Many state-of-the-art algorithms for solving Partially Observable Markov Decision Processes (POMDPs) rely on turning the problem into a "fully observable" problem--a belief MDP--and exploiting the piece-wise linearity and convexity (PWLC) of the optimal value function in this new state space (the belief simplex). This approach has been extended to solving ρ-POMDPs--i.e., for information-oriented criteria--when the reward ρ is convex in . General ρ-POMDPs can also be turned into "fully observable" problems, but with no means to exploit the PWLC property. In this paper, we focus on POMDPs and ρ-POMDPs with λ ρ -Lipschitz reward function, and demonstrate that, for finite horizons, the optimal value function is Lipschitz-continuous. Then, value function approximators are proposed for both upper-and lower-bounding the optimal value function, which are shown to provide uniformly improvable bounds. This allows proposing two algorithms derived from HSVI which are empirically evaluated on various benchmark problems.

Autonomous agents operating in real-world scenarios frequently encounter uncertainty and make decisions based on incomplete information.

While a number of RL methods have been proposed to boost exploration by designing an intrinsic reward signal as exploration bonus.

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However, Asynchronous VI still requires a maximization over the entire action space, making it impractical for domains with large action space.

Temporal-difference learning is a popular algorithm for policy evaluation. In this paper, we study the convergence of the regularized non-parametric TD(0) algorithm, in both the independent and Markovian observation settings. In particular, when TD is performed in a universal reproducing kernel Hilbert space (RKHS), we prove convergence of the averaged iterates to the optimal value function, even when it does not belong to the RKHS. We provide explicit convergence rates that depend on a source condition relating the regularity of the optimal value function to the RKHS. We illustrate this convergence numerically on a simple continuous-state Markov reward process.

Since the 1990s, considerable empirical work has been carried out to train statistical models, such as neural networks (NNs), as learned heuristics for combinatorial optimization (CO) problems. When successful, such an approach eliminates the need for experts to design heuristics per problem type. Due to their structure, many hard CO problems are amenable to treatment through reinforcement learning (RL). Indeed, we find a wealth of literature training NNs using value-based, policy gradient, or actor-critic approaches, with promising results, both in terms of empirical optimality gaps and inference runtimes. Nevertheless, there has been a paucity of theoretical work undergirding the use of RL for CO problems. To this end, we introduce a unified framework to model CO problems through Markov decision processes (MDPs) and solve them using RL techniques. We provide easy-to-test assumptions under which CO problems can be formulated as equivalent undiscounted MDPs that provide optimal solutions to the original CO problems. Moreover, we establish conditions under which value-based RL techniques converge to approximate solutions of the CO problem with a guarantee on the associated optimality gap. Our convergence analysis provides: (1) a sufficient rate of increase in batch size and projected gradient descent steps at each RL iteration; (2) the resulting optimality gap in terms of problem parameters and targeted RL accuracy; and (3) the importance of a choice of state-space embedding. Together, our analysis illuminates the success (and limitations) of the celebrated deep Q-learning algorithm in this problem context.

This paper studies the problem of risk-sensitive reinforcement learning (RSRL) in continuous time, where the environment is characterized by a controllable stochastic differential equation (SDE) and the objective is a potentially nonlinear functional of cumulative rewards. We prove that when the functional is an optimized certainty equivalent (OCE), the optimal policy is Markovian with respect to an augmented environment. We also propose \textit{CT-RS-q}, a risk-sensitive q-learning algorithm based on a novel martingale characterization approach. Finally, we run a simulation study on a dynamic portfolio selection problem and illustrate the effectiveness of our algorithm.