This paper initiates a systematic development of a theory of non-commutative optimization. It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesically convex optimization problems on Riemannian manifolds that arise from the symmetries of non-commutative groups. These algorithms minimize the moment map (a non-commutative notion of the usual gradient) and test membership in null cones and moment polytopes (a vast class of polytopes, typically of exponential vertex and facet complexity, which arise from this a priori non-convex, non-linear setting). This setting captures a diverse set of problems in different areas of computer science, mathematics, and physics. Several of them were solved efficiently for the first time using non-commutative methods; the corresponding algorithms also lead to solutions of purely structural problems and to many new connections between disparate fields. In the spirit of standard convex optimization, we develop two general methods in the geodesic setting, a first order and a second order method, which respectively receive first and second order information on the "derivatives" of the function to be optimized. These in particular subsume all past results. The main technical work goes into identifying the key parameters of the underlying group actions which control convergence to the optimum in each of these methods. These non-commutative analogues of "smoothness" are far more complex and require significant algebraic and analytic machinery. Despite this complexity, the way in which these parameters control convergence in both methods is quite simple and elegant. We show how to bound these parameters and hence obtain efficient algorithms for null cone membership in several concrete situations. Our work points to intriguing open problems and suggests further research directions.
翻译:本文启动系统开发非对调优化理论。 它旨在统一和概括过去几年中越来越多的工作, 包括开发和分析来自非对称组的对称性产生的里曼形元体上自然大地测量 Convex优化问题的算法。 这些算法将瞬时图( 通常梯度的非对称概念) 和空锥体和瞬时多元形体( 一种庞大的多面形参数, 通常是指数性垂直参数和表面复杂性, 产生于这一前置非相交的、 非线性趋同设置 ) 。 设置这样的设置可以捕捉到计算机科学、 数学和物理 不同领域的多种问题。 其中一些是首次使用非对称方法有效解决的; 相应的算法还导致纯结构问题的解决, 以及不同字段之间的许多新联系。 在标准的对称正向优化的状态下, 我们开发了两种一般的方法, 一种是简单的正序, 一种是非序的, 一种是非序法系, 两种方法, 分别在计算机科学、 数学和后导法 中, 需要先行的对着这些对等的系统进行最优化的操作, 。 这些对着这些对等系进行最精确的动作, 。 这些技术的动作, 这些对等系的动作, 这些对着, 这些对着的动作, 将显示的动作的动作, 这些对着的动作, 这些操作的动作, 都进行着, 将进行着这些对着,, 这些操作法, 这些操作的每个操作法, 向, 都进行着这些对着这些操作法, 。