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A Stochastic Interpretation of Stochastic Mirror Descent: Risk-Sensitive Optimality

Azizan, Navid and Hassibi, Babak (2019) A Stochastic Interpretation of Stochastic Mirror Descent: Risk-Sensitive Optimality. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20190628-084200896

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Abstract

Stochastic mirror descent (SMD) is a fairly new family of algorithms that has recently found a wide range of applications in optimization, machine learning, and control. It can be considered a generalization of the classical stochastic gradient algorithm (SGD), where instead of updating the weight vector along the negative direction of the stochastic gradient, the update is performed in a "mirror domain" defined by the gradient of a (strictly convex) potential function. This potential function, and the mirror domain it yields, provides considerable flexibility in the algorithm compared to SGD. While many properties of SMD have already been obtained in the literature, in this paper we exhibit a new interpretation of SMD, namely that it is a risk-sensitive optimal estimator when the unknown weight vector and additive noise are non-Gaussian and belong to the exponential family of distributions. The analysis also suggests a modified version of SMD, which we refer to as symmetric SMD (SSMD). The proofs rely on some simple properties of Bregman divergence, which allow us to extend results from quadratics and Gaussians to certain convex functions and exponential families in a rather seamless way.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1904.01855arXivDiscussion Paper
ORCID:
AuthorORCID
Azizan, Navid0000-0002-4299-2963
Additional Information:This work was supported in part by the National Science Foundation under grants CCF-1423663, CCF-1409204 and ECCS-1509977, by a grant from Qualcomm Inc., by NASA’s Jet Propulsion Laboratory through the President and Director’s Fund, and by an Amazon (AWS) AI Fellowship.
Funders:
Funding AgencyGrant Number
NSFCCF-1423663
NSFCCF-1409204
NSFECCS-1509977
Qualcomm Inc.UNSPECIFIED
JPL President and Director's FundUNSPECIFIED
Amazon Web ServicesUNSPECIFIED
Record Number:CaltechAUTHORS:20190628-084200896
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190628-084200896
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:96809
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:28 Jun 2019 17:10
Last Modified:03 Oct 2019 21:25

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