Shangtong Zhang

Michaël Mathieu, Sherjil Ozair, Srivatsan Srinivasan et al.

StarCraft II is one of the most challenging simulated reinforcement learning environments; it is partially observable, stochastic, multi-agent, and mastering StarCraft II requires strategic planning over long time horizons with real-time low-level execution. It also has an active professional competitive scene. StarCraft II is uniquely suited for advancing offline RL algorithms, both because of its challenging nature and because Blizzard has released a massive dataset of millions of StarCraft II games played by human players. This paper leverages that and establishes a benchmark, called AlphaStar Unplugged, introducing unprecedented challenges for offline reinforcement learning. We define a dataset (a subset of Blizzard's release), tools standardizing an API for machine learning methods, and an evaluation protocol. We also present baseline agents, including behavior cloning, offline variants of actor-critic and MuZero. We improve the state of the art of agents using only offline data, and we achieve 90% win rate against previously published AlphaStar behavior cloning agent.

Vagul Mahadevan, Claire Chen, Shuze Daniel Liu et al.

This work studies the convergence of two-timescale stochastic approximations (SA), a class of iterative algorithms that update two sets of parameters in fast and slow timescales respectively. Notable examples of two-timescale SA in reinforcement learning (RL) include temporal difference learning with gradient correction (TDC) and actor-critic methods. Previously, the stability (i.e., boundedness) and convergence of two-timescale SA were only established under i.i.d. noise. This work instead establishes the stability and convergence of two-timescale SA under Markovian noise, a setup that is more realistic in RL. Notably, we do not need to use any projection operator and the noise does not need to live in a compact space. Our key technical novelty is to control the fast timescale parameter with the running max of the slow timescale parameter, instead of with the current slow timescale parameter, as most prior works do. As a key application, we establish the first almost sure convergence of TDC with eligibility traces under off-policy learning with linear function approximation.

Minjae Kwon, Amir Moeini, Shangtong Zhang et al.

Safe in-context reinforcement learning (ICRL) adapts online from interaction history without test-time parameter updates while controlling episode cost under a safety budget. Under out-of-distribution (OOD) deployment shifts, pretraining-only safe ICRL can give poor reward-safety tradeoffs because the remaining budget affects behavior only through frozen policy conditioning, not an explicit action-level check against predicted future cost. We propose a latent Q-Barrier shield that learns a context representation, latent dynamics, and an ensemble cost critic before deployment. Without parameter updates, the shield infers context from history and filters or softly reweights candidate actions using the remaining budget and predicted future cost. We prove a conditional, error-decomposed barrier-margin result: a Q-Barrier-satisfying action leaves the next latent-budget state with an approximately budget-safe continuation under the learned critic, up to Bellman and latent-prediction errors. Across five safe ICRL benchmarks, the shield improves deployment-time reward-safety tradeoffs over a strong safe-ICRL baseline: after a short context window, it achieves higher return in four of five benchmarks while matching or lowering average episode cost in all five.

Vagul Mahadevan, Shangtong Zhang, Rohan Chandra

Safe, agile, and socially compliant multi-robot navigation in cluttered and constrained environments remains a critical challenge. This is especially difficult with self-interested agents with unique, unknown priorities in decentralized settings, where there is no central authority to resolve conflicts induced by spatial symmetry. We address this challenge by proposing an intuitive, but very effective approach, GameChat, which facilitates safe, agile, and deadlock-free navigation for both cooperative and self-interested agents in cluttered environments. Key to our approach is the idea that agents should resolve conflicts on their own using natural language to communicate, much like humans. We evaluate GameChat in simulated environments with doorways and intersections. The results show that even in the worst case, GameChat reduces the time for all agents to reach their goals by over 35% from a naive baseline and by over 20% from a state of the art baseline in the intersection scenario, while doubling the rate of ensuring the agent with a higher priority task reaches the goal first, from 50% (equivalent to random chance) to 100%. We also demonstrate how GameChat can be extended to more than two agents.

Xiaochi Qian, Shangtong Zhang

Off-policy learning enables a reinforcement learning (RL) agent to reason counterfactually about policies that are not executed and is one of the most important ideas in RL. It, however, can lead to instability when combined with function approximation and bootstrapping, two arguably indispensable ingredients for large-scale reinforcement learning. This is the notorious deadly triad. The seminal work Sutton et al. (2008) pioneers Gradient Temporal Difference learning (GTD) as the first solution to the deadly triad, which has enjoyed massive success thereafter. During the derivation of GTD, some intermediate algorithm, called $A^\top$TD, was invented but soon deemed inferior. In this paper, we revisit this $A^\top$TD and prove that a variant of $A^\top$TD, called $A_t^\top$TD, is also an effective solution to the deadly triad. Furthermore, this $A_t^\top$TD only needs one set of parameters and one learning rate. By contrast, GTD has two sets of parameters and two learning rates, making it hard to tune in practice. We provide asymptotic analysis for $A^\top_t$TD and finite sample analysis for a variant of $A^\top_t$TD that additionally involves a projection operator. The convergence rate of this variant is on par with the canonical on-policy temporal difference learning.

Shangtong Zhang

In this paper, we formalize the almost sure convergence of $Q$-learning and linear temporal difference (TD) learning with Markovian samples using the Lean 4 theorem prover based on the Mathlib library. $Q$-learning and linear TD are among the earliest and most influential reinforcement learning (RL) algorithms. The investigation of their convergence properties is not only a major research topic during the early development of the RL field but also receives increasing attention nowadays. This paper formally verifies their almost sure convergence in a unified framework based on the Robbins-Siegmund theorem. The framework developed in this work can be easily extended to convergence rates and other modes of convergence. This work thus makes an important step towards fully formalizing convergent RL results. The code is available at https://github.com/ShangtongZhang/rl-theory-in-lean.

Shuze Liu, Shangtong Zhang

Most reinforcement learning practitioners evaluate their policies with online Monte Carlo estimators for either hyperparameter tuning or testing different algorithmic design choices, where the policy is repeatedly executed in the environment to get the average outcome. Such massive interactions with the environment are prohibitive in many scenarios. In this paper, we propose novel methods that improve the data efficiency of online Monte Carlo estimators while maintaining their unbiasedness. We first propose a tailored closed-form behavior policy that provably reduces the variance of an online Monte Carlo estimator. We then design efficient algorithms to learn this closed-form behavior policy from previously collected offline data. Theoretical analysis is provided to characterize how the behavior policy learning error affects the amount of reduced variance. Compared with previous works, our method achieves better empirical performance in a broader set of environments, with fewer requirements for offline data.

Zixuan Xie, Xinyu Liu, Claire Chen et al.

In-context reinforcement learning (ICRL) studies agents that, after pretraining, adapt to new tasks by conditioning on additional context without parameter updates. Existing theoretical analyses of ICRL largely rely on linear attention, which replaces the softmax function in the standard attention with an identity mapping. This paper provides the first theoretical understanding of ICRL without making the unrealistic linear attention simplification. In particular, we consider the standard softmax attention used in practice. We show that, with certain parameters, the layerwise forward pass of a Transformer with such softmax attention is equivalent to iterative updates of a weighted softmax temporal difference (TD) learning algorithm. Here, weighted softmax TD is a new RL algorithm that performs policy evaluation in kernel space and adopts both linear TD and tabular TD as special cases. We also prove that under a certain contraction condition, the policy evaluation error decays as the number of layers grows, with the identified parameters above. Finally, we prove that those parameters are a global minimizer of a pretraining loss, explaining their emergence in our numerical experiments.

Shuze Daniel Liu, Claire Chen, Shangtong Zhang

To unbiasedly evaluate multiple target policies, the dominant approach among RL practitioners is to run and evaluate each target policy separately. However, this evaluation method is far from efficient because samples are not shared across policies, and running target policies to evaluate themselves is actually not optimal. In this paper, we address these two weaknesses by designing a tailored behavior policy to reduce the variance of estimators across all target policies. Theoretically, we prove that executing this behavior policy with manyfold fewer samples outperforms on-policy evaluation on every target policy under characterized conditions. Empirically, we show our estimator has a substantially lower variance compared with previous best methods and achieves state-of-the-art performance in a broad range of environments.

Xinyu Liu, Zixuan Xie, Amir Moeini et al.

While the ecosystem of Lean and Mathlib has enjoyed celebrated success in formal mathematical reasoning with the help of large language models (LLMs), the absence of many folklore lemmas in Mathlib remains a persistent barrier that limits Lean's usability as an everyday tool for mathematicians like LaTeX or Maple. To address this, we introduce MathlibLemma, the first LLM-based multi-agent system to automate the discovery and formalization of mathematical folklore lemmas. This framework constitutes our primary contribution, proactively mining the missing connective tissue of mathematics. Its efficacy is demonstrated by the production of a verified library of folklore lemmas, a subset of which has already been formally merged into the latest build of Mathlib, thereby validating the system's real-world utility and alignment with expert standards. Leveraging this pipeline, we further construct the MathlibLemma benchmark, a suite of 4,028 type-checked Lean statements spanning a broad range of mathematical domains. By transforming the role of LLMs from passive consumers to active contributors, this work establishes a constructive methodology for the self-evolution of formal mathematical libraries.

Jiuqi Wang, Jayanth Srinivasa, Claire Chen et al.

Deep continual learning requires models to adapt to new tasks without retraining from scratch. However, neural networks can lose their ability to adapt to new tasks after training on previous ones, a phenomenon known as loss of plasticity. There have been several explanations and diagnostics proposed for plasticity loss. Motivated by the philosophy that "all models are wrong, but some are useful", we ask: can existing diagnostics predict a neural network's plasticity? In this work, we take a practical view to interpret plasticity as trainability, i.e., a neural network's future optimization gain on a target task. We first take a theoretical approach, showing, by constructing a few counterexamples, that some widely adopted diagnostics of plasticity, including representation rank and neural tangent kernel rank, can fail to predict the loss of trainability in both regression and classification settings. We instead propose a novel metric, called optimization readiness, which combines gradient strength and gradient reliability. We prove that optimization readiness lower bounds one-step optimization gain under standard smoothness assumptions, providing a theoretical guarantee for its predictive power. Empirically, we show that across commonly used deep continual learning settings, such as Slowly-Changing Regression and Permuted MNIST, optimization readiness more reliably ranks checkpoints by trainability than prior diagnostics, even with substantially fewer samples.

Jiuqi Wang, Shangtong Zhang

Temporal difference (TD) learning with linear function approximation, abbreviated as linear TD, is a classic and powerful prediction algorithm in reinforcement learning. While it is well understood that linear TD converges almost surely to a unique point, this convergence traditionally requires the assumption that the features used by the approximator are linearly independent. However, this linear independence assumption does not hold in many practical scenarios. This work is the first to establish the almost sure convergence of linear TD without requiring linearly independent features. In fact, we do not make any assumptions on the features. We prove that the approximated value function converges to a unique point and the weight iterates converge to a set. We also establish a notion of local stability of the weight iterates. Importantly, we do not need to introduce any other additional assumptions and do not need to make any modification to the linear TD algorithm. Key to our analysis is a novel characterization of bounded invariant sets of the mean ODE of linear TD.

Zixuan Xie, Xinyu Liu, Rohan Chandra et al.

In-context reinforcement learning (ICRL) refers to the ability of RL agents to adapt to new tasks at inference time without parameter updates by conditioning on additional context. Recent empirical studies further demonstrate that Chain-of-Thought (CoT) generation can amplify this ICRL capability. This paper is the first to provide a theoretical understanding on how CoT interacts with ICRL. We conduct our analysis in a policy evaluation setup with linear Transformer. We prove that with specific Transformer parameters, the CoT generation process is equivalent to repeatedly executing temporal difference learning updates. Additionally, we provide finite sample convergence analysis showing that the policy evaluation error decreases geometrically with CoT length and eventually saturates at a statistical floor determined by the context length. We also prove that the desired Transformer parameters are a global minimizer of the pretraining loss, providing a theoretical understanding on the empirical emergence of those parameters.

Xinyu Liu, Zixuan Xie, Shangtong Zhang

Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-η})$ with $η\in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2η})$. For a harmonic learning rate $O(n^{-1})$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{-1})$, which we argue is a strong result because it is close to the optimal rate $O(n^{-1}\log\log n)$ given by the law of the iterated logarithm (for a special case of i.i.d. noise). Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.

Zixuan Xie, Xinyu Liu, Shangtong Zhang

The ecosystem of Lean and Mathlib has become the de facto standard for large language model (LLM) assisted formal reasoning with remarkable successes in recent years. Those successes, however, only consume Mathlib as an essential dependency but do not directly contribute to it. In the meantime, the growth of Mathlib has recently been bottlenecked by the review process, which requires human reviewers to judge whether proposed pull requests (PRs) follow the Mathlib's conventions and are worth integrating as part of a shared mathematical infrastructure. This leads to our central question: can LLMs help review Mathlib PRs? To this end, we introduce MathlibPR, a benchmark built from real Mathlib4 PR histories. We further propose a staged evaluation protocol and use it to evaluate both LLM models (e.g., DeepSeek, Qwen, Goedel, and Kimina) and LLM agents (e.g., Codex and Claude Code). Surprisingly, both LLM models and LLM agents struggle to distinguish merge-ready PRs from build-passing PRs that were revised or never merged. By turning Mathlib PR histories into a supervised signal, MathlibPR provides a step toward reviewer assistants and reward models that could help evaluate PRs and steer LLMs toward producing merge-ready Mathlib contributions.

Ethan Blaser, Shangtong Zhang

Stochastic approximation is a powerful class of algorithms with celebrated success. However, a large body of previous analysis focuses on stochastic approximations driven by contractive operators, which is not applicable in some important reinforcement learning settings like the average reward setting. This work instead investigates stochastic approximations with merely nonexpansive operators. In particular, we study nonexpansive stochastic approximations with Markovian noise, providing both asymptotic and finite sample analysis. Key to our analysis are novel bounds of noise terms resulting from the Poisson equation. As an application, we prove for the first time that classical tabular average reward temporal difference learning converges to a sample-path dependent fixed point.

David Antrobius, Shangtong Zhang

Learning rate is a critical component of reinforcement learning (RL). This work uses global and local clocks to distinguish two types of learning rates. The former is of the standard form $α_t$ that depends only on the time step $t$ (i.e., a global clock). The latter is of the form $α_{ν(S_t, t)}$, where $ν(s, t)$ counts the number of visits to state $s$ until time $t$ (i.e., a local clock). In discounted RL, an RL algorithm that is convergent with a local clock is always also convergent with a global clock, and vice versa. We are not aware of any counterexample. The key contribution of this work is to show that this nice correspondence breaks down in average-reward RL. Specifically, we construct a counterexample showing that although differential temporal difference learning is convergent with a local clock, it can diverge with a global clock. This counterexample closes the open problem in Wan et al. [2021], Blaser et al. [2026].

Alper Kamil Bozkurt, Xiaoan Xu, Shangtong Zhang et al.

In offline-to-online reinforcement learning (O2O-RL), policies are first safely trained offline using previously collected datasets and then further fine-tuned for tasks via limited online interactions. In a typical O2O-RL pipeline, candidate policies trained with offline RL are evaluated via either off-policy evaluation (OPE) or online evaluation (OE). The policy with the highest estimated value is then deployed and continually fine-tuned. However, this setup has two main issues. First, OPE can be unreliable, making it risky to deploy a policy based solely on those estimates, whereas OE may identify a viable policy with substantial online interaction, which could have been used for fine-tuning. Second--and more importantly--it is also often not possible to determine a priori whether a pretrained policy will improve with post-deployment fine-tuning, especially in non-stationary environments. As a result, procedures committing to a single deployed policy are impractical in many real-world settings. Moreover, a naive remedy that exhaustively fine-tunes all candidates would violate interaction budget constraints and is likewise infeasible. In this paper, we propose a novel adaptive approach for policy selection and fine-tuning under online interaction budgets in O2O-RL. Following the standard pipeline, we first train a set of candidate policies with different offline RL algorithms and hyperparameters; we then perform OPE to obtain initial performance estimates. We next adaptively select and fine-tune the policies based on their predicted performance via an upper-confidence-bound approach thereby making efficient use of online interactions. We demonstrate that our approach improves upon O2O-RL baselines with various benchmarks.

Zeyu Mu, Shangtong Zhang, B. Brian Park

Connected automated vehicles (CAVs) possess the ability to communicate and coordinate with one another, enabling cooperative platooning that enhances both energy efficiency and traffic flow. However, during the initial stage of CAV deployment, the sparse distribution of CAVs among human-driven vehicles reduces the likelihood of forming effective cooperative platoons. To address this challenge, this study proposes a hybrid multi-agent lane change decision model aimed at increasing CAV participation in cooperative platooning and maximizing its associated benefits. The proposed model employs the QMIX framework, integrating traffic data processed through a convolutional neural network (CNN-QMIX). This architecture addresses a critical issue in dynamic traffic scenarios by enabling CAVs to make optimal decisions irrespective of the varying number of CAVs present in mixed traffic. Additionally, a trajectory planner and a model predictive controller are designed to ensure smooth and safe lane-change execution. The proposed model is trained and evaluated within a microsimulation environment under varying CAV market penetration rates. The results demonstrate that the proposed model efficiently manages fluctuating traffic agent numbers, significantly outperforming the baseline rule-based models. Notably, it enhances cooperative platooning rates up to 26.2\%, showcasing its potential to optimize CAV cooperation and traffic dynamics during the early stage of deployment.

Ethan Blaser, Jiuqi Wang, Shangtong Zhang

The average reward is a fundamental performance metric in reinforcement learning (RL) focusing on the long-run performance of an agent. Differential temporal difference (TD) learning algorithms are a major advance for average reward RL as they provide an efficient online method to learn the value functions associated with the average reward in both on-policy and off-policy settings. However, existing convergence guarantees require a local clock in learning rates tied to state visit counts, which practitioners do not use and does not extend beyond tabular settings. We address this limitation by proving the almost sure convergence of on-policy $n$-step differential TD for any $n$ using standard diminishing learning rates without a local clock. We then derive three sufficient conditions under which off-policy $n$-step differential TD also converges without a local clock. These results strengthen the theoretical foundations of differential TD and bring its convergence analysis closer to practical implementations.

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.

Kefan Song, Amir Moeini, Peng Wang et al.

Reinforcement learning (RL) is a human-designed framework for solving sequential decision making problems. In this work, we demonstrate that, surprisingly, RL emerges in LLM's (Large Language Model) inference time -- a phenomenon known as in-context RL (ICRL). Specifically, we propose a novel multi-round prompting framework called ICRL prompting. The goal is to prompt the LLM to complete a task. After the LLM generates a response at the current round, we give numerical scalar feedbacks for the response, called the rewards. At the next round, we prompt the LLM again with the same task and a context consisting of all previous responses and rewards. We observe that the quality of the LLM's response increases as the context grows. In other words, the LLM is able to maximize the scalar reward signal in the inference time, just like an RL algorithm. We evaluate ICRL prompting in three benchmarks (Game of 24, creative writing, and ScienceWorld) and demonstrate significant performance improvements over baseline methods such as Self-Refine and Reflexion. Surprisingly, in some experiments the reward signals are generated by the LLM itself, yet performance improvements are still observed from ICRL prompting, offering a promising paradigm for scaling test-time compute.

Jiuqi Wang, Ethan Blaser, Hadi Daneshmand et al.

Traditionally, reinforcement learning (RL) agents learn to solve new tasks by updating their neural network parameters through interactions with the task environment. However, recent works demonstrate that some RL agents, after certain pretraining procedures, can learn to solve unseen new tasks without parameter updates, a phenomenon known as in-context reinforcement learning (ICRL). The empirical success of ICRL is widely attributed to the hypothesis that the forward pass of the pretrained agent neural network implements an RL algorithm. In this paper, we support this hypothesis by showing, both empirically and theoretically, that when a transformer is trained for policy evaluation tasks, it can discover and learn to implement temporal difference learning in its forward pass.

Shuze Daniel Liu, Shuhang Chen, Shangtong Zhang

Stochastic approximation is a class of algorithms that update a vector iteratively, incrementally, and stochastically, including, e.g., stochastic gradient descent and temporal difference learning. One fundamental challenge in analyzing a stochastic approximation algorithm is to establish its stability, i.e., to show that the stochastic vector iterates are bounded almost surely. In this paper, we extend the celebrated Borkar-Meyn theorem for stability from the Martingale difference noise setting to the Markovian noise setting, which greatly improves its applicability in reinforcement learning, especially in those off-policy reinforcement learning algorithms with linear function approximation and eligibility traces. Central to our analysis is the diminishing asymptotic rate of change of a few functions, which is implied by both a form of the strong law of large numbers and a form of the law of the iterated logarithm.

Xiaochi Qian, Zixuan Xie, Xinyu Liu et al.

This paper establishes the first almost sure convergence rate and the first maximal concentration bound with exponential tails for general contractive stochastic approximation algorithms with Markovian noise. As a corollary, we also obtain convergence rates in $L^p$. Key to our successes is a novel discretization of the mean ODE of stochastic approximation algorithms using intervals with diminishing (instead of constant) length. As applications, we provide the first almost sure convergence rate for $Q$-learning with Markovian samples without count-based learning rates. We also provide the first concentration bound for off-policy temporal difference learning with Markovian samples.

Xinyu Liu, Zixuan Xie, Shangtong Zhang

$Q$-learning is one of the most fundamental reinforcement learning algorithms. It is widely believed that $Q$-learning with linear function approximation (i.e., linear $Q$-learning) suffers from possible divergence until the recent work Meyn (2024) which establishes the ultimate almost sure boundedness of the iterates of linear $Q$-learning. Building on this success, this paper further establishes the first $L^2$ convergence rate of linear $Q$-learning iterates (to a bounded set). Similar to Meyn (2024), we do not make any modification to the original linear $Q$-learning algorithm, do not make any Bellman completeness assumption, and do not make any near-optimality assumption on the behavior policy. All we need is an $ε$-softmax behavior policy with an adaptive temperature. The key to our analysis is the general result of stochastic approximations under Markovian noise with fast-changing transition functions. As a side product, we also use this general result to establish the $L^2$ convergence rate of tabular $Q$-learning with an $ε$-softmax behavior policy, for which we rely on a novel pseudo-contraction property of the weighted Bellman optimality operator.

Amar Kulkarni, Shangtong Zhang, Madhur Behl

Ensuring the safety of autonomous vehicles (AVs) requires identifying rare but critical failure cases that on-road testing alone cannot discover. High-fidelity simulations provide a scalable alternative, but automatically generating realistic and diverse traffic scenarios that can effectively stress test AV motion planners remains a key challenge. This paper introduces CRASH - Challenging Reinforcement-learning based Adversarial scenarios for Safety Hardening - an adversarial deep reinforcement learning framework to address this issue. First CRASH can control adversarial Non Player Character (NPC) agents in an AV simulator to automatically induce collisions with the Ego vehicle, falsifying its motion planner. We also propose a novel approach, that we term safety hardening, which iteratively refines the motion planner by simulating improvement scenarios against adversarial agents, leveraging the failure cases to strengthen the AV stack. CRASH is evaluated on a simplified two-lane highway scenario, demonstrating its ability to falsify both rule-based and learning-based planners with collision rates exceeding 90%. Additionally, safety hardening reduces the Ego vehicle's collision rate by 26%. While preliminary, these results highlight RL-based safety hardening as a promising approach for scenario-driven simulation testing for autonomous vehicles.

Xinyu Liu, Zixuan Xie, Shangtong Zhang

The Robbins-Siegmund theorem establishes the convergence of stochastic processes that are almost supermartingales and is foundational for analyzing a wide range of stochastic iterative algorithms in stochastic approximation and reinforcement learning (RL). However, its original form has a significant limitation as it requires the zero-order term to be summable. In many important RL applications, this summable condition, however, cannot be met. This limitation motivates us to extend the Robbins-Siegmund theorem for almost supermartingales where the zero-order term is not summable but only square summable. Particularly, we introduce a novel and mild assumption on the increments of the stochastic processes. This together with the square summable condition enables an almost sure convergence to a bounded set. Additionally, we further provide almost sure convergence rates, high probability concentration bounds, and $L^p$ convergence rates. We then apply the new results in stochastic approximation and RL. Notably, we obtain the first almost sure convergence rate, the first high probability concentration bound, and the first $L^p$ convergence rate for $Q$-learning with linear function approximation.

Jiuqi Wang, Rohan Chandra, Shangtong Zhang

Loss of plasticity is one of the main challenges in continual learning with deep neural networks, where neural networks trained via backpropagation gradually lose their ability to adapt to new tasks and perform significantly worse than their freshly initialized counterparts. The main contribution of this paper is to propose a new hypothesis that experience replay addresses the loss of plasticity in continual learning. Here, experience replay is a form of memory. We provide supporting evidence for this hypothesis. In particular, we demonstrate in multiple different tasks, including regression, classification, and policy evaluation, that by simply adding an experience replay and processing the data in the experience replay with Transformers, the loss of plasticity disappears. Notably, we do not alter any standard components of deep learning. For example, we do not change backpropagation. We do not modify the activation functions. And we do not use any regularization. We conjecture that experience replay and Transformers can address the loss of plasticity because of the in-context learning phenomenon.

Amir Moeini, Minjae Kwon, Alper Kamil Bozkurt et al.

In-context reinforcement learning (ICRL) is an emerging RL paradigm where the agent, after some pretraining procedure, is able to adapt to out-of-distribution test tasks without any parameter updates. The agent achieves this by continually expanding the input (i.e., the context) to its policy neural networks. For example, the input could be all the history experience that the agent has access to until the current time step. The agent's performance improves as the input grows, without any parameter updates. In this work, we propose the first method that promotes the safety of ICRL's adaptation process in the framework of constrained Markov Decision Processes. In other words, during the parameter-update-free adaptation process, the agent not only maximizes the reward but also minimizes an additional cost function. We also demonstrate that our agent actively reacts to the threshold (i.e., budget) of the cost tolerance. With a higher cost budget, the agent behaves more aggressively, and with a lower cost budget, the agent behaves more conservatively.

Jiuqi Wang, Rohan Chandra, Shangtong Zhang

Typically, a modern reinforcement learning (RL) agent solves a task by updating its neural network parameters to adapt its policy to the task. Recently, it has been observed that some RL agents can solve a wide range of new out-of-distribution tasks without parameter updates after pretraining on some task distribution. When evaluated in a new task, instead of making parameter updates, the pretrained agent conditions its policy on additional input called the context, e.g., the agent's interaction history in the new task. The agent's performance increases as the information in the context increases, with the agent's parameters fixed. This phenomenon is typically called in-context RL (ICRL). The pretrained parameters of the agent network enable the remarkable ICRL phenomenon. However, many ICRL works perform the pretraining with standard RL algorithms. This raises the central question this paper aims to address: Why can the RL pretraining algorithm generate network parameters that enable ICRL? We hypothesize that the parameters capable of ICRL are minimizers of the pretraining loss. This work provides initial support for this hypothesis through a case study. In particular, we prove that when a Transformer is pretrained for policy evaluation, one of the global minimizers of the pretraining loss can enable in-context temporal difference learning.

Zixuan Xie, Xinyu Liu, Rohan Chandra et al.

Linear TD($λ$) is one of the most fundamental reinforcement learning algorithms for policy evaluation. Previously, convergence rates are typically established under the assumption of linearly independent features, which does not hold in many practical scenarios. This paper instead establishes the first $L^2$ convergence rates for linear TD($λ$) operating under arbitrary features, without making any algorithmic modification or additional assumptions. Our results apply to both the discounted and average-reward settings. To address the potential non-uniqueness of solutions resulting from arbitrary features, we develop a novel stochastic approximation result featuring convergence rates to the solution set instead of a single point.

Shuyang Dong, Shangtong Zhang, Lu Feng

Reinforcement Learning (RL) has shown great promise in domains like healthcare and robotics but often struggles with adoption due to its lack of interpretability. Counterfactual explanations, which address "what if" scenarios, provide a promising avenue for understanding RL decisions but remain underexplored for continuous action spaces. We propose a novel approach for generating counterfactual explanations in continuous action RL by computing alternative action sequences that improve outcomes while minimizing deviations from the original sequence. Our approach leverages a distance metric for continuous actions and accounts for constraints such as adhering to predefined policies in specific states. Evaluations in two RL domains, Diabetes Control and Lunar Lander, demonstrate the effectiveness, efficiency, and generalization of our approach, enabling more interpretable and trustworthy RL applications.

Aleesha Khurram, Amir Moeini, Shangtong Zhang et al.

Despite significant progress and advances in autonomous driving, many end-to-end systems still struggle with domain adaptation (DA), such as transferring a policy trained under clear weather to adverse weather conditions. Typical DA strategies in the literature include collecting additional data in the target domain or re-training the model, or both. Both these strategies quickly become impractical as we increase scale and complexity of driving. These limitations have encouraged investigation into few-shot and zero-shot prompt-driven DA at inference time involving LLMs and VLMs. These methods work by adding a few state-action trajectories during inference to the prompt (similar to in-context learning). However, there are two limitations of such an approach: $(i)$ prompt-driven DA methods are currently restricted to perception tasks such as detection and segmentation and $(ii)$ they require expert few-shot data. In this work, we present a new approach to inference-time few-shot prompt-driven DA for closed-loop autonomous driving in adverse weather condition using in-context reinforcement learning (ICRL). Similar to other prompt-driven DA methods, our approach does not require any updates to the model parameters nor does it require additional data collection in adversarial weather regime. Furthermore, our approach advances the state-of-the-art in prompt-driven DA by extending to closed driving using general trajectories observed during inference. Our experiments using the CARLA simulator show that ICRL results in safer, more efficient, and more comfortable driving policies in the target domain compared to state-of-the-art prompt-driven DA baselines.

Kefan Song, Runnan Jiang, Rohan Chandra et al.

This paper addresses a critical societal consideration in the application of Reinforcement Learning (RL): ensuring equitable outcomes across different demographic groups in multi-task settings. While previous work has explored fairness in single-task RL, many real-world applications are multi-task in nature and require policies to maintain fairness across all tasks. We introduce a novel formulation of multi-task group fairness in RL and propose a constrained optimization algorithm that explicitly enforces fairness constraints across multiple tasks simultaneously. We have shown that our proposed algorithm does not violate fairness constraints with high probability and with sublinear regret in the finite-horizon episodic setting. Through experiments in RiverSwim and MuJoCo environments, we demonstrate that our approach better ensures group fairness across multiple tasks compared to previous methods that lack explicit multi-task fairness constraints in both the finite-horizon setting and the infinite-horizon setting. Our results show that the proposed algorithm achieves smaller fairness gaps while maintaining comparable returns across different demographic groups and tasks, suggesting its potential for addressing fairness concerns in real-world multi-task RL applications.

Kefan Song, Jin Yao, Runnan Jiang et al.

As Large Language Models (LLMs) become increasingly powerful and accessible to human users, ensuring fairness across diverse demographic groups, i.e., group fairness, is a critical ethical concern. However, current fairness and bias research in LLMs is limited in two aspects. First, compared to traditional group fairness in machine learning classification, it requires that the non-sensitive attributes, in this case, the prompt questions, be the same across different groups. In many practical scenarios, different groups, however, may prefer different prompt questions and this requirement becomes impractical. Second, it evaluates group fairness only for the LLM's final output without identifying the source of possible bias. Namely, the bias in LLM's output can result from both the pretraining and the finetuning. For finetuning, the bias can result from both the RLHF procedure and the learned reward model. Arguably, evaluating the group fairness of each component in the LLM pipeline could help develop better methods to mitigate the possible bias. Recognizing those two limitations, this work benchmarks the group fairness of learned reward models. By using expert-written text from arXiv, we are able to benchmark the group fairness of reward models without requiring the same prompt questions across different demographic groups. Surprisingly, our results demonstrate that all the evaluated reward models (e.g., Nemotron-4-340B-Reward, ArmoRM-Llama3-8B-v0.1, and GRM-llama3-8B-sftreg) exhibit statistically significant group unfairness. We also observed that top-performing reward models (w.r.t. canonical performance metrics) tend to demonstrate better group fairness.

Shangtong Zhang, Remi Tachet, Romain Laroche

SARSA, a classical on-policy control algorithm for reinforcement learning, is known to chatter when combined with linear function approximation: SARSA does not diverge but oscillates in a bounded region. However, little is known about how fast SARSA converges to that region and how large the region is. In this paper, we make progress towards this open problem by showing the convergence rate of projected SARSA to a bounded region. Importantly, the region is much smaller than the region that we project into, provided that the magnitude of the reward is not too large. Existing works regarding the convergence of linear SARSA to a fixed point all require the Lipschitz constant of SARSA's policy improvement operator to be sufficiently small; our analysis instead applies to arbitrary Lipschitz constants and thus characterizes the behavior of linear SARSA for a new regime.

Shangtong Zhang, Remi Tachet, Romain Laroche

In this paper, we establish the global optimality and convergence rate of an off-policy actor critic algorithm in the tabular setting without using density ratio to correct the discrepancy between the state distribution of the behavior policy and that of the target policy. Our work goes beyond existing works on the optimality of policy gradient methods in that existing works use the exact policy gradient for updating the policy parameters while we use an approximate and stochastic update step. Our update step is not a gradient update because we do not use a density ratio to correct the state distribution, which aligns well with what practitioners do. Our update is approximate because we use a learned critic instead of the true value function. Our update is stochastic because at each step the update is done for only the current state action pair. Moreover, we remove several restrictive assumptions from existing works in our analysis. Central to our work is the finite sample analysis of a generic stochastic approximation algorithm with time-inhomogeneous update operators on time-inhomogeneous Markov chains, based on its uniform contraction properties.

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.

Ray Jiang, Shangtong Zhang, Veronica Chelu et al.

Off-policy sampling and experience replay are key for improving sample efficiency and scaling model-free temporal difference learning methods. When combined with function approximation, such as neural networks, this combination is known as the deadly triad and is potentially unstable. Recently, it has been shown that stability and good performance at scale can be achieved by combining emphatic weightings and multi-step updates. This approach, however, is generally limited to sampling complete trajectories in order, to compute the required emphatic weighting. In this paper we investigate how to combine emphatic weightings with non-sequential, off-line data sampled from a replay buffer. We develop a multi-step emphatic weighting that can be combined with replay, and a time-reversed $n$-step TD learning algorithm to learn the required emphatic weighting. We show that these state weightings reduce variance compared with prior approaches, while providing convergence guarantees. We tested the approach at scale on Atari 2600 video games, and observed that the new X-ETD($n$) agent improved over baseline agents, highlighting both the scalability and broad applicability of our approach.

Shangtong Zhang, Hengshuai Yao, Shimon Whiteson

The deadly triad refers to the instability of a reinforcement learning algorithm when it employs off-policy learning, function approximation, and bootstrapping simultaneously. In this paper, we investigate the target network as a tool for breaking the deadly triad, providing theoretical support for the conventional wisdom that a target network stabilizes training. We first propose and analyze a novel target network update rule which augments the commonly used Polyak-averaging style update with two projections. We then apply the target network and ridge regularization in several divergent algorithms and show their convergence to regularized TD fixed points. Those algorithms are off-policy with linear function approximation and bootstrapping, spanning both policy evaluation and control, as well as both discounted and average-reward settings. In particular, we provide the first convergent linear $Q$-learning algorithms under nonrestrictive and changing behavior policies without bi-level optimization.

Shangtong Zhang, Yi Wan, Richard S. Sutton et al.

We consider off-policy policy evaluation with function approximation (FA) in average-reward MDPs, where the goal is to estimate both the reward rate and the differential value function. For this problem, bootstrapping is necessary and, along with off-policy learning and FA, results in the deadly triad (Sutton & Barto, 2018). To address the deadly triad, we propose two novel algorithms, reproducing the celebrated success of Gradient TD algorithms in the average-reward setting. In terms of estimating the differential value function, the algorithms are the first convergent off-policy linear function approximation algorithms. In terms of estimating the reward rate, the algorithms are the first convergent off-policy linear function approximation algorithms that do not require estimating the density ratio. We demonstrate empirically the advantage of the proposed algorithms, as well as their nonlinear variants, over a competitive density-ratio-based approach, in a simple domain as well as challenging robot simulation tasks.

Shangtong Zhang, Romain Laroche, Harm van Seijen et al.

We investigate the discounting mismatch in actor-critic algorithm implementations from a representation learning perspective. Theoretically, actor-critic algorithms usually have discounting for both actor and critic, i.e., there is a $γ^t$ term in the actor update for the transition observed at time $t$ in a trajectory and the critic is a discounted value function. Practitioners, however, usually ignore the discounting ($γ^t$) for the actor while using a discounted critic. We investigate this mismatch in two scenarios. In the first scenario, we consider optimizing an undiscounted objective $(γ= 1)$ where $γ^t$ disappears naturally $(1^t = 1)$. We then propose to interpret the discounting in critic in terms of a bias-variance-representation trade-off and provide supporting empirical results. In the second scenario, we consider optimizing a discounted objective ($γ< 1$) and propose to interpret the omission of the discounting in the actor update from an auxiliary task perspective and provide supporting empirical results.

Shangtong Zhang, Vivek Veeriah, Shimon Whiteson

We present a Reverse Reinforcement Learning (Reverse RL) approach for representing retrospective knowledge. General Value Functions (GVFs) have enjoyed great success in representing predictive knowledge, i.e., answering questions about possible future outcomes such as "how much fuel will be consumed in expectation if we drive from A to B?". GVFs, however, cannot answer questions like "how much fuel do we expect a car to have given it is at B at time $t$?". To answer this question, we need to know when that car had a full tank and how that car came to B. Since such questions emphasize the influence of possible past events on the present, we refer to their answers as retrospective knowledge. In this paper, we show how to represent retrospective knowledge with Reverse GVFs, which are trained via Reverse RL. We demonstrate empirically the utility of Reverse GVFs in both representation learning and anomaly detection.

Shangtong Zhang, Bo Liu, Shimon Whiteson

We present a mean-variance policy iteration (MVPI) framework for risk-averse control in a discounted infinite horizon MDP optimizing the variance of a per-step reward random variable. MVPI enjoys great flexibility in that any policy evaluation method and risk-neutral control method can be dropped in for risk-averse control off the shelf, in both on- and off-policy settings. This flexibility reduces the gap between risk-neutral control and risk-averse control and is achieved by working on a novel augmented MDP directly. We propose risk-averse TD3 as an example instantiating MVPI, which outperforms vanilla TD3 and many previous risk-averse control methods in challenging Mujoco robot simulation tasks under a risk-aware performance metric. This risk-averse TD3 is the first to introduce deterministic policies and off-policy learning into risk-averse reinforcement learning, both of which are key to the performance boost we show in Mujoco domains.

Shangtong Zhang, Bo Liu, Shimon Whiteson

We present GradientDICE for estimating the density ratio between the state distribution of the target policy and the sampling distribution in off-policy reinforcement learning. GradientDICE fixes several problems of GenDICE (Zhang et al., 2020), the state-of-the-art for estimating such density ratios. Namely, the optimization problem in GenDICE is not a convex-concave saddle-point problem once nonlinearity in optimization variable parameterization is introduced to ensure positivity, so any primal-dual algorithm is not guaranteed to converge or find the desired solution. However, such nonlinearity is essential to ensure the consistency of GenDICE even with a tabular representation. This is a fundamental contradiction, resulting from GenDICE's original formulation of the optimization problem. In GradientDICE, we optimize a different objective from GenDICE by using the Perron-Frobenius theorem and eliminating GenDICE's use of divergence. Consequently, nonlinearity in parameterization is not necessary for GradientDICE, which is provably convergent under linear function approximation.

Shangtong Zhang, Bo Liu, Hengshuai Yao et al.

We present the first provably convergent two-timescale off-policy actor-critic algorithm (COF-PAC) with function approximation. Key to COF-PAC is the introduction of a new critic, the emphasis critic, which is trained via Gradient Emphasis Learning (GEM), a novel combination of the key ideas of Gradient Temporal Difference Learning and Emphatic Temporal Difference Learning. With the help of the emphasis critic and the canonical value function critic, we show convergence for COF-PAC, where the critics are linear and the actor can be nonlinear.

Borislav Mavrin, Shangtong Zhang, Hengshuai Yao et al.

In distributional reinforcement learning (RL), the estimated distribution of value function models both the parametric and intrinsic uncertainties. We propose a novel and efficient exploration method for deep RL that has two components. The first is a decaying schedule to suppress the intrinsic uncertainty. The second is an exploration bonus calculated from the upper quantiles of the learned distribution. In Atari 2600 games, our method outperforms QR-DQN in 12 out of 14 hard games (achieving 483 \% average gain across 49 games in cumulative rewards over QR-DQN with a big win in Venture). We also compared our algorithm with QR-DQN in a challenging 3D driving simulator (CARLA). Results show that our algorithm achieves near-optimal safety rewards twice faster than QRDQN.

Yuhang Song, Jianyi Wang, Thomas Lukasiewicz et al.

Intrinsic rewards were introduced to simulate how human intelligence works; they are usually evaluated by intrinsically-motivated play, i.e., playing games without extrinsic rewards but evaluated with extrinsic rewards. However, none of the existing intrinsic reward approaches can achieve human-level performance under this very challenging setting of intrinsically-motivated play. In this work, we propose a novel megalomania-driven intrinsic reward (called mega-reward), which, to our knowledge, is the first approach that achieves human-level performance in intrinsically-motivated play. Intuitively, mega-reward comes from the observation that infants' intelligence develops when they try to gain more control on entities in an environment; therefore, mega-reward aims to maximize the control capabilities of agents on given entities in a given environment. To formalize mega-reward, a relational transition model is proposed to bridge the gaps between direct and latent control. Experimental studies show that mega-reward (i) can greatly outperform all state-of-the-art intrinsic reward approaches, (ii) generally achieves the same level of performance as Ex-PPO and professional human-level scores, and (iii) has also a superior performance when it is incorporated with extrinsic rewards.

Shangtong Zhang, Wendelin Boehmer, Shimon Whiteson

We revisit residual algorithms in both model-free and model-based reinforcement learning settings. We propose the bidirectional target network technique to stabilize residual algorithms, yielding a residual version of DDPG that significantly outperforms vanilla DDPG in the DeepMind Control Suite benchmark. Moreover, we find the residual algorithm an effective approach to the distribution mismatch problem in model-based planning. Compared with the existing TD($k$) method, our residual-based method makes weaker assumptions about the model and yields a greater performance boost.