Convergence rate analysis for general state-space Markov chains is fundamentally important in areas such as Markov chain Monte Carlo and algorithmic analysis (for computing explicit convergence bounds). This problem, however, is notoriously difficult because traditional analytical methods often do not generate practically useful convergence bounds for realistic Markov chains. We propose the Deep Contractive Drift Calculator (DCDC), the first general-purpose sample-based algorithm for bounding the convergence of Markov chains to stationarity in Wasserstein distance. The DCDC has two components. First, inspired by the new convergence analysis framework in (Qu et al., 2023), we introduce the Contractive Drift Equation (CDE), the solution of which leads to an explicit convergence bound. Second, we develop an efficient neural-network-based CDE solver. Equipped with these two components, DCDC solves the CDE and converts the solution into a convergence bound. We analyze the sample complexity of the algorithm and further demonstrate the effectiveness of the DCDC by generating convergence bounds for realistic Markov chains arising from stochastic processing networks as well as constant step-size stochastic optimization.

We explore online learning in episodic loop-free Markov decision processes on non-stationary environments (changing losses and probability transitions). Our focus is on the Concave Utility Reinforcement Learning problem (CURL), an extension of classical RL for handling convex performance criteria in state-action distributions induced by agent policies. While various machine learning problems can be written as CURL, its non-linearity invalidates traditional Bellman equations. Despite recent solutions to classical CURL, none address non-stationary MDPs. This paper introduces MetaCURL, the first CURL algorithm for non-stationary MDPs. It employs a meta-algorithm running multiple black-box algorithms instances over different intervals, aggregating outputs via a sleeping expert framework. The key hurdle is partial information due to MDP uncertainty. Under partial information on the probability transitions (uncertainty and non-stationarity coming only from external noise, independent of agent state-action pairs), we achieve optimal dynamic regret without prior knowledge of MDP changes. Unlike approaches for RL, MetaCURL handles full adversarial losses, not just stochastic ones. We believe our approach for managing non-stationarity with experts can be of interest to the RL community.

We study reinforcement learning with multinomial logistic (MNL) function approximation where the underlying transition probability kernel of the Markov decision processes (MDPs) is parametrized by an unknown transition core with features of state and action. For the finite horizon episodic setting with inhomogeneous state transitions, we propose provably efficient algorithms with randomized exploration having frequentist regret guarantees. For our first algorithm, $\texttt{RRL-MNL}$, we adapt optimistic sampling to ensure the optimism of the estimated value function with sufficient frequency and establish that $\texttt{RRL-MNL}$ is both statistically and computationally efficient, achieving a $\tilde{O}(\kappa^{-1} d^{\frac{3}{2}} H^{\frac{3}{2}} \sqrt{T})$ frequentist regret bound with constant-time computational cost per episode. Here, $d$ is the dimension of the transition core, $H$ is the horizon length, $T$ is the total number of steps, and $\kappa$ is a problem-dependent constant. Despite the simplicity and practicality of $\texttt{RRL-MNL}$, its regret bound scales with $\kappa^{-1}$, which is potentially large in the worst case. To improve the dependence on $\kappa^{-1}$, we propose $\texttt{ORRL-MNL}$, which estimates the value function using local gradient information of the MNL transition model. We show that its frequentist regret bound is $\tilde{O}(d^{\frac{3}{2}} H^{\frac{3}{2}} \sqrt{T} + \kappa^{-1} d^2 H^2)$. To the best of our knowledge, these are the first randomized RL algorithms for the MNL transition model that achieve both computational and statistical efficiency. Numerical experiments demonstrate the superior performance of the proposed algorithms.

We present a new algorithm based on posterior sampling for learning in Constrained Markov Decision Processes (CMDP) in the infinite-horizon undiscounted setting. The algorithm achieves near-optimal regret bounds while being advantageous empirically compared to the existing algorithms. Our main theoretical result is a Bayesian regret bound for each cost component of $\tilde{O} (DS\sqrt{AT})$ for any communicating CMDP with $S$ states, $A$ actions, and diameter $D$. This regret bound matches the lower bound in order of time horizon $T$ and is the best-known regret bound for communicating CMDPs achieved by a computationally tractable algorithm. Empirical results show that our posterior sampling algorithm outperforms the existing algorithms for constrained reinforcement learning.

We prove bounds on the variance of a function $f$ under the empirical measure of the samples obtained by the Sequential Monte Carlo (SMC) algorithm, with time complexity depending on local rather than global Markov chain mixing dynamics. SMC is a Markov Chain Monte Carlo (MCMC) method, which starts by drawing $N$ particles from a known distribution, and then, through a sequence of distributions, re-weights and re-samples the particles, at each instance applying a Markov chain for smoothing. In principle, SMC tries to alleviate problems from multi-modality. However, most theoretical guarantees for SMC are obtained by assuming global mixing time bounds, which are only efficient in the uni-modal setting. We show that bounds can be obtained in the truly multi-modal setting, with mixing times that depend only on local MCMC dynamics.

Proteins are complex molecules responsible for different functions in nature. Enhancing the functionality of proteins and cellular fitness can significantly impact various industries. However, protein optimization using computational methods remains challenging, especially when starting from low-fitness sequences. We propose LatProtRL, an optimization method to efficiently traverse a latent space learned by an encoder-decoder leveraging a large protein language model. To escape local optima, our optimization is modeled as a Markov decision process using reinforcement learning acting directly in latent space. We evaluate our approach on two important fitness optimization tasks, demonstrating its ability to achieve comparable or superior fitness over baseline methods. Our findings and in vitro evaluation show that the generated sequences can reach high-fitness regions, suggesting a substantial potential of LatProtRL in lab-in-the-loop scenarios.

In this paper, we investigate the feature encoding process in a prototypical energy-based generative model, the Restricted Boltzmann Machine (RBM). We start with an analytical investigation using simplified architectures and data structures, and end with numerical analysis of real trainings on real datasets. Our study tracks the evolution of the model's weight matrix through its singular value decomposition, revealing a series of phase transitions associated to a progressive learning of the principal modes of the empirical probability distribution. The model first learns the center of mass of the modes and then progressively resolve all modes through a cascade of phase transitions. We first describe this process analytically in a controlled setup that allows us to study analytically the training dynamics. We then validate our theoretical results by training the Bernoulli-Bernoulli RBM on real data sets. By using data sets of increasing dimension, we show that learning indeed leads to sharp phase transitions in the high-dimensional limit. Moreover, we propose and test a mean-field finite-size scaling hypothesis. This shows that the first phase transition is in the same universality class of the one we studied analytically, and which is reminiscent of the mean-field paramagnetic-to-ferromagnetic phase transition.

Reinforcement learning is commonly concerned with problems of maximizing accumulated rewards in Markov decision processes. Oftentimes, a certain goal state or a subset of the state space attain maximal reward. In such a case, the environment may be considered solved when the goal is reached. Whereas numerous techniques, learning or non-learning based, exist for solving environments, doing so optimally is the biggest challenge. Say, one may choose a reward rate which penalizes the action effort. Reinforcement learning is currently among the most actively developed frameworks for solving environments optimally by virtue of maximizing accumulated reward, in other words, returns. Yet, tuning agents is a notoriously hard task as reported in a series of works. Our aim here is to help the agent learn a near-optimal policy efficiently while ensuring a goal reaching property of some basis policy that merely solves the environment. We suggest an algorithm, which is fairly flexible, and can be used to augment practically any agent as long as it comprises of a critic. A formal proof of a goal reaching property is provided. Simulation experiments on six problems under five agents, including the benchmarked one, provided an empirical evidence that the learning can indeed be boosted while ensuring goal reaching property.

We study non-episodic Reinforcement Learning for Lipschitz MDPs in which state-action space is a metric space, and the transition kernel and rewards are Lipschitz functions. We develop computationally efficient UCB-based algorithm, $\textit{ZoRL-}\epsilon$ that adaptively discretizes the state-action space and show that their regret as compared with $\epsilon$-optimal policy is bounded as $\mathcal{O}(\epsilon^{-(2 d_\mathcal{S} + d^\epsilon_z + 1)}\log{(T)})$, where $d^\epsilon_z$ is the $\epsilon$-zooming dimension. In contrast, if one uses the vanilla $\textit{UCRL-}2$ on a fixed discretization of the MDP, the regret w.r.t. a $\epsilon$-optimal policy scales as $\mathcal{O}(\epsilon^{-(2 d_\mathcal{S} + d + 1)}\log{(T)})$ so that the adaptivity gains are huge when $d^\epsilon_z \ll d$. Note that the absolute regret of any 'uniformly good' algorithm for a large family of continuous MDPs asymptotically scales as at least $\Omega(\log{(T)})$. Though adaptive discretization has been shown to yield $\mathcal{\tilde{O}}(H^{2.5}K^\frac{d_z + 1}{d_z + 2})$ regret in episodic RL, an attempt to extend this to the non-episodic case by employing constant duration episodes whose duration increases with $T$, is futile since $d_z \to d$ as $T \to \infty$. The current work shows how to obtain adaptivity gains for non-episodic RL. The theoretical results are supported by simulations on two systems where the performance of $\textit{ZoRL-}\epsilon$ is compared with that of '$\textit{UCRL-C}$,' the fixed discretization-based extension of $\textit{UCRL-}2$ for systems with continuous state-action spaces.

Acquiring a multi-task imitation policy in 3D manipulation poses challenges in terms of scene understanding and action prediction. Current methods employ both 3D representation and multi-view 2D representation to predict the poses of the robot's end-effector. However, they still require a considerable amount of high-quality robot trajectories, and suffer from limited generalization in unseen tasks and inefficient execution in long-horizon reasoning. In this paper, we propose SAM-E, a novel architecture for robot manipulation by leveraging a vision-foundation model for generalizable scene understanding and sequence imitation for long-term action reasoning. Specifically, we adopt Segment Anything (SAM) pre-trained on a huge number of images and promptable masks as the foundation model for extracting task-relevant features, and employ parameter-efficient fine-tuning on robot data for a better understanding of embodied scenarios. To address long-horizon reasoning, we develop a novel multi-channel heatmap that enables the prediction of the action sequence in a single pass, notably enhancing execution efficiency. Experimental results from various instruction-following tasks demonstrate that SAM-E achieves superior performance with higher execution efficiency compared to the baselines, and also significantly improves generalization in few-shot adaptation to new tasks.