Shimon Whiteson

Alexander Rutherford, Benjamin Ellis, Matteo Gallici et al. · deepmind, meta-ai

Benchmarks are crucial in the development of machine learning algorithms, with available environments significantly influencing reinforcement learning (RL) research. Traditionally, RL environments run on the CPU, which limits their scalability with typical academic compute. However, recent advancements in JAX have enabled the wider use of hardware acceleration, enabling massively parallel RL training pipelines and environments. While this has been successfully applied to single-agent RL, it has not yet been widely adopted for multi-agent scenarios. In this paper, we present JaxMARL, the first open-source, Python-based library that combines GPU-enabled efficiency with support for a large number of commonly used MARL environments and popular baseline algorithms. Our experiments show that, in terms of wall clock time, our JAX-based training pipeline is around 14 times faster than existing approaches, and up to 12500x when multiple training runs are vectorized. This enables efficient and thorough evaluations, potentially alleviating the evaluation crisis in the field. We also introduce and benchmark SMAX, a JAX-based approximate reimplementation of the popular StarCraft Multi-Agent Challenge, which removes the need to run the StarCraft II game engine. This not only enables GPU acceleration, but also provides a more flexible MARL environment, unlocking the potential for self-play, meta-learning, and other future applications in MARL. The code is available at https://github.com/flairox/jaxmarl.

Benjamin Ellis, Jonathan Cook, Skander Moalla et al. · deepmind

The availability of challenging benchmarks has played a key role in the recent progress of machine learning. In cooperative multi-agent reinforcement learning, the StarCraft Multi-Agent Challenge (SMAC) has become a popular testbed for centralised training with decentralised execution. However, after years of sustained improvement on SMAC, algorithms now achieve near-perfect performance. In this work, we conduct new analysis demonstrating that SMAC lacks the stochasticity and partial observability to require complex *closed-loop* policies. In particular, we show that an *open-loop* policy conditioned only on the timestep can achieve non-trivial win rates for many SMAC scenarios. To address this limitation, we introduce SMACv2, a new version of the benchmark where scenarios are procedurally generated and require agents to generalise to previously unseen settings (from the same distribution) during evaluation. We also introduce the extended partial observability challenge (EPO), which augments SMACv2 to ensure meaningful partial observability. We show that these changes ensure the benchmark requires the use of *closed-loop* policies. We evaluate state-of-the-art algorithms on SMACv2 and show that it presents significant challenges not present in the original benchmark. Our analysis illustrates that SMACv2 addresses the discovered deficiencies of SMAC and can help benchmark the next generation of MARL methods. Videos of training are available at https://sites.google.com/view/smacv2.

Jacob Beck, Risto Vuorio, Zheng Xiong et al.

Deep reinforcement learning (RL) is notoriously impractical to deploy due to sample inefficiency. Meta-RL directly addresses this sample inefficiency by learning to perform few-shot learning when a distribution of related tasks is available for meta-training. While many specialized meta-RL methods have been proposed, recent work suggests that end-to-end learning in conjunction with an off-the-shelf sequential model, such as a recurrent network, is a surprisingly strong baseline. However, such claims have been controversial due to limited supporting evidence, particularly in the face of prior work establishing precisely the opposite. In this paper, we conduct an empirical investigation. While we likewise find that a recurrent network can achieve strong performance, we demonstrate that the use of hypernetworks is crucial to maximizing their potential. Surprisingly, when combined with hypernetworks, the recurrent baselines that are far simpler than existing specialized methods actually achieve the strongest performance of all methods evaluated. We provide code at https://github.com/jacooba/hyper.

Yiren Lu, Justin Fu, George Tucker et al.

Imitation learning (IL) is a simple and powerful way to use high-quality human driving data, which can be collected at scale, to produce human-like behavior. However, policies based on imitation learning alone often fail to sufficiently account for safety and reliability concerns. In this paper, we show how imitation learning combined with reinforcement learning using simple rewards can substantially improve the safety and reliability of driving policies over those learned from imitation alone. In particular, we train a policy on over 100k miles of urban driving data, and measure its effectiveness in test scenarios grouped by different levels of collision likelihood. Our analysis shows that while imitation can perform well in low-difficulty scenarios that are well-covered by the demonstration data, our proposed approach significantly improves robustness on the most challenging scenarios (over 38% reduction in failures). To our knowledge, this is the first application of a combined imitation and reinforcement learning approach in autonomous driving that utilizes large amounts of real-world human driving data.

Matthew Thomas Jackson, Minqi Jiang, Jack Parker-Holder et al. · oxford

The past decade has seen vast progress in deep reinforcement learning (RL) on the back of algorithms manually designed by human researchers. Recently, it has been shown that it is possible to meta-learn update rules, with the hope of discovering algorithms that can perform well on a wide range of RL tasks. Despite impressive initial results from algorithms such as Learned Policy Gradient (LPG), there remains a generalization gap when these algorithms are applied to unseen environments. In this work, we examine how characteristics of the meta-training distribution impact the generalization performance of these algorithms. Motivated by this analysis and building on ideas from Unsupervised Environment Design (UED), we propose a novel approach for automatically generating curricula to maximize the regret of a meta-learned optimizer, in addition to a novel approximation of regret, which we name algorithmic regret (AR). The result is our method, General RL Optimizers Obtained Via Environment Design (GROOVE). In a series of experiments, we show that GROOVE achieves superior generalization to LPG, and evaluate AR against baseline metrics from UED, identifying it as a critical component of environment design in this setting. We believe this approach is a step towards the discovery of truly general RL algorithms, capable of solving a wide range of real-world environments.

Jacob Beck, Risto Vuorio, Evan Zheran Liu et al.

While deep reinforcement learning (RL) has fueled multiple high-profile successes in machine learning, it is held back from more widespread adoption by its often poor data efficiency and the limited generality of the policies it produces. A promising approach for alleviating these limitations is to cast the development of better RL algorithms as a machine learning problem itself in a process called meta-RL. Meta-RL is most commonly studied in a problem setting where, given a distribution of tasks, the goal is to learn a policy that is capable of adapting to any new task from the task distribution with as little data as possible. In this survey, we describe the meta-RL problem setting in detail as well as its major variations. We discuss how, at a high level, meta-RL research can be clustered based on the presence of a task distribution and the learning budget available for each individual task. Using these clusters, we then survey meta-RL algorithms and applications. We conclude by presenting the open problems on the path to making meta-RL part of the standard toolbox for a deep RL practitioner.

Alexander D. Goldie, Zilin Wang, Adrian Hayler et al.

Automating the development of machine learning algorithms has the potential to unlock new breakthroughs. However, our ability to improve and evaluate algorithm discovery systems has thus far been limited by existing task suites. They suffer from many issues, such as: poor evaluation methodologies; data contamination; and containing saturated or very similar problems. Here, we introduce DiscoGen, a procedural generator of algorithm discovery tasks for machine learning, such as developing optimisers for reinforcement learning or loss functions for image classification. Motivated by the success of procedural generation in reinforcement learning, DiscoGen spans millions of tasks of varying difficulty and complexity from a range of machine learning fields. These tasks are specified by a small number of configuration parameters and can be used to optimise algorithm discovery agents (ADAs). We present DiscoBench, a benchmark consisting of a fixed, small subset of DiscoGen tasks for principled evaluation of ADAs. Finally, we propose a number of ambitious, impactful research directions enabled by DiscoGen, in addition to experiments demonstrating its use for prompt optimisation of an ADA. DiscoGen is released open-source at https://github.com/AlexGoldie/discogen.

Maximilian Igl, Daewoo Kim, Alex Kuefler et al.

Simulation is a crucial tool for accelerating the development of autonomous vehicles. Making simulation realistic requires models of the human road users who interact with such cars. Such models can be obtained by applying learning from demonstration (LfD) to trajectories observed by cars already on the road. However, existing LfD methods are typically insufficient, yielding policies that frequently collide or drive off the road. To address this problem, we propose Symphony, which greatly improves realism by combining conventional policies with a parallel beam search. The beam search refines these policies on the fly by pruning branches that are unfavourably evaluated by a discriminator. However, it can also harm diversity, i.e., how well the agents cover the entire distribution of realistic behaviour, as pruning can encourage mode collapse. Symphony addresses this issue with a hierarchical approach, factoring agent behaviour into goal generation and goal conditioning. The use of such goals ensures that agent diversity neither disappears during adversarial training nor is pruned away by the beam search. Experiments on both proprietary and open Waymo datasets confirm that Symphony agents learn more realistic and diverse behaviour than several baselines.

Eli Bronstein, Mark Palatucci, Dominik Notz et al.

We demonstrate the first large-scale application of model-based generative adversarial imitation learning (MGAIL) to the task of dense urban self-driving. We augment standard MGAIL using a hierarchical model to enable generalization to arbitrary goal routes, and measure performance using a closed-loop evaluation framework with simulated interactive agents. We train policies from expert trajectories collected from real vehicles driving over 100,000 miles in San Francisco, and demonstrate a steerable policy that can navigate robustly even in a zero-shot setting, generalizing to synthetic scenarios with novel goals that never occurred in real-world driving. We also demonstrate the importance of mixing closed-loop MGAIL losses with open-loop behavior cloning losses, and show our best policy approaches the performance of the expert. We evaluate our imitative model in both average and challenging scenarios, and show how it can serve as a useful prior to plan successful trajectories.

Jacob Beck, Matthew Thomas Jackson, Risto Vuorio et al.

Training a reinforcement learning (RL) agent on a real-world robotics task remains generally impractical due to sample inefficiency. Multi-task RL and meta-RL aim to improve sample efficiency by generalizing over a distribution of related tasks. However, doing so is difficult in practice: In multi-task RL, state of the art methods often fail to outperform a degenerate solution that simply learns each task separately. Hypernetworks are a promising path forward since they replicate the separate policies of the degenerate solution while also allowing for generalization across tasks, and are applicable to meta-RL. However, evidence from supervised learning suggests hypernetwork performance is highly sensitive to the initialization. In this paper, we 1) show that hypernetwork initialization is also a critical factor in meta-RL, and that naive initializations yield poor performance; 2) propose a novel hypernetwork initialization scheme that matches or exceeds the performance of a state-of-the-art approach proposed for supervised settings, as well as being simpler and more general; and 3) use this method to show that hypernetworks can improve performance in meta-RL by evaluating on multiple simulated robotics benchmarks.

Zheng Xiong, Jacob Beck, Shimon Whiteson

Learning a universal policy across different robot morphologies can significantly improve learning efficiency and generalization in continuous control. However, it poses a challenging multi-task reinforcement learning problem, as the optimal policy may be quite different across robots and critically depend on the morphology. Existing methods utilize graph neural networks or transformers to handle heterogeneous state and action spaces across different morphologies, but pay little attention to the dependency of a robot's control policy on its morphology context. In this paper, we propose a hierarchical architecture to better model this dependency via contextual modulation, which includes two key submodules: (1) Instead of enforcing hard parameter sharing across robots, we use hypernetworks to generate morphology-dependent control parameters; (2) We propose a fixed attention mechanism that solely depends on the morphology to modulate the interactions between different limbs in a robot. Experimental results show that our method not only improves learning performance on a diverse set of training robots, but also generalizes better to unseen morphologies in a zero-shot fashion.

Darius Muglich, Christian Schroeder de Witt, Elise van der Pol et al.

Successful coordination in Dec-POMDPs requires agents to adopt robust strategies and interpretable styles of play for their partner. A common failure mode is symmetry breaking, when agents arbitrarily converge on one out of many equivalent but mutually incompatible policies. Commonly these examples include partial observability, e.g. waving your right hand vs. left hand to convey a covert message. In this paper, we present a novel equivariant network architecture for use in Dec-POMDPs that effectively leverages environmental symmetry for improving zero-shot coordination, doing so more effectively than prior methods. Our method also acts as a ``coordination-improvement operator'' for generic, pre-trained policies, and thus may be applied at test-time in conjunction with any self-play algorithm. We provide theoretical guarantees of our work and test on the AI benchmark task of Hanabi, where we demonstrate our methods outperforming other symmetry-aware baselines in zero-shot coordination, as well as able to improve the coordination ability of a variety of pre-trained policies. In particular, we show our method can be used to improve on the state of the art for zero-shot coordination on the Hanabi benchmark.

Eli Bronstein, Sirish Srinivasan, Supratik Paul et al.

ML-based motion planning is a promising approach to produce agents that exhibit complex behaviors, and automatically adapt to novel environments. In the context of autonomous driving, it is common to treat all available training data equally. However, this approach produces agents that do not perform robustly in safety-critical settings, an issue that cannot be addressed by simply adding more data to the training set - we show that an agent trained using only a 10% subset of the data performs just as well as an agent trained on the entire dataset. We present a method to predict the inherent difficulty of a driving situation given data collected from a fleet of autonomous vehicles deployed on public roads. We then demonstrate that this difficulty score can be used in a zero-shot transfer to generate curricula for an imitation-learning based planning agent. Compared to training on the entire unbiased training dataset, we show that prioritizing difficult driving scenarios both reduces collisions by 15% and increases route adherence by 14% in closed-loop evaluation, all while using only 10% of the training data.

Alexander David Goldie, Chris Lu, Matthew Thomas Jackson et al.

While reinforcement learning (RL) holds great potential for decision making in the real world, it suffers from a number of unique difficulties which often need specific consideration. In particular: it is highly non-stationary; suffers from high degrees of plasticity loss; and requires exploration to prevent premature convergence to local optima and maximize return. In this paper, we consider whether learned optimization can help overcome these problems. Our method, Learned Optimization for Plasticity, Exploration and Non-stationarity (OPEN), meta-learns an update rule whose input features and output structure are informed by previously proposed solutions to these difficulties. We show that our parameterization is flexible enough to enable meta-learning in diverse learning contexts, including the ability to use stochasticity for exploration. Our experiments demonstrate that when meta-trained on single and small sets of environments, OPEN outperforms or equals traditionally used optimizers. Furthermore, OPEN shows strong generalization characteristics across a range of environments and agent architectures.

Darius Muglich, Luisa Zintgraf, Christian Schroeder de Witt et al.

Self-play is a common paradigm for constructing solutions in Markov games that can yield optimal policies in collaborative settings. However, these policies often adopt highly-specialized conventions that make playing with a novel partner difficult. To address this, recent approaches rely on encoding symmetry and convention-awareness into policy training, but these require strong environmental assumptions and can complicate policy training. We therefore propose moving the learning of conventions to the belief space. Specifically, we propose a belief learning model that can maintain beliefs over rollouts of policies not seen at training time, and can thus decode and adapt to novel conventions at test time. We show how to leverage this model for both search and training of a best response over various pools of policies to greatly improve ad-hoc teamplay. We also show how our setup promotes explainability and interpretability of nuanced agent conventions.

Mattie Fellows, Matthew J. A. Smith, Shimon Whiteson

Integral to recent successes in deep reinforcement learning has been a class of temporal difference methods that use infrequently updated target values for policy evaluation in a Markov Decision Process. Yet a complete theoretical explanation for the effectiveness of target networks remains elusive. In this work, we provide an analysis of this popular class of algorithms, to finally answer the question: `why do target networks stabilise TD learning'? To do so, we formalise the notion of a partially fitted policy evaluation method, which describes the use of target networks and bridges the gap between fitted methods and semigradient temporal difference algorithms. Using this framework we are able to uniquely characterise the so-called deadly triad - the use of TD updates with (nonlinear) function approximation and off-policy data - which often leads to nonconvergent algorithms. This insight leads us to conclude that the use of target networks can mitigate the effects of poor conditioning in the Jacobian of the TD update. Instead, we show that under mild regularity conditions and a well tuned target network update frequency, convergence can be guaranteed even in the extremely challenging off-policy sampling and nonlinear function approximation setting.

Risto Vuorio, Jacob Beck, Shimon Whiteson et al.

Meta-gradients provide a general approach for optimizing the meta-parameters of reinforcement learning (RL) algorithms. Estimation of meta-gradients is central to the performance of these meta-algorithms, and has been studied in the setting of MAML-style short-horizon meta-RL problems. In this context, prior work has investigated the estimation of the Hessian of the RL objective, as well as tackling the problem of credit assignment to pre-adaptation behavior by making a sampling correction. However, we show that Hessian estimation, implemented for example by DiCE and its variants, always adds bias and can also add variance to meta-gradient estimation. Meanwhile, meta-gradient estimation has been studied less in the important long-horizon setting, where backpropagation through the full inner optimization trajectories is not feasible. We study the bias and variance tradeoff arising from truncated backpropagation and sampling correction, and additionally compare to evolution strategies, which is a recently popular alternative strategy to long-horizon meta-learning. While prior work implicitly chooses points in this bias-variance space, we disentangle the sources of bias and variance and present an empirical study that relates existing estimators to each other.

Angad Singh, Omar Makhlouf, Maximilian Igl et al.

Multi-object state estimation is a fundamental problem for robotic applications where a robot must interact with other moving objects. Typically, other objects' relevant state features are not directly observable, and must instead be inferred from observations. Particle filtering can perform such inference given approximate transition and observation models. However, these models are often unknown a priori, yielding a difficult parameter estimation problem since observations jointly carry transition and observation noise. In this work, we consider learning maximum-likelihood parameters using particle methods. Recent methods addressing this problem typically differentiate through time in a particle filter, which requires workarounds to the non-differentiable resampling step, that yield biased or high variance gradient estimates. By contrast, we exploit Fisher's identity to obtain a particle-based approximation of the score function (the gradient of the log likelihood) that yields a low variance estimate while only requiring stepwise differentiation through the transition and observation models. We apply our method to real data collected from autonomous vehicles (AVs) and show that it learns better models than existing techniques and is more stable in training, yielding an effective smoother for tracking the trajectories of vehicles around an AV.

Yat Long Lo, Christian Schroeder de Witt, Samuel Sokota et al.

By enabling agents to communicate, recent cooperative multi-agent reinforcement learning (MARL) methods have demonstrated better task performance and more coordinated behavior. Most existing approaches facilitate inter-agent communication by allowing agents to send messages to each other through free communication channels, i.e., cheap talk channels. Current methods require these channels to be constantly accessible and known to the agents a priori. In this work, we lift these requirements such that the agents must discover the cheap talk channels and learn how to use them. Hence, the problem has two main parts: cheap talk discovery (CTD) and cheap talk utilization (CTU). We introduce a novel conceptual framework for both parts and develop a new algorithm based on mutual information maximization that outperforms existing algorithms in CTD/CTU settings. We also release a novel benchmark suite to stimulate future research in CTD/CTU.

Mattie Fellows, Brandon Kaplowitz, Christian Schroeder de Witt et al.

Bayesian reinforcement learning (RL) offers a principled and elegant approach for sequential decision making under uncertainty. Most notably, Bayesian agents do not face an exploration/exploitation dilemma, a major pathology of frequentist methods. However theoretical understanding of model-free approaches is lacking. In this paper, we introduce a novel Bayesian model-free formulation and the first analysis showing that model-free approaches can yield Bayes-optimal policies. We show all existing model-free approaches make approximations that yield policies that can be arbitrarily Bayes-suboptimal. As a first step towards model-free Bayes optimality, we introduce the Bayesian exploration network (BEN) which uses normalising flows to model both the aleatoric uncertainty (via density estimation) and epistemic uncertainty (via variational inference) in the Bellman operator. In the limit of complete optimisation, BEN learns true Bayes-optimal policies, but like in variational expectation-maximisation, partial optimisation renders our approach tractable. Empirical results demonstrate that BEN can learn true Bayes-optimal policies in tasks where existing model-free approaches fail.

Maximilian Igl, Punit Shah, Paul Mougin et al.

Many applications of imitation learning require the agent to generate the full distribution of behaviour observed in the training data. For example, to evaluate the safety of autonomous vehicles in simulation, accurate and diverse behaviour models of other road users are paramount. Existing methods that improve this distributional realism typically rely on hierarchical policies. These condition the policy on types such as goals or personas that give rise to multi-modal behaviour. However, such methods are often inappropriate for stochastic environments where the agent must also react to external factors: because agent types are inferred from the observed future trajectory during training, these environments require that the contributions of internal and external factors to the agent behaviour are disentangled and only internal factors, i.e., those under the agent's control, are encoded in the type. Encoding future information about external factors leads to inappropriate agent reactions during testing, when the future is unknown and types must be drawn independently from the actual future. We formalize this challenge as distribution shift in the conditional distribution of agent types under environmental stochasticity. We propose Robust Type Conditioning (RTC), which eliminates this shift with adversarial training under randomly sampled types. Experiments on two domains, including the large-scale Waymo Open Motion Dataset, show improved distributional realism while maintaining or improving task performance compared to state-of-the-art baselines.

Mingfei Sun, Benjamin Ellis, Anuj Mahajan et al.

Trust Region Policy Optimization (TRPO) is an iterative method that simultaneously maximizes a surrogate objective and enforces a trust region constraint over consecutive policies in each iteration. The combination of the surrogate objective maximization and the trust region enforcement has been shown to be crucial to guarantee a monotonic policy improvement. However, solving a trust-region-constrained optimization problem can be computationally intensive as it requires many steps of conjugate gradient and a large number of on-policy samples. In this paper, we show that the trust region constraint over policies can be safely substituted by a trust-region-free constraint without compromising the underlying monotonic improvement guarantee. The key idea is to generalize the surrogate objective used in TRPO in a way that a monotonic improvement guarantee still emerges as a result of constraining the maximum advantage-weighted ratio between policies. This new constraint outlines a conservative mechanism for iterative policy optimization and sheds light on practical ways to optimize the generalized surrogate objective. We show that the new constraint can be effectively enforced by being conservative when optimizing the generalized objective function in practice. We call the resulting algorithm Trust-REgion-Free Policy Optimization (TREFree) as it is free of any explicit trust region constraints. Empirical results show that TREFree outperforms TRPO and Proximal Policy Optimization (PPO) in terms of policy performance and sample efficiency.

Matthew Thomas Jackson, Uljad Berdica, Jarek Liesen et al.

Progress in offline reinforcement learning (RL) has been impeded by ambiguous problem definitions and entangled algorithmic designs, resulting in inconsistent implementations, insufficient ablations, and unfair evaluations. Although offline RL explicitly avoids environment interaction, prior methods frequently employ extensive, undocumented online evaluation for hyperparameter tuning, complicating method comparisons. Moreover, existing reference implementations differ significantly in boilerplate code, obscuring their core algorithmic contributions. We address these challenges by first introducing a rigorous taxonomy and a transparent evaluation protocol that explicitly quantifies online tuning budgets. To resolve opaque algorithmic design, we provide clean, minimalistic, single-file implementations of various model-free and model-based offline RL methods, significantly enhancing clarity and achieving substantial speed-ups. Leveraging these streamlined implementations, we propose Unifloral, a unified algorithm that encapsulates diverse prior approaches within a single, comprehensive hyperparameter space, enabling algorithm development in a shared hyperparameter space. Using Unifloral with our rigorous evaluation protocol, we develop two novel algorithms - TD3-AWR (model-free) and MoBRAC (model-based) - which substantially outperform established baselines. Our implementation is publicly available at https://github.com/EmptyJackson/unifloral.

Jacob Beck, Matthew Jackson, Risto Vuorio et al.

A core ambition of reinforcement learning (RL) is the creation of agents capable of rapid learning in novel tasks. Meta-RL aims to achieve this by directly learning such agents. Black box methods do so by training off-the-shelf sequence models end-to-end. By contrast, task inference methods explicitly infer a posterior distribution over the unknown task, typically using distinct objectives and sequence models designed to enable task inference. Recent work has shown that task inference methods are not necessary for strong performance. However, it remains unclear whether task inference sequence models are beneficial even when task inference objectives are not. In this paper, we present evidence that task inference sequence models are indeed still beneficial. In particular, we investigate sequence models with permutation invariant aggregation, which exploit the fact that, due to the Markov property, the task posterior does not depend on the order of data. We empirically confirm the advantage of permutation invariant sequence models without the use of task inference objectives. However, we also find, surprisingly, that there are multiple conditions under which permutation variance remains useful. Therefore, we propose SplAgger, which uses both permutation variant and invariant components to achieve the best of both worlds, outperforming all baselines evaluated on continuous control and memory environments. Code is provided at https://github.com/jacooba/hyper.

Zheng Xiong, Kang Li, Zilin Wang et al.

Built upon language and vision foundation models with strong generalization ability and trained on large-scale robotic data, Vision-Language-Action (VLA) models have recently emerged as a promising approach to learning generalist robotic policies. However, a key drawback of existing VLAs is their extremely high inference costs. In this paper, we propose HyperVLA to address this problem. Unlike existing monolithic VLAs that activate the whole model during both training and inference, HyperVLA uses a novel hypernetwork (HN)-based architecture that activates only a small task-specific policy during inference, while still retaining the high model capacity needed to accommodate diverse multi-task behaviors during training. Successfully training an HN-based VLA is nontrivial so HyperVLA contains several key algorithm design features that improve its performance, including properly utilizing the prior knowledge from existing vision foundation models, HN normalization, and an action generation strategy. Compared to monolithic VLAs, HyperVLA achieves a similar or even higher success rate for both zero-shot generalization and few-shot adaptation, while significantly reducing inference costs. Compared to OpenVLA, a state-of-the-art VLA model, HyperVLA reduces the number of activated parameters at test time by $90\times$, and accelerates inference speed by $120\times$. Code is publicly available at https://github.com/MasterXiong/HyperVLA

Clémence Grislain, Risto Vuorio, Cong Lu et al.

Developing autonomous vehicles that can navigate complex environments with human-level safety and efficiency is a central goal in self-driving research. A common approach to achieving this is imitation learning, where agents are trained to mimic human expert demonstrations collected from real-world driving scenarios. However, discrepancies between human perception and the self-driving car's sensors can introduce an $\textit{imitation}$ gap, leading to imitation learning failures. In this work, we introduce $\textbf{IGDrivSim}$, a benchmark built on top of the Waymax simulator, designed to investigate the effects of the imitation gap in learning autonomous driving policy from human expert demonstrations. Our experiments show that this perception gap between human experts and self-driving agents can hinder the learning of safe and effective driving behaviors. We further show that combining imitation with reinforcement learning, using a simple penalty reward for prohibited behaviors, effectively mitigates these failures. Our code is open-sourced at: https://github.com/clemgris/IGDrivSim.git.

Mikayel Samvelyan, Tabish Rashid, Christian Schroeder de Witt et al.

In the last few years, deep multi-agent reinforcement learning (RL) has become a highly active area of research. A particularly challenging class of problems in this area is partially observable, cooperative, multi-agent learning, in which teams of agents must learn to coordinate their behaviour while conditioning only on their private observations. This is an attractive research area since such problems are relevant to a large number of real-world systems and are also more amenable to evaluation than general-sum problems. Standardised environments such as the ALE and MuJoCo have allowed single-agent RL to move beyond toy domains, such as grid worlds. However, there is no comparable benchmark for cooperative multi-agent RL. As a result, most papers in this field use one-off toy problems, making it difficult to measure real progress. In this paper, we propose the StarCraft Multi-Agent Challenge (SMAC) as a benchmark problem to fill this gap. SMAC is based on the popular real-time strategy game StarCraft II and focuses on micromanagement challenges where each unit is controlled by an independent agent that must act based on local observations. We offer a diverse set of challenge maps and recommendations for best practices in benchmarking and evaluations. We also open-source a deep multi-agent RL learning framework including state-of-the-art algorithms. We believe that SMAC can provide a standard benchmark environment for years to come. Videos of our best agents for several SMAC scenarios are available at: https://youtu.be/VZ7zmQ_obZ0.

Jakob Foerster, Gregory Farquhar, Maruan Al-Shedivat et al.

The score function estimator is widely used for estimating gradients of stochastic objectives in stochastic computation graphs (SCG), eg, in reinforcement learning and meta-learning. While deriving the first-order gradient estimators by differentiating a surrogate loss (SL) objective is computationally and conceptually simple, using the same approach for higher-order derivatives is more challenging. Firstly, analytically deriving and implementing such estimators is laborious and not compliant with automatic differentiation. Secondly, repeatedly applying SL to construct new objectives for each order derivative involves increasingly cumbersome graph manipulations. Lastly, to match the first-order gradient under differentiation, SL treats part of the cost as a fixed sample, which we show leads to missing and wrong terms for estimators of higher-order derivatives. To address all these shortcomings in a unified way, we introduce DiCE, which provides a single objective that can be differentiated repeatedly, generating correct estimators of derivatives of any order in SCGs. Unlike SL, DiCE relies on automatic differentiation for performing the requisite graph manipulations. We verify the correctness of DiCE both through a proof and numerical evaluation of the DiCE derivative estimates. We also use DiCE to propose and evaluate a novel approach for multi-agent learning. Our code is available at https://www.github.com/alshedivat/lola.

Jakob N. Foerster, Richard Y. Chen, Maruan Al-Shedivat et al.

Multi-agent settings are quickly gathering importance in machine learning. This includes a plethora of recent work on deep multi-agent reinforcement learning, but also can be extended to hierarchical RL, generative adversarial networks and decentralised optimisation. In all these settings the presence of multiple learning agents renders the training problem non-stationary and often leads to unstable training or undesired final results. We present Learning with Opponent-Learning Awareness (LOLA), a method in which each agent shapes the anticipated learning of the other agents in the environment. The LOLA learning rule includes a term that accounts for the impact of one agent's policy on the anticipated parameter update of the other agents. Results show that the encounter of two LOLA agents leads to the emergence of tit-for-tat and therefore cooperation in the iterated prisoners' dilemma, while independent learning does not. In this domain, LOLA also receives higher payouts compared to a naive learner, and is robust against exploitation by higher order gradient-based methods. Applied to repeated matching pennies, LOLA agents converge to the Nash equilibrium. In a round robin tournament we show that LOLA agents successfully shape the learning of a range of multi-agent learning algorithms from literature, resulting in the highest average returns on the IPD. We also show that the LOLA update rule can be efficiently calculated using an extension of the policy gradient estimator, making the method suitable for model-free RL. The method thus scales to large parameter and input spaces and nonlinear function approximators. We apply LOLA to a grid world task with an embedded social dilemma using recurrent policies and opponent modelling. By explicitly considering the learning of the other agent, LOLA agents learn to cooperate out of self-interest. The code is at github.com/alshedivat/lola.

Matthew Thomas Jackson, Michael Tryfan Matthews, Cong Lu et al. · deepmind

In many real-world settings, agents must learn from an offline dataset gathered by some prior behavior policy. Such a setting naturally leads to distribution shift between the behavior policy and the target policy being trained - requiring policy conservatism to avoid instability and overestimation bias. Autoregressive world models offer a different solution to this by generating synthetic, on-policy experience. However, in practice, model rollouts must be severely truncated to avoid compounding error. As an alternative, we propose policy-guided diffusion. Our method uses diffusion models to generate entire trajectories under the behavior distribution, applying guidance from the target policy to move synthetic experience further on-policy. We show that policy-guided diffusion models a regularized form of the target distribution that balances action likelihood under both the target and behavior policies, leading to plausible trajectories with high target policy probability, while retaining a lower dynamics error than an offline world model baseline. Using synthetic experience from policy-guided diffusion as a drop-in substitute for real data, we demonstrate significant improvements in performance across a range of standard offline reinforcement learning algorithms and environments. Our approach provides an effective alternative to autoregressive offline world models, opening the door to the controllable generation of synthetic training data.

Matthew Thomas Jackson, Chris Lu, Louis Kirsch et al.

Recent advancements in meta-learning have enabled the automatic discovery of novel reinforcement learning algorithms parameterized by surrogate objective functions. To improve upon manually designed algorithms, the parameterization of this learned objective function must be expressive enough to represent novel principles of learning (instead of merely recovering already established ones) while still generalizing to a wide range of settings outside of its meta-training distribution. However, existing methods focus on discovering objective functions that, like many widely used objective functions in reinforcement learning, do not take into account the total number of steps allowed for training, or "training horizon". In contrast, humans use a plethora of different learning objectives across the course of acquiring a new ability. For instance, students may alter their studying techniques based on the proximity to exam deadlines and their self-assessed capabilities. This paper contends that ignoring the optimization time horizon significantly restricts the expressive potential of discovered learning algorithms. We propose a simple augmentation to two existing objective discovery approaches that allows the discovered algorithm to dynamically update its objective function throughout the agent's training procedure, resulting in expressive schedules and increased generalization across different training horizons. In the process, we find that commonly used meta-gradient approaches fail to discover such adaptive objective functions while evolution strategies discover highly dynamic learning rules. We demonstrate the effectiveness of our approach on a wide range of tasks and analyze the resulting learned algorithms, which we find effectively balance exploration and exploitation by modifying the structure of their learning rules throughout the agent's lifetime.

Amir Moeini, Jiuqi Wang, Jacob Beck et al.

Reinforcement learning (RL) agents typically optimize their policies by performing expensive backward passes to update their network parameters. However, some agents can solve new tasks without updating any parameters by simply conditioning on additional context such as their action-observation histories. This paper surveys work on such behavior, known as in-context reinforcement learning.

Benjamin Ellis, Matthew T. Jackson, Andrei Lupu et al. · meta-ai, oxford

In reinforcement learning (RL), it is common to apply techniques used broadly in machine learning such as neural network function approximators and momentum-based optimizers. However, such tools were largely developed for supervised learning rather than nonstationary RL, leading practitioners to adopt target networks, clipped policy updates, and other RL-specific implementation tricks to combat this mismatch, rather than directly adapting this toolchain for use in RL. In this paper, we take a different approach and instead address the effect of nonstationarity by adapting the widely used Adam optimiser. We first analyse the impact of nonstationary gradient magnitude -- such as that caused by a change in target network -- on Adam's update size, demonstrating that such a change can lead to large updates and hence sub-optimal performance. To address this, we introduce Adam-Rel. Rather than using the global timestep in the Adam update, Adam-Rel uses the local timestep within an epoch, essentially resetting Adam's timestep to 0 after target changes. We demonstrate that this avoids large updates and reduces to learning rate annealing in the absence of such increases in gradient magnitude. Evaluating Adam-Rel in both on-policy and off-policy RL, we demonstrate improved performance in both Atari and Craftax. We then show that increases in gradient norm occur in RL in practice, and examine the differences between our theoretical model and the observed data.

Zheng Xiong, Risto Vuorio, Jacob Beck et al.

Learning a universal policy across different robot morphologies can significantly improve learning efficiency and enable zero-shot generalization to unseen morphologies. However, learning a highly performant universal policy requires sophisticated architectures like transformers (TF) that have larger memory and computational cost than simpler multi-layer perceptrons (MLP). To achieve both good performance like TF and high efficiency like MLP at inference time, we propose HyperDistill, which consists of: (1) A morphology-conditioned hypernetwork (HN) that generates robot-wise MLP policies, and (2) A policy distillation approach that is essential for successful training. We show that on UNIMAL, a benchmark with hundreds of diverse morphologies, HyperDistill performs as well as a universal TF teacher policy on both training and unseen test robots, but reduces model size by 6-14 times, and computational cost by 67-160 times in different environments. Our analysis attributes the efficiency advantage of HyperDistill at inference time to knowledge decoupling, i.e., the ability to decouple inter-task and intra-task knowledge, a general principle that could also be applied to improve inference efficiency in other domains.

Alexander David Goldie, Zilin Wang, Jaron Cohen et al.

The process of meta-learning algorithms from data, instead of relying on manual design, is growing in popularity as a paradigm for improving the performance of machine learning systems. Meta-learning shows particular promise for reinforcement learning (RL), where algorithms are often adapted from supervised or unsupervised learning despite their suboptimality for RL. However, until now there has been a severe lack of comparison between different meta-learning algorithms, such as using evolution to optimise over black-box functions or LLMs to propose code. In this paper, we carry out this empirical comparison of the different approaches when applied to a range of meta-learned algorithms which target different parts of the RL pipeline. In addition to meta-train and meta-test performance, we also investigate factors including the interpretability, sample cost and train time for each meta-learning algorithm. Based on these findings, we propose several guidelines for meta-learning new RL algorithms which will help ensure that future learned algorithms are as performant as possible.

Alexey Zakharov, Shimon Whiteson

Natural language can offer a concise and human-interpretable means of specifying reinforcement learning (RL) tasks. The ability to extract rewards from a language instruction can enable the development of robotic systems that can learn from human guidance; however, it remains a challenging problem, especially in visual environments. Existing approaches that employ large, pretrained language models either rely on non-visual environment representations, require prohibitively large amounts of feedback, or generate noisy, ill-shaped reward functions. In this paper, we propose a novel method, $\textbf{GoalLadder}$, that leverages vision-language models (VLMs) to train RL agents from a single language instruction in visual environments. GoalLadder works by incrementally discovering states that bring the agent closer to completing a task specified in natural language. To do so, it queries a VLM to identify states that represent an improvement in agent's task progress and to rank them using pairwise comparisons. Unlike prior work, GoalLadder does not trust VLM's feedback completely; instead, it uses it to rank potential goal states using an ELO-based rating system, thus reducing the detrimental effects of noisy VLM feedback. Over the course of training, the agent is tasked with minimising the distance to the top-ranked goal in a learned embedding space, which is trained on unlabelled visual data. This key feature allows us to bypass the need for abundant and accurate feedback typically required to train a well-shaped reward function. We demonstrate that GoalLadder outperforms existing related methods on classic control and robotic manipulation environments with the average final success rate of $\sim$95% compared to only $\sim$45% of the best competitor.

Bidipta Sarkar, Mattie Fellows, Juan Agustin Duque et al.

We introduce Evolution Guided General Optimization via Low-rank Learning (EGGROLL), an evolution strategies (ES) algorithm designed to scale backprop-free optimization to large population sizes for modern large neural network architectures with billions of parameters. ES is a set of powerful blackbox optimisation methods that can handle non-differentiable or noisy objectives with excellent scaling potential through parallelisation. Na{ï}ve ES becomes prohibitively expensive at scale due to the computational and memory costs associated with generating matrix perturbations $E\in\mathbb{R}^{m\times n}$ and the batched matrix multiplications needed to compute per-member forward passes. EGGROLL overcomes these bottlenecks by generating random matrices $A\in \mathbb{R}^{m\times r},\ B\in \mathbb{R}^{n\times r}$ with $r\ll \min(m,n)$ to form a low-rank matrix perturbation $A B^\top$ that are used in place of the full-rank perturbation $E$. As the overall update is an average across a population of $N$ workers, this still results in a high-rank update but with significant memory and computation savings, reducing the auxiliary storage from $mn$ to $r(m+n)$ per layer and the cost of a forward pass from $\mathcal{O}(mn)$ to $\mathcal{O}(r(m+n))$ when compared to full-rank ES. A theoretical analysis reveals our low-rank update converges to the full-rank update at a fast $\mathcal{O}\left(\frac{1}{r}\right)$ rate. Our experiments show that (1) EGGROLL does not compromise the performance of ES in tabula-rasa RL settings, despite being faster, (2) it is competitive with GRPO as a technique for improving LLM reasoning, and (3) EGGROLL enables stable pre-training of nonlinear recurrent language models that operate purely in integer datatypes.

Aman Sinha, Payam Nikdel, Supratik Paul et al.

Ensuring the safety of autonomous vehicles (AVs) requires both accurate estimation of their performance and efficient discovery of potential failure cases. This paper introduces Bayesian adaptive multifidelity sampling (BAMS), which leverages the power of adaptive Bayesian sampling to achieve efficient discovery while simultaneously estimating the rate of adverse events. BAMS prioritizes exploration of regions with potentially low performance, leading to the identification of novel and critical scenarios that traditional methods might miss. Using real-world AV data we demonstrate that BAMS discovers 10 times as many issues as Monte Carlo (MC) and importance sampling (IS) baselines, while at the same time generating rate estimates with variances 15 and 6 times narrower than MC and IS baselines respectively.

Risto Vuorio, Mattie Fellows, Cong Lu et al.

In many real-world settings, an agent must learn to act in environments where no reward signal can be specified, but a set of expert demonstrations is available. Imitation learning (IL) is a popular framework for learning policies from such demonstrations. However, in some cases, differences in observability between the expert and the agent can give rise to an imitation gap such that the expert's policy is not optimal for the agent and a naive application of IL can fail catastrophically. In particular, if the expert observes the Markov state and the agent does not, then the expert will not demonstrate the information-gathering behavior needed by the agent but not the expert. In this paper, we propose a Bayesian solution to the Imitation Gap (BIG), first using the expert demonstrations, together with a prior specifying the cost of exploratory behavior that is not demonstrated, to infer a posterior over rewards with Bayesian inverse reinforcement learning (IRL). BIG then uses the reward posterior to learn a Bayes-optimal policy. Our experiments show that BIG, unlike IL, allows the agent to explore at test time when presented with an imitation gap, whilst still learning to behave optimally using expert demonstrations when no such gap exists.

Reza Mahjourian, Rongbing Mu, Valerii Likhosherstov et al.

This paper introduces UniGen, a novel approach to generating new traffic scenarios for evaluating and improving autonomous driving software through simulation. Our approach models all driving scenario elements in a unified model: the position of new agents, their initial state, and their future motion trajectories. By predicting the distributions of all these variables from a shared global scenario embedding, we ensure that the final generated scenario is fully conditioned on all available context in the existing scene. Our unified modeling approach, combined with autoregressive agent injection, conditions the placement and motion trajectory of every new agent on all existing agents and their trajectories, leading to realistic scenarios with low collision rates. Our experimental results show that UniGen outperforms prior state of the art on the Waymo Open Motion Dataset.

Nico Montali, John Lambert, Paul Mougin et al.

Simulation with realistic, interactive agents represents a key task for autonomous vehicle software development. In this work, we introduce the Waymo Open Sim Agents Challenge (WOSAC). WOSAC is the first public challenge to tackle this task and propose corresponding metrics. The goal of the challenge is to stimulate the design of realistic simulators that can be used to evaluate and train a behavior model for autonomous driving. We outline our evaluation methodology, present results for a number of different baseline simulation agent methods, and analyze several submissions to the 2023 competition which ran from March 16, 2023 to May 23, 2023. The WOSAC evaluation server remains open for submissions and we discuss open problems for the task.

Anuj Mahajan, Mikayel Samvelyan, Tarun Gupta et al.

Collective intelligence is a fundamental trait shared by several species of living organisms. It has allowed them to thrive in the diverse environmental conditions that exist on our planet. From simple organisations in an ant colony to complex systems in human groups, collective intelligence is vital for solving complex survival tasks. As is commonly observed, such natural systems are flexible to changes in their structure. Specifically, they exhibit a high degree of generalization when the abilities or the total number of agents changes within a system. We term this phenomenon as Combinatorial Generalization (CG). CG is a highly desirable trait for autonomous systems as it can increase their utility and deployability across a wide range of applications. While recent works addressing specific aspects of CG have shown impressive results on complex domains, they provide no performance guarantees when generalizing towards novel situations. In this work, we shed light on the theoretical underpinnings of CG for cooperative multi-agent systems (MAS). Specifically, we study generalization bounds under a linear dependence of the underlying dynamics on the agent capabilities, which can be seen as a generalization of Successor Features to MAS. We then extend the results first for Lipschitz and then arbitrary dependence of rewards on team capabilities. Finally, empirical analysis on various domains using the framework of multi-agent reinforcement learning highlights important desiderata for multi-agent algorithms towards ensuring CG.

Mingfei Sun, Sam Devlin, Jacob Beck et al.

We present trust region bounds for optimizing decentralized policies in cooperative Multi-Agent Reinforcement Learning (MARL), which holds even when the transition dynamics are non-stationary. This new analysis provides a theoretical understanding of the strong performance of two recent actor-critic methods for MARL, which both rely on independent ratios, i.e., computing probability ratios separately for each agent's policy. We show that, despite the non-stationarity that independent ratios cause, a monotonic improvement guarantee still arises as a result of enforcing the trust region constraint over all decentralized policies. We also show this trust region constraint can be effectively enforced in a principled way by bounding independent ratios based on the number of agents in training, providing a theoretical foundation for proximal ratio clipping. Finally, our empirical results support the hypothesis that the strong performance of IPPO and MAPPO is a direct result of enforcing such a trust region constraint via clipping in centralized training, and tuning the hyperparameters with regards to the number of agents, as predicted by our theoretical analysis.

Mingfei Sun, Vitaly Kurin, Guoqing Liu et al.

Proximal Policy Optimization (PPO) methods learn a policy by iteratively performing multiple mini-batch optimization epochs of a surrogate objective with one set of sampled data. Ratio clipping PPO is a popular variant that clips the probability ratios between the target policy and the policy used to collect samples. Ratio clipping yields a pessimistic estimate of the original surrogate objective, and has been shown to be crucial for strong performance. We show in this paper that such ratio clipping may not be a good option as it can fail to effectively bound the ratios. Instead, one can directly optimize the original surrogate objective for multiple epochs; the key is to find a proper condition to early stop the optimization epoch in each iteration. Our theoretical analysis sheds light on how to determine when to stop the optimization epoch, and call the resulting algorithm Early Stopping Policy Optimization (ESPO). We compare ESPO with PPO across many continuous control tasks and show that ESPO significantly outperforms PPO. Furthermore, we show that ESPO can be easily scaled up to distributed training with many workers, delivering strong performance as well.

Vitaly Kurin, Alessandro De Palma, Ilya Kostrikov et al.

Recent multi-task learning research argues against unitary scalarization, where training simply minimizes the sum of the task losses. Several ad-hoc multi-task optimization algorithms have instead been proposed, inspired by various hypotheses about what makes multi-task settings difficult. The majority of these optimizers require per-task gradients, and introduce significant memory, runtime, and implementation overhead. We show that unitary scalarization, coupled with standard regularization and stabilization techniques from single-task learning, matches or improves upon the performance of complex multi-task optimizers in popular supervised and reinforcement learning settings. We then present an analysis suggesting that many specialized multi-task optimizers can be partly interpreted as forms of regularization, potentially explaining our surprising results. We believe our results call for a critical reevaluation of recent research in the area.

Mingfei Sun, Sam Devlin, Katja Hofmann et al.

Sample efficiency is crucial for imitation learning methods to be applicable in real-world applications. Many studies improve sample efficiency by extending adversarial imitation to be off-policy regardless of the fact that these off-policy extensions could either change the original objective or involve complicated optimization. We revisit the foundation of adversarial imitation and propose an off-policy sample efficient approach that requires no adversarial training or min-max optimization. Our formulation capitalizes on two key insights: (1) the similarity between the Bellman equation and the stationary state-action distribution equation allows us to derive a novel temporal difference (TD) learning approach; and (2) the use of a deterministic policy simplifies the TD learning. Combined, these insights yield a practical algorithm, Deterministic and Discriminative Imitation (D2-Imitation), which operates by first partitioning samples into two replay buffers and then learning a deterministic policy via off-policy reinforcement learning. Our empirical results show that D2-Imitation is effective in achieving good sample efficiency, outperforming several off-policy extension approaches of adversarial imitation on many control tasks.

Zheng Xiong, Luisa Zintgraf, Jacob Beck et al.

Consistency is the theoretical property of a meta learning algorithm that ensures that, under certain assumptions, it can adapt to any task at test time. An open question is whether and how theoretical consistency translates into practice, in comparison to inconsistent algorithms. In this paper, we empirically investigate this question on a set of representative meta-RL algorithms. We find that theoretically consistent algorithms can indeed usually adapt to out-of-distribution (OOD) tasks, while inconsistent ones cannot, although they can still fail in practice for reasons like poor exploration. We further find that theoretically inconsistent algorithms can be made consistent by continuing to update all agent components on the OOD tasks, and adapt as well or better than originally consistent ones. We conclude that theoretical consistency is indeed a desirable property, and inconsistent meta-RL algorithms can easily be made consistent to enjoy the same benefits.

Anuj Mahajan, Mikayel Samvelyan, Lei Mao et al.

We present an extended abstract for the previously published work TESSERACT [Mahajan et al., 2021], which proposes a novel solution for Reinforcement Learning (RL) in large, factored action spaces using tensor decompositions. The goal of this abstract is twofold: (1) To garner greater interest amongst the tensor research community for creating methods and analysis for approximate RL, (2) To elucidate the generalised setting of factored action spaces where tensor decompositions can be used. We use cooperative multi-agent reinforcement learning scenario as the exemplary setting where the action space is naturally factored across agents and learning becomes intractable without resorting to approximation on the underlying hypothesis space for candidate solutions.

Pascal Van Der Vaart, Anuj Mahajan, Shimon Whiteson

A challenge in multi-agent reinforcement learning is to be able to generalize over intractable state-action spaces. Inspired from Tesseract [Mahajan et al., 2021], this position paper investigates generalisation in state-action space over unexplored state-action pairs by modelling the transition and reward functions as tensors of low CP-rank. Initial experiments on synthetic MDPs show that using tensor decompositions in a model-based reinforcement learning algorithm can lead to much faster convergence if the true transition and reward functions are indeed of low rank.

Shangtong Zhang, Shimon Whiteson

Emphatic Temporal Difference (TD) methods are a class of off-policy Reinforcement Learning (RL) methods involving the use of followon traces. Despite the theoretical success of emphatic TD methods in addressing the notorious deadly triad of off-policy RL, there are still two open problems. First, followon traces typically suffer from large variance, making them hard to use in practice. Second, though Yu (2015) confirms the asymptotic convergence of some emphatic TD methods for prediction problems, there is still no finite sample analysis for any emphatic TD method for prediction, much less control. In this paper, we address those two open problems simultaneously via using truncated followon traces in emphatic TD methods. Unlike the original followon traces, which depend on all previous history, truncated followon traces depend on only finite history, reducing variance and enabling the finite sample analysis of our proposed emphatic TD methods for both prediction and control.