2024/06/04 by Quoc Tran-Dinh, Tran-Dinh, Quoc · 1 citation
Computer Science · Mathematics · #90-08 #90C06 #90C25 #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Matrix Theory and Algorithms #Numerical Methods and Algorithms #Numerical methods for differential equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2406.02413
openalex publication_date 2024/06/04 · openalex created_date 2024/06/08 · openalex updated_date 2026/07/28
We propose a new class of fast Krasnoselkii--Mann methods with variance reduction to solve a finite-sum co-coercive equation Gx = 0. Our algorithm is single-loop and leverages a new family of unbiased variance-reduced estimators specifically designed for a wider class of root-finding algorithms. Our method achieves both O(1/k2) and o(1/k2) last-iterate convergence rates in terms of 𝔼[‖ Gxk‖2], where k is the iteration counter and 𝔼[⋅] is the total expectation. We also establish almost sure o(1/k2) convergence rates and the almost sure convergence of iterates \xk\ to a solution of Gx=0. We instantiate our framework for two prominent estimators: SVRG and SAGA. By an appropriate choice of parameters, both variants attain an oracle complexity of O(n + n2/3ε-1) to reach an ε-solution, where n represents the number of summands in the finite-sum operator G. Furthermore, under σ-strong quasi-monotonicity, our method achieves a linear convergence rate and an oracle complexity of O(n+ max\n, n2/3κ\ log(\frac1ε)), where κ:= L/σ. We extend our approach to solve a class of finite-sum inclusions (possibly nonmonotone), demonstrating that our schemes retain the same theoretical guarantees as in the equation setting. Finally, numerical experiments validate our algorithms and demonstrate their promising performance compared to state-of-the-art methods.