We propose a new iteration scheme, the Cauchy-Simplex, to optimize convex problems over the probability simplex $\{w\in\mathbb{R}^n\ |\ \sum_i w_i=1\ \textrm{and}\ w_i\geq0\}$. Other works have taken steps to enforce positivity or unit normalization automatically but never simultaneously within a unified setting. This paper presents a natural framework for manifestly requiring the probability condition. Specifically, we map the simplex to the positive quadrant of a unit sphere, envisage gradient descent in latent variables, and map the result back in a way that only depends on the simplex variable. Moreover, proving rigorous convergence results in this formulation leads inherently to tools from information theory (e.g. cross entropy and KL divergence). Each iteration of the Cauchy-Simplex consists of simple operations, making it well-suited for high-dimensional problems. We prove that it has a convergence rate of ${O}(1/T)$ for convex functions, and numerical experiments of projection onto convex hulls show faster convergence than similar algorithms. Finally, we apply our algorithm to online learning problems and prove the convergence of the average regret for (1) Prediction with expert advice and (2) Universal Portfolios.

翻译：我们提出了一种新的迭代方案，即柯西-单纯形法，用于在概率单纯形$\{w\in\mathbb{R}^n\ |\ \sum_i w_i=1\ \textrm{且}\ w_i\geq0\}$上优化凸问题。已有工作曾尝试自动强制实现非负性或单位归一化，但从未在统一框架中同时实现这两者。本文提出了一种自然框架，以显式满足概率条件：我们将单纯形映射到单位球的正象限，在潜在变量中实施梯度下降，并以仅依赖于单纯形变量的方式将结果映射回来。此外，在该框架下建立严格的收敛性结果必然依赖于信息论工具（如交叉熵与KL散度）。柯西-单纯形法的每次迭代仅包含简单运算，因此特别适合高维问题。我们证明了其对于凸函数具有${O}(1/T)$的收敛速率，且凸包投影的数值实验表明其收敛速度快于同类算法。最后，我们将该算法应用于在线学习问题，并证明了以下场景中平均遗憾值的收敛性：（1）专家意见预测；（2）通用投资组合。