Statistical Inference for Polyak-Ruppert Averaged Zeroth-order Stochastic Gradient Algorithm

Abstract

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.

References in corpus (23)

• Practical Bayesian Optimization of Machine Learning Algorithms
• ZOO: Zeroth Order Optimization based Black-box Attacks to Deep Neural Networks without Training Substitute Models
• SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator
• An Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback
• Zeroth-Order Stochastic Variance Reduction for Nonconvex Optimization
• Stochastic (Approximate) Proximal Point Methods: Convergence, Optimality, and Adaptivity
• A Comprehensive Linear Speedup Analysis for Asynchronous Stochastic Parallel Optimization from Zeroth-Order to First-Order
• A companion for the Kiefer--Wolfowitz--Blum stochastic approximation algorithm
• Risk and parameter convergence of logistic regression
• Stochastic Zeroth-order Optimization in High Dimensions
• Gradientless Descent: High-Dimensional Zeroth-Order Optimization
• ZO-AdaMM: Zeroth-Order Adaptive Momentum Method for Black-Box Optimization
• Zeroth-Order Regularized Optimization (ZORO): Approximately Sparse Gradients and Adaptive Sampling
• HiGrad: Uncertainty Quantification for Online Learning and Stochastic Approximation
• A Primer on Zeroth-Order Optimization in Signal Processing and Machine Learning
• Improved Zeroth-Order Variance Reduced Algorithms and Analysis for Nonconvex Optimization
• Towards Gradient Free and Projection Free Stochastic Optimization
• An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias
• On the Design of Black-box Adversarial Examples by Leveraging Gradient-free Optimization and Operator Splitting Method
• Accelerated Stochastic Gradient-free and Projection-free Methods
• Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT
• Online Covariance Matrix Estimation in Stochastic Gradient Descent
• Approximate Newton-based statistical inference using only stochastic gradients