Ian Gemp

Marco Jiralerspong, Avishek Joey Bose, Ian Gemp et al.

The past few years have seen impressive progress in the development of deep generative models capable of producing high-dimensional, complex, and photo-realistic data. However, current methods for evaluating such models remain incomplete: standard likelihood-based metrics do not always apply and rarely correlate with perceptual fidelity, while sample-based metrics, such as FID, are insensitive to overfitting, i.e., inability to generalize beyond the training set. To address these limitations, we propose a new metric called the Feature Likelihood Divergence (FLD), a parametric sample-based metric that uses density estimation to provide a comprehensive trichotomic evaluation accounting for novelty (i.e., different from the training samples), fidelity, and diversity of generated samples. We empirically demonstrate the ability of FLD to identify overfitting problem cases, even when previously proposed metrics fail. We also extensively evaluate FLD on various image datasets and model classes, demonstrating its ability to match intuitions of previous metrics like FID while offering a more comprehensive evaluation of generative models. Code is available at https://github.com/marcojira/fld.

Ian Gemp, Thomas Anthony, Yoram Bachrach et al. · deepmind

The Game Theory & Multi-Agent team at DeepMind studies several aspects of multi-agent learning ranging from computing approximations to fundamental concepts in game theory to simulating social dilemmas in rich spatial environments and training 3-d humanoids in difficult team coordination tasks. A signature aim of our group is to use the resources and expertise made available to us at DeepMind in deep reinforcement learning to explore multi-agent systems in complex environments and use these benchmarks to advance our understanding. Here, we summarise the recent work of our team and present a taxonomy that we feel highlights many important open challenges in multi-agent research.

Luke Marris, Ian Gemp, Thomas Anthony et al.

Solution concepts such as Nash Equilibria, Correlated Equilibria, and Coarse Correlated Equilibria are useful components for many multiagent machine learning algorithms. Unfortunately, solving a normal-form game could take prohibitive or non-deterministic time to converge, and could fail. We introduce the Neural Equilibrium Solver which utilizes a special equivariant neural network architecture to approximately solve the space of all games of fixed shape, buying speed and determinism. We define a flexible equilibrium selection framework, that is capable of uniquely selecting an equilibrium that minimizes relative entropy, or maximizes welfare. The network is trained without needing to generate any supervised training data. We show remarkable zero-shot generalization to larger games. We argue that such a network is a powerful component for many possible multiagent algorithms.

Zun Li, Marc Lanctot, Kevin R. McKee et al.

Opponent modeling methods typically involve two crucial steps: building a belief distribution over opponents' strategies, and exploiting this opponent model by playing a best response. However, existing approaches typically require domain-specific heurstics to come up with such a model, and algorithms for approximating best responses are hard to scale in large, imperfect information domains. In this work, we introduce a scalable and generic multiagent training regime for opponent modeling using deep game-theoretic reinforcement learning. We first propose Generative Best Respoonse (GenBR), a best response algorithm based on Monte-Carlo Tree Search (MCTS) with a learned deep generative model that samples world states during planning. This new method scales to large imperfect information domains and can be plug and play in a variety of multiagent algorithms. We use this new method under the framework of Policy Space Response Oracles (PSRO), to automate the generation of an \emph{offline opponent model} via iterative game-theoretic reasoning and population-based training. We propose using solution concepts based on bargaining theory to build up an opponent mixture, which we find identifying profiles that are near the Pareto frontier. Then GenBR keeps updating an \emph{online opponent model} and reacts against it during gameplay. We conduct behavioral studies where human participants negotiate with our agents in Deal-or-No-Deal, a class of bilateral bargaining games. Search with generative modeling finds stronger policies during both training time and test time, enables online Bayesian co-player prediction, and can produce agents that achieve comparable social welfare and Nash bargaining score negotiating with humans as humans trading among themselves.

Luke Marris, Marc Lanctot, Ian Gemp et al.

Rating strategies in a game is an important area of research in game theory and artificial intelligence, and can be applied to any real-world competitive or cooperative setting. Traditionally, only transitive dependencies between strategies have been used to rate strategies (e.g. Elo), however recent work has expanded ratings to utilize game theoretic solutions to better rate strategies in non-transitive games. This work generalizes these ideas and proposes novel algorithms suitable for N-player, general-sum rating of strategies in normal-form games according to the payoff rating system. This enables well-established solution concepts, such as equilibria, to be leveraged to efficiently rate strategies in games with complex strategic interactions, which arise in multiagent training and real-world interactions between many agents. We empirically validate our methods on real world normal-form data (Premier League) and multiagent reinforcement learning agent evaluation.

Elise van der Pol, Ian Gemp, Yoram Bachrach et al.

Large graphs commonly appear in social networks, knowledge graphs, recommender systems, life sciences, and decision making problems. Summarizing large graphs by their high level properties is helpful in solving problems in these settings. In spectral clustering, we aim to identify clusters of nodes where most edges fall within clusters and only few edges fall between clusters. This task is important for many downstream applications and exploratory analysis. A core step of spectral clustering is performing an eigendecomposition of the corresponding graph Laplacian matrix (or equivalently, a singular value decomposition, SVD, of the incidence matrix). The convergence of iterative singular value decomposition approaches depends on the eigengaps of the spectrum of the given matrix, i.e., the difference between consecutive eigenvalues. For a graph Laplacian corresponding to a well-clustered graph, the eigenvalues will be non-negative but very small (much less than $1$) slowing convergence. This paper introduces a parallelizable approach to dilating the spectrum in order to accelerate SVD solvers and in turn, spectral clustering. This is accomplished via polynomial approximations to matrix operations that favorably transform the spectrum of a matrix without changing its eigenvectors. Experiments demonstrate that this approach significantly accelerates convergence, and we explain how this transformation can be parallelized and stochastically approximated to scale with available compute.

Ian Gemp, Charlie Chen, Brian McWilliams

The symmetric generalized eigenvalue problem (SGEP) is a fundamental concept in numerical linear algebra. It captures the solution of many classical machine learning problems such as canonical correlation analysis, independent components analysis, partial least squares, linear discriminant analysis, principal components and others. Despite this, most general solvers are prohibitively expensive when dealing with streaming data sets (i.e., minibatches) and research has instead concentrated on finding efficient solutions to specific problem instances. In this work, we develop a game-theoretic formulation of the top-$k$ SGEP whose Nash equilibrium is the set of generalized eigenvectors. We also present a parallelizable algorithm with guaranteed asymptotic convergence to the Nash. Current state-of-the-art methods require $O(d^2k)$ runtime complexity per iteration which is prohibitively expensive when the number of dimensions ($d$) is large. We show how to modify this parallel approach to achieve $O(dk)$ runtime complexity. Empirically we demonstrate that this resulting algorithm is able to solve a variety of SGEP problem instances including a large-scale analysis of neural network activations.

Marc Lanctot, Kate Larson, Ian Gemp et al.

As intelligent agents become more generally-capable, i.e. able to master a wide variety of tasks, the complexity and cost of properly evaluating them rises significantly. Tasks that assess specific capabilities of the agents can be correlated and stochastic, requiring many samples for accurate comparisons, leading to added costs. In this paper, we propose a formal definition and a conceptual framework for active evaluation of agents across multiple tasks, which assesses the performance of ranking algorithms as a function of number of evaluation data samples. Rather than curating, filtering, or compressing existing data sets as a preprocessing step, we propose an online framing: on every iteration, the ranking algorithm chooses the task and agents to sample scores from. Then, evaluation algorithms report a ranking of agents on each iteration and their performance is assessed with respect to the ground truth ranking over time. Several baselines are compared under different experimental contexts, with synthetic generated data and simulated online access to real evaluation data from Atari game-playing agents. We find that the classical Elo rating system -- while it suffers from well-known failure modes, in theory -- is a consistently reliable choice for efficient reduction of ranking error in practice. A recently-proposed method, Soft Condorcet Optimization, shows comparable performance to Elo on synthetic data and significantly outperforms Elo on real Atari agent evaluation. When task variation from the ground truth is high, selecting tasks based on proportional representation leads to higher rate of ranking error reduction.

Gheorghe Comanici, Eric Bieber, Mike Schaekermann et al. · amazon-science, baidu

In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal understanding and it is now able to process up to 3 hours of video content. Its unique combination of long context, multimodal and reasoning capabilities can be combined to unlock new agentic workflows. Gemini 2.5 Flash provides excellent reasoning abilities at a fraction of the compute and latency requirements and Gemini 2.0 Flash and Flash-Lite provide high performance at low latency and cost. Taken together, the Gemini 2.X model generation spans the full Pareto frontier of model capability vs cost, allowing users to explore the boundaries of what is possible with complex agentic problem solving.

Kevin Du, Ian Gemp, Yi Wu et al.

Reinforcement learning has recently been used to approach well-known NP-hard combinatorial problems in graph theory. Among these problems, Hamiltonian cycle problems are exceptionally difficult to analyze, even when restricted to individual instances of structurally complex graphs. In this paper, we use Monte Carlo Tree Search (MCTS), the search algorithm behind many state-of-the-art reinforcement learning algorithms such as AlphaZero, to create autonomous agents that learn to play the game of Snake, a game centered on properties of Hamiltonian cycles on grid graphs. The game of Snake can be formulated as a single-player discounted Markov Decision Process (MDP) where the agent must behave optimally in a stochastic environment. Determining the optimal policy for Snake, defined as the policy that maximizes the probability of winning - or win rate - with higher priority and minimizes the expected number of time steps to win with lower priority, is conjectured to be NP-hard. Performance-wise, compared to prior work in the Snake game, our algorithm is the first to achieve a win rate over $0.5$ (a uniform random policy achieves a win rate $< 2.57 \times 10^{-15}$), demonstrating the versatility of AlphaZero in approaching NP-hard environments.

Ian Gemp, Crystal Qian, Marc Lanctot et al.

Nash equilibrium serves as a fundamental mathematical tool in economics and game theory. However, it classically assumes knowledge of player utilities, whereas economics generally regards preferences as more fundamental. To leverage equilibrium analysis in strategic scenarios, one must first elicit numerical utilities consistent with player preferences, a delicate and time-consuming process. In this work, we forgo precise utilities and generalize the Nash equilibrium to a setting where we only assume a player is capable of providing an ordinal ranking of their actions within the context of other players' joint actions. The key technical challenge is to rethink the definition of a best-response. While the classical definition identifies actions maximizing expected payoff, we naturally look towards social choice theory for how to aggregate preferences to identify the most preferred actions. We define this generalized notion of a context-ordinal Nash equilibrium, establish its existence under mild conditions on aggregation methods, introduce notions of regularization, approximation, and regret, explore complexity for simple settings, and develop learning rules for computing such equilibria. In doing so, we provide a generalization of Nash equilibrium and demonstrate its direct applicability to elicited preferences in human experiments.

Siqi Liu, Ian Gemp, Luke Marris et al.

Evaluation has traditionally focused on ranking candidates for a specific skill. Modern generalist models, such as Large Language Models (LLMs), decidedly outpace this paradigm. Open-ended evaluation systems, where candidate models are compared on user-submitted prompts, have emerged as a popular solution. Despite their many advantages, we show that the current Elo-based rating systems can be susceptible to and even reinforce biases in data, intentional or accidental, due to their sensitivity to redundancies. To address this issue, we propose evaluation as a 3-player game, and introduce novel game-theoretic solution concepts to ensure robustness to redundancy. We show that our method leads to intuitive ratings and provide insights into the competitive landscape of LLM development.

Fivos Kalogiannis, Emmanouil-Vasileios Vlatakis-Gkaragkounis, Ian Gemp et al.

We contribute the first provable guarantees of global convergence to Nash equilibria (NE) in two-player zero-sum convex Markov games (cMGs) by using independent policy gradient methods. Convex Markov games, recently defined by Gemp et al. (2024), extend Markov decision processes to multi-agent settings with preferences that are convex over occupancy measures, offering a broad framework for modeling generic strategic interactions. However, even the fundamental min-max case of cMGs presents significant challenges, including inherent nonconvexity, the absence of Bellman consistency, and the complexity of the infinite horizon. We follow a two-step approach. First, leveraging properties of hidden-convex--hidden-concave functions, we show that a simple nonconvex regularization transforms the min-max optimization problem into a nonconvex-proximal Polyak-Lojasiewicz (NC-pPL) objective. Crucially, this regularization can stabilize the iterates of independent policy gradient methods and ultimately lead them to converge to equilibria. Second, building on this reduction, we address the general constrained min-max problems under NC-pPL and two-sided pPL conditions, providing the first global convergence guarantees for stochastic nested and alternating gradient descent-ascent methods, which we believe may be of independent interest.

Constantinos Daskalakis, Ian Gemp, Yanchen Jiang et al.

Stories are records of our experiences and their analysis reveals insights into the nature of being human. Successful analyses are often interdisciplinary, leveraging mathematical tools to extract structure from stories and insights from structure. Historically, these tools have been restricted to one dimensional charts and dynamic social networks; however, modern AI offers the possibility of identifying more fully the plot structure, character incentives, and, importantly, counterfactual plot lines that the story could have taken but did not take. In this work, we use AI to model the structure of stories as game-theoretic objects, amenable to quantitative analysis. This allows us to not only interrogate each character's decision making, but also possibly peer into the original author's conception of the characters' world. We demonstrate our proposed technique on Shakespeare's famous Romeo and Juliet. We conclude with a discussion of how our analysis could be replicated in broader contexts, including real-life scenarios.

Philipp D. Siedler, Ian Gemp

In complex multi-agent environments, achieving efficient learning and desirable behaviours is a significant challenge for Multi-Agent Reinforcement Learning (MARL) systems. This work explores the potential of combining MARL with Large Language Model (LLM)-mediated interventions to guide agents toward more desirable behaviours. Specifically, we investigate how LLMs can be used to interpret and facilitate interventions that shape the learning trajectories of multiple agents. We experimented with two types of interventions, referred to as controllers: a Natural Language (NL) Controller and a Rule-Based (RB) Controller. The NL Controller, which uses an LLM to simulate human-like interventions, showed a stronger impact than the RB Controller. Our findings indicate that agents particularly benefit from early interventions, leading to more efficient training and higher performance. Both intervention types outperform the baseline without interventions, highlighting the potential of LLM-mediated guidance to accelerate training and enhance MARL performance in challenging environments.

Luke Marris, Siqi Liu, Ian Gemp et al.

Many real-world multi-agent or multi-task evaluation scenarios can be naturally modelled as normal-form games due to inherent strategic (adversarial, cooperative, and mixed motive) interactions. These strategic interactions may be agentic (e.g. players trying to win), fundamental (e.g. cost vs quality), or complementary (e.g. niche finding and specialization). In such a formulation, it is the strategies (actions, policies, agents, models, tasks, prompts, etc.) that are rated. However, the rating problem is complicated by redundancy and complexity of N-player strategic interactions. Repeated or similar strategies can distort ratings for those that counter or complement them. Previous work proposed ``clone invariant'' ratings to handle such redundancies, but this was limited to two-player zero-sum (i.e. strictly competitive) interactions. This work introduces the first N-player general-sum clone invariant rating, called deviation ratings, based on coarse correlated equilibria. The rating is explored on several domains including LLMs evaluation.

Marc Lanctot, Kate Larson, Michael Kaisers et al.

Driving progress of AI models and agents requires comparing their performance on standardized benchmarks; for general agents, individual performances must be aggregated across a potentially wide variety of different tasks. In this paper, we describe a novel ranking scheme inspired by social choice frameworks, called Soft Condorcet Optimization (SCO), to compute the optimal ranking of agents: the one that makes the fewest mistakes in predicting the agent comparisons in the evaluation data. This optimal ranking is the maximum likelihood estimate when evaluation data (which we view as votes) are interpreted as noisy samples from a ground truth ranking, a solution to Condorcet's original voting system criteria. SCO ratings are maximal for Condorcet winners when they exist, which we show is not necessarily true for the classical rating system Elo. We propose three optimization algorithms to compute SCO ratings and evaluate their empirical performance. When serving as an approximation to the Kemeny-Young voting method, SCO rankings are on average 0 to 0.043 away from the optimal ranking in normalized Kendall-tau distance across 865 preference profiles from the PrefLib open ranking archive. In a simulated noisy tournament setting, SCO achieves accurate approximations to the ground truth ranking and the best among several baselines when 59\% or more of the preference data is missing. Finally, SCO ranking provides the best approximation to the optimal ranking, measured on held-out test sets, in a problem containing 52,958 human players across 31,049 games of the classic seven-player game of Diplomacy.

Ian Gemp, Andreas Haupt, Luke Marris et al.

Behavioral diversity, expert imitation, fairness, safety goals and others give rise to preferences in sequential decision making domains that do not decompose additively across time. We introduce the class of convex Markov games that allow general convex preferences over occupancy measures. Despite infinite time horizon and strictly higher generality than Markov games, pure strategy Nash equilibria exist. Furthermore, equilibria can be approximated empirically by performing gradient descent on an upper bound of exploitability. Our experiments reveal novel solutions to classic repeated normal-form games, find fair solutions in a repeated asymmetric coordination game, and prioritize safe long-term behavior in a robot warehouse environment. In the prisoner's dilemma, our algorithm leverages transient imitation to find a policy profile that deviates from observed human play only slightly, yet achieves higher per-player utility while also being three orders of magnitude less exploitable.

Wolfgang Lehrach, Daniel Hennes, Miguel Lazaro-Gredilla et al.

Large Language Models (LLMs) reasoning abilities are increasingly being applied to classical board and card games, but the dominant approach -- involving prompting for direct move generation -- has significant drawbacks. It relies on the model's implicit fragile pattern-matching capabilities, leading to frequent illegal moves and strategically shallow play. Here we introduce an alternative approach: We use the LLM to translate natural language rules and game trajectories into a formal, executable world model represented as Python code. This generated model -- comprising functions for state transition, legal move enumeration, and termination checks -- serves as a verifiable simulation engine for high-performance planning algorithms like Monte Carlo tree search (MCTS). In addition, we prompt the LLM to generate heuristic value functions (to make MCTS more efficient), and inference functions (to estimate hidden states in imperfect information games). Our method offers three distinct advantages compared to directly using the LLM as a policy: (1) Verifiability: The generated CWM serves as a formal specification of the game's rules, allowing planners to algorithmically enumerate valid actions and avoid illegal moves, contingent on the correctness of the synthesized model; (2) Strategic Depth: We combine LLM semantic understanding with the deep search power of classical planners; and (3) Generalization: We direct the LLM to focus on the meta-task of data-to-code translation, enabling it to adapt to new games more easily. We evaluate our agent on 10 different games, of which 4 are novel and created for this paper. 5 of the games are fully observed (perfect information), and 5 are partially observed (imperfect information). We find that our method outperforms or matches Gemini 2.5 Pro in 9 out of the 10 considered games.

Ian Gemp

This work proposes a novel set of techniques for approximating a Nash equilibrium in a finite, normal-form game. It achieves this by constructing a new reformulation as solving a parameterized system of multivariate polynomials with tunable complexity. In doing so, it forges an itinerant loop from game theory to machine learning and back. We show a Nash equilibrium can be approximated with purely calls to stochastic, iterative variants of singular value decomposition and power iteration, with implications for biological plausibility. We provide pseudocode and experiments demonstrating solving for all equilibria of a general-sum game using only these readily available linear algebra tools.

Ian Gemp, Roma Patel, Yoram Bachrach et al.

Mathematical models of interactions among rational agents have long been studied in game theory. However these interactions are often over a small set of discrete game actions which is very different from how humans communicate in natural language. To bridge this gap, we introduce a framework that allows equilibrium solvers to work over the space of natural language dialogue generated by large language models (LLMs). Specifically, by modelling the players, strategies and payoffs in a "game" of dialogue, we create a binding from natural language interactions to the conventional symbolic logic of game theory. Given this binding, we can ask existing game-theoretic algorithms to provide us with strategic solutions (e.g., what string an LLM should generate to maximize payoff in the face of strategic partners or opponents), giving us predictors of stable, rational conversational strategies. We focus on three domains that require different negotiation strategies: scheduling meetings, trading fruit and debate, and evaluate an LLM's generated language when guided by solvers. We see that LLMs that follow game-theory solvers result in dialogue generations that are less exploitable than the control (no guidance from solvers), and the language generated results in higher rewards, in all negotiation domains. We discuss future implications of this work, and how game-theoretic solvers that can leverage the expressivity of natural language can open up a new avenue of guiding language research.

Ian Gemp, Brian McWilliams, Claire Vernade et al.

We build on the recently proposed EigenGame that views eigendecomposition as a competitive game. EigenGame's updates are biased if computed using minibatches of data, which hinders convergence and more sophisticated parallelism in the stochastic setting. In this work, we propose an unbiased stochastic update that is asymptotically equivalent to EigenGame, enjoys greater parallelism allowing computation on datasets of larger sample sizes, and outperforms EigenGame in experiments. We present applications to finding the principal components of massive datasets and performing spectral clustering of graphs. We analyze and discuss our proposed update in the context of EigenGame and the shift in perspective from optimization to games.

Ian Gemp, Brian McWilliams, Claire Vernade et al.

We present a novel view on principal component analysis (PCA) as a competitive game in which each approximate eigenvector is controlled by a player whose goal is to maximize their own utility function. We analyze the properties of this PCA game and the behavior of its gradient based updates. The resulting algorithm -- which combines elements from Oja's rule with a generalized Gram-Schmidt orthogonalization -- is naturally decentralized and hence parallelizable through message passing. We demonstrate the scalability of the algorithm with experiments on large image datasets and neural network activations. We discuss how this new view of PCA as a differentiable game can lead to further algorithmic developments and insights.

Thomas Anthony, Tom Eccles, Andrea Tacchetti et al.

Recent advances in deep reinforcement learning (RL) have led to considerable progress in many 2-player zero-sum games, such as Go, Poker and Starcraft. The purely adversarial nature of such games allows for conceptually simple and principled application of RL methods. However real-world settings are many-agent, and agent interactions are complex mixtures of common-interest and competitive aspects. We consider Diplomacy, a 7-player board game designed to accentuate dilemmas resulting from many-agent interactions. It also features a large combinatorial action space and simultaneous moves, which are challenging for RL algorithms. We propose a simple yet effective approximate best response operator, designed to handle large combinatorial action spaces and simultaneous moves. We also introduce a family of policy iteration methods that approximate fictitious play. With these methods, we successfully apply RL to Diplomacy: we show that our agents convincingly outperform the previous state-of-the-art, and game theoretic equilibrium analysis shows that the new process yields consistent improvements.

Bo Liu, Ian Gemp, Mohammad Ghavamzadeh et al.

In this paper, we introduce proximal gradient temporal difference learning, which provides a principled way of designing and analyzing true stochastic gradient temporal difference learning algorithms. We show how gradient TD (GTD) reinforcement learning methods can be formally derived, not by starting from their original objective functions, as previously attempted, but rather from a primal-dual saddle-point objective function. We also conduct a saddle-point error analysis to obtain finite-sample bounds on their performance. Previous analyses of this class of algorithms use stochastic approximation techniques to prove asymptotic convergence, and do not provide any finite-sample analysis. We also propose an accelerated algorithm, called GTD2-MP, that uses proximal ``mirror maps'' to yield an improved convergence rate. The results of our theoretical analysis imply that the GTD family of algorithms are comparable and may indeed be preferred over existing least squares TD methods for off-policy learning, due to their linear complexity. We provide experimental results showing the improved performance of our accelerated gradient TD methods.

Kevin R. McKee, Ian Gemp, Brian McWilliams et al.

Recent research on reinforcement learning in pure-conflict and pure-common interest games has emphasized the importance of population heterogeneity. In contrast, studies of reinforcement learning in mixed-motive games have primarily leveraged homogeneous approaches. Given the defining characteristic of mixed-motive games--the imperfect correlation of incentives between group members--we study the effect of population heterogeneity on mixed-motive reinforcement learning. We draw on interdependence theory from social psychology and imbue reinforcement learning agents with Social Value Orientation (SVO), a flexible formalization of preferences over group outcome distributions. We subsequently explore the effects of diversity in SVO on populations of reinforcement learning agents in two mixed-motive Markov games. We demonstrate that heterogeneity in SVO generates meaningful and complex behavioral variation among agents similar to that suggested by interdependence theory. Empirical results in these mixed-motive dilemmas suggest agents trained in heterogeneous populations develop particularly generalized, high-performing policies relative to those trained in homogeneous populations.

Ian Gemp, Sridhar Mahadevan

In optimization, the negative gradient of a function denotes the direction of steepest descent. Furthermore, traveling in any direction orthogonal to the gradient maintains the value of the function. In this work, we show that these orthogonal directions that are ignored by gradient descent can be critical in equilibrium problems. Equilibrium problems have drawn heightened attention in machine learning due to the emergence of the Generative Adversarial Network (GAN). We use the framework of Variational Inequalities to analyze popular training algorithms for a fundamental GAN variant: the Wasserstein Linear-Quadratic GAN. We show that the steepest descent direction causes divergence from the equilibrium, and convergence to the equilibrium is achieved through following a particular orthogonal direction. We call this successful technique Crossing-the-Curl, named for its mathematical derivation as well as its intuition: identify the game's axis of rotation and move "across" space in the direction towards smaller "curling".

Ian Gemp, Sridhar Mahadevan

Algorithmic game theory (AGT) focuses on the design and analysis of algorithms for interacting agents, with interactions rigorously formalized within the framework of games. Results from AGT find applications in domains such as online bidding auctions for web advertisements and network routing protocols. Monotone games are games where agent strategies naturally converge to an equilibrium state. Previous results in AGT have been obtained for convex, socially-convex, or smooth games, but not monotone games. Our primary theoretical contributions are defining the monotone game setting and its extension to the online setting, a new notion of regret for this setting, and accompanying algorithms that achieve sub-linear regret. We demonstrate the utility of online monotone game theory on a variety of problem domains including variational inequalities, reinforcement learning, and generative adversarial networks.

Ishan Durugkar, Ian Gemp, Sridhar Mahadevan

Generative adversarial networks (GANs) are a framework for producing a generative model by way of a two-player minimax game. In this paper, we propose the \emph{Generative Multi-Adversarial Network} (GMAN), a framework that extends GANs to multiple discriminators. In previous work, the successful training of GANs requires modifying the minimax objective to accelerate training early on. In contrast, GMAN can be reliably trained with the original, untampered objective. We explore a number of design perspectives with the discriminator role ranging from formidable adversary to forgiving teacher. Image generation tasks comparing the proposed framework to standard GANs demonstrate GMAN produces higher quality samples in a fraction of the iterations when measured by a pairwise GAM-type metric.

Ian Gemp, Sridhar Mahadevan

This paper presents a new framework for analyzing and designing no-regret algorithms for dynamic (possibly adversarial) systems. The proposed framework generalizes the popular online convex optimization framework and extends it to its natural limit allowing it to capture a notion of regret that is intuitive for more general problems such as those encountered in game theory and variational inequalities. The framework hinges on a special choice of a system-wide loss function we have developed. Using this framework, we prove that a simple update scheme provides a no-regret algorithm for monotone systems. While previous results in game theory prove individual agents can enjoy unilateral no-regret guarantees, our result proves monotonicity sufficient for guaranteeing no-regret when considering the adjustments of multiple agent strategies in parallel. Furthermore, to our knowledge, this is the first framework to provide a suitable notion of regret for variational inequalities. Most importantly, our proposed framework ensures monotonicity a sufficient condition for employing multiple online learners safely in parallel.

Ian Gemp, Ishan Durugkar, Mario Parente et al.

Recent advances in semi-supervised learning with deep generative models have shown promise in generalizing from small labeled datasets ($\mathbf{x},\mathbf{y}$) to large unlabeled ones ($\mathbf{x}$). In the case where the codomain has known structure, a large unfeatured dataset ($\mathbf{y}$) is potentially available. We develop a parameter-efficient, deep semi-supervised generative model for the purpose of exploiting this untapped data source. Empirical results show improved performance in disentangling latent variable semantics as well as improved discriminative prediction on Martian spectroscopic and handwritten digit domains.

Sridhar Mahadevan, Bo Liu, Philip Thomas et al.

In this paper, we set forth a new vision of reinforcement learning developed by us over the past few years, one that yields mathematically rigorous solutions to longstanding important questions that have remained unresolved: (i) how to design reliable, convergent, and robust reinforcement learning algorithms (ii) how to guarantee that reinforcement learning satisfies pre-specified "safety" guarantees, and remains in a stable region of the parameter space (iii) how to design "off-policy" temporal difference learning algorithms in a reliable and stable manner, and finally (iv) how to integrate the study of reinforcement learning into the rich theory of stochastic optimization. In this paper, we provide detailed answers to all these questions using the powerful framework of proximal operators. The key idea that emerges is the use of primal dual spaces connected through the use of a Legendre transform. This allows temporal difference updates to occur in dual spaces, allowing a variety of important technical advantages. The Legendre transform elegantly generalizes past algorithms for solving reinforcement learning problems, such as natural gradient methods, which we show relate closely to the previously unconnected framework of mirror descent methods. Equally importantly, proximal operator theory enables the systematic development of operator splitting methods that show how to safely and reliably decompose complex products of gradients that occur in recent variants of gradient-based temporal difference learning. This key technical innovation makes it possible to finally design "true" stochastic gradient methods for reinforcement learning. Finally, Legendre transforms enable a variety of other benefits, including modeling sparsity and domain geometry. Our work builds extensively on recent work on the convergence of saddle-point algorithms, and on the theory of monotone operators.