Romuald Élie

Julien Perolat, Bart de Vylder, Daniel Hennes et al.

We introduce DeepNash, an autonomous agent capable of learning to play the imperfect information game Stratego from scratch, up to a human expert level. Stratego is one of the few iconic board games that Artificial Intelligence (AI) has not yet mastered. This popular game has an enormous game tree on the order of $10^{535}$ nodes, i.e., $10^{175}$ times larger than that of Go. It has the additional complexity of requiring decision-making under imperfect information, similar to Texas hold'em poker, which has a significantly smaller game tree (on the order of $10^{164}$ nodes). Decisions in Stratego are made over a large number of discrete actions with no obvious link between action and outcome. Episodes are long, with often hundreds of moves before a player wins, and situations in Stratego can not easily be broken down into manageably-sized sub-problems as in poker. For these reasons, Stratego has been a grand challenge for the field of AI for decades, and existing AI methods barely reach an amateur level of play. DeepNash uses a game-theoretic, model-free deep reinforcement learning method, without search, that learns to master Stratego via self-play. The Regularised Nash Dynamics (R-NaD) algorithm, a key component of DeepNash, converges to an approximate Nash equilibrium, instead of 'cycling' around it, by directly modifying the underlying multi-agent learning dynamics. DeepNash beats existing state-of-the-art AI methods in Stratego and achieved a yearly (2022) and all-time top-3 rank on the Gravon games platform, competing with human expert players.

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.

Zhe Wang, Petar Veličković, Daniel Hennes et al.

Identifying key patterns of tactics implemented by rival teams, and developing effective responses, lies at the heart of modern football. However, doing so algorithmically remains an open research challenge. To address this unmet need, we propose TacticAI, an AI football tactics assistant developed and evaluated in close collaboration with domain experts from Liverpool FC. We focus on analysing corner kicks, as they offer coaches the most direct opportunities for interventions and improvements. TacticAI incorporates both a predictive and a generative component, allowing the coaches to effectively sample and explore alternative player setups for each corner kick routine and to select those with the highest predicted likelihood of success. We validate TacticAI on a number of relevant benchmark tasks: predicting receivers and shot attempts and recommending player position adjustments. The utility of TacticAI is validated by a qualitative study conducted with football domain experts at Liverpool FC. We show that TacticAI's model suggestions are not only indistinguishable from real tactics, but also favoured over existing tactics 90% of the time, and that TacticAI offers an effective corner kick retrieval system. TacticAI achieves these results despite the limited availability of gold-standard data, achieving data efficiency through geometric deep learning.

Mathieu Laurière, Sarah Perrin, Sertan Girgin et al.

Mean Field Games (MFGs) have been introduced to efficiently approximate games with very large populations of strategic agents. Recently, the question of learning equilibria in MFGs has gained momentum, particularly using model-free reinforcement learning (RL) methods. One limiting factor to further scale up using RL is that existing algorithms to solve MFGs require the mixing of approximated quantities such as strategies or $q$-values. This is far from being trivial in the case of non-linear function approximation that enjoy good generalization properties, e.g. neural networks. We propose two methods to address this shortcoming. The first one learns a mixed strategy from distillation of historical data into a neural network and is applied to the Fictitious Play algorithm. The second one is an online mixing method based on regularization that does not require memorizing historical data or previous estimates. It is used to extend Online Mirror Descent. We demonstrate numerically that these methods efficiently enable the use of Deep RL algorithms to solve various MFGs. In addition, we show that these methods outperform SotA baselines from the literature.

Khaled Kahouli, Romuald Elie, Klaus-Robert Müller et al.

Diffusion models offer a robust framework for sampling from unnormalized probability densities, which requires accurately estimating the score of the noise-perturbed target distribution. While the standard Denoising Score Identity (DSI) relies on data samples, access to the target energy function enables an alternative formulation via the Target Score Identity (TSI). However, these estimators face a fundamental variance trade-off: DSI exhibits high variance in low-noise regimes, whereas TSI suffers from high variance at high noise levels. In this work, we reconcile these approaches by unifying both estimators within the principled framework of control variates. We introduce the Control Variate Score Identity (CVSI), deriving an optimal, time-dependent control coefficient that theoretically guarantees variance minimization across the entire noise spectrum. We demonstrate that CVSI serves as a robust, low-variance plug-in estimator that significantly enhances sample efficiency in both data-free sampler learning and inference-time diffusion sampling.

Valentin de Bortoli, Romuald Elie, Anna Kazeykina et al.

Diffusion generative models have emerged as powerful tools for producing synthetic data from an empirically observed distribution. A common approach involves simulating the time-reversal of an Ornstein-Uhlenbeck (OU) process initialized at the true data distribution. Since the score function associated with the OU process is typically unknown, it is approximated using a trained neural network. This approximation, along with finite time simulation, time discretization and statistical approximation, introduce several sources of error whose impact on the generated samples must be carefully understood. Previous analyses have quantified the error between the generated and the true data distributions in terms of Wasserstein distance or Kullback-Leibler (KL) divergence. However, both metrics present limitations: KL divergence requires absolute continuity between distributions, while Wasserstein distance, though more general, leads to error bounds that scale poorly with dimension, rendering them impractical in high-dimensional settings. In this work, we derive an explicit, dimension-free bound on the discrepancy between the generated and the true data distributions. The bound is expressed in terms of a smooth test functional with bounded first and second derivatives. The key novelty lies in the use of this weaker, functional metric to obtain dimension-independent guarantees, at the cost of higher regularity on the test functions. As an application, we formulate and solve a variational problem to minimize the time-discretization error, leading to the derivation of an optimal time-scheduling strategy for the reverse-time diffusion. Interestingly, this scheduler has appeared previously in the literature in a different context; our analysis provides a new justification for its optimality, now grounded in minimizing the discretization bias in generative sampling.

Arthur Charpentier, Christophe Denis, Romuald Elie et al.

Demographic parity (DP) is a widely used group fairness criterion requiring predictive distributions to be invariant across sensitive groups. While natural in classification, full distributional DP is often overly restrictive in regression and can lead to substantial accuracy loss. We propose a relaxation of DP tailored to regression, enforcing parity only at a finite set of quantile levels and/or score thresholds. Concretely, we introduce a novel (${\ell}$, Z)-fair predictor, which imposes groupwise CDF constraints of the form F f |S=s (z m ) = ${\ell}$ m for prescribed pairs (${\ell}$ m , z m ). For this setting, we derive closed-form characterizations of the optimal fair discretized predictor via a Lagrangian dual formulation and quantify the discretization cost, showing that the risk gap to the continuous optimum vanishes as the grid is refined. We further develop a model-agnostic post-processing algorithm based on two samples (labeled for learning a base regressor and unlabeled for calibration), and establish finite-sample guarantees on constraint violation and excess penalized risk. In addition, we introduce two alternative frameworks where we match group and marginal CDF values at selected score thresholds. In both settings, we provide closed-form solutions for the optimal fair discretized predictor. Experiments on synthetic and real datasets illustrate an interpretable fairness-accuracy trade-off, enabling targeted corrections at decision-relevant quantiles or thresholds while preserving predictive performance.

Daniel Jarrett, Miruna Pîslar, Michiel A. Bakker et al.

Consider the process of collective decision-making, in which a group of individuals interactively select a preferred outcome from among a universe of alternatives. In this context, "representation" is the activity of making an individual's preferences present in the process via participation by a proxy agent -- i.e. their "representative". To this end, learned models of human behavior have the potential to fill this role, with practical implications for multi-agent scenario studies and mechanism design. In this work, we investigate the possibility of training \textit{language agents} to behave in the capacity of representatives of human agents, appropriately expressing the preferences of those individuals whom they stand for. First, we formalize the setting of \textit{collective decision-making} -- as the episodic process of interaction between a group of agents and a decision mechanism. On this basis, we then formalize the problem of \textit{digital representation} -- as the simulation of an agent's behavior to yield equivalent outcomes from the mechanism. Finally, we conduct an empirical case study in the setting of \textit{consensus-finding} among diverse humans, and demonstrate the feasibility of fine-tuning large language models to act as digital representatives.

Raphael Koster, Miruna Pîslar, Andrea Tacchetti et al.

A canonical social dilemma arises when finite resources are allocated to a group of people, who can choose to either reciprocate with interest, or keep the proceeds for themselves. What resource allocation mechanisms will encourage levels of reciprocation that sustain the commons? Here, in an iterated multiplayer trust game, we use deep reinforcement learning (RL) to design an allocation mechanism that endogenously promotes sustainable contributions from human participants to a common pool resource. We first trained neural networks to behave like human players, creating a stimulated economy that allowed us to study how different mechanisms influenced the dynamics of receipt and reciprocation. We then used RL to train a social planner to maximise aggregate return to players. The social planner discovered a redistributive policy that led to a large surplus and an inclusive economy, in which players made roughly equal gains. The RL agent increased human surplus over baseline mechanisms based on unrestricted welfare or conditional cooperation, by conditioning its generosity on available resources and temporarily sanctioning defectors by allocating fewer resources to them. Examining the AI policy allowed us to develop an explainable mechanism that performed similarly and was more popular among players. Deep reinforcement learning can be used to discover mechanisms that promote sustainable human behaviour.

Romuald Elie, Caroline Hillairet, François Hu et al.

This paper addresses significant obstacles that arise from the widespread use of machine learning models in the insurance industry, with a specific focus on promoting fairness. The initial challenge lies in effectively leveraging unlabeled data in insurance while reducing the labeling effort and emphasizing data relevance through active learning techniques. The paper explores various active learning sampling methodologies and evaluates their impact on both synthetic and real insurance datasets. This analysis highlights the difficulty of achieving fair model inferences, as machine learning models may replicate biases and discrimination found in the underlying data. To tackle these interconnected challenges, the paper introduces an innovative fair active learning method. The proposed approach samples informative and fair instances, achieving a good balance between model predictive performance and fairness, as confirmed by numerical experiments on insurance datasets.

Sarah Perrin, Mathieu Laurière, Julien Pérolat et al.

Mean Field Games (MFGs) can potentially scale multi-agent systems to extremely large populations of agents. Yet, most of the literature assumes a single initial distribution for the agents, which limits the practical applications of MFGs. Machine Learning has the potential to solve a wider diversity of MFG problems thanks to generalizations capacities. We study how to leverage these generalization properties to learn policies enabling a typical agent to behave optimally against any population distribution. In reference to the Master equation in MFGs, we coin the term ``Master policies'' to describe them and we prove that a single Master policy provides a Nash equilibrium, whatever the initial distribution. We propose a method to learn such Master policies. Our approach relies on three ingredients: adding the current population distribution as part of the observation, approximating Master policies with neural networks, and training via Reinforcement Learning and Fictitious Play. We illustrate on numerical examples not only the efficiency of the learned Master policy but also its generalization capabilities beyond the distributions used for training.

Georgios Piliouras, Mark Rowland, Shayegan Omidshafiei et al.

Regret has been established as a foundational concept in online learning, and likewise has important applications in the analysis of learning dynamics in games. Regret quantifies the difference between a learner's performance against a baseline in hindsight. It is well-known that regret-minimizing algorithms converge to certain classes of equilibria in games; however, traditional forms of regret used in game theory predominantly consider baselines that permit deviations to deterministic actions or strategies. In this paper, we revisit our understanding of regret from the perspective of deviations over partitions of the full \emph{mixed} strategy space (i.e., probability distributions over pure strategies), under the lens of the previously-established $Φ$-regret framework, which provides a continuum of stronger regret measures. Importantly, $Φ$-regret enables learning agents to consider deviations from and to mixed strategies, generalizing several existing notions of regret such as external, internal, and swap regret, and thus broadening the insights gained from regret-based analysis of learning algorithms. We prove here that the well-studied evolutionary learning algorithm of replicator dynamics (RD) seamlessly minimizes the strongest possible form of $Φ$-regret in generic $2 \times 2$ games, without any modification of the underlying algorithm itself. We subsequently conduct experiments validating our theoretical results in a suite of 144 $2 \times 2$ games wherein RD exhibits a diverse set of behaviors. We conclude by providing empirical evidence of $Φ$-regret minimization by RD in some larger games, hinting at further opportunity for $Φ$-regret based study of such algorithms from both a theoretical and empirical perspective.

Shayegan Omidshafiei, Daniel Hennes, Marta Garnelo et al.

In multiagent environments, several decision-making individuals interact while adhering to the dynamics constraints imposed by the environment. These interactions, combined with the potential stochasticity of the agents' decision-making processes, make such systems complex and interesting to study from a dynamical perspective. Significant research has been conducted on learning models for forward-direction estimation of agent behaviors, for example, pedestrian predictions used for collision-avoidance in self-driving cars. However, in many settings, only sporadic observations of agents may be available in a given trajectory sequence. For instance, in football, subsets of players may come in and out of view of broadcast video footage, while unobserved players continue to interact off-screen. In this paper, we study the problem of multiagent time-series imputation, where available past and future observations of subsets of agents are used to estimate missing observations for other agents. Our approach, called the Graph Imputer, uses forward- and backward-information in combination with graph networks and variational autoencoders to enable learning of a distribution of imputed trajectories. We evaluate our approach on a dataset of football matches, using a projective camera module to train and evaluate our model for the off-screen player state estimation setting. We illustrate that our method outperforms several state-of-the-art approaches, including those hand-crafted for football.

Matthieu Geist, Julien Pérolat, Mathieu Laurière et al.

Concave Utility Reinforcement Learning (CURL) extends RL from linear to concave utilities in the occupancy measure induced by the agent's policy. This encompasses not only RL but also imitation learning and exploration, among others. Yet, this more general paradigm invalidates the classical Bellman equations, and calls for new algorithms. Mean-field Games (MFGs) are a continuous approximation of many-agent RL. They consider the limit case of a continuous distribution of identical agents, anonymous with symmetric interests, and reduce the problem to the study of a single representative agent in interaction with the full population. Our core contribution consists in showing that CURL is a subclass of MFGs. We think this important to bridge together both communities. It also allows to shed light on aspects of both fields: we show the equivalence between concavity in CURL and monotonicity in the associated MFG, between optimality conditions in CURL and Nash equilibrium in MFG, or that Fictitious Play (FP) for this class of MFGs is simply Frank-Wolfe, bringing the first convergence rate for discrete-time FP for MFGs. We also experimentally demonstrate that, using algorithms recently introduced for solving MFGs, we can address the CURL problem more efficiently.

Sarah Perrin, Mathieu Laurière, Julien Pérolat et al.

We present a method enabling a large number of agents to learn how to flock, which is a natural behavior observed in large populations of animals. This problem has drawn a lot of interest but requires many structural assumptions and is tractable only in small dimensions. We phrase this problem as a Mean Field Game (MFG), where each individual chooses its acceleration depending on the population behavior. Combining Deep Reinforcement Learning (RL) and Normalizing Flows (NF), we obtain a tractable solution requiring only very weak assumptions. Our algorithm finds a Nash Equilibrium and the agents adapt their velocity to match the neighboring flock's average one. We use Fictitious Play and alternate: (1) computing an approximate best response with Deep RL, and (2) estimating the next population distribution with NF. We show numerically that our algorithm learn multi-group or high-dimensional flocking with obstacles.

Julien Perolat, Sarah Perrin, Romuald Elie et al.

We address scaling up equilibrium computation in Mean Field Games (MFGs) using Online Mirror Descent (OMD). We show that continuous-time OMD provably converges to a Nash equilibrium under a natural and well-motivated set of monotonicity assumptions. This theoretical result nicely extends to multi-population games and to settings involving common noise. A thorough experimental investigation on various single and multi-population MFGs shows that OMD outperforms traditional algorithms such as Fictitious Play (FP). We empirically show that OMD scales up and converges significantly faster than FP by solving, for the first time to our knowledge, examples of MFGs with hundreds of billions states. This study establishes the state-of-the-art for learning in large-scale multi-agent and multi-population games.

Carl Remlinger, Joseph Mikael, Romuald Elie

We introduce three new generative models for time series that are based on Euler discretization of Stochastic Differential Equations (SDEs) and Wasserstein metrics. Two of these methods rely on the adaptation of generative adversarial networks (GANs) to time series. The third algorithm, called Conditional Euler Generator (CEGEN), minimizes a dedicated distance between the transition probability distributions over all time steps. In the context of Ito processes, we provide theoretical guarantees that minimizing this criterion implies accurate estimations of the drift and volatility parameters. We demonstrate empirically that CEGEN outperforms state-of-the-art and GAN generators on both marginal and temporal dynamics metrics. Besides, it identifies accurate correlation structures in high dimension. When few data points are available, we verify the effectiveness of CEGEN, when combined with transfer learning methods on Monte Carlo simulations. Finally, we illustrate the robustness of our method on various real-world datasets.

Karl Tuyls, Shayegan Omidshafiei, Paul Muller et al.

The rapid progress in artificial intelligence (AI) and machine learning has opened unprecedented analytics possibilities in various team and individual sports, including baseball, basketball, and tennis. More recently, AI techniques have been applied to football, due to a huge increase in data collection by professional teams, increased computational power, and advances in machine learning, with the goal of better addressing new scientific challenges involved in the analysis of both individual players' and coordinated teams' behaviors. The research challenges associated with predictive and prescriptive football analytics require new developments and progress at the intersection of statistical learning, game theory, and computer vision. In this paper, we provide an overarching perspective highlighting how the combination of these fields, in particular, forms a unique microcosm for AI research, while offering mutual benefits for professional teams, spectators, and broadcasters in the years to come. We illustrate that this duality makes football analytics a game changer of tremendous value, in terms of not only changing the game of football itself, but also in terms of what this domain can mean for the field of AI. We review the state-of-the-art and exemplify the types of analysis enabled by combining the aforementioned fields, including illustrative examples of counterfactual analysis using predictive models, and the combination of game-theoretic analysis of penalty kicks with statistical learning of player attributes. We conclude by highlighting envisioned downstream impacts, including possibilities for extensions to other sports (real and virtual).

Sarah Perrin, Julien Perolat, Mathieu Laurière et al.

In this paper, we deepen the analysis of continuous time Fictitious Play learning algorithm to the consideration of various finite state Mean Field Game settings (finite horizon, $γ$-discounted), allowing in particular for the introduction of an additional common noise. We first present a theoretical convergence analysis of the continuous time Fictitious Play process and prove that the induced exploitability decreases at a rate $O(\frac{1}{t})$. Such analysis emphasizes the use of exploitability as a relevant metric for evaluating the convergence towards a Nash equilibrium in the context of Mean Field Games. These theoretical contributions are supported by numerical experiments provided in either model-based or model-free settings. We provide hereby for the first time converging learning dynamics for Mean Field Games in the presence of common noise.

Arthur Charpentier, Romuald Elie, Carl Remlinger

Reinforcement learning algorithms describe how an agent can learn an optimal action policy in a sequential decision process, through repeated experience. In a given environment, the agent policy provides him some running and terminal rewards. As in online learning, the agent learns sequentially. As in multi-armed bandit problems, when an agent picks an action, he can not infer ex-post the rewards induced by other action choices. In reinforcement learning, his actions have consequences: they influence not only rewards, but also future states of the world. The goal of reinforcement learning is to find an optimal policy -- a mapping from the states of the world to the set of actions, in order to maximize cumulative reward, which is a long term strategy. Exploring might be sub-optimal on a short-term horizon but could lead to optimal long-term ones. Many problems of optimal control, popular in economics for more than forty years, can be expressed in the reinforcement learning framework, and recent advances in computational science, provided in particular by deep learning algorithms, can be used by economists in order to solve complex behavioral problems. In this article, we propose a state-of-the-art of reinforcement learning techniques, and present applications in economics, game theory, operation research and finance.

Romuald Elie, Julien Pérolat, Mathieu Laurière et al.

Learning by experience in Multi-Agent Systems (MAS) is a difficult and exciting task, due to the lack of stationarity of the environment, whose dynamics evolves as the population learns. In order to design scalable algorithms for systems with a large population of interacting agents (e.g. swarms), this paper focuses on Mean Field MAS, where the number of agents is asymptotically infinite. Recently, a very active burgeoning field studies the effects of diverse reinforcement learning algorithms for agents with no prior information on a stationary Mean Field Game (MFG) and learn their policy through repeated experience. We adopt a high perspective on this problem and analyze in full generality the convergence of a fictitious iterative scheme using any single agent learning algorithm at each step. We quantify the quality of the computed approximate Nash equilibrium, in terms of the accumulated errors arising at each learning iteration step. Notably, we show for the first time convergence of model free learning algorithms towards non-stationary MFG equilibria, relying only on classical assumptions on the MFG dynamics. We illustrate our theoretical results with a numerical experiment in a continuous action-space environment, where the approximate best response of the iterative fictitious play scheme is computed with a deep RL algorithm.