An incremental mirror descent subgradient algorithm with random sweeping and proximal step

Abstract

We investigate the convergence properties of incremental mirror descent type subgradient algorithms for minimizing the sum of convex functions. In each step, we only evaluate the subgradient of a single component function and mirror it back to the feasible domain, which makes iterations very cheap to compute. The analysis is made for a randomized selection of the component functions, which yields the deterministic algorithm as a special case. Under supplementary differentiability assumptions on the function which induces the mirror map, we are also able to deal with the presence of another term in the objective function, which is evaluated via a proximal type step. In both cases, we derive convergence rates of in expectation for the k th best objective function value and illustrate our theoretical findings by numerical experiments in positron emission tomography and machine learning.

Stochastic incremental mirror descent algorithms with Nesterov smoothing

“…As noted in [10], this scheme generalizes the classical subgradient method and is close to the subgradient projection algorithm.…”

“…Taking into consideration this remark, only the maximum in the construction (2) needs to be attained in the case of such compositions when the involved functions are proper, convex and semicontinuous, and the operators linear, in which case we say that they fulfill the property (2 ′ ). Unlike the construction proposed in [10], our approach is flexible enough to allow modifying Algorithm 3.10 in order to solve such problems as well.…”

“…Moreover, our approach does not require knowledge of the Lipschitz constants or the subgradients of the involved functions, which sometimes can be computationally expensive to determine. We also show that in case some of the involved functions are Lipschitz continuous our methods can be easily combined with the ones proposed in [10]. In [11] one can find a variable smoothing approach to minimize convex optimization problems with stochastic gradients, so that large scale problems can be addressed, where, different to our work, the Moreau-envelope, a special case of Nesterov smoothing, is used.…”

“…Generalizations of mirror descent methods can be found, for instance, in [2,32], while for continuous versions (by means of dynamical systems) we refer to [34,36]. The mirror descent type algorithms are usually employed for minimizing a single function, however in works like [5,9,10,14,15,20,21,23,31,50,51] such methods were used for minimizing sums of (convex) functions by considering splitting techniques, in order to solve problems arising from various applications from fields such as machine learning or imaging. A specific feature of mirror descent type algorithms is that the convergence statements are provided in terms of values of objective functions, however in papers like [14,37,51] the convergence of the generated iterative sequence is investigated, too.…”

“…The resulting algorithm is a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing, and this is further modified in order to minimize the sum over a given nonempty closed convex set in an Euclidean space of finitely many proper, convex and lower semicontinuous functions composed with linear operators, and the mentioned prox-friendly proper, convex and lower semicontinuous function. Different to the previous contributions from the literature on designing mirror descent methods for minimizing sums of functions mentioned above (in particular [10,14,20,21,23]), the functions we consider need not be (Lipschitz) continuous or differentiable. Moreover, our approach does not require knowledge of the Lipschitz constants or the subgradients of the involved functions, which sometimes can be computationally expensive to determine.…”