Approximate linear programming (ALP) represents one of the major algorithmic families to solve large-scale Markov decision processes (MDP). In this work, we study a primal-dual formulation of the ALP, and develop a scalable, model-free algorithm called bilinear $\pi$ learning for reinforcement learning when a sampling oracle is provided. This algorithm enjoys a number of advantages. First, it adopts (bi)linear models to represent the high-dimensional value function and state-action distributions, using given state and action features. Its run-time complexity depends on the number of features, not the size of the underlying MDPs. Second, it operates in a fully online fashion without having to store any sample, thus having minimal memory footprint. Third, we prove that it is sample-efficient, solving for the optimal policy to high precision with a sample complexity linear in the dimension of the parameter space.
In many practical uses of reinforcement learning (RL) the set of actions available at a given state is a random variable, with realizations governed by an exogenous stochastic process. Somewhat surprisingly, the foundations for such sequential decision processes have been unaddressed. In this work, we formalize and investigate MDPs with stochastic action sets (SAS-MDPs) to provide these foundations. We show that optimal policies and value functions in this model have a structure that admits a compact representation. From an RL perspective, we show that Q-learning with sampled action sets is sound. In model-based settings, we consider two important special cases: when individual actions are available with independent probabilities; and a sampling-based model for unknown distributions. We develop poly-time value and policy iteration methods for both cases; and in the first, we offer a poly-time linear programming solution.
We consider off-policy evaluation and optimization with continuous action spaces. We focus on observational data where the data collection policy is unknown and needs to be estimated from data. We take a semi-parametric approach where the value function takes a known parametric form in the treatment, but we are agnostic on how it depends on the observed contexts. We propose a doubly robust off-policy estimate for this setting and show that off-policy optimization based on this doubly robust estimate is robust to estimation errors of the policy function or the regression model. We also show that the variance of our off-policy estimate achieves the semi-parametric efficiency bound.
Policy optimization methods have shown great promise in solving complex reinforcement and imitation learning tasks. While model-free methods are broadly applicable, they often require many samples to optimize complex policies. Model-based methods greatly improve sample-efficiency but at the cost of poor generalization, requiring a carefully handcrafted model of the system dynamics for each task. Recently, hybrid methods have been successful in trading off applicability for improved sample-complexity. However, these have been limited to continuous action spaces. In this work, we present a new hybrid method based on an approximation of the dynamics as an expectation over the next state under the current policy. This relaxation allows us to derive a novel hybrid policy gradient estimator, combining score function and pathwise derivative estimators, that is applicable to discrete action spaces. We show significant gains in sample complexity, ranging between 1.7 and 25 times, when learning parameterized policies on Cart Pole, Acrobot, Mountain Car and Hand Mass. Our method is applicable to both discrete and continuous action spaces, when competing pathwise methods are limited to the latter.