接下来,我分两个情况来讨论收敛性:1.Convex。2. Strongly convex。 1.1.convex case 定理1.1(Nonsmooth + convex)如果函数 f 是凸的且是Lipschitzness的。对于迭代方法(1.1),步长选择策略为: \alpha_k =\frac{f(x^k) - f^*}{\|g^k\|^2} 如果g^k \neq 0 ,否则 \alpha_k = 1 。那么我们有:...
注意到前面的基本假设可以分成目标函数的基本假设(梯度lipschitz连续、Strongly Convex)以及抽样梯度 g 的几个条件。在这些条件下,我们可以对随机梯度下降法的收敛性进行分析。 首先,我们假设第 k 步的时候,优化变量的值 w_k,采样得到的 \xi 为\xi_k。于是当前步的随机梯度为 g(w_k, \xi_k)。令学习率固定...
严格凸(Strictly convex):对于任意$x \neq y$,且$0<t<1$,有$f(tx+(1-t)y)<tf(x)+(1-t)f(y)$。简而言之,就是$f$比线性函数要更弯曲 强凸(Strongly convex):对于参数$m>0$:$f-\frac{m}{2}||x||^2_2$依旧是一个凸函数。简而言之就是$f$要比一般的二次函数要弯曲。 强凸$\Righta...
In this paper a robust second-order method is developed for the solution of strongly convex l1-regularized problems. The main aim is to make the proposed method as inexpensive as possible, while even difficult problems can be efficiently solved. The proposed approach is a primal-dual Newton Conj...
We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard st... Nemirovski,Arkadi - 《Siam Journal on Optimization》 被引量: 506发表: 2004年 Accelerated Gradient Methods for Nonconvex Nonlinear and...
Aaron Defazio, Francis Bach, Simon Lacoste-Julien: SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives. https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm ...
Convex optimizationWe propose a simple variant of the generalized Frank-Wolfe method for solving strongly convex composite optimization problems, by introducing an additional averaging step on the dual variables. We show that in this variant, one can choose a simple constant step-size and obtain a ...
Geometry, Convex hull Discrete math Machine Learning Additional Detail on Some Subjects Video Series Computer Science Courses Why use it? When I started this project, I didn't know a stack from a heap, didn't know Big-O anything, anything about trees, or how to traverse a graph. If I ...
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定义1.2:称函数 f 是m-强凸(m-strongly convex)的,如果有 f(x)≥f(y)+∇f(y)T(x−y)+m2||x−y||2. 注意到, f(x)≥f(y)+∇f(y)T(x−y) 是凸函数的定义,强凸则说明下界可以更强一些。关于L-光滑性,我们还有下面的一些结论。 定理1.2:如果函数 f(x) 是L-光滑的,且 x∗...