Linesearch-Free Nonlinearly Preconditioned GD with Semiautomatic Geometry Design

Abstract: Nonlinear preconditioning makes it possible to adapt gradient updates to the growth of the objective's curvature, but combining its design with suitable stepsize selection is challenging. We propose adaptive dual preconditioned gradient descent (adaptive DPGD), which combines curvature-based preconditioner design with a linesearch-free stepsize rule based on local information. For convex objectives, we establish linear convergence under local strong convexity of the objective. By adapting the stepsize rule, we extend the method to weakly convex objectives and establish asymptotic stationarity without global Lipschitz smoothness. Moreover, our local assumptions give greater freedom in preconditioner design. We exploit this freedom to develop a semiautomatic recipe guided by the objective's curvature, yielding a family of preconditioners that includes those underlying normalized and hyperbolic gradient descent. Experiments on convex and weakly convex problems with real data demonstrate the effectiveness of the preconditioner design and adaptive stepsize selection.