Enhanced Balancing of Bias-Variance Tradeoff in Stochastic Estimation: A Minimax Perspective

TL;DR

This paper develops a minimax framework for constructing biased stochastic estimators that outperform traditional methods by optimally balancing bias and variance, especially when model parameters are unknown.

Contribution

Introduces a new class of estimators using combined simulation runs, minimizing worst-case asymptotic risk ratios to improve bias-variance tradeoff.

Findings

Derived the minimax risk ratio for weighted estimators

Identified regimes where the new estimators outperform conventional ones

Characterized the optimal weighting scheme with two decay components

Abstract

Biased stochastic estimators, such as finite-differences for noisy gradient estimation, often contain parameters that need to be properly chosen to balance impacts from the bias and the variance. While the optimal order of these parameters in terms of the simulation budget can be readily established, the precise best values depend on model characteristics that are typically unknown in advance. We introduce a framework to construct new classes of estimators, based on judicious combinations of simulation runs on sequences of tuning parameter values, such that the estimators consistently outperform a given tuning parameter choice in the conventional approach, regardless of the unknown model characteristics. We argue the outperformance via what we call the asymptotic minimax risk ratio, obtained by minimizing the worst-case asymptotic ratio between the mean square errors of our estimators and the conventional one, where the worst case is over any possible values of the model unknowns. In particular, when the minimax ratio is less than 1, the calibrated estimator is guaranteed to perform better asymptotically. We identify this minimax ratio for general classes of weighted estimators, and the regimes where this ratio is less than 1. Moreover, we show that the best weighting scheme is characterized by a sum of two components with distinct decay rates. We explain how this arises from bias-variance balancing that combats the adversarial selection of the model constants, which can be analyzed via a tractable reformulation of a non-convex optimization problem.