Stochastic gradient methods are central to large-scale learning, but they treat mini-batch gradients as unbiased estimators, which classical decision theory shows are inadmissible in high dimensions. We formulate gradient computation as a high-dimensional estimation problem and introduce a framework based on Stein-rule shrinkage. We construct a gradient estimator that adaptively contracts noisy mini-batch gradients toward a stable estimator derived from historical momentum. The shrinkage intensity is determined in a data-driven manner using an online estimate of gradient noise variance, leveraging statistics from adaptive optimizers. Under a Gaussian noise model, we show our estimator uniformly dominates the standard stochastic gradient under squared error loss and is minimax-optimal. We incorporate this into the Adam optimizer, yielding SR-Adam, a practical algorithm with negligible computational cost. Empirical evaluations on CIFAR10 and CIFAR100 across multiple levels of input noise show consistent improvements over Adam in the large-batch regime. Ablation studies indicate that 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 approach to improving stochastic gradient estimation in deep learning.

Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures.

Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME and captures additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to pick up information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causaldiscovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories.

Towardsthis end,wedesigned apreconditioned gradient solverforkernelmethods exploiting both GPU acceleration and parallelization with multiple GPUs, implementing out-of-core variants of common linear algebra operations to guarantee optimal hardware utilization.

Finally, we extend Min-AM to solve stochastic programming problems.