Submodular Maximization via Gradient Ascent: The Case of Deep Submodular Functions.

We study the problem of maximizing deep submodular functions (DSFs) [13, 3] subject to a matroid constraint. DSFs are an expressive class of submodular functions that include, as strict subfamilies, the facility location, weighted coverage, and sums of concave composed with modular functions. We use a strategy similar to the continuous greedy approach [6], but we show that the multilinear extension of any DSF has a natural and computationally attainable concave relaxation that we can optimize using gradient ascent. Our results show a guarantee of max 0 < δ < 1 ( 1 - ϵ - δ - e - δ 2 Ω ( k ) ) with a running time of O(n 2 /ϵ 2 ) plus time for pipage rounding [6] to recover a discrete solution, where k is the rank of the matroid constraint. This bound is often better than the standard 1 - 1/e guarantee of the continuous greedy algorithm, but runs much faster. Our bound also holds even for fully curved (c = 1) functions where the guarantee of 1 - c/e degenerates to 1 - 1/e where c is the curvature of f [37]. We perform computational experiments that support our theoretical results.