Andre Carlon | Searcharxiv

arXiv2022

Approximating Hessian matrices using Bayesian inference: a new approach for quasi-Newton methods in stochastic optimization

Using quasi-Newton methods in stochastic optimization is not a trivial task given the difficulty of extracting curvature information from the noisy gradients. Moreover, pre-conditioning noisy gradient observations tend to amplify the noise. We propose a Bayesian approach to obtain a Hessian matrix approximation for stochastic optimization that minimizes the secant equations residue while retaining the extreme eigenvalues between a specified range. Thus, the proposed approach assists stochastic gradient descent to converge to local minima without augmenting gradient noise. We propose maximizing the log posterior using the Newton-CG method. Numerical results on a stochastic quadratic function and an $\ell_2$-regularized logistic regression problem are presented. In all the cases tested, our approach improves the convergence of stochastic gradient descent, compensating for the overhead of solving the log posterior maximization. In particular, pre-conditioning the stochastic gradient with the inverse of our Hessian approximation becomes more advantageous the larger the condition number of the problem is.

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arXiv2020

Multi-Iteration Stochastic Optimizers

We introduce Multi-Iteration Stochastic Optimizers, a novel class of first-order stochastic methods that control the relative $L^2$ error using successive control variates along the iteration path. By exploiting correlations between iterates, these control variates reduce the estimator's variance, making an accurate mean gradient estimation computationally affordable. Our approach centers on the Multi-Iteration stochastiC Estimator (MICE), which can be seamlessly coupled with any first-order stochastic optimizer due to its non-intrusive design. The algorithm adaptively selects which iterates to include in its index set. We provide both an error analysis of MICE and a convergence analysis for Multi-Iteration Stochastic Optimizers across various problem classes, including some non-convex cases. In the smooth, strongly convex setting, we demonstrate that to approximate a minimizer within a tolerance $tol$, SGD-MICE requires, on average, $O(tol^{-1})$ stochastic gradient evaluations, compared to $O(tol^{-1}\log(tol^{-1}))$ for SGD with adaptive batch sizes. In numerical experiments, SGD-MICE achieved the desired tolerance with fewer than 3\% of the gradient evaluations required by adaptive batch SGD. Additionally, MICE offers a straightforward stopping criterion based on the gradient norm, validated through consistency tests. To assess its efficiency, we present examples using both SGD-MICE and Adam-MICE, including a stochastic adaptation of the Rosenbrock function and logistic regression on various datasets. Compared to SGD, SAG, SAGA, SVRG, and SARAH, our approach consistently reduces the gradient sampling cost without the need for extensive parameter tuning.

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