The objectives of this technical report is to provide additional results on the generalized conditional gradient methods introduced by [BLM05]. Indeed , when the objective function is smooth, we provide a novel certificate of optimality and we show that the algorithm has a linear convergence rate. Applications of this algorithm are also discussed.
Finally our approach is also closely related to projected gradient descent and its spectral variant @cite_21 . As discussed more in detail in section , when @math is an euclidean distance, the solving problem boils down to a projection onto the convex @math . In practice the method is more general since it can be used for any problem as long as problem can be efficiently solved.
@cite_21
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The mirror descent algorithm (MDA) was introduced by Nemirovsky and Yudin for solving convex optimization problems. This method exhibits an efficiency estimate that is mildly dependent in the decision variables dimension, and thus suitable for solving very large scale optimization problems. We present a new derivation and analysis of this algorithm. We show that the MDA can be viewed as a nonlinear projected-subgradient type method, derived from using a general distance-like function instead of the usual Euclidean squared distance. Within this interpretation, we derive in a simple way convergence and efficiency estimates. We then propose an Entropic mirror descent algorithm for convex minimization over the unit simplex, with a global efficiency estimate proven to be mildly dependent in the dimension of the problem.