In reinforcement learning, the classic objectives of maximizing discounted and finite-horizon cumulative rewards are PAC-learnable: There are algorithms that learn a near-optimal policy with high probability using a finite amount of samples and computation. In recent years, researchers have introduced objectives and corresponding reinforcement-learning algorithms beyond the classic cumulative rewards, such as objectives specified as linear temporal logic formulas. However, questions about the PAC-learnability of these new objectives have remained open.This work demonstrates the PAC-learnability of general reinforcement-learning objectives through sufficient conditions for PAC-learnability in two analysis settings. In particular, for the analysis that considers only sample complexity, we prove that if an objective given as an oracle is uniformly continuous, then it is PAC-learnable. Further, for the analysis that considers computational complexity, we prove that if an objective is computable, then it is PAC-learnable. In other words, if a procedure computes successive approximations of the objective’s value, then the objective is PAC-learnable.We give three applications of our condition on objectives from the literature with previously unknown PAC-learnability and prove that these objectives are PAC-learnable. Overall, our result helps verify existing objectives’ PAC-learnability. Also, as some studied objectives that are not uniformly continuous have been shown to be not PAC-learnable, our results could guide the design of new PAC-learnable objectives.

In recent years, researchers have made significant progress in devising reinforcement-learning algorithms for optimizing linear temporal logic (LTL) objectives and LTL-like objectives. Despite these advancements, there are fundamental limitations to how well this problem can be solved. Previous studies have alluded to this fact but have not examined it in depth. In this paper, we address the tractability of reinforcement learning for general LTL objectives from a theoretical perspective. We formalize the problem under the probably approximately correct learning in Markov decision processes (PAC-MDP) framework, a standard framework for measuring sample complexity in reinforcement learning. In this formalization, we prove that the optimal policy for any LTL formula is PAC-MDP-learnable if and only if the formula is in the most limited class in the LTL hierarchy, consisting of formulas that are decidable within a finite horizon. Practically, our result implies that it is impossible for a reinforcement-learning algorithm to obtain a PAC-MDP guarantee on the performance of its learned policy after finitely many interactions with an unconstrained environment for LTL objectives that are not decidable within a finite horizon.

A Reduction – an accumulation over a set of values, using an associative and commutative operator – is a common computation in many numerical computations, including scientific computations, machine learning, computer vision, and financial analytics. Contemporary polyhedral-based compilation techniques make it possible to optimize reductions, such as prefix sums, in which each component of the reduction’s output potentially shares computation with another component in the reduction. Therefore an optimizing compiler can identify the computation shared between multiple components and generate code that computes the shared computation only once. These techniques, however, do not support reductions that – when phrased in the language of the polyhedral model – span multiple dependent statements. In such cases, existing approaches can generate incorrect code that violates the data dependences of the original, unoptimized program. In this work, we identify and formalize the optimization of dependent reductions as an integer bilinear program. We present a heuristic optimization algorithm that uses an affine sequential schedule of the program to determine how to simplfy reductions yet still preserve the program’s dependences. We demonstrate that the algorithm provides optimal complexity for a set of benchmark programs from the literature on probabilistic inference algorithms, whose performance critically relies on simplifying these reductions. The complexities for 10 of the 11 programs improve siginifcantly by factors at least of the sizes of the input data, which are in the range of 104 to 106 for typical real application inputs. We also confirm the significance of the improvement by showing speedups in wall-clock time that range from 1.1x to over 106x.