing social welfare functions.

Chanwoo Park, Xiangyu Liu, Asuman Ozdaglar et al.

Large language models (LLMs) have been increasingly employed for (interactive) decision-making, via the development of LLM-based autonomous agents. Despite their emerging successes, the performance of LLM agents in decision-making has not been fully investigated through quantitative metrics, especially in the multi-agent setting when they interact with each other, a typical scenario in real-world LLM-agent applications. To better understand the limits of LLM agents in these interactive environments, we propose to study their interactions in benchmark decision-making settings in online learning and game theory, through the performance metric of \emph{regret}. We first empirically study the {no-regret} behaviors of LLMs in canonical (non-stationary) online learning problems, as well as the emergence of equilibria when LLM agents interact through playing repeated games. We then provide some theoretical insights into the no-regret behaviors of LLM agents, under certain assumptions on the supervised pre-training and the rationality model of human decision-makers who generate the data. Notably, we also identify (simple) cases where advanced LLMs such as GPT-4 fail to be no-regret. To promote the no-regret behaviors, we propose a novel \emph{unsupervised} training loss of \emph{regret-loss}, which, in contrast to the supervised pre-training loss, does not require the labels of (optimal) actions. We then establish the statistical guarantee of generalization bound for regret-loss minimization, followed by the optimization guarantee that minimizing such a loss may automatically lead to known no-regret learning algorithms. Our further experiments demonstrate the effectiveness of our regret-loss, especially in addressing the above ``regrettable'' cases.

Chanwoo Park, Seungju Han, Xingzhi Guo et al. · stanford

Leveraging multiple large language models (LLMs) to build collaborative multi-agentic workflows has demonstrated significant potential. However, most previous studies focus on prompting the out-of-the-box LLMs, relying on their innate capability for collaboration, which may not improve LLMs' performance as shown recently. In this paper, we introduce a new post-training paradigm MAPoRL (Multi-Agent Post-co-training for collaborative LLMs with Reinforcement Learning), to explicitly elicit the collaborative behaviors and further unleash the power of multi-agentic LLM frameworks. In MAPoRL, multiple LLMs first generate their own responses independently and engage in a multi-turn discussion to collaboratively improve the final answer. In the end, a MAPoRL verifier evaluates both the answer and the discussion, by assigning a score that verifies the correctness of the answer, while adding incentives to encourage corrective and persuasive discussions. The score serves as the co-training reward, and is then maximized through multi-agent RL. Unlike existing LLM post-training paradigms, MAPoRL advocates the co-training of multiple LLMs together using RL for better generalization. Accompanied by analytical insights, our experiments demonstrate that training individual LLMs alone is insufficient to induce effective collaboration. In contrast, multi-agent co-training can boost the collaboration performance across benchmarks, with generalization to unseen domains.

Yuhan Ye, Saurabh Amin, Asuman Ozdaglar

We study how to construct compressed datasets that suffice to recover optimal decisions in linear programs with an unknown cost vector $c$ lying in a prior set $\mathcal{C}$. Recent work by Bennouna et al. provides an exact geometric characterization of sufficient decision datasets (SDDs) via an intrinsic decision-relevant dimension $d^\star$. However, their algorithm for constructing minimum-size SDDs requires solving mixed-integer programs. In this paper, we establish hardness results showing that computing $d^\star$ is NP-hard and deciding whether a dataset is globally sufficient is coNP-hard, thereby resolving a recent open problem posed by Bennouna et al. To address this worst-case intractability, we introduce pointwise sufficiency, a relaxation that requires sufficiency for an individual cost vector. Under nondegeneracy, we provide a polynomial-time cutting-plane algorithm for constructing pointwise-sufficient decision datasets. In a data-driven regime with i.i.d.\ costs, we further propose a cumulative algorithm that aggregates decision-relevant directions across samples, yielding a stable compression scheme of size at most $d^\star$. This leads to a distribution-free PAC guarantee: with high probability over the training sample, the pointwise sufficiency failure probability on a fresh draw is at most $\tilde{O}(d^\star/n)$, and this rate is tight up to logarithmic factors. Finally, we apply decision-sufficient representations to contextual linear optimization, obtaining compressed predictors with generalization bounds scaling as $\tilde{O}(\sqrt{d^\star/n})$ rather than $\tilde{O}(\sqrt{d/n})$, where $d$ is the ambient cost dimension.

Mingyang Liu, Gabriele Farina, Asuman Ozdaglar

Post-training has demonstrated its importance in enhancing the reasoning capabilities of large language models (LLMs). The primary post-training methods can be categorized into supervised fine-tuning (SFT) and reinforcement fine-tuning (RFT). SFT is efficient and well-suited for small language models, but it may lead to overfitting and limit the reasoning abilities of larger models. In contrast, RFT generally yields better generalization but depends heavily on the strength of the base model. To address the limitations of SFT and RFT, we propose Unified Fine-Tuning (UFT), a novel post-training paradigm that unifies SFT and RFT into a single, integrated process. UFT enables the model to effectively explore solutions while incorporating informative supervision signals, bridging the gap between memorizing and thinking underlying existing methods. Notably, UFT outperforms both SFT and RFT in general, regardless of model sizes. Furthermore, we theoretically prove that UFT breaks RFT's inherent exponential sample complexity bottleneck, showing for the first time that unified training can exponentially accelerate convergence on long-horizon reasoning tasks.

Mingyang Liu, Gabriele Farina, Asuman Ozdaglar

Most familiar equilibrium concepts, such as Nash and correlated equilibrium, guarantee only that no single player can improve their utility by deviating unilaterally. They offer no guarantees against profitable coordinated deviations by coalitions. Although the literature proposes solution concepts that provide stability against multilateral deviations (\emph{e.g.}, strong Nash and coalition-proof equilibrium), these generally fail to exist. In this paper, we study an alternative solution concept that minimizes coalitional deviation incentives, rather than requiring them to vanish, and is therefore guaranteed to exist. Specifically, we focus on minimizing the average gain of a deviating coalition, and extend the framework to weighted-average and maximum-within-coalition gains. In contrast, the minimum-gain analogue is shown to be computationally intractable. For the average-gain and maximum-gain objectives, we prove a lower bound on the complexity of computing such an equilibrium and present an algorithm that matches this bound. Finally, we use our framework to solve the \emph{Exploitability Welfare Frontier} (EWF), the maximum attainable social welfare subject to a given exploitability (the maximum gain over all unilateral deviations).

Daniel Huttenlocher, Hannah Li, Liang Lyu et al.

Many online platforms of today, including social media sites, are two-sided markets bridging content creators and users. Most of the existing literature on platform recommendation algorithms largely focuses on user preferences and decisions, and does not simultaneously address creator incentives. We propose a model of content recommendation that explicitly focuses on the dynamics of user-content matching, with the novel property that both users and creators may leave the platform permanently if they do not experience sufficient engagement. In our model, each player decides to participate at each time step based on utilities derived from the current match: users based on alignment of the recommended content with their preferences, and creators based on their audience size. We show that a user-centric greedy algorithm that does not consider creator departures can result in arbitrarily poor total engagement, relative to an algorithm that maximizes total engagement while accounting for two-sided departures. Moreover, in stark contrast to the case where only users or only creators leave the platform, we prove that with two-sided departures, approximating maximum total engagement within any constant factor is NP-hard. We present two practical algorithms, one with performance guarantees under mild assumptions on user preferences, and another that tends to outperform algorithms that ignore two-sided departures in practice.

Zaiwei Chen, Kaiqing Zhang, Eric Mazumdar et al.

We consider two-player zero-sum stochastic games and propose a two-timescale $Q$-learning algorithm with function approximation that is payoff-based, convergent, rational, and symmetric between the two players. In two-timescale $Q$-learning, the fast-timescale iterates are updated in spirit to the stochastic gradient descent and the slow-timescale iterates (which we use to compute the policies) are updated by taking a convex combination between its previous iterate and the latest fast-timescale iterate. Introducing the slow timescale as well as its update equation marks as our main algorithmic novelty. In the special case of linear function approximation, we establish, to the best of our knowledge, the first last-iterate finite-sample bound for payoff-based independent learning dynamics of these types. The result implies a polynomial sample complexity to find a Nash equilibrium in such stochastic games. To establish the results, we model our proposed algorithm as a two-timescale stochastic approximation and derive the finite-sample bound through a Lyapunov-based approach. The key novelty lies in constructing a valid Lyapunov function to capture the evolution of the slow-timescale iterates. Specifically, through a change of variable, we show that the update equation of the slow-timescale iterates resembles the classical smoothed best-response dynamics, where the regularized Nash gap serves as a valid Lyapunov function. This insight enables us to construct a valid Lyapunov function via a generalized variant of the Moreau envelope of the regularized Nash gap. The construction of our Lyapunov function might be of broad independent interest in studying the behavior of stochastic approximation algorithms.

Daron Acemoglu, Tianyi Lin, Asuman Ozdaglar et al.

Artificial intelligence (AI) changes social learning when aggregated outputs become training data for future predictions. To study this, we extend the DeGroot model by introducing an AI aggregator that trains on population beliefs and feeds synthesized signals back to agents. We define the learning gap as the deviation of long-run beliefs from the efficient benchmark, allowing us to capture how AI aggregation affects learning. Our main result identifies a threshold in the speed of updating: when the aggregator updates too quickly, there is no positive-measure set of training weights that robustly improves learning across a broad class of environments, whereas such weights exist when updating is sufficiently slow. We then compare global and local architectures. Local aggregators trained on proximate or topic-specific data robustly improve learning in all environments. Consequently, replacing specialized local aggregators with a single global aggregator worsens learning in at least one dimension of the state.

Jiancong Xiao, Jiawei Zhang, Zhi-Quan Luo et al.

In adversarial machine learning, neural networks suffer from a significant issue known as robust overfitting, where the robust test accuracy decreases over epochs (Rice et al., 2020). Recent research conducted by Xing et al.,2021; Xiao et al., 2022 has focused on studying the uniform stability of adversarial training. Their investigations revealed that SGD-based adversarial training fails to exhibit uniform stability, and the derived stability bounds align with the observed phenomenon of robust overfitting in experiments. This motivates us to develop uniformly stable algorithms specifically tailored for adversarial training. To this aim, we introduce Moreau envelope-$\mathcal{A}$, a variant of the Moreau Envelope-type algorithm. We employ a Moreau envelope function to reframe the original problem as a min-min problem, separating the non-strong convexity and non-smoothness of the adversarial loss. Then, this approach alternates between solving the inner and outer minimization problems to achieve uniform stability without incurring additional computational overhead. In practical scenarios, we show the efficacy of ME-$\mathcal{A}$ in mitigating the issue of robust overfitting. Beyond its application in adversarial training, this represents a fundamental result in uniform stability analysis, as ME-$\mathcal{A}$ is the first algorithm to exhibit uniform stability for weakly-convex, non-smooth problems.

Kihyun Kim, Jiawei Zhang, Asuman Ozdaglar et al.

Conventional preference learning methods often prioritize opinions held more widely when aggregating preferences from multiple evaluators. This may result in policies that are biased in favor of some types of opinions or groups and susceptible to strategic manipulation. To address this issue, we develop a novel preference learning framework capable of aligning aggregate opinions and policies proportionally with the true population distribution of evaluator preferences. Grounded in social choice theory, our approach infers the feasible set of evaluator population distributions directly from pairwise comparison data. Using these estimates, the algorithm constructs a policy that satisfies foundational axioms from social choice theory, namely monotonicity and Pareto efficiency, as well as our newly-introduced axioms of population-proportional alignment and population-bounded manipulability. Moreover, we propose a soft-max relaxation method that smoothly trade-offs population-proportional alignment with the selection of the Condorcet winner (which beats all other options in pairwise comparisons). Finally, we validate the effectiveness and scalability of our approach through experiments on both tabular recommendation tasks and large language model alignment.

Runyu Zhang, Na Li, Asuman Ozdaglar et al.

Risk sensitivity has become a central theme in reinforcement learning (RL), where convex risk measures and robust formulations provide principled ways to model preferences beyond expected return. Recent extensions to multi-agent RL (MARL) have largely emphasized the risk-averse setting, prioritizing robustness to uncertainty. In cooperative MARL, however, such conservatism often leads to suboptimal equilibria, and a parallel line of work has shown that optimism can promote cooperation. Existing optimistic methods, though effective in practice, are typically heuristic and lack theoretical grounding. Building on the dual representation for convex risk measures, we propose a principled framework that interprets risk-seeking objectives as optimism. We introduce optimistic value functions, which formalize optimism as divergence-penalized risk-seeking evaluations. Building on this foundation, we derive a policy-gradient theorem for optimistic value functions, including explicit formulas for the entropic risk/KL-penalty setting, and develop decentralized optimistic actor-critic algorithms that implement these updates. Empirical results on cooperative benchmarks demonstrate that risk-seeking optimism consistently improves coordination over both risk-neutral baselines and heuristic optimistic methods. Our framework thus unifies risk-sensitive learning and optimism, offering a theoretically grounded and practically effective approach to cooperation in MARL.

Omar Bennouna, Amine Bennouna, Saurabh Amin et al.

We study the fundamental question of how informative a dataset is for solving a given decision-making task. In our setting, the dataset provides partial information about unknown parameters that influence task outcomes. Focusing on linear programs, we characterize when a dataset is sufficient to recover an optimal decision, given an uncertainty set on the cost vector. Our main contribution is a sharp geometric characterization that identifies the directions of the cost vector that matter for optimality, relative to the task constraints and uncertainty set. We further develop a practical algorithm that, for a given task, constructs a minimal or least-costly sufficient dataset. Our results reveal that small, well-chosen datasets can often fully determine optimal decisions -- offering a principled foundation for task-aware data selection.

Mingyang Liu, Gabriele Farina, Asuman Ozdaglar

We study equilibrium finding in polymatrix games under differential privacy constraints. To start, we show that high accuracy and asymptotically vanishing differential privacy budget (as the number of players goes to infinity) cannot be achieved simultaneously under either of the two settings: (i) We seek to establish equilibrium approximation guarantees in terms of Euclidean distance to the equilibrium set, and (ii) the adversary has access to all communication channels. Then, assuming the adversary has access to a constant number of communication channels, we develop a novel distributed algorithm that recovers strategies with simultaneously vanishing Nash gap (in expected utility, also referred to as exploitability and privacy budget as the number of players increases.

Alireza Fallah, Ali Makhdoumi, Azarakhsh Malekian et al.

We consider a platform's problem of collecting data from privacy sensitive users to estimate an underlying parameter of interest. We formulate this question as a Bayesian-optimal mechanism design problem, in which an individual can share her (verifiable) data in exchange for a monetary reward or services, but at the same time has a (private) heterogeneous privacy cost which we quantify using differential privacy. We consider two popular differential privacy settings for providing privacy guarantees for the users: central and local. In both settings, we establish minimax lower bounds for the estimation error and derive (near) optimal estimators for given heterogeneous privacy loss levels for users. Building on this characterization, we pose the mechanism design problem as the optimal selection of an estimator and payments that will elicit truthful reporting of users' privacy sensitivities. Under a regularity condition on the distribution of privacy sensitivities we develop efficient algorithmic mechanisms to solve this problem in both privacy settings. Our mechanism in the central setting can be implemented in time $\mathcal{O}(n \log n)$ where $n$ is the number of users and our mechanism in the local setting admits a Polynomial Time Approximation Scheme (PTAS).

Asuman Ozdaglar, Muhammed O. Sayin, Kaiqing Zhang

Reinforcement learning (RL) has recently achieved tremendous successes in many artificial intelligence applications. Many of the forefront applications of RL involve multiple agents, e.g., playing chess and Go games, autonomous driving, and robotics. Unfortunately, the framework upon which classical RL builds is inappropriate for multi-agent learning, as it assumes an agent's environment is stationary and does not take into account the adaptivity of other agents. In this review paper, we present the model of stochastic games for multi-agent learning in dynamic environments. We focus on the development of simple and independent learning dynamics for stochastic games: each agent is myopic and chooses best-response type actions to other agents' strategy without any coordination with her opponent. There has been limited progress on developing convergent best-response type independent learning dynamics for stochastic games. We present our recently proposed simple and independent learning dynamics that guarantee convergence in zero-sum stochastic games, together with a review of other contemporaneous algorithms for dynamic multi-agent learning in this setting. Along the way, we also reexamine some classical results from both the game theory and RL literature, to situate both the conceptual contributions of our independent learning dynamics, and the mathematical novelties of our analysis. We hope this review paper serves as an impetus for the resurgence of studying independent and natural learning dynamics in game theory, for the more challenging settings with a dynamic environment.

Theo Diamandis, Yonina C. Eldar, Alireza Fallah et al.

Multi-modal distributions are commonly used to model clustered data in statistical learning tasks. In this paper, we consider the Mixed Linear Regression (MLR) problem. We propose an optimal transport-based framework for MLR problems, Wasserstein Mixed Linear Regression (WMLR), which minimizes the Wasserstein distance between the learned and target mixture regression models. Through a model-based duality analysis, WMLR reduces the underlying MLR task to a nonconvex-concave minimax optimization problem, which can be provably solved to find a minimax stationary point by the Gradient Descent Ascent (GDA) algorithm. In the special case of mixtures of two linear regression models, we show that WMLR enjoys global convergence and generalization guarantees. We prove that WMLR's sample complexity grows linearly with the dimension of data. Finally, we discuss the application of WMLR to the federated learning task where the training samples are collected by multiple agents in a network. Unlike the Expectation Maximization algorithm, WMLR directly extends to the distributed, federated learning setting. We support our theoretical results through several numerical experiments, which highlight our framework's ability to handle the federated learning setting with mixture models.

Muhammed O. Sayin, Kaiqing Zhang, David S. Leslie et al.

We study multi-agent reinforcement learning (MARL) in infinite-horizon discounted zero-sum Markov games. We focus on the practical but challenging setting of decentralized MARL, where agents make decisions without coordination by a centralized controller, but only based on their own payoffs and local actions executed. The agents need not observe the opponent's actions or payoffs, possibly being even oblivious to the presence of the opponent, nor be aware of the zero-sum structure of the underlying game, a setting also referred to as radically uncoupled in the literature of learning in games. In this paper, we develop a radically uncoupled Q-learning dynamics that is both rational and convergent: the learning dynamics converges to the best response to the opponent's strategy when the opponent follows an asymptotically stationary strategy; when both agents adopt the learning dynamics, they converge to the Nash equilibrium of the game. The key challenge in this decentralized setting is the non-stationarity of the environment from an agent's perspective, since both her own payoffs and the system evolution depend on the actions of other agents, and each agent adapts her policies simultaneously and independently. To address this issue, we develop a two-timescale learning dynamics where each agent updates her local Q-function and value function estimates concurrently, with the latter happening at a slower timescale.

Alireza Fallah, Aryan Mokhtari, Asuman Ozdaglar

In this paper, we study the generalization properties of Model-Agnostic Meta-Learning (MAML) algorithms for supervised learning problems. We focus on the setting in which we train the MAML model over $m$ tasks, each with $n$ data points, and characterize its generalization error from two points of view: First, we assume the new task at test time is one of the training tasks, and we show that, for strongly convex objective functions, the expected excess population loss is bounded by ${\mathcal{O}}(1/mn)$. Second, we consider the MAML algorithm's generalization to an unseen task and show that the resulting generalization error depends on the total variation distance between the underlying distributions of the new task and the tasks observed during the training process. Our proof techniques rely on the connections between algorithmic stability and generalization bounds of algorithms. In particular, we propose a new definition of stability for meta-learning algorithms, which allows us to capture the role of both the number of tasks $m$ and number of samples per task $n$ on the generalization error of MAML.

Farzan Farnia, Asuman Ozdaglar

The success of minimax learning problems of generative adversarial networks (GANs) has been observed to depend on the minimax optimization algorithm used for their training. This dependence is commonly attributed to the convergence speed and robustness properties of the underlying optimization algorithm. In this paper, we show that the optimization algorithm also plays a key role in the generalization performance of the trained minimax model. To this end, we analyze the generalization properties of standard gradient descent ascent (GDA) and proximal point method (PPM) algorithms through the lens of algorithmic stability under both convex concave and non-convex non-concave minimax settings. While the GDA algorithm is not guaranteed to have a vanishing excess risk in convex concave problems, we show the PPM algorithm enjoys a bounded excess risk in the same setup. For non-convex non-concave problems, we compare the generalization performance of stochastic GDA and GDmax algorithms where the latter fully solves the maximization subproblem at every iteration. Our generalization analysis suggests the superiority of GDA provided that the minimization and maximization subproblems are solved simultaneously with similar learning rates. We discuss several numerical results indicating the role of optimization algorithms in the generalization of the learned minimax models.

Manxi Wu, Saurabh Amin, Asuman Ozdaglar

We study learning dynamics induced by strategic agents who repeatedly play a game with an unknown payoff-relevant parameter. In each step, an information system estimates a belief distribution of the parameter based on the players' strategies and realized payoffs using Bayes' rule. Players adjust their strategies by accounting for an equilibrium strategy or a best response strategy based on the updated belief. We prove that beliefs and strategies converge to a fixed point with probability 1. We also provide conditions that guarantee local and global stability of fixed points. Any fixed point belief consistently estimates the payoff distribution given the fixed point strategy profile. However, convergence to a complete information Nash equilibrium is not always guaranteed. We provide a sufficient and necessary condition under which fixed point belief recovers the unknown parameter. We also provide a sufficient condition for convergence to complete information equilibrium even when parameter learning is incomplete.

Muhammed O. Sayin, Francesca Parise, Asuman Ozdaglar

We present a novel variant of fictitious play dynamics combining classical fictitious play with Q-learning for stochastic games and analyze its convergence properties in two-player zero-sum stochastic games. Our dynamics involves players forming beliefs on the opponent strategy and their own continuation payoff (Q-function), and playing a greedy best response by using the estimated continuation payoffs. Players update their beliefs from observations of opponent actions. A key property of the learning dynamics is that update of the beliefs on Q-functions occurs at a slower timescale than update of the beliefs on strategies. We show both in the model-based and model-free cases (without knowledge of player payoff functions and state transition probabilities), the beliefs on strategies converge to a stationary mixed Nash equilibrium of the zero-sum stochastic game.

Farzan Farnia, Asuman Ozdaglar

Generative adversarial networks (GANs) represent a zero-sum game between two machine players, a generator and a discriminator, designed to learn the distribution of data. While GANs have achieved state-of-the-art performance in several benchmark learning tasks, GAN minimax optimization still poses great theoretical and empirical challenges. GANs trained using first-order optimization methods commonly fail to converge to a stable solution where the players cannot improve their objective, i.e., the Nash equilibrium of the underlying game. Such issues raise the question of the existence of Nash equilibrium solutions in the GAN zero-sum game. In this work, we show through several theoretical and numerical results that indeed GAN zero-sum games may not have any local Nash equilibria. To characterize an equilibrium notion applicable to GANs, we consider the equilibrium of a new zero-sum game with an objective function given by a proximal operator applied to the original objective, a solution we call the proximal equilibrium. Unlike the Nash equilibrium, the proximal equilibrium captures the sequential nature of GANs, in which the generator moves first followed by the discriminator. We prove that the optimal generative model in Wasserstein GAN problems provides a proximal equilibrium. Inspired by these results, we propose a new approach, which we call proximal training, for solving GAN problems. We discuss several numerical experiments demonstrating the existence of proximal equilibrium solutions in GAN minimax problems.

Alireza Fallah, Aryan Mokhtari, Asuman Ozdaglar

In Federated Learning, we aim to train models across multiple computing units (users), while users can only communicate with a common central server, without exchanging their data samples. This mechanism exploits the computational power of all users and allows users to obtain a richer model as their models are trained over a larger set of data points. However, this scheme only develops a common output for all the users, and, therefore, it does not adapt the model to each user. This is an important missing feature, especially given the heterogeneity of the underlying data distribution for various users. In this paper, we study a personalized variant of the federated learning in which our goal is to find an initial shared model that current or new users can easily adapt to their local dataset by performing one or a few steps of gradient descent with respect to their own data. This approach keeps all the benefits of the federated learning architecture, and, by structure, leads to a more personalized model for each user. We show this problem can be studied within the Model-Agnostic Meta-Learning (MAML) framework. Inspired by this connection, we study a personalized variant of the well-known Federated Averaging algorithm and evaluate its performance in terms of gradient norm for non-convex loss functions. Further, we characterize how this performance is affected by the closeness of underlying distributions of user data, measured in terms of distribution distances such as Total Variation and 1-Wasserstein metric.

Alireza Fallah, Asuman Ozdaglar, Sarath Pattathil

In this paper, we study the minimax optimization problem in the smooth and strongly convex-strongly concave setting when we have access to noisy estimates of gradients. In particular, we first analyze the stochastic Gradient Descent Ascent (GDA) method with constant stepsize, and show that it converges to a neighborhood of the solution of the minimax problem. We further provide tight bounds on the convergence rate and the size of this neighborhood. Next, we propose a multistage variant of stochastic GDA (M-GDA) that runs in multiple stages with a particular learning rate decay schedule and converges to the exact solution of the minimax problem. We show M-GDA achieves the lower bounds in terms of noise dependence without any assumptions on the knowledge of noise characteristics. We also show that M-GDA obtains a linear decay rate with respect to the error's dependence on the initial error, although the dependence on condition number is suboptimal. In order to improve this dependence, we apply the multistage machinery to the stochastic Optimistic Gradient Descent Ascent (OGDA) algorithm and propose the M-OGDA algorithm which also achieves the optimal linear decay rate with respect to the initial error. To the best of our knowledge, this method is the first to simultaneously achieve the best dependence on noise characteristic as well as the initial error and condition number.

Alireza Fallah, Kristian Georgiev, Aryan Mokhtari et al.

We consider Model-Agnostic Meta-Learning (MAML) methods for Reinforcement Learning (RL) problems, where the goal is to find a policy using data from several tasks represented by Markov Decision Processes (MDPs) that can be updated by one step of stochastic policy gradient for the realized MDP. In particular, using stochastic gradients in MAML update steps is crucial for RL problems since computation of exact gradients requires access to a large number of possible trajectories. For this formulation, we propose a variant of the MAML method, named Stochastic Gradient Meta-Reinforcement Learning (SG-MRL), and study its convergence properties. We derive the iteration and sample complexity of SG-MRL to find an $ε$-first-order stationary point, which, to the best of our knowledge, provides the first convergence guarantee for model-agnostic meta-reinforcement learning algorithms. We further show how our results extend to the case where more than one step of stochastic policy gradient method is used at test time. Finally, we empirically compare SG-MRL and MAML in several deep RL environments.

Noah Golowich, Sarath Pattathil, Constantinos Daskalakis et al.

In this paper we study the smooth convex-concave saddle point problem. Specifically, we analyze the last iterate convergence properties of the Extragradient (EG) algorithm. It is well known that the ergodic (averaged) iterates of EG converge at a rate of $O(1/T)$ (Nemirovski, 2004). In this paper, we show that the last iterate of EG converges at a rate of $O(1/\sqrt{T})$. To the best of our knowledge, this is the first paper to provide a convergence rate guarantee for the last iterate of EG for the smooth convex-concave saddle point problem. Moreover, we show that this rate is tight by proving a lower bound of $Ω(1/\sqrt{T})$ for the last iterate. This lower bound therefore shows a quadratic separation of the convergence rates of ergodic and last iterates in smooth convex-concave saddle point problems.

Weijie Liu, Aryan Mokhtari, Asuman Ozdaglar et al.

In this paper, we focus on solving a class of constrained non-convex non-concave saddle point problems in a decentralized manner by a group of nodes in a network. Specifically, we assume that each node has access to a summand of a global objective function and nodes are allowed to exchange information only with their neighboring nodes. We propose a decentralized variant of the proximal point method for solving this problem. We show that when the objective function is $ρ$-weakly convex-weakly concave the iterates converge to approximate stationarity with a rate of $\mathcal{O}(1/\sqrt{T})$ where the approximation error depends linearly on $\sqrtρ$. We further show that when the objective function satisfies the Minty VI condition (which generalizes the convex-concave case) we obtain convergence to stationarity with a rate of $\mathcal{O}(1/\sqrt{T})$. To the best of our knowledge, our proposed method is the first decentralized algorithm with theoretical guarantees for solving a non-convex non-concave decentralized saddle point problem. Our numerical results for training a general adversarial network (GAN) in a decentralized manner match our theoretical guarantees.

Alireza Fallah, Mert Gurbuzbalaban, Asuman Ozdaglar et al.

We study distributed stochastic gradient (D-SG) method and its accelerated variant (D-ASG) for solving decentralized strongly convex stochastic optimization problems where the objective function is distributed over several computational units, lying on a fixed but arbitrary connected communication graph, subject to local communication constraints where noisy estimates of the gradients are available. We develop a framework which allows to choose the stepsize and the momentum parameters of these algorithms in a way to optimize performance by systematically trading off the bias, variance, robustness to gradient noise and dependence to network effects. When gradients do not contain noise, we also prove that distributed accelerated methods can \emph{achieve acceleration}, requiring $\mathcal{O}(κ\log(1/\varepsilon))$ gradient evaluations and $\mathcal{O}(κ\log(1/\varepsilon))$ communications to converge to the same fixed point with the non-accelerated variant where $κ$ is the condition number and $\varepsilon$ is the target accuracy. To our knowledge, this is the first acceleration result where the iteration complexity scales with the square root of the condition number in the context of \emph{primal} distributed inexact first-order methods. For quadratic functions, we also provide finer performance bounds that are tight with respect to bias and variance terms. Finally, we study a multistage version of D-ASG with parameters carefully varied over stages to ensure exact $\mathcal{O}(-k/\sqrtκ)$ linear decay in the bias term as well as optimal $\mathcal{O}(σ^2/k)$ in the variance term. We illustrate through numerical experiments that our approach results in practical algorithms that are robust to gradient noise and that can outperform existing methods.

Alireza Fallah, Aryan Mokhtari, Asuman Ozdaglar

We study the convergence of a class of gradient-based Model-Agnostic Meta-Learning (MAML) methods and characterize their overall complexity as well as their best achievable accuracy in terms of gradient norm for nonconvex loss functions. We start with the MAML method and its first-order approximation (FO-MAML) and highlight the challenges that emerge in their analysis. By overcoming these challenges not only we provide the first theoretical guarantees for MAML and FO-MAML in nonconvex settings, but also we answer some of the unanswered questions for the implementation of these algorithms including how to choose their learning rate and the batch size for both tasks and datasets corresponding to tasks. In particular, we show that MAML can find an $ε$-first-order stationary point ($ε$-FOSP) for any positive $ε$ after at most $\mathcal{O}(1/ε^2)$ iterations at the expense of requiring second-order information. We also show that FO-MAML which ignores the second-order information required in the update of MAML cannot achieve any small desired level of accuracy, i.e., FO-MAML cannot find an $ε$-FOSP for any $ε>0$. We further propose a new variant of the MAML algorithm called Hessian-free MAML which preserves all theoretical guarantees of MAML, without requiring access to second-order information.