Linear Convergence for Natural Policy Gradient with Log-linear Policy Parametrization

TL;DR

This paper proves that the natural policy gradient algorithm with log-linear policies converges linearly in both deterministic and sample-based settings for infinite-horizon discounted MDPs, extending previous results.

Contribution

It establishes linear convergence guarantees for natural policy gradient with log-linear policies, including in the presence of estimation and bias errors.

Findings

Linear convergence in deterministic case with known Q-value.

Linear convergence in sample-based case up to an error term.

Extension of previous softmax tabular results to log-linear policies.

Abstract

We analyze the convergence rate of the unregularized natural policy gradient algorithm with log-linear policy parametrizations in infinite-horizon discounted Markov decision processes. In the deterministic case, when the Q-value is known and can be approximated by a linear combination of a known feature function up to a bias error, we show that a geometrically-increasing step size yields a linear convergence rate towards an optimal policy. We then consider the sample-based case, when the best representation of the Q- value function among linear combinations of a known feature function is known up to an estimation error. In this setting, we show that the algorithm enjoys the same linear guarantees as in the deterministic case up to an error term that depends on the estimation error, the bias error, and the condition number of the feature covariance matrix. Our results build upon the general framework of policy mirror descent and extend previous findings for the softmax tabular parametrization to the log-linear policy class.