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Conformal mirror descent with logarithmic divergences.

Amanjit Singh Kainth1,2, Ting-Kam Leonard Wong3, Frank Rudzicz1,2

  • 1Department of Computer Science, University of Toronto, Toronto, Canada.

Information Geometry
|January 1, 2024
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Summary

We introduce conformal mirror descent, a novel algorithm extending Bregman divergence for optimal transport and convex duality. This method offers new insights into continuous-time optimization and online estimation for generalized exponential families.

Keywords:
Conformal Hessian metricGradient flowLogarithmic divergenceMirror descent

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Area of Science:

  • Optimization Theory
  • Machine Learning
  • Information Geometry

Background:

  • Logarithmic divergence extends Bregman divergence, drawing from optimal transport and generalized convex duality.
  • This divergence possesses notable mathematical properties and induces a unique geometric structure.

Purpose of the Study:

  • To introduce conformal mirror descent, a generalization of continuous-time mirror descent.
  • To analyze the dynamics and convergence properties of this new algorithm.
  • To apply the method to online estimation and gradient flow construction.

Main Methods:

  • Utilizing the geometry induced by logarithmic divergence.
  • Deriving the dynamics of conformal mirror descent under a generalized mirror map.
  • Proving continuous-time convergence results.
  • Applying the algorithm to generalized exponential family estimation.

Main Results:

  • Conformal mirror descent is shown to be a time change of a Hessian gradient flow.
  • Convergence in continuous time is established.
  • The method is successfully applied to online estimation and constructing gradient flows on the unit simplex via Dirichlet optimal transport.

Conclusions:

  • Conformal mirror descent provides a powerful new tool for continuous-time optimization problems.
  • The framework offers novel approaches to online learning and geometric analysis in statistical models.
  • The connection to optimal transport and gradient flows opens avenues for further research.