Log-Sum-Exponential Estimator for Off-Policy Evaluation and Learning

Off-policy learning and evaluation leverage logged bandit feedback datasets, which contain context, action, propensity score, and feedback for each data point. These scenarios face significant challenges due to high variance and poor performance with low-quality propensity scores and heavy-tailed reward distributions. We address these issues by introducing a novel estimator based on the log-sum-exponential (LSE) operator, which outperforms traditional inverse propensity score estimators. Our LSE estimator demonstrates variance reduction and robustness under heavy-tailed conditions. For off-policy evaluation, we derive upper bounds on the estimator's bias and variance. In the off-policy learning scenario, we establish bounds on the regret -- the performance gap between our LSE estimator and the optimal policy -- assuming bounded $(1+\epsilon)$-th moment of weighted reward. Notably, we achieve a convergence rate of $O(n^{-\epsilon/(1+ \epsilon)})$ for the regret bounds, where $\epsilon \in [0,1]$ and $n$ is the size of logged bandit feedback dataset. Theoretical analysis is complemented by comprehensive empirical evaluations in both off-policy learning and evaluation scenarios, confirming the practical advantages of our approach. The code for our estimator is available at the following link: https://github.com/armin-behnamnia/lse-offpolicy-learning.

Demystifying the Paradox of Importance Sampling with an Estimated History-Dependent Behavior Policy in Off-Policy Evaluation

This paper studies off-policy evaluation (OPE) in reinforcement learning with a focus on behavior policy estimation for importance sampling. Prior work has shown empirically that estimating a history-dependent behavior policy can lead to lower mean squared error (MSE) even when the true behavior policy is Markovian. However, the question of why the use of history should lower MSE remains open. In this paper, we theoretically demystify this paradox by deriving a bias-variance decomposition of the MSE of ordinary importance sampling (IS) estimators, demonstrating that history-dependent behavior policy estimation decreases their asymptotic variances while increasing their finite-sample biases. Additionally, as the estimated behavior policy conditions on a longer history, we show a consistent decrease in variance. We extend these findings to a range of other OPE estimators, including the sequential IS estimator, the doubly robust estimator and the marginalized IS estimator, with the behavior policy estimated either parametrically or non-parametrically.

Semi-pessimistic Reinforcement Learning

Offline reinforcement learning (RL) aims to learn an optimal policy from pre-collected data. However, it faces challenges of distributional shift, where the learned policy may encounter unseen scenarios not covered in the offline data. Additionally, numerous applications suffer from a scarcity of labeled reward data. Relying on labeled data alone often leads to a narrow state-action distribution, further amplifying the distributional shift, and resulting in suboptimal policy learning. To address these issues, we first recognize that the volume of unlabeled data is typically substantially larger than that of labeled data. We then propose a semi-pessimistic RL method to effectively leverage abundant unlabeled data. Our approach offers several advantages. It considerably simplifies the learning process, as it seeks a lower bound of the reward function, rather than that of the Q-function or state transition function. It is highly flexible, and can be integrated with a range of model-free and model-based RL algorithms. It enjoys the guaranteed improvement when utilizing vast unlabeled data, but requires much less restrictive conditions. We compare our method with a number of alternative solutions, both analytically and numerically, and demonstrate its clear competitiveness. We further illustrate with an application to adaptive deep brain stimulation for Parkinson's disease.

Deep Distributional Learning with Non-crossing Quantile Network

In this paper, we introduce a non-crossing quantile (NQ) network for conditional distribution learning. By leveraging non-negative activation functions, the NQ network ensures that the learned distributions remain monotonic, effectively addressing the issue of quantile crossing. Furthermore, the NQ network-based deep distributional learning framework is highly adaptable, applicable to a wide range of applications, from classical non-parametric quantile regression to more advanced tasks such as causal effect estimation and distributional reinforcement learning (RL). We also develop a comprehensive theoretical foundation for the deep NQ estimator and its application to distributional RL, providing an in-depth analysis that demonstrates its effectiveness across these domains. Our experimental results further highlight the robustness and versatility of the NQ network.

Statistical Inference in Reinforcement Learning: A Selective Survey

Reinforcement learning (RL) is concerned with how intelligence agents take actions in a given environment to maximize the cumulative reward they receive. In healthcare, applying RL algorithms could assist patients in improving their health status. In ride-sharing platforms, applying RL algorithms could increase drivers' income and customer satisfaction. Over the past decade, RL has been arguably one of the most vibrant research frontiers in machine learning. Nevertheless, statistics as a field, as opposed to computer science, has only recently begun to engage with RL both in depth and in breadth. This paper present a selective review of statistical inferential tools for RL, covering both hypothesis testing and confidence interval construction. Our goal is to highlight the value of statistical inference in RL for both the statistics and machine learning communities, and to promote the broader application of classical statistical inference tools in this vibrant area of research.

Counterfactually Fair Reinforcement Learning via Sequential Data Preprocessing

When applied in healthcare, reinforcement learning (RL) seeks to dynamically match the right interventions to subjects to maximize population benefit. However, the learned policy may disproportionately allocate efficacious actions to one subpopulation, creating or exacerbating disparities in other socioeconomically-disadvantaged subgroups. These biases tend to occur in multi-stage decision making and can be self-perpetuating, which if unaccounted for could cause serious unintended consequences that limit access to care or treatment benefit. Counterfactual fairness (CF) offers a promising statistical tool grounded in causal inference to formulate and study fairness. In this paper, we propose a general framework for fair sequential decision making. We theoretically characterize the optimal CF policy and prove its stationarity, which greatly simplifies the search for optimal CF policies by leveraging existing RL algorithms. The theory also motivates a sequential data preprocessing algorithm to achieve CF decision making under an additive noise assumption. We prove and then validate our policy learning approach in controlling unfairness and attaining optimal value through simulations. Analysis of a digital health dataset designed to reduce opioid misuse shows that our proposal greatly enhances fair access to counseling.

Two-way Deconfounder for Off-policy Evaluation in Causal Reinforcement Learning

This paper studies off-policy evaluation (OPE) in the presence of unmeasured confounders. Inspired by the two-way fixed effects regression model widely used in the panel data literature, we propose a two-way unmeasured confounding assumption to model the system dynamics in causal reinforcement learning and develop a two-way deconfounder algorithm that devises a neural tensor network to simultaneously learn both the unmeasured confounders and the system dynamics, based on which a model-based estimator can be constructed for consistent policy value estimation. We illustrate the effectiveness of the proposed estimator through theoretical results and numerical experiments.

Dual Active Learning for Reinforcement Learning from Human Feedback

Aligning large language models (LLMs) with human preferences is critical to recent advances in generative artificial intelligence. Reinforcement learning from human feedback (RLHF) is widely applied to achieve this objective. A key step in RLHF is to learn the reward function from human feedback. However, human feedback is costly and time-consuming, making it essential to collect high-quality conversation data for human teachers to label. Additionally, different human teachers have different levels of expertise. It is thus critical to query the most appropriate teacher for their opinions. In this paper, we use offline reinforcement learning (RL) to formulate the alignment problem. Motivated by the idea of $D$-optimal design, we first propose a dual active reward learning algorithm for the simultaneous selection of conversations and teachers. Next, we apply pessimistic RL to solve the alignment problem, based on the learned reward estimator. Theoretically, we show that the reward estimator obtained through our proposed adaptive selection strategy achieves minimal generalized variance asymptotically, and prove that the sub-optimality of our pessimistic policy scales as $O(1/\sqrt{T})$ with a given sample budget $T$. Through simulations and experiments on LLMs, we demonstrate the effectiveness of our algorithm and its superiority over state-of-the-arts.

Causal Deepsets for Off-policy Evaluation under Spatial or Spatio-temporal Interferences

Off-policy evaluation (OPE) is widely applied in sectors such as pharmaceuticals and e-commerce to evaluate the efficacy of novel products or policies from offline datasets. This paper introduces a causal deepset framework that relaxes several key structural assumptions, primarily the mean-field assumption, prevalent in existing OPE methodologies that handle spatio-temporal interference. These traditional assumptions frequently prove inadequate in real-world settings, thereby restricting the capability of current OPE methods to effectively address complex interference effects. In response, we advocate for the implementation of the permutation invariance (PI) assumption. This innovative approach enables the data-driven, adaptive learning of the mean-field function, offering a more flexible estimation method beyond conventional averaging. Furthermore, we present novel algorithms that incorporate the PI assumption into OPE and thoroughly examine their theoretical foundations. Our numerical analyses demonstrate that this novel approach yields significantly more precise estimations than existing baseline algorithms, thereby substantially improving the practical applicability and effectiveness of OPE methodologies. A Python implementation of our proposed method is available at https://github.com/BIG-S2/Causal-Deepsets.

Forward and Backward State Abstractions for Off-policy Evaluation

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.