Many tasks in imaging can be modeled via the minimization of a nonconvex composite function. A popular class of algorithms for solving such problems are majorization-minimization techniques which iteratively approximate the composite nonconvex function by a majorizing function that is easy to minimize. Most techniques, e.g. gradient descent, utilize convex majorizers in order guarantee that the majorizer is easy to minimize. In our work we consider a natural class of nonconvex majorizers for these functions, and show that these majorizers are still sufficient for a provably convergent optimization scheme under quite general assumptions. Numerical results illustrate that by applying this scheme, one can often obtain superior local optima compared to previous majorization-minimization methods, when the nonconvex majorizers are solved to global optimality. Finally, we illustrate the behavior of our algorithm for depth super-resolution from raw time-of-flight data.
Recent works have also proposed general frameworks for iteratively replacing the original minimization problem with simple approximation functions @math beyond majorization-minimization. @cite_2 analyzed approximation functions @math , satisfying @math for a proper growth function @math . A minimization scheme of these approximation functions exhibits subsequential convergence to critical points, even if the subproblem evaluations are inexact. These approximation functions need not necessarily be convex, but the distance of their subsequent evaluations must tend to zero. A slightly different generalization is discussed in @cite_44 , where approximation functions satisfying @math are examined. Here @math is a growth function and @math a Bregman distance generated by a 'kernel' function similarly to the previously discussed eq:intro_fb . Subsequence convergence can again be shown here, under relatively weak conditions. However the approximation function @math is taken to be convex in this work to, among other properties, guarantee the success of a backtracking scheme and reach an implementable algorithm.
@cite_44 @cite_2
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We propose a unifying algorithm for non-smooth non-convex optimization. The algorithm approximates the objective function by a convex model function and finds an approximate (Bregman) proximal point of the convex model. This approximate minimizer of the model function yields a descent direction, along which the next iterate is found. Complemented with an Armijo-like line search strategy, we obtain a flexible algorithm for which we prove (subsequential) convergence to a stationary point under weak assumptions on the growth of the model function error. Special instances of the algorithm with a Euclidean distance function are, for example, Gradient Descent, Forward--Backward Splitting, ProxDescent, without the common requirement of a "Lipschitz continuous gradient". In addition, we consider a broad class of Bregman distance functions (generated by Legendre functions) replacing the Euclidean distance. The algorithm has a wide range of applications including many linear and non-linear inverse problems in signal image processing and machine learning. We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result is an explicit relationship between the step-size of any such algorithm and the slope of the function at a nearby point. Consequently, we (1) show that the step-sizes can be reliably used to terminate the algorithm, (2) prove that as long as the step-sizes tend to zero, every limit point of the iterates is stationary, and (3) show that conditions, akin to classical quadratic growth, imply that the step-sizes linearly bound the distance of the iterates to the solution set. The latter so-called error bound property is typically used to establish linear (or faster) convergence guarantees. Analogous results hold when the step-size is replaced by the square root of the decrease in the model's value. We complete the paper with extensions to when the models are minimized only inexactly.