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

Carlon, André Gustavo; Espath, Luis; Tempone, Raúl

Abstract

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 ℓ2-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.

Citation

Carlon, A. G., Espath, L., & Tempone, R. (2024). Approximating Hessian matrices using Bayesian inference: a new approach for quasi-Newton methods in stochastic optimization. Optimization Methods and Software, 39(6), 1352-1382. https://doi.org/10.1080/10556788.2024.2339226

• Journal Article Type: Article
• Acceptance Date: Mar 13, 2024
• Online Publication Date: Apr 29, 2024
• Publication Date: Apr 29, 2024
• Deposit Date: May 30, 2024
• Publicly Available Date: May 31, 2024
• Journal: Optimization Methods and Software
• Print ISSN: 1055-6788
• Electronic ISSN: 1029-4937
• Publisher: Taylor and Francis
• Peer Reviewed: Peer Reviewed
• Volume: 39
• Issue: 6
• Pages: 1352-1382
• DOI: https://doi.org/10.1080/10556788.2024.2339226
• Keywords: Stochastic optimization; quasi-Newton; Monte Carlo; variance reduction; control variates; machine learning
• Public URL: https://nottingham-repository.worktribe.com/output/34850137
• Publisher URL: https://www.tandfonline.com/doi/full/10.1080/10556788.2024.2339226
• Additional Information: Peer Review Statement: The publishing and review policy for this title is described in its Aims & Scope.; Aim & Scope: http://www.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=goms20; Received: 2022-09-08; Accepted: 2024-03-13; Published: 2024-04-29