Stochastic gradient descent (SGD) is a foundational algorithm for large-scale statistical learning and stochastic optimization. However, statistical inference based on SGD iterates remains challenging when stochastic gradients have infinite variance, as the relevant limiting distributions depend on unknown nuisance parameters. In this paper, we develop an efficient, model-agnostic methodology for constructing confidence regions from SGD trajectories that applies in both finite- and infinite-variance regimes. The procedure is based on a joint weak convergence result for the Polyak-Ruppert averaged estimator and an empirical second-moment normalizer constructed from stochastic gradients along the SGD trajectory. This joint limit yields a self-normalized statistic in which the leading tail-dependent scaling terms cancel. We then use a subsampling calibration scheme to estimate the relevant critical values, avoiding explicit estimation of tail indices, slowly varying functions, or stable-law parameters. The resulting confidence regions are straightforward to implement and are asymptotically valid under both the finite- and infinite-second-moment regimes. Simulation studies show reliable coverage in various settings, supporting the proposed method as a practical tool for uncertainty quantification in stochastic optimization.

Asynchronous stochastic gradient descent (ASGD) is a standard way to exploit heterogeneous compute resources in distributed learning: instead of forcing fast workers to wait for slow ones, the server updates the model whenever a gradient arrives. Vanilla ASGD applies each arriving gradient with the same weight. When local data distributions are heterogeneous, this becomes problematic: faster workers contribute more updates, and we show theoretically that the method is biased toward a frequency-weighted average of the local objectives rather than the desired global objective. Existing remedies typically move away from the simple ASGD template by introducing gathering phases, buffering, or extra memory. We show that this is unnecessary. Keeping the standard ASGD mechanism, we recover the correct objective by rescaling worker-specific stepsizes in proportion to their computation times, so that each worker contributes the same aggregate learning rate over a cycle. In the non-convex setting, under smoothness and bounded heterogeneity assumptions, we prove that the resulting method, Rescaled ASGD, converges to stationary points of the correct global objective in the fixed-computation model. Its time complexity matches the known lower bound in the leading term, while the effects of staleness and data heterogeneity appear only in lower-order terms. Experiments confirm that the method converges to the correct objective and is competitive with state-of-the-art baselines.

Large-scale machine learning models are trained on clusters of machines that exhibit heterogeneous performance due to hardware variability, network delays, and system-level instabilities. In such environments, time complexity rather than iteration complexity becomes the relevant performance metric for optimization algorithms. Recent work by Tyurin and Richtárik [2023] established the first time complexity analysis for parallel first-order stochastic optimization, proposing Rennala SGD as a time-optimal method for smooth nonconvex optimization. However, Rennala SGD is fundamentally a modification of SGD, and variance reduction techniques are known to improve the iteration complexity of SGD. In this work, we investigate whether variance reduction can also improve time complexity in heterogeneous systems. We show that, under a mean-squared smoothness assumption, variance reduction can improve time complexity in relevant parameter regimes. To this end, we propose Rennala MVR, a variance-reduced extension of Rennala SGD based on momentum-based variance reduction, and analyze its oracle and time complexity. We establish lower bounds for time complexity under these assumptions.

Due to the high communication overhead when training machine learning models in a distributed environment, modern algorithms invariably rely on lossy communication compression. However, when untreated, the errors caused by compression propagate, and can lead to severely unstable behavior, including exponential divergence. Almost a decade ago, Seide et al. [2014] proposed an error feedback (EF) mechanism, which we refer to as EF14, as an immensely effective heuristic for mitigating this issue. However, despite steady algorithmic and theoretical advances in the EF field in the last decade, our understanding is far from complete. In this work we address one of the most pressing issues.

Knowledge distillation is a popular approach for enhancing the performance of "student" models, with lower representational capacity, by taking advantage of more powerful "teacher" models. Despite its apparent simplicity and widespread use, the underlying mechanics behind knowledge distillation (KD) are still not fully understood. In this work, we shed new light on the inner workings of this method, by examining it from an optimization perspective. We show that, in the context of linear and deep linear models, KD can be interpreted as a novel type of stochastic variance reduction mechanism. We provide a detailed convergence analysis of the resulting dynamics, which hold under standard assumptions for both strongly-convex and non-convex losses, showing that KD acts as a form of partial variance reduction, which can reduce the stochastic gradient noise, but may not eliminate it completely, depending on the properties of the "teacher" model. Our analysis puts further emphasis on the need for careful parametrization of KD, in particular w.r.t. the weighting of the distillation loss, and is validated empirically on both linear models and deep neural networks.

We consider the problem of scalable sampling algorithms to fit Bayesian generalized linear mixed models on large datasets. Stochastic gradient Langevin dynamics, coupled with smooth re-parameterizations of variance parameters, produces divergent Markov chains and cannot be reliably used for sampling covariance parameters of random effects. We advocate the use of a mirror Langevin dynamics algorithm, propose the novel stochastic mirror Langevin dynamics based on data subsampling, and provide concrete guidelines for its use in a Bayesian inference framework. Based on an explicit Wasserstein distance error bound between the posterior and its algorithmic approximation, we propose a post-processing step that yields an asymptotic, order-wise correct estimation of the posterior variance, eliminating the irreducible posterior variance estimation bias due to subsampling. Empirical performance of the method is evaluated through simulated experiments and a longitudinal study of pain trajectories in a study of breast cancer survivors.

Stochastic gradient descent (SGD) is a prevalent optimization technique for largescale distributed machine learning. While SGD computation can be efficiently divided between multiple machines, communication typically becomes a bottleneck in the distributed setting. Gradient compression methods can be used to alleviate this problem, and a recent line of work shows that SGD augmented with gradient compression converges to an ε-first-order stationary point. In this paper we extend these results to convergence to an ε-second-order stationary point (ε-SOSP), which is to the best of our knowledge the first result of this type. In addition, we show that, when the stochastic gradient is not Lipschitz, compressed SGD with RANDOMK compressor converges to an ε-SOSP with the same number of iterations as uncompressed SGD [25], while improving the total communication by a factor of Θ( dε 3/4), where dis the dimension of the optimization problem. We present additional results for the cases when the compressor is arbitrary and when the stochastic gradient is Lipschitz.