arXiv subjects

Haimei Huo

Publications and source records attributed to Haimei Huo.

4 recordsLinked to original sources

arXiv2024

Alternating Stochastic Variance-Reduced Algorithms with Optimal Complexity for Bilevel Optimization

This paper studies the unconstrained nonconvex-strongly-convex bilevel optimization problem. A common approach to solving this problem is to alternately update the upper-level and lower-level variables using (biased) stochastic gradients or their variants, with the lower-level variable updated either one step or multiple steps. In this context, we propose two alternating stochastic variance-reduced algorithms, namely ALS-SPIDER and ALS-STORM, which introduce an auxiliary variable to estimate the hypergradient for updating the upper-level variable. ALS-SPIDER employs the SPIDER estimator for updating variables, while ALS-STORM is a modification of ALS-SPIDER designed to avoid using large batch sizes in every iteration. Theoretically, both algorithms can find an $\epsilon$-stationary point of the bilevel problem with a sample complexity of $O(\epsilon^{-1.5})$ for arbitrary constant number of lower-level variable updates. To the best of our knowledge, they are the first algorithms to achieve the optimal complexity of $O(\epsilon^{-1.5})$ when performing multiple updates on the lower-level variable. Numerical experiments are conducted to illustrate the efficiency of our algorithms.

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arXiv2024

A Perturbed Value-Function-Based Interior-Point Method for Perturbed Pessimistic Bilevel Problems

Bilevel optimizaiton serves as a powerful tool for many machine learning applications. Perturbed pessimistic bilevel problem PBP$ε$, with $ε$ being an arbitrary positive number, is a variant of the bilevel problem to deal with the case where there are multiple solutions in the lower level problem. However, the provably convergent algorithms for PBP$ε$ with a nonlinear lower level problem are lacking. To fill the gap, we consider in the paper the problem PBP$ε$ with a nonlinear lower level problem. By introducing a log-barrier function to replace the inequality constraint associated with the value function of the lower level problem, and approximating this value function, an algorithm named Perturbed Value-Function-based Interior-point Method(PVFIM) is proposed. We present a stationary condition for PBP$ε$, which has not been given before, and we show that PVFIM can converge to a stationary point of PBP$ε$. Finally, experiments are presented to verify the theoretical results and to show the application of the algorithm to GAN.

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arXiv2023

A New Simple Stochastic Gradient Descent Type Algorithm With Lower Computational Complexity for Bilevel Optimization

Bilevel optimization has been widely used in many machine learning applications such as hyperparameter optimization and meta learning. Recently, many simple stochastic gradient descent(SGD) type algorithms(without using momentum and variance techniques) have been proposed to solve the bilevel optimization problems. However, all the existing simple SGD type algorithms estimate the hypergradient via stochastic estimation of Neumann series. In the paper, we propose to estimate the hypergradient via SGD-based Estimation(i.e., solving the linear system with SGD). By using warm start initialization strategy, a new simple SGD type algorithm SSGD based on SGD-based Estimation is proposed. We provide the convergence rate guarantee for SSGD and show that SSGD outperforms the best known computational complexity achieved by the existing simple SGD type algorithms. Our experiments validate our theoretical results and demonstrate the efficiency of our proposed algorithm SSGD in hyperparameter optimization applications.

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arXiv2021

Homotopy Methods for Eigenvector-Dependent Nonlinear Eigenvalue Problems

Eigenvector-dependent nonlinear eigenvalue problems are considered which arise from the finite difference discretizations of the Gross-Pitaevskii equation. Existence and uniqueness of positive eigenvector for both one and two dimensional cases and existence of antisymmetric eigenvector for one dimensional case are proved. In order to compute eigenpairs corresponding to excited states as well as ground state, homotopies for both one and two dimensional problems are constructed respectively and the homotopy paths are proved to be regular and bounded. Numerical results are presented to verify the theories derived for both one and two dimensional problems.

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