Dictionary-Restricted First-Order Descent Methods: Bounds and Convergence Rates

TL;DR

This paper introduces a unified theory for first-order descent methods with restricted dictionaries in Banach spaces, providing new convergence bounds and rates applicable to various approximation and optimization problems.

Contribution

It develops a geometric framework allowing analysis of descent methods with nonlinear or parameterized dictionaries, extending classical results to broader settings.

Findings

Derives explicit descent bounds and convergence rates under mild assumptions.

Achieves algebraic, polynomial, and exponential convergence regimes.

Applicable to convex variational problems and high-dimensional approximation.

Abstract

This paper develops a general theory for first-order descent methods whose search directions are restricted to a prescribed dictionary in a reflexive Banach space. Instead of assuming that the linear span of the dictionary is dense, as in the classical Proper Generalized Decomposition framework of Falc\'o and Nouy or in the universality approach of Bern\'a and Falc\'o, we introduce a geometric condition based on norming sets that guarantees density through a duality argument. This makes it possible to treat dictionaries arising from tensor formats, neural network units, and other nonlinear or parameterized approximation families within a unified setting. On the algorithmic side, we analyze a simple greedy update rule in which each iterate is obtained by minimizing the energy functional along one direction from the dictionary. Under mild differentiability, Lipschitz continuity, and ellipticity assumptions on the objective, we derive explicit quantitative descent bounds and sharp convergence rates. These include algebraic rates that improve those of classical steepest-descent schemes in Banach spaces, as well as arbitrarily high polynomial rates and exponential convergence in a critical regime. The results apply broadly to convex variational problems, high-dimensional approximation, and structured optimization methods that rely on restricted or compressed search directions.