The LSE function is convex and strictly increasing. Its gradient corresponds to the softmax function: $$ \frac{\partial}{\partial x_i} \text{LSE}(\mathbf{x}) = \frac{\exp(x_i)}{\sum_{j} \exp(x_j)} $$ This ensures smooth gradients and better propagation through the network. Lic...
convex optimizationmax-functionExamples of the Problem In a classic nonlinear program (NLP) a smooth objective function is minimized on a feasible set defined by finitely many smooth constraints. However, many optimization problems...Claudia Sagastizábal...
摘要: Examples of the ProblemIn a classic nonlinear program (NLP) a smooth objective function is minimized on a feasible set defined by finitely many smooth constraints. However, many optimization problems关键词: 49K35 49M27 65K10 90C25 minimax problem Convex max-functions Minimax: Directional ...
7.3Polynomial Activation Functions Smooth Adaptive AF (SAAF) is defined as the piecewise polynomial function[105]. Two power functions symmetric to the linear part of ReLU are combined in[106]to improve the performance of ReLU. A piecewise polynomial approximation based AF is also learnt from the...
Vaidya, P.M.: A new algorithm for minimizing convex functions over convex sets. In: 30th Annual Symposium on Foundations of Computer Science, pp. 338–343. IEEE Computer Society (1989) Google Scholar Vaidya, P.M.: A new algorithm for minimizing convex functions over convex sets. Math. Pr...
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... NL Roux,M Schmidt,F Bach - 《Advances in Neural Information Processing Systems》 被引量: 595发表: 2013年 Game theory, maximum entr...
We provide several algorithms that are effective for very large scale problems, and we demonstrate the power of the max-norm regularizer using examples from a variety of applications. In particular, we study convex programs of the form min f(X) + μ kXkmax (2) where f is a smooth ...
that a variant of the widely used Gradient Descent/Ascent procedure, called "Optimistic Gradient Descent/Ascent (OGDA)", exhibits last-iterate convergence to saddle points in {\\em unconstrained} convex-concave min-max optimization ... C Daskalakis,I Panageas 被引量: 1发表: 2018年 Conic Geome...
Of particular importance is the epigraph of such functions: \({\text{ epi }\, f}:=\left\{ (x,\mu )\,\left| \,f(x)\le \mu \right. \right\} \). We have that f is lsc if and only if \({\text{ epi }\, f}\) is closed, and f is convex if and only if \({\text...
A natural candidate for this loss is the convex closure of function (6) in Rp. In general, computing the convex closure of set functions is NP-hard. However, the Jaccard set function (6) has been shown to be submodular [27, Proposition 11]. Definition 1 [9]. A set function ∆ :...