Convergence of the Forward-Backward Algorithm: Beyond the Worst Case with the Help of Geometry
This work addresses convergence issues in optimization for inverse problems and signal processing, offering incremental improvements by unifying and extending existing geometric frameworks.
The paper tackles the convergence analysis of the forward-backward algorithm by extending geometric conditions like Łojasiewicz properties to arbitrary sets, enabling new results such as a first Łojasiewicz inequality for quadratic functions with compact operators and linear rates for inverse problems with low-complexity priors.
We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.