Duality Gap Estimates for Weak Chebyshev Greedy Algorithms in Banach Spaces

Abstract

The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept of duality gap, the values of which are implicitly calculated at the step of choosing the direction of the fastest descent at each iteration of the greedy algorithm. We show that these values give upper bounds for the difference between the values of the objective function in the current state and the optimal point. Since the value of the objective function at the optimal point is not known in advance, the current values of the duality gap can be used, for example, in the stopping criteria for the greedy algorithm. In the paper, we find estimates of the duality gap values depending on the number of iterations for the weak greedy algorithms under consideration.