Entropy Regularized Reinforcement Learning

Inverse reinforcement learning (IRL) learns a reward function and a corresponding policy that best fit the demonstration data of an expert. However, in the current IRL setting, the learner is isolated from the expert and can only passively observe the expert demonstrations. This limits the applicability of IRL to interactive settings, where the learner actively interacts with the expert and needs to infer the expert's reward function from the interactions. To bridge the gap, this paper studies interactive IRL (IIRL) where a learner aims to learn the reward function of an expert and a policy to interact with the expert during its interactions with the expert. We formulate IIRL as a stochastic bi-level optimization problem where the lower level learns a reward function to explain the behaviors of the expert, and the upper level learns a policy to interact with the expert. We develop a double-loop algorithm, Bi-level Interactive Scenarios Inverse Reinforcement Learning (BISIRL), which solves the lower-level problem in the inner loop and the upper-level problem in the outer loop. We formally guarantee that BISIRL converges and validate our algorithm through extensive experiments.

Exponential families encompass the distributions central to modern machine learning -- softmax, Gaussians, and Boltzmann distributions -- and underlie the theory of variational inference, entropy-regularized reinforcement learning, and RLHF. We isolate a simple identity for exponential families that expresses the KL difference $\mathrm{KL}(q \| p_{λ_2}) - \mathrm{KL}(q \| p_{λ_1})$ in terms of the log-partition function $A(λ)$ and the moment $μ_q$. Remarkably, this identity together with the single fact that $\mathrm{KL} \geq 0$ (with equality iff $p = q$) suffices, by direct substitution and rearrangement, to derive a cluster of results that are classically obtained by separate, heavier arguments: a generalized three-point identity for arbitrary reference distributions, Pythagorean theorems for I-projections and reverse I-projections, convexity of the log-partition function, identification of its Legendre dual in KL terms, the Gibbs variational principle, and the explicit optimizer in KL-regularized reward maximization, including the exponential tilting formula underlying entropy-regularized control and RLHF. Beyond these purely algebraic consequences, standard analytic arguments recover the gradient formula for the log-partition function, the Bregman representation of within-family KL divergence, and the surjectivity of the moment map. The note is self-contained.

Model-Based Reinforcement Learning distinguishes between physical dynamics models operating on proprioceptive inputs and latent dynamics models operating on high-dimensional image observations. A prominent latent approach is the Recurrent State Space Model used in the Dreamer family. While epistemic uncertainty quantification to inform exploration and mitigate model exploitation is well established for physical dynamics models, its transfer to latent dynamics models has received limited scrutiny. We empirically demonstrate that latent transitions are biased toward well-represented regions of latent space, exhibiting an attractor behavior that can deviate from true environment dynamics. As a result, discrepancies in environment dynamics may not manifest in latent space, undermining the reliability of epistemic uncertainty estimates. Because these attractors often lie in high-reward regions, latent rollouts systematically overestimate predicted rewards. Our findings highlight key limitations of epistemic uncertainty estimation in latent dynamics models and motivate more critical evaluation of this method.

We present a framework using Relative Entropy Inverse Reinforcement Learning (RE-IRL) to recover investor reward functions from observed investment actions and market conditions. Unlike traditional IRL algorithms, RE-IRL is employed to account for environments where transition probabilities are unknown or inaccessible. To address the challenge of data sparsity, we utilize a $K$-nearest neighbor approach to estimate the observed behavior policy. Furthermore, we propose a statistical testing framework to evaluate the validity and robustness of the estimated results.

A popular approach to apprenticeship learning (AL) is to formulate it as an inverse reinforcement learning (IRL) problem. The MaxEnt-IRL algorithm successfully integrates the maximum entropy principle into IRL and unlike its predecessors, it resolves the ambiguity arising from the fact that a possibly large number of policies could match the expert's behavior. In this paper, we study an AL setting in which in addition to the expert's trajectories, a number of unsupervised trajectories is available. We introduce MESSI, a novel algorithm that combines MaxEnt-IRL with principles coming from semi-supervised learning. In particular, MESSI integrates the unsupervised data into the MaxEnt-IRL framework using a pairwise penalty on trajectories. Empirical results in a highway driving and grid-world problems indicate that MESSI is able to take advantage of the unsupervised trajectories and improve the performance of MaxEnt-IRL.

We propose SmoothCruiser, a new planning algorithm for estimating the value function in entropy-regularized Markov decision processes and two-player games, given a generative model of the environment. SmoothCruiser makes use of the smoothness of the Bellman operator promoted by the regularization to achieve problem-independent sample complexity of order O~(1/epsilon^4) for a desired accuracy epsilon, whereas for non-regularized settings there are no known algorithms with guaranteed polynomial sample complexity in the worst case.

Embodiment is a significant keyword in recent machine learning fields. This study focused on the passive nature of the body of a biped robot to generate walking and running locomotion using model-based deep reinforcement learning. We constructed two models in a simulator, one with passive elements (e.g., springs) and the other, which is similar to general humanoids, without passive elements. The training of the model with passive elements was highly affected by the attractor of the system. This lead that although the trajectories quickly converged to limit cycles, it took a long time to obtain large rewards. However, thanks to the attractor-driven learning, the acquired locomotion was robust and energy-efficient. The results revealed that robots with passive elements could efficiently acquire high-performance locomotion by utilizing stable limit cycles generated through dynamic interaction between the body and ground. This study demonstrates the importance of implementing passive properties in the body for future embodied AI.

We analyze the Bayesian regret of the Gaussian process posterior sampling reinforcement learning (GP-PSRL) algorithm. Posterior sampling is a heuristic for decision-making under uncertainty that has been used to develop successful algorithms for a variety of continuous control problems. However, theoretical work on GP-PSRL is limited. All known regret bounds either have a sub-optimal growth rate, require strong smoothness assumptions, or fail to properly account for the fact that the set of possible system states is unbounded. Through a recursive application of the Borell-Tsirelson-Ibragimov-Sudakov inequality, we show that, with high probability, the states actually visited by the algorithm are contained within a ball of near-constant radius. We then use the chaining method to control the regret suffered by GP-PSRL under weak smoothness conditions. Our main result is a Bayesian regret bound of the order $\widetilde{\mathcal{O}}(H\sqrt{γ_TT})$, where $H$ is the horizon, $T$ is the number of time steps and $γ_T$ is the expected information gain. With this result, we resolve the limitations with prior theoretical work on PSRL, and provide the theoretical foundation and tools for analyzing PSRL in complex settings.

We consider the policy evaluation and control in a finite horizon reinforcement learning (RL) setting under an expectile-based objective. First, we derive the mean-squared error (MSE) and concentration bounds for the classic estimator of expectiles based on independent and identically distributed (i.i.d.) samples. To the best of our knowledge, expectiles have not been analyzed in the non-asymptotic regime and the bounds we derive may be of independent interest. Next, we analyze a Monte Carlo type estimator of expectile of the Markov chain underlying a given policy. We derive upper bounds that hold in expectation as well as with high probability for this estimator. Further, we show the order-optimality of our estimator by deriving a lower bound on expectile-based policy evaluation. For the problem of control, we adopt a policy gradient approach and derive a policy gradient theorem for expectiles. Using this result, we propose a gradient estimator with a $O\left(1/m\right)$ mean-squared error bounds, where $m$ is the number of trajectories. Further, under standard assumptions for policy gradient-type algorithms, we establish smoothness of the expectile-sensitive objective, in turn leading to stationary convergence rate bounds for the overall risk-sensitive policy gradient algorithm that we propose. Finally, we conduct numerical experiments to show the utility of expectiles on popular RL benchmarks.

Process rewards have been widely used in deep reinforcement learning to improve training efficiency, reduce variance, and prevent reward hacking. In LLM reasoning, existing works also explore various solutions for learning effective process reward models (PRM) with or without the help of an expert policy. However, existing methods either rely on strong assumptions about the expert policies (e.g., requiring their reward functions) or suffer intrinsic limitations (e.g., entropy collapse), resulting in weak PRMs or limited generalizability. In this paper, we introduce rePIRL, an inverse RL-inspired framework that learns effective PRMs with minimal assumptions about expert policies. Specifically, we design a dual learning process that updates the policy and the PRM interchangeably. Our learning algorithm has customized techniques to address the challenges of scaling traditional inverse RL to LLMs. We theoretically show that our proposed learning framework can unify both online and offline PRM learning methods, justifying that rePIRL can learn PRMs with minimal assumptions. Empirical evaluations on standardized math and coding reasoning datasets demonstrate the effectiveness of rePIRL over existing methods. We further show the application of our trained PRM in test-time training, test-time scaling, and providing an early signal for training hard problems. Finally, we validate our training recipe and key design choices via a detailed ablation study.

Inverse reinforcement learning aims to infer the reward function that explains expert behavior observed through trajectories of state--action pairs. A long-standing difficulty in classical IRL is the non-uniqueness of the recovered reward: many reward functions can induce the same optimal policy, rendering the inverse problem ill-posed. In this paper, we develop a statistical framework for Inverse Entropy-regularized Reinforcement Learning that resolves this ambiguity by combining entropy regularization with a least-squares reconstruction of the reward from the soft Bellman residual. This combination yields a unique and well-defined so-called least-squares reward consistent with the expert policy. We model the expert demonstrations as a Markov chain with the invariant distribution defined by an unknown expert policy $π^\star$ and estimate the policy by a penalized maximum-likelihood procedure over a class of conditional distributions on the action space. We establish high-probability bounds for the excess Kullback--Leibler divergence between the estimated policy and the expert policy, accounting for statistical complexity through covering numbers of the policy class. These results lead to non-asymptotic minimax optimal convergence rates for the least-squares reward function, revealing the interplay between smoothing (entropy regularization), model complexity, and sample size. Our analysis bridges the gap between behavior cloning, inverse reinforcement learning, and modern statistical learning theory.

Offline RL (ORL) promises safe and sample-efficient deployment but existing methods rely on undocumented online interactions for hyperparameter tuning and lack reliable fully offline estimates of initial online performance. We introduce SOReL, a fully offline Bayesian model-based RL method that learns a posterior over dynamics, estimates policy value via predictive uncertainty, and enables complete offline hyperparameter selection. We further propose TOReL, which extends this tuning framework to arbitrary model-free and model-based ORL algorithms. We provide a regret analysis showing that Bayesian offline RL achieves the minimax-optimal parametric rate under standard regularity conditions. Together, our methods establish a practical and theoretically grounded framework for fully offline RL.

Reinforcement learning agents depend on reward signals whose density is rarely under the designer's control, and when such signals are absent, an agent must generate its own drive to explore. State entropy maximization offers a principled objective for this, but existing methods break down at scale in two ways: the intrinsic reward vanishes once a state has been visited, discouraging revisits to the very gateways that lead onward, and estimating entropy over millions of accumulated observations becomes computationally prohibitive. We address both with Episodic and Lifelong Exploration via Maximum Entropy (ELEMENT), a multiscale intrinsically motivated framework for reward-free exploration that transfers to downstream tasks. ELEMENT couples lifelong entropy maximization with a complementary episodic term acting on a faster timescale. For the episodic term, we derive average episodic state entropy, an intrinsic reward that is the exact minimizer of a tractable upper bound on the reward-decomposition objective; for the lifelong term, we propose a $k$NN graph-based estimator that keeps entropy tractable without forgetting. ELEMENT consistently outperforms state-of-the-art intrinsic reward baselines on state coverage and unsupervised pre-training. Videos, code, and supplementary material: https://sites.google.com/view/element-rl.

Identifying the most suitable variables to represent the state is a fundamental challenge in Reinforcement Learning (RL). These variables must efficiently capture the information necessary for making optimal decisions. In order to address this problem, in this paper, we introduce the Transfer Entropy Redundancy Criterion (TERC), an information-theoretic criterion, which determines if there is entropy transferred from observable state variables to actions during training. We define an algorithm based on TERC that provably excludes variables from the observable state that do not affect the agent's policy during learning. This yields compact state representations that reduce inference time by up to 2.6 times. Our approach is policy-dependent, making it agnostic to the underlying learning algorithm. The efficiency gains we demonstrate arise at retraining and inference time on the reduced state. Our method improves both retraining and inference efficiency. We demonstrate its effectiveness across three distinct algorithm classes, namely tabular Q-learning, Actor-Critic, and Proximal Policy Optimization (PPO), evaluated in a range of environments. Furthermore, to highlight the differences between the proposed methodology and the current state-of-the-art feature selection approaches, we present a series of controlled experiments on synthetic data, before generalizing to real-world decision-making tasks. We also introduce a representation of the problem that compactly captures the transfer of information from observable state variables to actions as Bayesian networks.

Regularization is a cornerstone of modern reinforcement learning. Regularized policy iteration (RPI) provides a fundamental scheme for solving regularized Markov decision processes (RMDPs), and the widely used soft actor-critic algorithm arises as a special case when the regularizer is Shannon entropy. Despite its empirical success, the theoretical underpinnings of RPI remain unclear. In this paper, we address this gap by proving that RPI is formally equivalent to the standard Newton-Raphson method applied to the Bellman equation smoothed by strongly convex regularizers. This equivalence enables a unified convergence analysis of existing methods and supports the development of accelerated algorithms. We show that RPI enjoys local quadratic convergence; notably, for Shannon entropy, the guarantee is dimension-free. We further study RPI with inexact policy evaluation, establishing its equivalence to an inexact Newton method in which each Newton step is solved via truncated iterations, and derive an asymptotic linear convergence rate of $γ^{M}$, where $M$ denotes the number of operator steps used in policy evaluation. Finally, motivated by higher-order Newton schemes, we propose a new algorithm for RMDPs that achieves third-order local convergence. Numerical experiments corroborate our theory and demonstrate the practical advantages of the proposed algorithm. Overall, our results advance the theoretical understanding of regularization in reinforcement learning and suggest new directions for efficient algorithm design.

We analyze a neural semi-discrete method for high-dimensional first-order Hamilton-Jacobi-Bellman (HJB) equations with known or learned dynamics. Centered differences and an artificial viscosity $Nh=O(h)$ define a monotone operator evaluated through $2d+1$ shifted network queries; policy iteration solves the resulting Bellman equation without a tensor grid. At fixed $h$, the sharp componentwise condition $\max_i|f_i|\le2N$ turns every frozen-policy operator into a nearest-neighbor Markov-chain generator with a policy-independent total jump rate. Uniformization gives whole-space well-posedness for measurable feedbacks, an explicit Poisson-tail bound on the numerical domain of dependence, and boundary-free localization. The representation also yields a posteriori policy-evaluation bounds that account for residual and learned-model errors. A greedy-gap analysis controls inexact policy iteration at fixed $h$; a separate consistency estimate connects the semi-discrete equation to the continuous HJB equation. Experiments reproduce the extremal tail, show rates consistent with $O(\sqrt h)$ and nearly $h$-independent exact-policy-iteration decay, and assess empirical estimator effectivity. A nonsmooth example shows that the continuous residual can miss a non-viscosity solution, whereas the shifted residual detects the defect. Further tests provide a structured interval-verified certificate calibration, an early-budget benefit of policy freezing for bang-bang control, and learned-dynamics diagnostics. A structured nonlinear problem with active compact-control constraints is tested against a manufactured semi-discrete reference through $d=1024$.