Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent

Suriya Gunasekar, Blake Woodworth, Nathan Srebro
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:2305-2313, 2021.

Abstract

We present a direct (primal only) derivation of Mirror Descent as a “partial” discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential function. We contrast this discretization to Natural Gradient Descent, which is obtained by a “full” forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to any Riemannian geometry in $\mathbb{R}^d$, even when the metric tensor is not a Hessian, and thus there is no “dual.”

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-gunasekar21a, title = { Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent }, author = {Gunasekar, Suriya and Woodworth, Blake and Srebro, Nathan}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {2305--2313}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/gunasekar21a/gunasekar21a.pdf}, url = {https://proceedings.mlr.press/v130/gunasekar21a.html}, abstract = { We present a direct (primal only) derivation of Mirror Descent as a “partial” discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential function. We contrast this discretization to Natural Gradient Descent, which is obtained by a “full” forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to any Riemannian geometry in $\mathbb{R}^d$, even when the metric tensor is not a Hessian, and thus there is no “dual.” } }
Endnote
%0 Conference Paper %T Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent %A Suriya Gunasekar %A Blake Woodworth %A Nathan Srebro %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-gunasekar21a %I PMLR %P 2305--2313 %U https://proceedings.mlr.press/v130/gunasekar21a.html %V 130 %X We present a direct (primal only) derivation of Mirror Descent as a “partial” discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential function. We contrast this discretization to Natural Gradient Descent, which is obtained by a “full” forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to any Riemannian geometry in $\mathbb{R}^d$, even when the metric tensor is not a Hessian, and thus there is no “dual.”
APA
Gunasekar, S., Woodworth, B. & Srebro, N.. (2021). Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:2305-2313 Available from https://proceedings.mlr.press/v130/gunasekar21a.html.

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