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 approximating Bregman divergences. In this paper, we show how a broad class of convex function learning problems can be solved via a 2-block ADMM approach, where updates for 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 \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 $O(n^5d^2/\epsilon)$ 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 up to 30 times faster than the existing method, and produces results that are comparable to state-of-the-art.

翻译：类似任意 convex 函数的任务在几个学习问题中产生, 比如 convex 回归, 学习 convex (DC) 函数的差异, 以及相似的 Bregman 差异 。 在本文中, 我们展示了如何通过 2 块 ADMM 方法解决广泛的 convex 函数学习问题 。 在2 块 AdMM 方法中, 每个区块的更新可以以封闭的形式计算 。 关于 convex Lipschitz 回归的任务, 我们确定我们提议的算法与 $ O( n\ sqrt{d}/\ epsilon) 的循环复杂性相融合 。 这个新的算法可以改善 $X\ in mathbb R\ nn\ times d} $ 和 $\ eepslationalg roleg 的方法的状态。 与 $O3 d% d= rblead road 方法相比, road road road roads 提供 $n_ relegleglevelmental rences。