Matthieu Geist

Tadashi Kozuno, Wenhao Yang, Nino Vieillard et al. · deepmind

In this work, we consider and analyze the sample complexity of model-free reinforcement learning with a generative model. Particularly, we analyze mirror descent value iteration (MDVI) by Geist et al. (2019) and Vieillard et al. (2020a), which uses the Kullback-Leibler divergence and entropy regularization in its value and policy updates. Our analysis shows that it is nearly minimax-optimal for finding an $\varepsilon$-optimal policy when $\varepsilon$ is sufficiently small. This is the first theoretical result that demonstrates that a simple model-free algorithm without variance-reduction can be nearly minimax-optimal under the considered setting.

Markus Wulfmeier, Michael Bloesch, Nino Vieillard et al. · deepmind

The majority of language model training builds on imitation learning. It covers pretraining, supervised fine-tuning, and affects the starting conditions for reinforcement learning from human feedback (RLHF). The simplicity and scalability of maximum likelihood estimation (MLE) for next token prediction led to its role as predominant paradigm. However, the broader field of imitation learning can more effectively utilize the sequential structure underlying autoregressive generation. We focus on investigating the inverse reinforcement learning (IRL) perspective to imitation, extracting rewards and directly optimizing sequences instead of individual token likelihoods and evaluate its benefits for fine-tuning large language models. We provide a new angle, reformulating inverse soft-Q-learning as a temporal difference regularized extension of MLE. This creates a principled connection between MLE and IRL and allows trading off added complexity with increased performance and diversity of generations in the supervised fine-tuning (SFT) setting. We find clear advantages for IRL-based imitation, in particular for retaining diversity while maximizing task performance, rendering IRL a strong alternative on fixed SFT datasets even without online data generation. Our analysis of IRL-extracted reward functions further indicates benefits for more robust reward functions via tighter integration of supervised and preference-based LLM post-training.

Divyansh Garg, Joey Hejna, Matthieu Geist et al.

Modern Deep Reinforcement Learning (RL) algorithms require estimates of the maximal Q-value, which are difficult to compute in continuous domains with an infinite number of possible actions. In this work, we introduce a new update rule for online and offline RL which directly models the maximal value using Extreme Value Theory (EVT), drawing inspiration from economics. By doing so, we avoid computing Q-values using out-of-distribution actions which is often a substantial source of error. Our key insight is to introduce an objective that directly estimates the optimal soft-value functions (LogSumExp) in the maximum entropy RL setting without needing to sample from a policy. Using EVT, we derive our \emph{Extreme Q-Learning} framework and consequently online and, for the first time, offline MaxEnt Q-learning algorithms, that do not explicitly require access to a policy or its entropy. Our method obtains consistently strong performance in the D4RL benchmark, outperforming prior works by \emph{10+ points} on the challenging Franka Kitchen tasks while offering moderate improvements over SAC and TD3 on online DM Control tasks. Visualizations and code can be found on our website at https://div99.github.io/XQL/.

Rishabh Agarwal, Nino Vieillard, Yongchao Zhou et al.

Knowledge distillation (KD) is widely used for compressing a teacher model to reduce its inference cost and memory footprint, by training a smaller student model. However, current KD methods for auto-regressive sequence models suffer from distribution mismatch between output sequences seen during training and those generated by the student during inference. To address this issue, we introduce Generalized Knowledge Distillation (GKD). Instead of solely relying on a fixed set of output sequences, GKD trains the student on its self-generated output sequences by leveraging feedback from the teacher on such sequences. Unlike supervised KD approaches, GKD also offers the flexibility to employ alternative loss functions between the student and teacher, which can be useful when the student lacks the expressivity to mimic the teacher's distribution. Furthermore, GKD facilitates the seamless integration of distillation with RL fine-tuning (RLHF). We demonstrate the efficacy of GKD for distilling auto-regressive language models on summarization, translation, and arithmetic reasoning tasks, and task-agnostic distillation for instruction-tuning.

Navdeep Kumar, Esther Derman, Matthieu Geist et al.

Policy gradient methods have become a standard for training reinforcement learning agents in a scalable and efficient manner. However, they do not account for transition uncertainty, whereas learning robust policies can be computationally expensive. In this paper, we introduce robust policy gradient (RPG), a policy-based method that efficiently solves rectangular robust Markov decision processes (MDPs). We provide a closed-form expression for the worst occupation measure. Incidentally, we find that the worst kernel is a rank-one perturbation of the nominal. Combining the worst occupation measure with a robust Q-value estimation yields an explicit form of the robust gradient. Our resulting RPG can be estimated from data with the same time complexity as its non-robust equivalent. Hence, it relieves the computational burden of convex optimization problems required for training robust policies by current policy gradient approaches.

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.

Batuhan Yardim, Semih Cayci, Matthieu Geist et al.

Mean-field games have been used as a theoretical tool to obtain an approximate Nash equilibrium for symmetric and anonymous $N$-player games. However, limiting applicability, existing theoretical results assume variations of a "population generative model", which allows arbitrary modifications of the population distribution by the learning algorithm. Moreover, learning algorithms typically work on abstract simulators with population instead of the $N$-player game. Instead, we show that $N$ agents running policy mirror ascent converge to the Nash equilibrium of the regularized game within $\widetilde{\mathcal{O}}(\varepsilon^{-2})$ samples from a single sample trajectory without a population generative model, up to a standard $\mathcal{O}(\frac{1}{\sqrt{N}})$ error due to the mean field. Taking a divergent approach from the literature, instead of working with the best-response map we first show that a policy mirror ascent map can be used to construct a contractive operator having the Nash equilibrium as its fixed point. We analyze single-path TD learning for $N$-agent games, proving sample complexity guarantees by only using a sample path from the $N$-agent simulator without a population generative model. Furthermore, we demonstrate that our methodology allows for independent learning by $N$ agents with finite sample guarantees.

Pierre Clavier, Erwan Le Pennec, Matthieu Geist

We study the sample complexity of obtaining an $ε$-optimal policy in \emph{Robust} discounted Markov Decision Processes (RMDPs), given only access to a generative model of the nominal kernel. This problem is widely studied in the non-robust case, and it is known that any planning approach applied to an empirical MDP estimated with $\tilde{\mathcal{O}}(\frac{H^3 \mid S \mid\mid A \mid}{ε^2})$ samples provides an $ε$-optimal policy, which is minimax optimal. Results in the robust case are much more scarce. For $sa$- (resp $s$-)rectangular uncertainty sets, the best known sample complexity is $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid^2\mid A \mid}{ε^2})$ (resp. $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid^2\mid A \mid^2}{ε^2})$), for specific algorithms and when the uncertainty set is based on the total variation (TV), the KL or the Chi-square divergences. In this paper, we consider uncertainty sets defined with an $L_p$-ball (recovering the TV case), and study the sample complexity of \emph{any} planning algorithm (with high accuracy guarantee on the solution) applied to an empirical RMDP estimated using the generative model. In the general case, we prove a sample complexity of $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid\mid A \mid}{ε^2})$ for both the $sa$- and $s$-rectangular cases (improvements of $\mid S \mid$ and $\mid S \mid\mid A \mid$ respectively). When the size of the uncertainty is small enough, we improve the sample complexity to $\tilde{\mathcal{O}}(\frac{H^3 \mid S \mid\mid A \mid }{ε^2})$, recovering the lower-bound for the non-robust case for the first time and a robust lower-bound when the size of the uncertainty is small enough.

Alexis Jacq, Johan Ferret, Olivier Pietquin et al.

Traditionally, Reinforcement Learning (RL) aims at deciding how to act optimally for an artificial agent. We argue that deciding when to act is equally important. As humans, we drift from default, instinctive or memorized behaviors to focused, thought-out behaviors when required by the situation. To enhance RL agents with this aptitude, we propose to augment the standard Markov Decision Process and make a new mode of action available: being lazy, which defers decision-making to a default policy. In addition, we penalize non-lazy actions in order to encourage minimal effort and have agents focus on critical decisions only. We name the resulting formalism lazy-MDPs. We study the theoretical properties of lazy-MDPs, expressing value functions and characterizing optimal solutions. Then we empirically demonstrate that policies learned in lazy-MDPs generally come with a form of interpretability: by construction, they show us the states where the agent takes control over the default policy. We deem those states and corresponding actions important since they explain the difference in performance between the default and the new, lazy policy. With suboptimal policies as default (pretrained or random), we observe that agents are able to get competitive performance in Atari games while only taking control in a limited subset of states.

Esther Derman, Yevgeniy Men, Matthieu Geist et al.

Robust Markov decision processes (MDPs) aim to handle changing or partially known system dynamics. To solve them, one typically resorts to robust optimization methods. However, this significantly increases computational complexity and limits scalability in both learning and planning. On the other hand, regularized MDPs show more stability in policy learning without impairing time complexity. Yet, they generally do not encompass uncertainty in the model dynamics. In this work, we aim to learn robust MDPs using regularization. We first show that regularized MDPs are a particular instance of robust MDPs with uncertain reward. We thus establish that policy iteration on reward-robust MDPs can have the same time complexity as on regularized MDPs. We further extend this relationship to MDPs with uncertain transitions: this leads to a regularization term with an additional dependence on the value function. We then generalize regularized MDPs to twice regularized MDPs ($\text{R}^2$ MDPs), i.e., MDPs with $\textit{both}$ value and policy regularization. The corresponding Bellman operators enable us to derive planning and learning schemes with convergence and generalization guarantees, thus reducing robustness to regularization. We numerically show this two-fold advantage on tabular and physical domains, highlighting the fact that $\text{R}^2$ preserves its efficacy in continuous environments.

Giorgia Ramponi, Pavel Kolev, Olivier Pietquin et al.

We explore the problem of imitation learning (IL) in the context of mean-field games (MFGs), where the goal is to imitate the behavior of a population of agents following a Nash equilibrium policy according to some unknown payoff function. IL in MFGs presents new challenges compared to single-agent IL, particularly when both the reward function and the transition kernel depend on the population distribution. In this paper, departing from the existing literature on IL for MFGs, we introduce a new solution concept called the Nash imitation gap. Then we show that when only the reward depends on the population distribution, IL in MFGs can be reduced to single-agent IL with similar guarantees. However, when the dynamics is population-dependent, we provide a novel upper-bound that suggests IL is harder in this setting. To address this issue, we propose a new adversarial formulation where the reinforcement learning problem is replaced by a mean-field control (MFC) problem, suggesting progress in IL within MFGs may have to build upon MFC.

Mathieu Blondel, Felipe Llinares-López, Robert Dadashi et al.

Energy-based models, a.k.a. energy networks, perform inference by optimizing an energy function, typically parametrized by a neural network. This allows one to capture potentially complex relationships between inputs and outputs. To learn the parameters of the energy function, the solution to that optimization problem is typically fed into a loss function. The key challenge for training energy networks lies in computing loss gradients, as this typically requires argmin/argmax differentiation. In this paper, building upon a generalized notion of conjugate function, which replaces the usual bilinear pairing with a general energy function, we propose generalized Fenchel-Young losses, a natural loss construction for learning energy networks. Our losses enjoy many desirable properties and their gradients can be computed efficiently without argmin/argmax differentiation. We also prove the calibration of their excess risk in the case of linear-concave energies. We demonstrate our losses on multilabel classification and imitation learning tasks.

Laixi Shi, Robert Dadashi, Yuejie Chi et al.

The core challenge of offline reinforcement learning (RL) is dealing with the (potentially catastrophic) extrapolation error induced by the distribution shift between the history dataset and the desired policy. A large portion of prior work tackles this challenge by implicitly/explicitly regularizing the learning policy towards the behavior policy, which is hard to estimate reliably in practice. In this work, we propose to regularize towards the Q-function of the behavior policy instead of the behavior policy itself, under the premise that the Q-function can be estimated more reliably and easily by a SARSA-style estimate and handles the extrapolation error more straightforwardly. We propose two algorithms taking advantage of the estimated Q-function through regularizations, and demonstrate they exhibit strong performance on the D4RL benchmarks.

Alexis Jacq, Manu Orsini, Gabriel Dulac-Arnold et al.

Recent advances in ML suggest that the quantity of data available to a model is one of the primary bottlenecks to high performance. Although for language-based tasks there exist almost unlimited amounts of reasonably coherent data to train from, this is generally not the case for Reinforcement Learning, especially when dealing with a novel environment. In effect, even a relatively trivial continuous environment has an almost limitless number of states, but simply sampling random states and actions will likely not provide transitions that are interesting or useful for any potential downstream task. How should one generate massive amounts of useful data given only an MDP with no indication of downstream tasks? Are the quantity and quality of data truly transformative to the performance of a general controller? We propose to answer both of these questions. First, we introduce a principled unsupervised exploration method, ChronoGEM, which aims to achieve uniform coverage over the manifold of achievable states, which we believe is the most reasonable goal given no prior task information. Secondly, we investigate the effects of both data quantity and data quality on the training of a downstream goal-achievement policy, and show that both large quantities and high-quality of data are essential to train a general controller: a high-precision pose-achievement policy capable of attaining a large number of poses over numerous continuous control embodiments including humanoid.

Matteo El-Hariry, Antoine Richard, Vivek Muralidharan et al.

This investigation introduces a novel deep reinforcement learning-based suite to control floating platforms in both simulated and real-world environments. Floating platforms serve as versatile test-beds to emulate micro-gravity environments on Earth, useful to test autonomous navigation systems for space applications. Our approach addresses the system and environmental uncertainties in controlling such platforms by training policies capable of precise maneuvers amid dynamic and unpredictable conditions. Leveraging Deep Reinforcement Learning (DRL) techniques, our suite achieves robustness, adaptability, and good transferability from simulation to reality. Our deep reinforcement learning framework provides advantages such as fast training times, large-scale testing capabilities, rich visualization options, and ROS bindings for integration with real-world robotic systems. Being open access, our suite serves as a comprehensive platform for practitioners who want to replicate similar research in their own simulated environments and labs.

Andrej Orsula, Matthieu Geist, Miguel Olivares-Mendez et al.

The growing ambition for space exploration demands robust autonomous systems that can operate in unstructured environments under extreme extraterrestrial conditions. The adoption of robot learning in this domain is severely hindered by the prohibitive cost of technology demonstrations and the limited availability of data. To bridge this gap, we introduce the Space Robotics Bench, an open-source simulation framework for robot learning in space. It offers a modular architecture that integrates on-demand procedural generation with massively parallel simulation environments to support the creation of vast and diverse training distributions for learning-based agents. To ground research and enable direct comparison, the framework includes a comprehensive suite of benchmark tasks that span a wide range of mission-relevant scenarios. We establish performance baselines using standard reinforcement learning algorithms and present a series of experimental case studies that investigate key challenges in generalization, end-to-end learning, adaptive control, and sim-to-real transfer. Our results reveal insights into the limitations of current methods and demonstrate the utility of the framework in producing policies capable of real-world operation. These contributions establish the Space Robotics Bench as a valuable resource for developing, benchmarking, and deploying the robust autonomous systems required for the final frontier.

Matteo El-Hariry, Antoine Richard, Ricard M. Castan et al.

Autonomous robots must navigate and operate in diverse environments, from terrestrial and aquatic settings to aerial and space domains. While Reinforcement Learning (RL) has shown promise in training policies for specific autonomous robots, existing frameworks and benchmarks are often constrained to unique platforms, limiting generalization and fair comparisons across different mobility systems. In this paper, we present a multi-domain framework for training, evaluating and deploying RL-based navigation policies across diverse robotic platforms and operational environments. Our work presents four key contributions: (1) a scalable and modular framework, facilitating seamless robot-task interchangeability and reproducible training pipelines; (2) sim-to-real transfer demonstrated through real-world experiments with multiple robots, including a satellite robotic simulator, an unmanned surface vessel, and a wheeled ground vehicle; (3) the release of the first open-source API for deploying Isaac Lab-trained policies to real robots, enabling lightweight inference and rapid field validation; and (4) uniform tasks and metrics for cross-medium evaluation, through a unified evaluation testbed to assess performance of navigation tasks in diverse operational conditions (aquatic, terrestrial and space). By ensuring consistency between simulation and real-world deployment, RoboRAN lowers the barrier to developing adaptable RL-based navigation strategies. Its modular design enables straightforward integration of new robots and tasks through predefined templates, fostering reproducibility and extension to diverse domains. To support the community, we release RoboRAN as open-source.

Adil Zouitine, David Bertoin, Pierre Clavier et al.

Robust reinforcement learning is the problem of learning control policies that provide optimal worst-case performance against a span of adversarial environments. It is a crucial ingredient for deploying algorithms in real-world scenarios with prevalent environmental uncertainties and has been a long-standing object of attention in the community, without a standardized set of benchmarks. This contribution endeavors to fill this gap. We introduce the Robust Reinforcement Learning Suite (RRLS), a benchmark suite based on Mujoco environments. RRLS provides six continuous control tasks with two types of uncertainty sets for training and evaluation. Our benchmark aims to standardize robust reinforcement learning tasks, facilitating reproducible and comparable experiments, in particular those from recent state-of-the-art contributions, for which we demonstrate the use of RRLS. It is also designed to be easily expandable to new environments. The source code is available at \href{https://github.com/SuReLI/RRLS}{https://github.com/SuReLI/RRLS}.

Robert Dadashi, Léonard Hussenot, Damien Vincent et al.

In this paper, we propose a novel Reinforcement Learning (RL) framework for problems with continuous action spaces: Action Quantization from Demonstrations (AQuaDem). The proposed approach consists in learning a discretization of continuous action spaces from human demonstrations. This discretization returns a set of plausible actions (in light of the demonstrations) for each input state, thus capturing the priors of the demonstrator and their multimodal behavior. By discretizing the action space, any discrete action deep RL technique can be readily applied to the continuous control problem. Experiments show that the proposed approach outperforms state-of-the-art methods such as SAC in the RL setup, and GAIL in the Imitation Learning setup. We provide a website with interactive videos: https://google-research.github.io/aquadem/ and make the code available: https://github.com/google-research/google-research/tree/master/aquadem.

Geoffrey Cideron, Sertan Girgin, Mauro Verzetti et al.

We propose MusicRL, the first music generation system finetuned from human feedback. Appreciation of text-to-music models is particularly subjective since the concept of musicality as well as the specific intention behind a caption are user-dependent (e.g. a caption such as "upbeat work-out music" can map to a retro guitar solo or a techno pop beat). Not only this makes supervised training of such models challenging, but it also calls for integrating continuous human feedback in their post-deployment finetuning. MusicRL is a pretrained autoregressive MusicLM (Agostinelli et al., 2023) model of discrete audio tokens finetuned with reinforcement learning to maximise sequence-level rewards. We design reward functions related specifically to text-adherence and audio quality with the help from selected raters, and use those to finetune MusicLM into MusicRL-R. We deploy MusicLM to users and collect a substantial dataset comprising 300,000 pairwise preferences. Using Reinforcement Learning from Human Feedback (RLHF), we train MusicRL-U, the first text-to-music model that incorporates human feedback at scale. Human evaluations show that both MusicRL-R and MusicRL-U are preferred to the baseline. Ultimately, MusicRL-RU combines the two approaches and results in the best model according to human raters. Ablation studies shed light on the musical attributes influencing human preferences, indicating that text adherence and quality only account for a part of it. This underscores the prevalence of subjectivity in musical appreciation and calls for further involvement of human listeners in the finetuning of music generation models.

Team Cohere, Aakanksha, Arash Ahmadian et al. · mila

In this report we describe the development of Command A, a powerful large language model purpose-built to excel at real-world enterprise use cases. Command A is an agent-optimised and multilingual-capable model, with support for 23 languages of global business, and a novel hybrid architecture balancing efficiency with top of the range performance. It offers best-in-class Retrieval Augmented Generation (RAG) capabilities with grounding and tool use to automate sophisticated business processes. These abilities are achieved through a decentralised training approach, including self-refinement algorithms and model merging techniques. We also include results for Command R7B which shares capability and architectural similarities to Command A. Weights for both models have been released for research purposes. This technical report details our original training pipeline and presents an extensive evaluation of our models across a suite of enterprise-relevant tasks and public benchmarks, demonstrating excellent performance and efficiency.

Zhengyan Shi, Sander Land, Acyr Locatelli et al.

Direct Alignment Algorithms (DAAs), such as Direct Preference Optimisation (DPO) and Identity Preference Optimisation (IPO), have emerged as alternatives to online Reinforcement Learning from Human Feedback (RLHF) algorithms such as Proximal Policy Optimisation (PPO) for aligning language models to human preferences, without the need for explicit reward modelling. These methods generally aim to increase the likelihood of generating better (preferred) completions while discouraging worse (non-preferred) ones, while staying close to the original model's behaviour. In this work, we explore the relationship between completion likelihood and model performance in state-of-the-art DAAs, and identify a critical issue of likelihood over-optimisation. Contrary to expectations, we find that higher likelihood of better completions and larger margins between better and worse completion likelihoods do not necessarily lead to better performance, and may even degrade it. Our analysis reveals that while higher likelihood correlates with better memorisation of factual knowledge patterns, a slightly lower completion likelihood tends to improve output diversity, thus leading to better generalisation to unseen scenarios. Moreover, we identify two key indicators that signal when over-optimised output diversity begins to harm performance: Decreasing Entropy over Top-k Tokens and Diminishing Top-k Probability Mass. Our experimental results validate that these indicators are reliable signs of declining performance under different regularisations, helping prevent over-optimisation and improve alignment with human preferences.

Zida Wu, Mathieu Lauriere, Samuel Jia Cong Chua et al.

Mean Field Games (MFGs) have the ability to handle large-scale multi-agent systems, but learning Nash equilibria in MFGs remains a challenging task. In this paper, we propose a deep reinforcement learning (DRL) algorithm that achieves population-dependent Nash equilibrium without the need for averaging or sampling from history, inspired by Munchausen RL and Online Mirror Descent. Through the design of an additional inner-loop replay buffer, the agents can effectively learn to achieve Nash equilibrium from any distribution, mitigating catastrophic forgetting. The resulting policy can be applied to various initial distributions. Numerical experiments on four canonical examples demonstrate our algorithm has better convergence properties than SOTA algorithms, in particular a DRL version of Fictitious Play for population-dependent policies.

Kai Cui, Gökçe Dayanıklı, Mathieu Laurière et al.

Recent techniques based on Mean Field Games (MFGs) allow the scalable analysis of multi-player games with many similar, rational agents. However, standard MFGs remain limited to homogeneous players that weakly influence each other, and cannot model major players that strongly influence other players, severely limiting the class of problems that can be handled. We propose a novel discrete time version of major-minor MFGs (M3FGs), along with a learning algorithm based on fictitious play and partitioning the probability simplex. Importantly, M3FGs generalize MFGs with common noise and can handle not only random exogeneous environment states but also major players. A key challenge is that the mean field is stochastic and not deterministic as in standard MFGs. Our theoretical investigation verifies both the M3FG model and its algorithmic solution, showing firstly the well-posedness of the M3FG model starting from a finite game of interest, and secondly convergence and approximation guarantees of the fictitious play algorithm. Then, we empirically verify the obtained theoretical results, ablating some of the theoretical assumptions made, and show successful equilibrium learning in three example problems. Overall, we establish a learning framework for a novel and broad class of tractable games.

Andrej Orsula, Matthieu Geist, Miguel Olivares-Mendez et al.

The ability to autonomously assemble structures is crucial for the development of future space infrastructure. However, the unpredictable conditions of space pose significant challenges for robotic systems, necessitating the development of advanced learning techniques to enable autonomous assembly. In this study, we present a novel approach for learning autonomous peg-in-hole assembly in the context of space robotics. Our focus is on enhancing the generalization and adaptability of autonomous systems through deep reinforcement learning. By integrating procedural generation and domain randomization, we train agents in a highly parallelized simulation environment across a spectrum of diverse scenarios with the aim of acquiring a robust policy. The proposed approach is evaluated using three distinct reinforcement learning algorithms to investigate the trade-offs among various paradigms. We demonstrate the adaptability of our agents to novel scenarios and assembly sequences while emphasizing the potential of leveraging advanced simulation techniques for robot learning in space. Our findings set the stage for future advancements in intelligent robotic systems capable of supporting ambitious space missions and infrastructure development beyond Earth.

Till Freihaut, Luca Viano, Emanuele Nevali et al.

We close open theoretical gaps in Multi-Agent Imitation Learning (MAIL) by characterizing the limits of non-interactive MAIL and presenting the first interactive algorithm with near-optimal sample complexity. In the non-interactive setting, we prove a statistical lower bound that identifies the all-policy deviation concentrability coefficient as the fundamental complexity measure, and we show that Behavior Cloning (BC) is rate-optimal. For the interactive setting, we introduce a framework that combines reward-free reinforcement learning with interactive MAIL and instantiate it with an algorithm, MAIL-WARM. It improves the best previously known sample complexity from $\mathcal{O}(\varepsilon^{-8})$ to $\mathcal{O}(\varepsilon^{-2}),$ matching the dependence on $\varepsilon$ implied by our lower bound. Finally, we provide numerical results that support our theory and illustrate, in environments such as grid worlds, where Behavior Cloning fails to learn.

Andrej Orsula, Matthieu Geist, Miguel Olivares-Mendez et al.

Autonomous regolith excavation is a cornerstone of in-situ resource utilization for a sustained human presence beyond Earth. However, this task is fundamentally hindered by the complex interaction dynamics of granular media and the operational need for robots to use diverse tools. To address these challenges, this work introduces a framework where a model-based reinforcement learning agent learns within a parallelized simulation. This environment leverages high-fidelity particle physics and procedural generation to create a vast distribution of both lunar terrains and excavation tool geometries. To master this diversity, the agent learns an adaptive interaction strategy by dynamically modulating its own stiffness and damping at each control step through operational space control. Our experiments demonstrate that training with a procedural distribution of tools is critical for generalization and enables the development of sophisticated tool-aware behavior. Furthermore, we show that augmenting the agent with visual feedback significantly improves task success. These results represent a validated methodology for developing the robust and versatile autonomous systems required for the foundational tasks of future space missions.

Zida Wu, Mathieu Lauriere, Matthieu Geist et al.

Mean Field Games (MFGs) offer a powerful framework for studying large-scale multi-agent systems. Yet, learning Nash equilibria in MFGs remains a challenging problem, particularly when the initial distribution is unknown or when the population is subject to common noise. In this paper, we introduce an efficient deep reinforcement learning (DRL) algorithm designed to achieve population-dependent Nash equilibria without relying on averaging or historical sampling, inspired by Munchausen RL and Online Mirror Descent. The resulting policy is adaptable to various initial distributions and sources of common noise. Through numerical experiments on seven canonical examples, we demonstrate that our algorithm exhibits superior convergence properties compared to state-of-the-art algorithms, particularly a DRL version of Fictitious Play for population-dependent policies. The performance in the presence of common noise underscores the robustness and adaptability of our approach.

Amit Sinha, Matthieu Geist, Aditya Mahajan

In this paper, we present a framework to understand the convergence of commonly used Q-learning reinforcement learning algorithms in practice. Two salient features of such algorithms are: (i)~the Q-table is recursively updated using an agent state (such as the state of a recurrent neural network) which is not a belief state or an information state and (ii)~policy regularization is often used to encourage exploration and stabilize the learning algorithm. We investigate the simplest form of such Q-learning algorithms which we call regularized agent-state-based Q-learning (RASQL) and show that it converges under mild technical conditions to the fixed point of an appropriately defined regularized MDP, which depends on the stationary distribution induced by the behavioral policy. We also show that a similar analysis continues to work for a variant of RASQL that learns periodic policies. We present numerical examples to illustrate that the empirical convergence behavior matches with the proposed theoretical limit.

Nathan Grinsztajn, Yannis Flet-Berliac, Mohammad Gheshlaghi Azar et al.

To better align Large Language Models (LLMs) with human judgment, Reinforcement Learning from Human Feedback (RLHF) learns a reward model and then optimizes it using regularized RL. Recently, direct alignment methods were introduced to learn such a fine-tuned model directly from a preference dataset without computing a proxy reward function. These methods are built upon contrastive losses involving the log-likelihood of (dis)preferred completions according to the trained model. However, completions have various lengths, and the log-likelihood is not length-invariant. On the other side, the cross-entropy loss used in supervised training is length-invariant, as batches are typically averaged token-wise. To reconcile these approaches, we introduce a principled approach for making direct alignment length-invariant. Formally, we introduce a new averaging operator, to be composed with the optimality operator giving the best policy for the underlying RL problem. It translates into averaging the log-likelihood within the loss. We empirically study the effect of such averaging, observing a trade-off between the length of generations and their scores.

Yannis Flet-Berliac, Nathan Grinsztajn, Florian Strub et al.

Reinforcement Learning (RL) has been used to finetune Large Language Models (LLMs) using a reward model trained from preference data, to better align with human judgment. The recently introduced direct alignment methods, which are often simpler, more stable, and computationally lighter, can more directly achieve this. However, these approaches cannot optimize arbitrary rewards, and the preference-based ones are not the only rewards of interest for LLMs (eg., unit tests for code generation or textual entailment for summarization, among others). RL-finetuning is usually done with a variation of policy gradient, which calls for on-policy or near-on-policy samples, requiring costly generations. We introduce Contrastive Policy Gradient, or CoPG, a simple and mathematically principled new RL algorithm that can estimate the optimal policy even from off-policy data. It can be seen as an off-policy policy gradient approach that does not rely on important sampling techniques and highlights the importance of using (the right) state baseline. We show this approach to generalize the direct alignment method IPO (identity preference optimization) and classic policy gradient. We experiment with the proposed CoPG on a toy bandit problem to illustrate its properties, as well as for finetuning LLMs on a summarization task, using a learned reward function considered as ground truth for the purpose of the experiments.

Adil Zouitine, David Bertoin, Pierre Clavier et al.

Robust reinforcement learning is essential for deploying reinforcement learning algorithms in real-world scenarios where environmental uncertainty predominates. Traditional robust reinforcement learning often depends on rectangularity assumptions, where adverse probability measures of outcome states are assumed to be independent across different states and actions. This assumption, rarely fulfilled in practice, leads to overly conservative policies. To address this problem, we introduce a new time-constrained robust MDP (TC-RMDP) formulation that considers multifactorial, correlated, and time-dependent disturbances, thus more accurately reflecting real-world dynamics. This formulation goes beyond the conventional rectangularity paradigm, offering new perspectives and expanding the analytical framework for robust RL. We propose three distinct algorithms, each using varying levels of environmental information, and evaluate them extensively on continuous control benchmarks. Our results demonstrate that these algorithms yield an efficient tradeoff between performance and robustness, outperforming traditional deep robust RL methods in time-constrained environments while preserving robustness in classical benchmarks. This study revisits the prevailing assumptions in robust RL and opens new avenues for developing more practical and realistic RL applications.

Pierre Clavier, Emmanuel Rachelson, Erwan Le Pennec et al.

Many classic Reinforcement Learning (RL) algorithms rely on a Bellman operator, which involves an expectation over the next states, leading to the concept of bootstrapping. To introduce a form of pessimism, we propose to replace this expectation with an expectile. In practice, this can be very simply done by replacing the $L_2$ loss with a more general expectile loss for the critic. Introducing pessimism in RL is desirable for various reasons, such as tackling the overestimation problem (for which classic solutions are double Q-learning or the twin-critic approach of TD3) or robust RL (where transitions are adversarial). We study empirically these two cases. For the overestimation problem, we show that the proposed approach, ExpectRL, provides better results than a classic twin-critic. On robust RL benchmarks, involving changes of the environment, we show that our approach is more robust than classic RL algorithms. We also introduce a variation of ExpectRL combined with domain randomization which is competitive with state-of-the-art robust RL agents. Eventually, we also extend \ExpectRL with a mechanism for choosing automatically the expectile value, that is the degree of pessimism

Eugene Choi, Arash Ahmadian, Matthieu Geist et al.

Online and offline RLHF methods, such as PPO and DPO, have been highly successful in aligning AI with human preferences. Despite their success, however, these methods suffer from fundamental limitations: (a) Models trained with RLHF can learn from mistakes or negative examples through RL mechanism or contrastive loss during training. However, at inference time, they lack an innate self-improvement mechanism for error corrections. (b) The optimal solution of existing methods is highly task-dependent, making it difficult for them to generalize to new tasks. To address these challenges, we propose Self-Improving Robust Preference Optimization (SRPO), a practical and mathematically principled offline RLHF framework. The key idea behind SRPO is to cast the problem of learning from human preferences as a self-improvement process, mathematically formulated as a min-max objective that jointly optimizes a self-improvement policy and a generative policy in an adversarial fashion. Crucially, the solution for this optimization problem is independent of the training task, which makes it robust to its changes. We then show that this objective can be reformulated as a non-adversarial offline loss, which can be efficiently optimized using standard supervised learning techniques at scale. To demonstrate SRPO's effectiveness, we evaluate it using AI Win-Rate (WR) against human (GOLD) completions. When tested on the XSum dataset, SRPO outperforms DPO by a margin of 15% after 5 self revisions, achieving an impressive 90% WR. Moreover, on the challenging Arena-Hard prompts, SRPO outperforms both DPO and IPO (by 4% without revision and 6% after a single revision), reaching a 56% WR against against Llama-3.1-8B-Instruct.

Raj Ghugare, Matthieu Geist, Glen Berseth et al.

Some reinforcement learning (RL) algorithms can stitch pieces of experience to solve a task never seen before during training. This oft-sought property is one of the few ways in which RL methods based on dynamic-programming differ from RL methods based on supervised-learning (SL). Yet, certain RL methods based on off-the-shelf SL algorithms achieve excellent results without an explicit mechanism for stitching; it remains unclear whether those methods forgo this important stitching property. This paper studies this question for the problems of achieving a target goal state and achieving a target return value. Our main result is to show that the stitching property corresponds to a form of combinatorial generalization: after training on a distribution of (state, goal) pairs, one would like to evaluate on (state, goal) pairs not seen together in the training data. Our analysis shows that this sort of generalization is different from i.i.d. generalization. This connection between stitching and generalisation reveals why we should not expect SL-based RL methods to perform stitching, even in the limit of large datasets and models. Based on this analysis, we construct new datasets to explicitly test for this property, revealing that SL-based methods lack this stitching property and hence fail to perform combinatorial generalization. Nonetheless, the connection between stitching and combinatorial generalisation also suggests a simple remedy for improving generalisation in SL: data augmentation. We propose a temporal data augmentation and demonstrate that adding it to SL-based methods enables them to successfully complete tasks not seen together during training. On a high level, this connection illustrates the importance of combinatorial generalization for data efficiency in time-series data beyond tasks beyond RL, like audio, video, or text.

Benjamin Eysenbach, Matthieu Geist, Sergey Levine et al.

As with any machine learning problem with limited data, effective offline RL algorithms require careful regularization to avoid overfitting. One-step methods perform regularization by doing just a single step of policy improvement, while critic regularization methods do many steps of policy improvement with a regularized objective. These methods appear distinct. One-step methods, such as advantage-weighted regression and conditional behavioral cloning, truncate policy iteration after just one step. This ``early stopping'' makes one-step RL simple and stable, but can limit its asymptotic performance. Critic regularization typically requires more compute but has appealing lower-bound guarantees. In this paper, we draw a close connection between these methods: applying a multi-step critic regularization method with a regularization coefficient of 1 yields the same policy as one-step RL. While practical implementations violate our assumptions and critic regularization is typically applied with smaller regularization coefficients, our experiments nevertheless show that our analysis makes accurate, testable predictions about practical offline RL methods (CQL and one-step RL) with commonly-used hyperparameters. Our results that every problem can be solved with a single step of policy improvement, but rather that one-step RL might be competitive with critic regularization on RL problems that demand strong regularization.

Paul Roit, Johan Ferret, Lior Shani et al.

Despite the seeming success of contemporary grounded text generation systems, they often tend to generate factually inconsistent text with respect to their input. This phenomenon is emphasized in tasks like summarization, in which the generated summaries should be corroborated by their source article. In this work, we leverage recent progress on textual entailment models to directly address this problem for abstractive summarization systems. We use reinforcement learning with reference-free, textual entailment rewards to optimize for factual consistency and explore the ensuing trade-offs, as improved consistency may come at the cost of less informative or more extractive summaries. Our results, according to both automatic metrics and human evaluation, show that our method considerably improves the faithfulness, salience, and conciseness of the generated summaries.

Laixi Shi, Gen Li, Yuting Wei et al.

This paper investigates model robustness in reinforcement learning (RL) to reduce the sim-to-real gap in practice. We adopt the framework of distributionally robust Markov decision processes (RMDPs), aimed at learning a policy that optimizes the worst-case performance when the deployed environment falls within a prescribed uncertainty set around the nominal MDP. Despite recent efforts, the sample complexity of RMDPs remained mostly unsettled regardless of the uncertainty set in use. It was unclear if distributional robustness bears any statistical consequences when benchmarked against standard RL. Assuming access to a generative model that draws samples based on the nominal MDP, we provide a near-optimal characterization of the sample complexity of RMDPs when the uncertainty set is specified via either the total variation (TV) distance or chi-squared divergence. The algorithm studied here is a model-based method called distributionally robust value iteration, which is shown to be near-optimal for the full range of uncertainty levels. Somewhat surprisingly, our results uncover that RMDPs are not necessarily easier or harder to learn than standard MDPs. The statistical consequence incurred by the robustness requirement depends heavily on the size and shape of the uncertainty set: in the case w.r.t.~the TV distance, the minimax sample complexity of RMDPs is always smaller than that of standard MDPs; in the case w.r.t.~the chi-squared divergence, the sample complexity of RMDPs far exceeds the standard MDP counterpart.

Toshinori Kitamura, Tadashi Kozuno, Yunhao Tang et al.

Mirror descent value iteration (MDVI), an abstraction of Kullback-Leibler (KL) and entropy-regularized reinforcement learning (RL), has served as the basis for recent high-performing practical RL algorithms. However, despite the use of function approximation in practice, the theoretical understanding of MDVI has been limited to tabular Markov decision processes (MDPs). We study MDVI with linear function approximation through its sample complexity required to identify an $\varepsilon$-optimal policy with probability $1-δ$ under the settings of an infinite-horizon linear MDP, generative model, and G-optimal design. We demonstrate that least-squares regression weighted by the variance of an estimated optimal value function of the next state is crucial to achieving minimax optimality. Based on this observation, we present Variance-Weighted Least-Squares MDVI (VWLS-MDVI), the first theoretical algorithm that achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs. Furthermore, we propose a practical VWLS algorithm for value-based deep RL, Deep Variance Weighting (DVW). Our experiments demonstrate that DVW improves the performance of popular value-based deep RL algorithms on a set of MinAtar benchmarks.

Geoffrey Cideron, Baruch Tabanpour, Sebastian Curi et al.

We consider the Imitation Learning (IL) setup where expert data are not collected on the actual deployment environment but on a different version. To address the resulting distribution shift, we combine behavior cloning (BC) with a planner that is tasked to bring the agent back to states visited by the expert whenever the agent deviates from the demonstration distribution. The resulting algorithm, POIR, can be trained offline, and leverages online interactions to efficiently fine-tune its planner to improve performance over time. We test POIR on a variety of human-generated manipulation demonstrations in a realistic robotic manipulation simulator and show robustness of the learned policy to different initial state distributions and noisy dynamics.

Esther Derman, Matthieu Geist, Shie Mannor

Robust Markov decision processes (MDPs) aim to handle changing or partially known system dynamics. To solve them, one typically resorts to robust optimization methods. However, this significantly increases computational complexity and limits scalability in both learning and planning. On the other hand, regularized MDPs show more stability in policy learning without impairing time complexity. Yet, they generally do not encompass uncertainty in the model dynamics. In this work, we aim to learn robust MDPs using regularization. We first show that regularized MDPs are a particular instance of robust MDPs with uncertain reward. We thus establish that policy iteration on reward-robust MDPs can have the same time complexity as on regularized MDPs. We further extend this relationship to MDPs with uncertain transitions: this leads to a regularization term with an additional dependence on the value function. We finally generalize regularized MDPs to twice regularized MDPs (R${}^2$ MDPs), i.e., MDPs with $\textit{both}$ value and policy regularization. The corresponding Bellman operators enable developing policy iteration schemes with convergence and robustness guarantees. It also reduces planning and learning in robust MDPs to regularized MDPs.

Thibault Lahire, Matthieu Geist, Emmanuel Rachelson

Several algorithms have been proposed to sample non-uniformly the replay buffer of deep Reinforcement Learning (RL) agents to speed-up learning, but very few theoretical foundations of these sampling schemes have been provided. Among others, Prioritized Experience Replay appears as a hyperparameter sensitive heuristic, even though it can provide good performance. In this work, we cast the replay buffer sampling problem as an importance sampling one for estimating the gradient. This allows deriving the theoretically optimal sampling distribution, yielding the best theoretical convergence speed. Elaborating on the knowledge of the ideal sampling scheme, we exhibit new theoretical foundations of Prioritized Experience Replay. The optimal sampling distribution being intractable, we make several approximations providing good results in practice and introduce, among others, LaBER (Large Batch Experience Replay), an easy-to-code and efficient method for sampling the replay buffer. LaBER, which can be combined with Deep Q-Networks, distributional RL agents or actor-critic methods, yields improved performance over a diverse range of Atari games and PyBullet environments, compared to the base agent it is implemented on and to other prioritization schemes.

Nino Vieillard, Marcin Andrychowicz, Anton Raichuk et al.

The $Q$-function is a central quantity in many Reinforcement Learning (RL) algorithms for which RL agents behave following a (soft)-greedy policy w.r.t. to $Q$. It is a powerful tool that allows action selection without a model of the environment and even without explicitly modeling the policy. Yet, this scheme can only be used in discrete action tasks, with small numbers of actions, as the softmax cannot be computed exactly otherwise. Especially the usage of function approximation, to deal with continuous action spaces in modern actor-critic architectures, intrinsically prevents the exact computation of a softmax. We propose to alleviate this issue by parametrizing the $Q$-function implicitly, as the sum of a log-policy and of a value function. We use the resulting parametrization to derive a practical off-policy deep RL algorithm, suitable for large action spaces, and that enforces the softmax relation between the policy and the $Q$-value. We provide a theoretical analysis of our algorithm: from an Approximate Dynamic Programming perspective, we show its equivalence to a regularized version of value iteration, accounting for both entropy and Kullback-Leibler regularization, and that enjoys beneficial error propagation results. We then evaluate our algorithm on classic control tasks, where its results compete with state-of-the-art methods.

Sharan Vaswani, Olivier Bachem, Simone Totaro et al.

Common policy gradient methods rely on the maximization of a sequence of surrogate functions. In recent years, many such surrogate functions have been proposed, most without strong theoretical guarantees, leading to algorithms such as TRPO, PPO or MPO. Rather than design yet another surrogate function, we instead propose a general framework (FMA-PG) based on functional mirror ascent that gives rise to an entire family of surrogate functions. We construct surrogate functions that enable policy improvement guarantees, a property not shared by most existing surrogate functions. Crucially, these guarantees hold regardless of the choice of policy parameterization. Moreover, a particular instantiation of FMA-PG recovers important implementation heuristics (e.g., using forward vs reverse KL divergence) resulting in a variant of TRPO with additional desirable properties. Via experiments on simple bandit problems, we evaluate the algorithms instantiated by FMA-PG. The proposed framework also suggests an improved variant of PPO, whose robustness and efficiency we empirically demonstrate on the MuJoCo suite.

Nathan Grinsztajn, Johan Ferret, Olivier Pietquin et al.

We propose to learn to distinguish reversible from irreversible actions for better informed decision-making in Reinforcement Learning (RL). From theoretical considerations, we show that approximate reversibility can be learned through a simple surrogate task: ranking randomly sampled trajectory events in chronological order. Intuitively, pairs of events that are always observed in the same order are likely to be separated by an irreversible sequence of actions. Conveniently, learning the temporal order of events can be done in a fully self-supervised way, which we use to estimate the reversibility of actions from experience, without any priors. We propose two different strategies that incorporate reversibility in RL agents, one strategy for exploration (RAE) and one strategy for control (RAC). We demonstrate the potential of reversibility-aware agents in several environments, including the challenging Sokoban game. In synthetic tasks, we show that we can learn control policies that never fail and reduce to zero the side-effects of interactions, even without access to the reward function.

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.

Manu Orsini, Anton Raichuk, Léonard Hussenot et al.

Adversarial imitation learning has become a popular framework for imitation in continuous control. Over the years, several variations of its components were proposed to enhance the performance of the learned policies as well as the sample complexity of the algorithm. In practice, these choices are rarely tested all together in rigorous empirical studies. It is therefore difficult to discuss and understand what choices, among the high-level algorithmic options as well as low-level implementation details, matter. To tackle this issue, we implement more than 50 of these choices in a generic adversarial imitation learning framework and investigate their impacts in a large-scale study (>500k trained agents) with both synthetic and human-generated demonstrations. While many of our findings confirm common practices, some of them are surprising or even contradict prior work. In particular, our results suggest that artificial demonstrations are not a good proxy for human data and that the very common practice of evaluating imitation algorithms only with synthetic demonstrations may lead to algorithms which perform poorly in the more realistic scenarios with human demonstrations.

Leonard Hussenot, Marcin Andrychowicz, Damien Vincent et al.

We address the issue of tuning hyperparameters (HPs) for imitation learning algorithms in the context of continuous-control, when the underlying reward function of the demonstrating expert cannot be observed at any time. The vast literature in imitation learning mostly considers this reward function to be available for HP selection, but this is not a realistic setting. Indeed, would this reward function be available, it could then directly be used for policy training and imitation would not be necessary. To tackle this mostly ignored problem, we propose a number of possible proxies to the external reward. We evaluate them in an extensive empirical study (more than 10'000 agents across 9 environments) and make practical recommendations for selecting HPs. Our results show that while imitation learning algorithms are sensitive to HP choices, it is often possible to select good enough HPs through a proxy to the reward function.

Robert Dadashi, Shideh Rezaeifar, Nino Vieillard et al.

Offline Reinforcement Learning methods seek to learn a policy from logged transitions of an environment, without any interaction. In the presence of function approximation, and under the assumption of limited coverage of the state-action space of the environment, it is necessary to enforce the policy to visit state-action pairs close to the support of logged transitions. In this work, we propose an iterative procedure to learn a pseudometric (closely related to bisimulation metrics) from logged transitions, and use it to define this notion of closeness. We show its convergence and extend it to the function approximation setting. We then use this pseudometric to define a new lookup based bonus in an actor-critic algorithm: PLOFF. This bonus encourages the actor to stay close, in terms of the defined pseudometric, to the support of logged transitions. Finally, we evaluate the method on hand manipulation and locomotion tasks.

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.