Is Long Horizon Reinforcement Learning More Difficult Than Short Horizon Reinforcement Learning?

TL;DR

This paper demonstrates that, contrary to previous beliefs, long horizon reinforcement learning in tabular settings can be as sample-efficient as short horizon learning when normalized appropriately, with complexity only logarithmic in the horizon.

Contribution

The work refutes the conjecture that sample complexity must polynomially depend on the horizon, showing it can be logarithmic, and introduces new techniques for policy class analysis and evaluation.

Findings

Sample complexity scales logarithmically with the horizon.

Introduces an $oldsymbol{ ext{ε}}$-net for optimal policies with logarithmic size.

Proposes the Online Trajectory Synthesis algorithm for adaptive policy evaluation.

Abstract

Learning to plan for long horizons is a central challenge in episodic reinforcement learning problems. A fundamental question is to understand how the difficulty of the problem scales as the horizon increases. Here the natural measure of sample complexity is a normalized one: we are interested in the number of episodes it takes to provably discover a policy whose value is $\varepsilon$ near to that of the optimal value, where the value is measured by the normalized cumulative reward in each episode. In a COLT 2018 open problem, Jiang and Agarwal conjectured that, for tabular, episodic reinforcement learning problems, there exists a sample complexity lower bound which exhibits a polynomial dependence on the horizon -- a conjecture which is consistent with all known sample complexity upper bounds. This work refutes this conjecture, proving that tabular, episodic reinforcement learning is possible with a sample complexity that scales only logarithmically with the planning horizon. In other words, when the values are appropriately normalized (to lie in the unit interval), this results shows that long horizon RL is no more difficult than short horizon RL, at least in a minimax sense. Our analysis introduces two ideas: (i) the construction of an $\varepsilon$-net for optimal policies whose log-covering number scales only logarithmically with the planning horizon, and (ii) the Online Trajectory Synthesis algorithm, which adaptively evaluates all policies in a given policy class using sample complexity that scales with the log-covering number of the given policy class. Both may be of independent interest.