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An incremental mirror descent subgradient algorithm with random sweeping and proximal step.

Radu Ioan Boţ1, Axel Böhm1

  • 1Faculty of Mathematics, University of Vienna, Vienna, Austria.

Optimization
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PubMed
Summary
This summary is machine-generated.

This study introduces efficient incremental mirror descent algorithms for minimizing sums of convex functions. These methods offer fast convergence rates, demonstrated through applications in medical imaging and machine learning.

Keywords:
90C0690C2590C90Nonsmooth convex minimizationglobal rate of convergenceincremental mirror descent algorithmrandom sweeping

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Area of Science:

  • Optimization Algorithms
  • Convex Analysis
  • Machine Learning Theory

Background:

  • Minimizing sums of convex functions is crucial in many scientific fields.
  • Existing methods can be computationally expensive due to full gradient evaluations.
  • Incremental algorithms offer a more efficient alternative by processing components sequentially.

Purpose of the Study:

  • To develop and analyze novel incremental mirror descent subgradient algorithms.
  • To establish theoretical convergence guarantees for these algorithms.
  • To demonstrate the practical applicability of the proposed methods.

Main Methods:

  • Investigating convergence properties of incremental mirror descent subgradient algorithms.
  • Utilizing randomized component function selection for analysis.
  • Incorporating supplementary differentiability assumptions for proximal steps.

Main Results:

  • Derived convergence rates of O(1/k) in expectation for the kth best objective function value.
  • Showcased the computational efficiency of evaluating single component subgradients.
  • Validated theoretical findings through numerical experiments.

Conclusions:

  • The proposed incremental mirror descent algorithms provide an efficient approach for minimizing sums of convex functions.
  • These algorithms exhibit favorable convergence properties, making them suitable for large-scale problems.
  • Successful application in positron emission tomography and machine learning highlights their practical utility.