We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, which total cost depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions.

We show that the updating technique can be efficiently coupled only with the simplest subgradient methods. Similar results can be obtained for a new nonsmooth random variant of a coordinate descent scheme. We present also the promising results of preliminary computational experiments.

Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time (Iwata and Orlin 2009), however these algorithms are not practical. In 1976, Wolfe proposed an algorithm to find the minimum Euclidean norm point in a polytope, and in 1980, Fujishige showed how Wolfe's algorithm can be used for SFM. For general submodular functions, the Fujishige-Wolfe minimum norm algorithm seems to have the best empirical performance. Despite its good practical performance, theoretically very little is known about Wolfe's minimum norm algorithm -- to our knowledge the only result is an exponential time analysis due to Wolfe himself. In this paper we give a maiden convergence analysis of Wolfe's algorithm. We prove that in t iterations, Wolfe's algorithm returns a O(1/t)-approximate solution to the min-norm point. We also prove a robust version of Fujishige's theorem which shows that an O(1/n^2)-approximate solution to the min-norm point problem implies exact submodular minimization. As a corollary, we get the first pseudo-polynomial time guarantee for the Fujishige-Wolfe minimum norm algorithm for submodular function minimization. In particular, we show that the min-norm point algorithm solves SFM in O(n^7F^2)-time, where $F$ is an upper bound on the maximum change a single element can cause in the function value.