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Philip Amortila, Dylan J. Foster, Akshay Krishnamurthy

Exploration is a major challenge in reinforcement learning, especially for high-dimensional domains that require function approximation. We propose exploration objectives -- policy optimization objectives that enable downstream maximization of any reward function -- as a conceptual framework to systematize the study of exploration. Within this framework, we introduce a new objective, $L_1$-Coverage, which generalizes previous exploration schemes and supports three fundamental desiderata: 1. Intrinsic complexity control. $L_1$-Coverage is associated with a structural parameter, $L_1$-Coverability, which reflects the intrinsic statistical difficulty of the underlying MDP, subsuming Block and Low-Rank MDPs. 2. Efficient planning. For a known MDP, optimizing $L_1$-Coverage efficiently reduces to standard policy optimization, allowing flexible integration with off-the-shelf methods such as policy gradient and Q-learning approaches. 3. Efficient exploration. $L_1$-Coverage enables the first computationally efficient model-based and model-free algorithms for online (reward-free or reward-driven) reinforcement learning in MDPs with low coverability. Empirically, we find that $L_1$-Coverage effectively drives off-the-shelf policy optimization algorithms to explore the state space.

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Philip Amortila, Tongyi Cao, Akshay Krishnamurthy

A pervasive phenomenon in machine learning applications is distribution shift, where training and deployment conditions for a machine learning model differ. As distribution shift typically results in a degradation in performance, much attention has been devoted to algorithmic interventions that mitigate these detrimental effects. In this paper, we study the effect of distribution shift in the presence of model misspecification, specifically focusing on $L_{\infty}$-misspecified regression and adversarial covariate shift, where the regression target remains fixed while the covariate distribution changes arbitrarily. We show that empirical risk minimization, or standard least squares regression, can result in undesirable misspecification amplification where the error due to misspecification is amplified by the density ratio between the training and testing distributions. As our main result, we develop a new algorithm -- inspired by robust optimization techniques -- that avoids this undesirable behavior, resulting in no misspecification amplification while still obtaining optimal statistical rates. As applications, we use this regression procedure to obtain new guarantees in offline and online reinforcement learning with misspecification and establish new separations between previously studied structural conditions and notions of coverage.

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Philip Amortila, Dylan J. Foster, Nan Jiang, Ayush Sekhari, Tengyang Xie

The theories of offline and online reinforcement learning, despite having evolved in parallel, have begun to show signs of the possibility for a unification, with algorithms and analysis techniques for one setting often having natural counterparts in the other. However, the notion of density ratio modeling, an emerging paradigm in offline RL, has been largely absent from online RL, perhaps for good reason: the very existence and boundedness of density ratios relies on access to an exploratory dataset with good coverage, but the core challenge in online RL is to collect such a dataset without having one to start. In this work we show -- perhaps surprisingly -- that density ratio-based algorithms have online counterparts. Assuming only the existence of an exploratory distribution with good coverage, a structural condition known as coverability (Xie et al., 2023), we give a new algorithm (GLOW) that uses density ratio realizability and value function realizability to perform sample-efficient online exploration. GLOW addresses unbounded density ratios via careful use of truncation, and combines this with optimism to guide exploration. GLOW is computationally inefficient; we complement it with a more efficient counterpart, HyGLOW, for the Hybrid RL setting (Song et al., 2022) wherein online RL is augmented with additional offline data. HyGLOW is derived as a special case of a more general meta-algorithm that provides a provable black-box reduction from hybrid RL to offline RL, which may be of independent interest.

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Philip Amortila, Nan Jiang, Csaba Szepesvári

Theoretical guarantees in reinforcement learning (RL) are known to suffer multiplicative blow-up factors with respect to the misspecification error of function approximation. Yet, the nature of such \emph{approximation factors} -- especially their optimal form in a given learning problem -- is poorly understood. In this paper we study this question in linear off-policy value function estimation, where many open questions remain. We study the approximation factor in a broad spectrum of settings, such as with the weighted $L_2$-norm (where the weighting is the offline state distribution), the $L_\infty$ norm, the presence vs. absence of state aliasing, and full vs. partial coverage of the state space. We establish the optimal asymptotic approximation factors (up to constants) for all of these settings. In particular, our bounds identify two instance-dependent factors for the $L_2(\mu)$ norm and only one for the $L_\infty$ norm, which are shown to dictate the hardness of off-policy evaluation under misspecification.

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Philip Amortila, Nan Jiang, Dhruv Madeka, Dean P. Foster

The current paper studies sample-efficient Reinforcement Learning (RL) in settings where only the optimal value function is assumed to be linearly-realizable. It has recently been understood that, even under this seemingly strong assumption and access to a generative model, worst-case sample complexities can be prohibitively (i.e., exponentially) large. We investigate the setting where the learner additionally has access to interactive demonstrations from an expert policy, and we present a statistically and computationally efficient algorithm (Delphi) for blending exploration with expert queries. In particular, Delphi requires $\tilde{\mathcal{O}}(d)$ expert queries and a $\texttt{poly}(d,H,|\mathcal{A}|,1/\varepsilon)$ amount of exploratory samples to provably recover an $\varepsilon$-suboptimal policy. Compared to pure RL approaches, this corresponds to an exponential improvement in sample complexity with surprisingly-little expert input. Compared to prior imitation learning (IL) approaches, our required number of expert demonstrations is independent of $H$ and logarithmic in $1/\varepsilon$, whereas all prior work required at least linear factors of both in addition to the same dependence on $d$. Towards establishing the minimal amount of expert queries needed, we show that, in the same setting, any learner whose exploration budget is polynomially-bounded (in terms of $d,H,$ and $|\mathcal{A}|$) will require at least $\tilde\Omega(\sqrt{d})$ oracle calls to recover a policy competing with the expert's value function. Under the weaker assumption that the expert's policy is linear, we show that the lower bound increases to $\tilde\Omega(d)$.

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Gellert Weisz, Philip Amortila, Barnabás Janzer, Yasin Abbasi-Yadkori, Nan Jiang, Csaba Szepesvári

We consider the problem of local planning in fixed-horizon Markov Decision Processes (MDPs) with a generative model under the assumption that the optimal value function lies in the span of a feature map that is accessible through the generative model. As opposed to previous work where linear realizability of all policies was assumed, we consider the significantly relaxed assumption of a single linearly realizable (deterministic) policy. A recent lower bound established that the related problem when the action-value function of the optimal policy is linearly realizable requires an exponential number of queries, either in H (the horizon of the MDP) or d (the dimension of the feature mapping). Their construction crucially relies on having an exponentially large action set. In contrast, in this work, we establish that poly$(H, d)$ learning is possible (with state value function realizability) whenever the action set is small (i.e. O(1)). In particular, we present the TensorPlan algorithm which uses poly$((dH/\delta)^A)$ queries to find a $\delta$-optimal policy relative to any deterministic policy for which the value function is linearly realizable with a parameter from a fixed radius ball around zero. This is the first algorithm to give a polynomial query complexity guarantee using only linear-realizability of a single competing value function. Whether the computation cost is similarly bounded remains an interesting open question. The upper bound is complemented by a lower bound which proves that in the infinite-horizon episodic setting, planners that achieve constant suboptimality need exponentially many queries, either in the dimension or the number of actions.

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Philip Amortila, Nan Jiang, Tengyang Xie

Recently, Wang et al. (2020) showed a highly intriguing hardness result for batch reinforcement learning (RL) with linearly realizable value function and good feature coverage in the finite-horizon case. In this note we show that once adapted to the discounted setting, the construction can be simplified to a 2-state MDP with 1-dimensional features, such that learning is impossible even with an infinite amount of data.

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Gellert Weisz, Philip Amortila, Csaba Szepesvári

We consider the problem of local planning in fixed-horizon Markov Decision Processes (MDPs) with linear function approximation and a generative model under the assumption that the optimal action-value function lies in the span of a feature map that is available to the planner. Previous work has left open the question of whether there exists sound planners that need only poly(H, d) queries regardless of the MDP, where H is the horizon and d is the dimensionality of the features. We answer this question in the negative: we show that any sound planner must query at least min(exp({\Omega}(d)), {\Omega}(2^H)) samples. We also show that for any {\delta}>0, the least-squares value iteration algorithm with O(H^5d^(H+1)/{\delta}^2) queries can compute a {\delta}-optimal policy. We discuss implications and remaining open questions.

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Harsh Satija, Philip Amortila, Joelle Pineau

Although Reinforcement Learning (RL) algorithms have found tremendous success in simulated domains, they often cannot directly be applied to physical systems, especially in cases where there are hard constraints to satisfy (e.g. on safety or resources). In standard RL, the agent is incentivized to explore any behavior as long as it maximizes rewards, but in the real world, undesired behavior can damage either the system or the agent in a way that breaks the learning process itself. In this work, we model the problem of learning with constraints as a Constrained Markov Decision Process and provide a new on-policy formulation for solving it. A key contribution of our approach is to translate cumulative cost constraints into state-based constraints. Through this, we define a safe policy improvement method which maximizes returns while ensuring that the constraints are satisfied at every step. We provide theoretical guarantees under which the agent converges while ensuring safety over the course of training. We also highlight the computational advantages of this approach. The effectiveness of our approach is demonstrated on safe navigation tasks and in safety-constrained versions of MuJoCo environments, with deep neural networks.

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Philip Amortila, Doina Precup, Prakash Panangaden, Marc G. Bellemare

We present a distributional approach to theoretical analyses of reinforcement learning algorithms for constant step-sizes. We demonstrate its effectiveness by presenting simple and unified proofs of convergence for a variety of commonly-used methods. We show that value-based methods such as TD($\lambda$) and $Q$-Learning have update rules which are contractive in the space of distributions of functions, thus establishing their exponentially fast convergence to a stationary distribution. We demonstrate that the stationary distribution obtained by any algorithm whose target is an expected Bellman update has a mean which is equal to the true value function. Furthermore, we establish that the distributions concentrate around their mean as the step-size shrinks. We further analyse the optimistic policy iteration algorithm, for which the contraction property does not hold, and formulate a probabilistic policy improvement property which entails the convergence of the algorithm.

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