Stochastic Mirror Descent with Inexact Prox - Mapping in Density
نویسندگان
چکیده
Appendix A Strong convexity As we discussed, the posterior from Bayes’s rule could be viewed as the optimal of an optimization problem in Eq (1). We will show that the objective function is strongly convex w.r.t KL-divergence. Proof for Lemma 1. The lemma directly results from the generalized Pythagaras theorem for Bregman divergence. Particularly, for KL-divergence, we have KL(q 1 ||q) = KL(q 1 ||q 2 ) +KL(q 2 ||q) hq 1 q 2 ,r (q) r (q 2 )i 2
منابع مشابه
Provable Bayesian Inference via Particle Mirror Descent
Since the prox-mapping of stochastic mirror descent is intractable when directly being applied to the optimization problem (1), we propose the -inexact prox-mapping within the stochastic mirror descent framework in Section 3. Instead of solving the prox-mapping exactly, we approximate the solution with error. In this section, we will show as long as the approximation error is tolerate, the stoc...
متن کاملStochastic Optimization with Importance Sampling for Regularized Loss Minimization
Uniform sampling of training data has been commonly used in traditional stochastic optimization algorithms such as Proximal Stochastic Mirror Descent (prox-SMD) and Proximal Stochastic Dual Coordinate Ascent (prox-SDCA). Although uniform sampling can guarantee that the sampled stochastic quantity is an unbiased estimate of the corresponding true quantity, the resulting estimator may have a rath...
متن کاملVariance-Reduced Proximal Stochastic Gradient Descent for Non-convex Composite optimization
Here we study non-convex composite optimization: first, a finite-sum of smooth but non-convex functions, and second, a general function that admits a simple proximal mapping. Most research on stochastic methods for composite optimization assumes convexity or strong convexity of each function. In this paper, we extend this problem into the non-convex setting using variance reduction techniques, ...
متن کاملTheory of Convex Optimization for Machine Learning
This monograph presents the main mathematical ideas in convex optimization. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by the seminal book of Nesterov, includes the analysis of the Ellipsoid Method, as well a...
متن کاملStochastic Optimization with Importance Sampling
Uniform sampling of training data has been commonly used in traditional stochastic optimization algorithms such as Proximal Stochastic Gradient Descent (prox-SGD) and Proximal Stochastic Dual Coordinate Ascent (prox-SDCA). Although uniform sampling can guarantee that the sampled stochastic quantity is an unbiased estimate of the corresponding true quantity, the resulting estimator may have a ra...
متن کامل