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In this paper we consider the problem of obtaining sharp bounds for theperformance of temporal difference (TD) methods with linear functionapproximation for policy evaluation in discounted Markov decision processes. Weshow that a simple algorithm with a universal and instance-independent stepsize together with Polyak-Ruppert tail averaging is sufficient to obtainnear-optimal variance and bias terms. We also provide the respective samplecomplexity bounds. Our proof technique is based on refined error bounds forlinear stochastic approximation together with the novel stability result forthe product of random matrices that arise from the TD-type recurrence.

Improved High-Probability Bounds for the Temporal Difference Learning Algorithm via Exponential Stability