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We study covariate shift in the context of nonparametric regression. We introduce a new measure of distribution mismatch between the source and target distributions using the integrated ratio of probabilities of balls at a given radius. We use the scaling of this measure with respect to the radius to characterize the minimax rate of estimation over a family of H{ö}lder continuous functions under covariate shift. In comparison to the recently proposed notion of transfer exponent, this measure leads to a sharper rate of convergence and is more fine-grained. We accompany our theory with concrete instances of covariate shift that illustrate this sharp difference. 

We propose and analyze a reinforcement learning principle that approximates the Bellman equations by enforcing their validity only along an user-defined space of test functions. Focusing on applications to model-free offline RL with function approximation, we exploit this principle to derive confidence intervals for off-policy evaluation, as well as to optimize over policies within a prescribed policy class. We prove an oracle inequality on our policy optimization procedure in terms of a trade-off between the value and uncertainty of an arbitrary comparator policy. Different choices of test function spaces allow us to tackle different problems within a common framework. We characterize the loss of efficiency in moving from on-policy to off-policy data using our procedures, and establish connections to concentrability coefficients studied in past work. We examine in depth the implementation of our methods with linear function approximation, and provide theoretical guarantees with polynomial-time implementations even when Bellman closure does not hold. 

Actor-critic methods are widely used in offline reinforcement learning practice, but are not so well-understood theoretically. We propose a new offline actor-critic algorithm that naturally incorporates the pessimism principle, leading to several key advantages compared to the state of the art. The algorithm can operate when the Bellman evaluation operator is closed with respect to the action value function of the actor's policies; this is a more general setting than the low-rank MDP model. Despite the added generality, the procedure is computationally tractable as it involves the solution of a sequence of second-order programs. We prove an upper bound on the suboptimality gap of the policy returned by the procedure that depends on the data coverage of any arbitrary, possibly data dependent comparator policy. The achievable guarantee is complemented with a minimax lower bound that is matching up to logarithmic factors. 

We undertake a precise study of the asymptotic and non-asymptotic properties of stochastic approximation procedures with Polyak-Ruppert averaging for solving a linear system $\bar{A} \theta = \bar{b}$. When the matrix $\bar{A}$ is Hurwitz, we prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity. The CLT characterizes the exact asymptotic covariance matrix, which is the sum of the classical Polyak-Ruppert covariance and a correction term that scales with the step size. Under assumptions on the tail of the noise distribution, we prove a non-asymptotic concentration inequality whose main term matches the covariance in CLT in any direction, up to universal constants. When the matrix $\bar{A}$ is not Hurwitz but only has non-negative real parts in its eigenvalues, we prove that the averaged LSA procedure actually achieves an $O(1/T)$ rate in mean-squared error. Our results provide a more refined understanding of linear stochastic approximation in both the asymptotic and non-asymptotic settings. We also show various applications of the main results, including the study of momentum-based stochastic gradient methods as well as temporal difference algorithms in reinforcement learning.