Stein-Rule Shrinkage for Stochastic Gradient Estimation in High Dimensions

Abstract: Stochastic gradient methods are central to large-scale learning, yet their analysis typically treats mini-batch gradients as unbiased estimators of the population gradient. In high-dimensional settings, however, classical results from statistical decision theory show that unbiased estimators are generally inadmissible under quadratic loss, suggesting that standard stochastic gradients may be suboptimal from a risk perspective. In this work, we formulate stochastic gradient computation as a high-dimensional estimation problem and introduce a decision-theoretic framework based on Stein-rule shrinkage. We construct a shrinkage gradient estimator that adaptively contracts noisy mini-batch gradients toward a stable restricted estimator derived from historical momentum. The shrinkage intensity is determined in a data-driven manner using an online estimate of gradient noise variance, leveraging second-moment statistics commonly maintained by adaptive optimization methods. Under a Gaussian noise model and for dimension p>=3, we show that the proposed estimator uniformly dominates the standard stochastic gradient under squared error loss and is minimax-optimal in the classical decision-theoretic sense. We further demonstrate how this estimator can be incorporated into the Adam optimizer, yielding a practical algorithm with negligible additional computational cost. Empirical evaluations on CIFAR10 and CIFAR100, across multiple levels of label noise, show consistent improvements over Adam in the large-batch regime. Ablation studies indicate that the gains arise primarily from selectively applying shrinkage to high-dimensional convolutional layers, while indiscriminate shrinkage across all parameters degrades performance. These results illustrate that classical shrinkage principles provide a principled and effective approach to improving stochastic gradient estimation in modern deep learning.