Optimal higher-order methods improve solving monotone problems
Anchored Extra-Proximal Methods: Optimal Higher-Order Methods for Monotone Inclusion Problems
Machine Learning
Summary
The paper deals with a math problem called monotone inclusion, common in optimization and other fields. The authors propose a new method that combines past ideas to create faster ways to find approximate solutions. Their method works better than previous ones, especially when using more detailed information about the problem (higher-order derivatives). They also prove that their method is as fast as any method can possibly be under their assumptions.
What this means in practice
- •For optimization software developers: Improve the efficiency of algorithms used for complex composite monotone inclusion problems by integrating optimal higher-order proximal updates.
- •For machine learning engineers: Enhance training algorithms that rely on monotone operator models by applying faster higher-order methods with provable optimality.
A theory result. No direct application yet.
Authors
Ruichen Jiang, TaeHo Yoon
Abstract
We study the deterministic oracle complexity of finding approximate solutions to composite monotone inclusion problems, formed by the sum of a smooth single-valued monotone operator and a maximally monotone set-valued operator, under the tangent-residual criterion. We introduce the Anchored Extra-Proximal (AEP) framework, which combines an anchored extrapolation step with an inexact anchored proximal update satisfying a relative-error condition. The framework recovers the composite Fast Extragradient method in the first-order setting and yields natural second- and higher-order extensions by replacing the operator in the implicit update with its Taylor approximation at the extrapolated point. For every $p\geq 2$, assuming that the $(p-1)$th derivative of the single-valued operator is Lipschitz continuous, we combine this construction with a bisection line search to obtain a $p$th-order method that finds a point with tangent residual at most $\varepsilon$ in $\widetilde{O}(\varepsilon^{-2/(3p-1)})$ oracle calls. This improves all prior upper bounds for $p$th-order methods: in particular, it improves the previous best-known $\widetilde{O}(\varepsilon^{-1/p})$ tangent-residual complexity as well as the classical $O(\varepsilon^{-2/(p+1)})$ bound of higher-order hybrid proximal extragradient methods under the weaker duality-gap criterion. We complement this result with a worst-case lower bound of $Ω(\varepsilon^{-2/(3p-1)})$ for every deterministic algorithm in the $p$th-order oracle model, without restricting the algorithm to tensor steps or any other prescribed update structure. Thus, the proposed method attains the optimal dependence on $\varepsilon$, up to logarithmic factors, for all $p\geq2$.