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Faster Algorithms for Learning Convex Functions
Ali Siahkamari · Durmus Alp Emre Acar · Christopher Liao · Kelly Geyer · Venkatesh Saligrama · Brian Kulis

Tue Jul 19 11:35 AM -- 11:40 AM (PDT) @ Hall G
The task of approximating an arbitrary convex function arises in several learning problems such as convex regression, learning with a difference of convex (DC) functions, and learning Bregman or $f$-divergences. In this paper, we develop and analyze an approach for solving a broad range of convex function learning problems that is faster than state-of-the-art approaches. Our approach is based on a 2-block ADMM method where each block can be computed in closed form. For the task of convex Lipschitz regression, we establish that our proposed algorithm converges with iteration complexity of $ O(n\sqrt{d}/\epsilon)$ for a dataset $ X \in R^{n\times d}$ and $\epsilon > 0$. Combined with per-iteration computation complexity, our method converges with the rate $O(n^3 d^{1.5}/\epsilon+n^2 d^{2.5}/\epsilon+n d^3/\epsilon)$. This new rate improves the state of the art rate of $O(n^5d^2/\epsilon)$ available by interior point methods if $d = o( n^4)$. Further we provide similar solvers for DC regression and Bregman divergence learning. Unlike previous approaches, our method is amenable to the use of GPUs. We demonstrate on regression and metric learning experiments that our approach is over 100 times faster than existing approaches on some data sets, and produces results that are comparable to state of the art.

Author Information

Ali Siahkamari (Boston University)
Durmus Alp Emre Acar (Boston University)
Christopher Liao (Boston University)
Kelly Geyer (Boston University)
Venkatesh Saligrama (Boston University)
Brian Kulis (Boston University and Amazon)

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