Q-learning with Uniformly Bounded Variance: Large Discounting is Not a Barrier to Fast Learning
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It has been a trend in the Reinforcement Learning literature to derive sample complexity bounds: a bound on how many experiences with the environment are required to obtain an ε-optimal policy. In the discounted cost, infinite horizon setting, all of the known bounds have a factor that is a polynomial in 1/(1-β), where β < 1 is the discount factor. For a large discount factor, these bounds seem to imply that a very large number of samples is required to achieve an ε-optimal policy. The objective of the present work is to introduce a new class of algorithms that have sample complexity uniformly bounded for all β < 1. One may argue that this is impossible, due to a recent min-max lower bound. The explanation is that this previous lower bound is for a specific problem, which we modify, without compromising the ultimate objective of obtaining an ε-optimal policy. Specifically, we show that the asymptotic variance of the Q-learning algorithm, with an optimized step-size sequence, is a quadratic function of 1/(1-β); an expected, and essentially known result. The new relative Q-learning algorithm proposed here is shown to have asymptotic variance that is a quadratic in 1/(1- ρβ), where 1 - ρ > 0 is the spectral gap of an optimal transition matrix.
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