Variance Reduction for Score Functions Using Optimal Baselines

TL;DR

This paper derives the optimal state-dependent baseline for variance reduction in score function gradient estimators, providing theoretical insights into their effectiveness in reinforcement learning.

Contribution

It introduces the first expression for the optimal baseline that minimizes variance in score function estimators, and compares it with value function baselines.

Findings

Optimal baseline can significantly outperform value function baseline in some cases.

Value function baseline generally performs similarly to the optimal baseline in variance reduction.

Using the value function for bootstrapping further reduces variance.

Abstract

Many problems involve the use of models which learn probability distributions or incorporate randomness in some way. In such problems, because computing the true expected gradient may be intractable, a gradient estimator is used to update the model parameters. When the model parameters directly affect a probability distribution, the gradient estimator will involve score function terms. This paper studies baselines, a variance reduction technique for score functions. Motivated primarily by reinforcement learning, we derive for the first time an expression for the optimal state-dependent baseline, the baseline which results in a gradient estimator with minimum variance. Although we show that there exist examples where the optimal baseline may be arbitrarily better than a value function baseline, we find that the value function baseline usually performs similarly to an optimal baseline in terms of variance reduction. Moreover, the value function can also be used for bootstrapping estimators of the return, leading to additional variance reduction. Our results give new insight and justification for why value function baselines and the generalized advantage estimator (GAE) work well in practice.