小于一次通过：随机控制的随机梯度法

凸与非凸复合问题的随机控制随机梯度

Stochastically Controlled Stochastic Gradient for the Convex and Non-convex Composition problem

本文研究了具有$\frac{1}{n}\sum\nolimits_{i = 1}^n {{F_i}( {G( x )} )}$结构的凸与非凸复合问题，其中$G( x )=\frac{1}{n}\sum\nolimits_{j = 1}^n {{G_j}( x )} $为内函数，$F_i(\cdot)$为外函数.我们探索基于方差缩减的方法来解决组合优化。由于内函数和外函数的个数较多时，直接估计它们是不合理的，因此我们采用随机控制的随机梯度（SCSG）方法来估计复合函数的梯度和内函数的值.我们提出的方法的查询复杂度的凸和非凸的问题是等于或优于目前的方法的组合问题。此外，我们还提出了所提出的方法的小批量版本，这与小批量的大小有关的查询复杂度的改善。

Escaping Saddle Points with Stochastically Controlled Stochastic Gradient Methods

Stochastically controlled stochastic gradient (SCSG) methods have been proved to converge efficiently to first-order stationary points which, however, can be saddle points in nonconvex optimization. It has been observed that a stochastic gradient descent (SGD) step introduces anistropic noise around saddle points for deep learning and non-convex half space learning problems, which indicates that SGD satisfies the correlated negative curvature (CNC) condition for these problems. Therefore, we propose to use a separate SGD step to help the SCSG method escape from strict saddle points, resulting …

随机扰动动力学中的受控平衡选择

Controlled equilibrium selection in stochastically perturbed dynamics

我们考虑一个动力系统，除了被一个“昂贵的”控制，还具有许多平衡点和小噪声扰动。被控过程是最优的遍历标准的运行成本，包括控制努力和状态空间上的罚函数的总和。我们研究的最佳平稳分布的受控过程的方差的噪声变得非常小。它示出，根据噪声方差和“运行成本”的控制的相对大小，可以确定三个制度，在每一个其中的最优控制的力量的不变分布的过程中，可以根据政权的特点，集中在平衡附近。我们还获得了最佳平稳分布的矩界。此外，我们表明，在附近的点的浓度的最佳平稳分布的密度近似的高斯密度，我们明确解决其协方差阵。

Conditional gradient methods for stochastically constrained convex minimization

We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of combinatorial problems, which involve a number of constraints that is polynomial in the problem dimension. The most important feature of our framework is that only a subset of the constraints is processed at each iteration, thus gaining a computational advantage over prior works that require full passes. Our algorithms rely on variance reduction and smoothing used in c…

Escaping strict saddle points of the Moreau envelope in nonsmooth optimization

Recent work has shown that stochastically perturbed gradient methods can efficiently escape strict saddle points of smooth functions. We extend this body of work to nonsmooth optimization, by analyzing an inexact analogue of a stochastically perturbed gradient method applied to the Moreau envelope. The main conclusion is that a variety of algorithms for nonsmooth optimization can escape strict saddle points of the Moreau envelope at a controlled rate. The main technical insight is that typical algorithms applied to the proximal subproblem yield directions that approximate the gradient of the M…

信息约束下受控随机非线性系统的不变性

Invariance Properties of Controlled Stochastic Nonlinear Systems under Information Constraints

给定一个随机非线性系统，在一个可能存在噪声的通信信道上被控制，本文研究了最大类信道存在编码和控制策略，使得闭环系统是随机稳定的。考虑的稳定性标准是渐近平均平稳性（AMS）。我们开发了一个通用的方法的基础上遍历论和概率来获得基本的信息传输的要求，导致稳定的界限。通过这种方法，我们开发了一个新的熵的概念，这是量身定制的无噪声和有噪信道的渐近平均平稳性的下界。通过概率和遍历理论的分析得到的界限更精细，通过信息理论的方法得到的界限相比。此外，我们的方法是更通用的考虑模型，并允许更精细的下界时，AMS措施是已知的承认进一步的属性，如矩界。