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We present a new algorithm to solve min-max or min-min problems out of theconvex world. We use rigidity assumptions, ubiquitous in learning, making ourmethod applicable to many optimization problems. Our approach takes advantageof hidden regularity properties and allows us to devise a simple algorithm ofridge type. An original feature of our method is to come with automatic stepsize adaptation which departs from the usual overly cautious backtrackingmethods. In a general framework, we provide convergence theoretical guaranteesand rates. We apply our findings on simple GAN problems obtaining promisingnumerical results.

A H\"olderian backtracking method for min-max and min-min problems