Statistical Inference for Polyak-Ruppert Averaged Zeroth-order Stochastic Gradient Algorithm
This addresses a gap in statistical machine learning by enabling practical inference for derivative-free optimization, which is crucial for applications where gradients are expensive or unavailable.
The paper tackles the problem of uncertainty quantification for zeroth-order stochastic gradient algorithms, which are used when gradient computation is infeasible, by establishing a central limit theorem for Polyak-Ruppert averaged versions and providing online estimators for constructing asymptotically valid confidence sets.
Statistical machine learning models trained with stochastic gradient algorithms are increasingly being deployed in critical scientific applications. However, computing the stochastic gradient in several such applications is highly expensive or even impossible at times. In such cases, derivative-free or zeroth-order algorithms are used. An important question which has thus far not been addressed sufficiently in the statistical machine learning literature is that of equipping stochastic zeroth-order algorithms with practical yet rigorous inferential capabilities so that we not only have point estimates or predictions but also quantify the associated uncertainty via confidence intervals or sets. Towards this, in this work, we first establish a central limit theorem for Polyak-Ruppert averaged stochastic zeroth-order gradient algorithm. We then provide online estimators of the asymptotic covariance matrix appearing in the central limit theorem, thereby providing a practical procedure for constructing asymptotically valid confidence sets (or intervals) for parameter estimation (or prediction) in the zeroth-order setting.