Faster Algorithms for Learning Convex Functions

TL;DR

This paper introduces a faster algorithm for convex function learning using a 2-block ADMM method, significantly improving convergence rates and enabling GPU acceleration, with practical benefits demonstrated in regression and metric learning tasks.

Contribution

The paper develops a novel ADMM-based approach for convex function learning that outperforms existing methods in speed and scalability, especially with GPU compatibility.

Findings

Converges with rate $O(nrac{ ext{sqrt}(d)}{ ext{epsilon}})$ for convex Lipschitz regression.

Achieves faster overall convergence than previous methods, especially when $d = o(n^4)$.

Demonstrates over 100x speedup in experiments while maintaining comparable accuracy.

Abstract

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 $\bm X \in \mathbb R^{n\times d}$ 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 $O(n^5d^2/\epsilon)$ 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.