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Reinforcement learning (RL) -- finding the optimal behaviour (also referred to as policy) maximizing the collected long-term cumulative reward -- is among the most influential approaches in machine learning with a large number of successful applications. In several decision problems, however, one faces the possibility of policy switching -- changing from the current policy to a new one -- which incurs a non-negligible cost (examples include the shifting of the currently applied educational technology, modernization of a computing cluster, and the introduction of a new webpage design), and in the decision one is limited to using historical data without the availability for further online interaction. Despite the inevitable importance of this offline learning scenario, to our best knowledge, very little effort has been made to tackle the key problem of balancing between the gain and the cost of switching in a flexible and principled way. Leveraging ideas from the area of optimal transport, we initialize the systematic study of policy switching in offline RL. We establish fundamental properties and design a Net Actor-Critic algorithm for the proposed novel switching formulation. Numerical experiments demonstrate the efficiency of our approach on multiple benchmarks of the Gymnasium.

Off-policy evaluation (OPE) is crucial for evaluating a target policy's impact offline before its deployment. However, achieving accurate OPE in large state spaces remains challenging.This paper studies state abstractions-originally designed for policy learning-in the context of OPE. Our contributions are three-fold: (i) We define a set of irrelevance conditions central to learning state abstractions for OPE. (ii) We derive sufficient conditions for achieving irrelevance in Q-functions and marginalized importance sampling ratios, the latter obtained by constructing a time-reversed Markov decision process (MDP) based on the observed MDP. (iii) We propose a novel two-step procedure that sequentially projects the original state space into a smaller space, which substantially simplify the sample complexity of OPE arising from high cardinality., Comment: 42 pages, 5 figures

For each nonnegative integer m we show that any closed, oriented topological four-manifold with fundamental group Z_{4m+2} and odd intersection form, with possibly seven exceptions, either admits no smooth structure or admits infinitely many distinct smooth structures up to diffeomorphism. Moreover, we construct infinite families of non-complex irreducible fake projective planes with diverse fundamental groups., Comment: 32 pages

Kernel methods underpin many of the most successful approaches in data science and statistics, and they allow representing probability measures as elements of a reproducing kernel Hilbert space without loss of information. Recently, the kernel Stein discrepancy (KSD), which combines Stein's method with kernel techniques, gained considerable attention. Through the Stein operator, KSD allows the construction of powerful goodness-of-fit tests where it is sufficient to know the target distribution up to a multiplicative constant. However, the typical U- and V-statistic-based KSD estimators suffer from a quadratic runtime complexity, which hinders their application in large-scale settings. In this work, we propose a Nystr\"om-based KSD acceleration -- with runtime $\mathcal O\!\left(mn+m^3\right)$ for $n$ samples and $m\ll n$ Nystr\"om points -- , show its $\sqrt{n}$-consistency under the null with a classical sub-Gaussian assumption, and demonstrate its applicability for goodness-of-fit testing on a suite of benchmarks., Comment: Update proof of Lemma B.3, milder Assumption 1, more experiments

Kernel techniques are among the most influential approaches in data science and statistics. Under mild conditions, the reproducing kernel Hilbert space associated to a kernel is capable of encoding the independence of $M\ge 2$ random variables. Probably the most widespread independence measure relying on kernels is the so-called Hilbert-Schmidt independence criterion (HSIC; also referred to as distance covariance in the statistics literature). Despite various existing HSIC estimators designed since its introduction close to two decades ago, the fundamental question of the rate at which HSIC can be estimated is still open. In this work, we prove that the minimax optimal rate of HSIC estimation on $\mathbb R^d$ for Borel measures containing the Gaussians with continuous bounded translation-invariant characteristic kernels is $\mathcal O\!\left(n^{-1/2}\right)$. Specifically, our result implies the optimality in the minimax sense of many of the most-frequently used estimators (including the U-statistic, the V-statistic, and the Nystr\"om-based one) on $\mathbb R^d$.