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Bootstrapping provides a flexible and effective approach for assessing the quality of batch reinforcement learning, yet its theoretical property is less understood. In this paper, we study the use of bootstrapping in off-policy evaluation (OPE), and in particular, we focus on the fitted Q-evaluation (FQE) that is known to be minimax-optimal in the tabular and linear-model cases. We propose a bootstrapping FQE method for inferring the distribution of the policy evaluation error and show that this method is asymptotically efficient and distributionally consistent for off-policy statistical inference. To overcome the computation limit of bootstrapping, we further adapt a subsampling procedure that improves the runtime by an order of magnitude. We numerically evaluate the bootrapping method in classical RL environments for confidence interval estimation, estimating the variance of off-policy evaluator, and estimating the correlation between multiple off-policy evaluators.
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In reinforcement learning, it is typical to use the empirically observed transitions and rewards to estimate the value of a policy via either model-based or Q-fitting approaches. Although straightforward, these techniques in general yield biased estimates of the true value of the policy. In this work, we investigate the potential for statistical bootstrapping to be used as a way to take these biased estimates and produce calibrated confidence intervals for the true value of the policy. We identify conditions - specifically, sufficient data size and sufficient coverage - under which statistical bootstrapping in this setting is guaranteed to yield correct confidence intervals. In practical situations, these conditions often do not hold, and so we discuss and propose mechanisms that can be employed to mitigate their effects. We evaluate our proposed method and show that it can yield accurate confidence intervals in a variety of conditions, including challenging continuous control environments and small data regimes.
While machine learning (ML) methods have received a lot of attention in recent years, these methods are primarily for prediction. Empirical researchers conducting policy evaluations are, on the other hand, pre-occupied with causal problems, trying to answer counterfactual questions: what would have happened in the absence of a policy? Because these counterfactuals can never be directly observed (described as the fundamental problem of causal inference) prediction tools from the ML literature cannot be readily used for causal inference. In the last decade, major innovations have taken place incorporating supervised ML tools into estimators for causal parameters such as the average treatment effect (ATE). This holds the promise of attenuating model misspecification issues, and increasing of transparency in model selection. One particularly mature strand of the literature include approaches that incorporate supervised ML approaches in the estimation of the ATE of a binary treatment, under the textit{unconfoundedness} and positivity assumptions (also known as exchangeability and overlap assumptions). This article reviews popular supervised machine learning algorithms, including the Super Learner. Then, some specific uses of machine learning for treatment effect estimation are introduced and illustrated, namely (1) to create balance among treated and control groups, (2) to estimate so-called nuisance models (e.g. the propensity score, or conditional expectations of the outcome) in semi-parametric estimators that target causal parameters (e.g. targeted maximum likelihood estimation or the double ML estimator), and (3) the use of machine learning for variable selection in situations with a high number of covariates.
Logistic regression remains one of the most widely used tools in applied statistics, machine learning and data science. However, in moderately high-dimensional problems, where the number of features $d$ is a non-negligible fraction of the sample size $n$, the logistic regression maximum likelihood estimator (MLE), and statistical procedures based the large-sample approximation of its distribution, behave poorly. Recently, Sur and Cand`es (2019) showed that these issues can be corrected by applying a new approximation of the MLEs sampling distribution in this high-dimensional regime. Unfortunately, these corrections are difficult to implement in practice, because they require an estimate of the emph{signal strength}, which is a function of the underlying parameters $beta$ of the logistic regression. To address this issue, we propose SLOE, a fast and straightforward approach to estimate the signal strength in logistic regression. The key insight of SLOE is that the Sur and Cand`es (2019) correction can be reparameterized in terms of the emph{corrupted signal strength}, which is only a function of the estimated parameters $widehat beta$. We propose an estimator for this quantity, prove that it is consistent in the relevant high-dimensional regime, and show that dimensionality correction using SLOE is accurate in finite samples. Compared to the existing ProbeFrontier heuristic, SLOE is conceptually simpler and orders of magnitude faster, making it suitable for routine use. We demonstrate the importance of routine dimensionality correction in the Heart Disease dataset from the UCI repository, and a genomics application using data from the UK Biobank. We provide an open source package for this method, available at url{https://github.com/google-research/sloe-logistic}.
The recent emergence of reinforcement learning has created a demand for robust statistical inference methods for the parameter estimates computed using these algorithms. Existing methods for statistical inference in online learning are restricted to settings involving independently sampled observations, while existing statistical inference methods in reinforcement learning (RL) are limited to the batch setting. The online bootstrap is a flexible and efficient approach for statistical inference in linear stochastic approximation algorithms, but its efficacy in settings involving Markov noise, such as RL, has yet to be explored. In this paper, we study the use of the online bootstrap method for statistical inference in RL. In particular, we focus on the temporal difference (TD) learning and Gradient TD (GTD) learning algorithms, which are themselves special instances of linear stochastic approximation under Markov noise. The method is shown to be distributionally consistent for statistical inference in policy evaluation, and numerical experiments are included to demonstrate the effectiveness of this algorithm at statistical inference tasks across a range of real RL environments.
General Value Function (GVF) is a powerful tool to represent both the {em predictive} and {em retrospective} knowledge in reinforcement learning (RL). In practice, often multiple interrelated GVFs need to be evaluated jointly with pre-collected off-policy samples. In the literature, the gradient temporal difference (GTD) learning method has been adopted to evaluate GVFs in the off-policy setting, but such an approach may suffer from a large estimation error even if the function approximation class is sufficiently expressive. Moreover, none of the previous work have formally established the convergence guarantee to the ground truth GVFs under the function approximation settings. In this paper, we address both issues through the lens of a class of GVFs with causal filtering, which cover a wide range of RL applications such as reward variance, value gradient, cost in anomaly detection, stationary distribution gradient, etc. We propose a new algorithm called GenTD for off-policy GVFs evaluation and show that GenTD learns multiple interrelated multi-dimensional GVFs as efficiently as a single canonical scalar value function. We further show that unlike GTD, the learned GVFs by GenTD are guaranteed to converge to the ground truth GVFs as long as the function approximation power is sufficiently large. To our best knowledge, GenTD is the first off-policy GVF evaluation algorithm that has global optimality guarantee.
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