Extrapolation methods use the last few iterates of an optimization algorithm to produce a better estimate of the optimum. They were shown to achieve optimal convergence rates in a deterministic setting using simple gradient iterates. Here, we study extrapolation methods in a stochastic setting, where the iterates are produced by either a simple or an accelerated stochastic gradient algorithm. We first derive convergence bounds for arbitrary, potentially biased perturbations, then produce asymptotic bounds using the ratio between the variance of the noise and the accuracy of the current point. Finally, we apply this acceleration technique to stochastic algorithms such as SGD, SAGA, SVRG and Katyusha in different settings, and show significant performance gains.

We study the implicit regularization properties of optimization techniques by explicitly connecting their optimization paths to the regularization paths of ``corresponding'' regularized problems. This surprising connection shows that iterates of optimization techniques such as gradient descent and mirror descent are \emph{pointwise} close to solutions of appropriately regularized objectives. While such a tight connection between optimization and regularization is of independent intellectual interest, it also has important implications for machine learning: we can port results from regularized estimators to optimization, and vice versa. We investigate one key consequence, that borrows from the well-studied analysis of regularized estimators, to then obtain tight excess risk bounds of the iterates generated by optimization techniques.

To address this gap, we study deterministic/stochastic versions of SAM with practical configurations (i.e., constant

In this paper, we focus on solving Extensive-F orm Games (EFGs).

Knowledge distillation is a popular approach for enhancing the performance of "student" models, with lower representational capacity, by taking advantage of

The dynamical stability of the iterates during training plays a key role in determining the minima obtained by optimization algorithms. For example, stable solutions of gradient descent (GD) correspond to flat minima, which have been associated with favorable features. While prior work often relies on linearization to determine stability, it remains unclear whether linearized dynamics faithfully capture the full nonlinear behavior. Recent work has shown that GD may stably oscillate near a linearly unstable minimum and still converge once the step size decays, indicating that linear analysis can be misleading. In this work, we explicitly study the effect of nonlinear terms. Specifically, we derive an exact criterion for stable oscillations of GD near minima in the multivariate setting. Our condition depends on high-order derivatives, generalizing existing results. Extending the analysis to stochastic gradient descent (SGD), we show that nonlinear dynamics can diverge in expectation even if a single batch is unstable. This implies that stability can be dictated by a single batch that oscillates unstably, rather than an average effect, as linear analysis suggests. Finally, we prove that if all batches are linearly stable, the nonlinear dynamics of SGD are stable in expectation.