Matthew Zurek

Brahma S. Pavse, Matthew Zurek, Yudong Chen et al.

In many reinforcement learning (RL) applications, we want policies that reach desired states and then keep the controlled system within an acceptable region around the desired states over an indefinite period of time. This latter objective is called stability and is especially important when the state space is unbounded, such that the states can be arbitrarily far from each other and the agent can drift far away from the desired states. For example, in stochastic queuing networks, where queues of waiting jobs can grow without bound, the desired state is all-zero queue lengths. Here, a stable policy ensures queue lengths are finite while an optimal policy minimizes queue lengths. Since an optimal policy is also stable, one would expect that RL algorithms would implicitly give us stable policies. However, in this work, we find that deep RL algorithms that directly minimize the distance to the desired state during online training often result in unstable policies, i.e., policies that drift far away from the desired state. We attribute this instability to poor credit-assignment for destabilizing actions. We then introduce an approach based on two ideas: 1) a Lyapunov-based cost-shaping technique and 2) state transformations to the unbounded state space. We conduct an empirical study on various queueing networks and traffic signal control problems and find that our approach performs competitively against strong baselines with knowledge of the transition dynamics. Our code is available here: https://github.com/Badger-RL/STOP.

Gaurav R. Ghosal, Matthew Zurek, Daniel S. Brown et al.

When inferring reward functions from human behavior (be it demonstrations, comparisons, physical corrections, or e-stops), it has proven useful to model the human as making noisy-rational choices, with a "rationality coefficient" capturing how much noise or entropy we expect to see in the human behavior. Prior work typically sets the rationality level to a constant value, regardless of the type, or quality, of human feedback. However, in many settings, giving one type of feedback (e.g. a demonstration) may be much more difficult than a different type of feedback (e.g. answering a comparison query). Thus, we expect to see more or less noise depending on the type of human feedback. In this work, we advocate that grounding the rationality coefficient in real data for each feedback type, rather than assuming a default value, has a significant positive effect on reward learning. We test this in both simulated experiments and in a user study with real human feedback. We find that overestimating human rationality can have dire effects on reward learning accuracy and regret. We also find that fitting the rationality coefficient to human data enables better reward learning, even when the human deviates significantly from the noisy-rational choice model due to systematic biases. Further, we find that the rationality level affects the informativeness of each feedback type: surprisingly, demonstrations are not always the most informative -- when the human acts very suboptimally, comparisons actually become more informative, even when the rationality level is the same for both. Ultimately, our results emphasize the importance and advantage of paying attention to the assumed human-rationality level, especially when agents actively learn from multiple types of human feedback.

Matthew Zurek, Yudong Chen

We study graph clustering in the Stochastic Block Model (SBM) in the presence of both large clusters and small, unrecoverable clusters. Previous convex relaxation approaches achieving exact recovery do not allow any small clusters of size $o(\sqrt{n})$, or require a size gap between the smallest recovered cluster and the largest non-recovered cluster. We provide an algorithm based on semidefinite programming (SDP) which removes these requirements and provably recovers large clusters regardless of the remaining cluster sizes. Mid-sized clusters pose unique challenges to the analysis, since their proximity to the recovery threshold makes them highly sensitive to small noise perturbations and precludes a closed-form candidate solution. We develop novel techniques, including a leave-one-out-style argument which controls the correlation between SDP solutions and noise vectors even when the removal of one row of noise can drastically change the SDP solution. We also develop improved eigenvalue perturbation bounds of potential independent interest. Our results are robust to certain semirandom settings that are challenging for alternative algorithms. Using our gap-free clustering procedure, we obtain efficient algorithms for the problem of clustering with a faulty oracle with superior query complexities, notably achieving $o(n^2)$ sample complexity even in the presence of a large number of small clusters. Our gap-free clustering procedure also leads to improved algorithms for recursive clustering.

Matthew Zurek, Yudong Chen

We study the sample complexity of learning an $\varepsilon$-optimal policy in an average-reward Markov decision process (MDP) under a generative model. We establish the complexity bound $\widetilde{O}\left(SA\frac{H}{\varepsilon^2} \right)$, where $H$ is the span of the bias function of the optimal policy and $SA$ is the cardinality of the state-action space. Our result is the first that is minimax optimal (up to log factors) in all parameters $S,A,H$ and $\varepsilon$, improving on existing work that either assumes uniformly bounded mixing times for all policies or has suboptimal dependence on the parameters. Our result is based on reducing the average-reward MDP to a discounted MDP. To establish the optimality of this reduction, we develop improved bounds for $γ$-discounted MDPs, showing that $\widetilde{O}\left(SA\frac{H}{(1-γ)^2\varepsilon^2} \right)$ samples suffice to learn a $\varepsilon$-optimal policy in weakly communicating MDPs under the regime that $γ\geq 1 - \frac{1}{H}$, circumventing the well-known lower bound of $\widetildeΩ\left(SA\frac{1}{(1-γ)^3\varepsilon^2} \right)$ for general $γ$-discounted MDPs. Our analysis develops upper bounds on certain instance-dependent variance parameters in terms of the span parameter. These bounds are tighter than those based on the mixing time or diameter of the MDP and may be of broader use.

Guy Zamir, Matthew Zurek, Yudong Chen

Online reinforcement learning in infinite-horizon Markov decision processes (MDPs) remains less theoretically and algorithmically developed than its episodic counterpart, with many algorithms suffering from high ``burn-in'' costs and failing to adapt to benign instance-specific complexity. In this work, we address these shortcomings for two infinite-horizon objectives: the classical average-reward regret and the $γ$-regret. We develop a single tractable UCB-style algorithm applicable to both settings, which achieves the first optimal variance-dependent regret guarantees. Our regret bounds in both settings take the form $\tilde{O}( \sqrt{SA\,\text{Var}} + \text{lower-order terms})$, where $S,A$ are the state and action space sizes, and $\text{Var}$ captures cumulative transition variance. This implies minimax-optimal average-reward and $γ$-regret bounds in the worst case but also adapts to easier problem instances, for example yielding nearly constant regret in deterministic MDPs. Furthermore, our algorithm enjoys significantly improved lower-order terms for the average-reward setting. With prior knowledge of the optimal bias span $\Vert h^\star\Vert_\text{sp}$, our algorithm obtains lower-order terms scaling as $\Vert h^\star\Vert_\text{sp} S^2 A$, which we prove is optimal in both $\Vert h^\star\Vert_\text{sp}$ and $A$. Without prior knowledge, we prove that no algorithm can have lower-order terms smaller than $\Vert h^\star \Vert_\text{sp}^2 S A$, and we provide a prior-free algorithm whose lower-order terms scale as $\Vert h^\star\Vert_\text{sp}^2 S^3 A$, nearly matching this lower bound. Taken together, these results completely characterize the optimal dependence on $\Vert h^\star\Vert_\text{sp}$ in both leading and lower-order terms, and reveal a fundamental gap in what is achievable with and without prior knowledge.

Matthew Zurek, Yudong Chen

We study the sample complexity of learning an $\varepsilon$-optimal policy in an average-reward Markov decision process (MDP) under a generative model. For weakly communicating MDPs, we establish the complexity bound $\widetilde{O}(SA\frac{H}{\varepsilon^2} )$, where $H$ is the span of the bias function of the optimal policy and $SA$ is the cardinality of the state-action space. Our result is the first that is minimax optimal (up to log factors) in all parameters $S,A,H$, and $\varepsilon$, improving on existing work that either assumes uniformly bounded mixing times for all policies or has suboptimal dependence on the parameters. We also initiate the study of sample complexity in general (multichain) average-reward MDPs. We argue a new transient time parameter $B$ is necessary, establish an $\widetilde{O}(SA\frac{B + H}{\varepsilon^2})$ complexity bound, and prove a matching (up to log factors) minimax lower bound. Both results are based on reducing the average-reward MDP to a discounted MDP, which requires new ideas in the general setting. To optimally analyze this reduction, we develop improved bounds for $γ$-discounted MDPs, showing that $\widetilde{O}(SA\frac{H}{(1-γ)^2\varepsilon^2} )$ and $\widetilde{O}(SA\frac{B + H}{(1-γ)^2\varepsilon^2} )$ samples suffice to learn $\varepsilon$-optimal policies in weakly communicating and in general MDPs, respectively. Both these results circumvent the well-known minimax lower bound of $\widetildeΩ(SA\frac{1}{(1-γ)^3\varepsilon^2} )$ for $γ$-discounted MDPs, and establish a quadratic rather than cubic horizon dependence for a fixed MDP instance.

Matthew Zurek, Yudong Chen

We study the sample complexity of finding an $\varepsilon$-optimal policy in average-reward Markov Decision Processes (MDPs) with a generative model. The minimax optimal span-based complexity of $\widetilde{O}(SAH/\varepsilon^2)$, where $H$ is the span of the optimal bias function, has only been achievable with prior knowledge of the value of $H$. Prior-knowledge-free algorithms have been the objective of intensive research, but several natural approaches provably fail to achieve this goal. We resolve this problem, developing the first algorithms matching the optimal span-based complexity without $H$ knowledge, both when the dataset size is fixed and when the suboptimality level $\varepsilon$ is fixed. Our main technique combines the discounted reduction approach with a method for automatically tuning the effective horizon based on empirical confidence intervals or lower bounds on performance, which we term horizon calibration. We also develop an empirical span penalization approach, inspired by sample variance penalization, which satisfies an oracle inequality performance guarantee. In particular this algorithm can outperform the minimax complexity in benign settings such as when there exist near-optimal policies with span much smaller than $H$.

Matthew Zurek, Yudong Chen

We study value-iteration (VI) algorithms for solving general (a.k.a. multichain) Markov decision processes (MDPs) under the average-reward criterion, a fundamental but theoretically challenging setting. Beyond the difficulties inherent to all average-reward problems posed by the lack of contractivity and non-uniqueness of solutions to the Bellman operator, in the multichain setting an optimal policy must solve the navigation subproblem of steering towards the best connected component, in addition to optimizing long-run performance within each component. We develop algorithms which better solve this navigational subproblem in order to achieve faster convergence for multichain MDPs, obtaining improved rates of convergence and sharper measures of complexity relative to prior work. Many key components of our results are of potential independent interest, including novel connections between average-reward and discounted problems, optimal fixed-point methods for discounted VI which extend to general Banach spaces, new sublinear convergence rates for the discounted value error, and refined suboptimality decompositions for multichain MDPs. Overall our results yield faster convergence rates for discounted and average-reward problems and expand the theoretical foundations of VI approaches.

Matthew Zurek, Guy Zamir, Yudong Chen

We study offline reinforcement learning in average-reward MDPs, which presents increased challenges from the perspectives of distribution shift and non-uniform coverage, and has been relatively underexamined from a theoretical perspective. While previous work obtains performance guarantees under single-policy data coverage assumptions, such guarantees utilize additional complexity measures which are uniform over all policies, such as the uniform mixing time. We develop sharp guarantees depending only on the target policy, specifically the bias span and a novel policy hitting radius, yielding the first fully single-policy sample complexity bound for average-reward offline RL. We are also the first to handle general weakly communicating MDPs, contrasting restrictive structural assumptions made in prior work. To achieve this, we introduce an algorithm based on pessimistic discounted value iteration enhanced by a novel quantile clipping technique, which enables the use of a sharper empirical-span-based penalty function. Our algorithm also does not require any prior parameter knowledge for its implementation. Remarkably, we show via hard examples that learning under our conditions requires coverage assumptions beyond the stationary distribution of the target policy, distinguishing single-policy complexity measures from previously examined cases. We also develop lower bounds nearly matching our main result.

Matthew Zurek, Andreea Bobu, Daniel S. Brown et al.

Shared autonomy enables robots to infer user intent and assist in accomplishing it. But when the user wants to do a new task that the robot does not know about, shared autonomy will hinder their performance by attempting to assist them with something that is not their intent. Our key idea is that the robot can detect when its repertoire of intents is insufficient to explain the user's input, and give them back control. This then enables the robot to observe unhindered task execution, learn the new intent behind it, and add it to this repertoire. We demonstrate with both a case study and a user study that our proposed method maintains good performance when the human's intent is in the robot's repertoire, outperforms prior shared autonomy approaches when it isn't, and successfully learns new skills, enabling efficient lifelong learning for confidence-based shared autonomy.