Extending the Reach of First-Order Algorithms for Nonconvex Min-Max
Problems with Cohypomonotonicity
- URL: http://arxiv.org/abs/2402.05071v1
- Date: Wed, 7 Feb 2024 18:22:41 GMT
- Title: Extending the Reach of First-Order Algorithms for Nonconvex Min-Max
Problems with Cohypomonotonicity
- Authors: Ahmet Alacaoglu, Donghwan Kim, Stephen J. Wright
- Abstract summary: We conjecture on $fracKMLotonicity guarantees weak MVrhon$coords or weak MVrhonLotonicity or weak MVrhonKML$.
We also provide algorithms and complexity guarantees in the case with the same range on $$.
- Score: 20.710343135282123
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We focus on constrained, $L$-smooth, nonconvex-nonconcave min-max problems
either satisfying $\rho$-cohypomonotonicity or admitting a solution to the
$\rho$-weakly Minty Variational Inequality (MVI), where larger values of the
parameter $\rho>0$ correspond to a greater degree of nonconvexity. These
problem classes include examples in two player reinforcement learning,
interaction dominant min-max problems, and certain synthetic test problems on
which classical min-max algorithms fail. It has been conjectured that
first-order methods can tolerate value of $\rho$ no larger than $\frac{1}{L}$,
but existing results in the literature have stagnated at the tighter
requirement $\rho < \frac{1}{2L}$. With a simple argument, we obtain optimal or
best-known complexity guarantees with cohypomonotonicity or weak MVI conditions
for $\rho < \frac{1}{L}$. The algorithms we analyze are inexact variants of
Halpern and Krasnosel'ski\u{\i}-Mann (KM) iterations. We also provide
algorithms and complexity guarantees in the stochastic case with the same range
on $\rho$. Our main insight for the improvements in the convergence analyses is
to harness the recently proposed "conic nonexpansiveness" property of
operators. As byproducts, we provide a refined analysis for inexact Halpern
iteration and propose a stochastic KM iteration with a multilevel Monte Carlo
estimator.
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