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Tensor methods for finding approximate stationary points of convex functions.

G N Grapiglia1, Yurii Nesterov2

  • 1Departamento de Matemática, Universidade Federal do Paraná, Curitiba, Brazil.

Optimization Methods & Software
|February 27, 2025
PubMed
Summary

This study introduces tensor methods for finding approximate stationary points in convex functions. It establishes new iteration complexity bounds for both accelerated and non-accelerated schemes, improving efficiency in optimization.

Keywords:
49M1549M3758C1590C2590C30Hölder conditionUnconstrained minimizationhigh-order methodstensor methodsworst-case complexity

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Area of Science:

  • Optimization Theory
  • Numerical Analysis
  • Convex Analysis

Background:

  • Finding approximate stationary points is crucial for solving optimization problems.
  • Convex functions with p-times differentiable and nu-Hölder continuous pth derivatives are common in machine learning and applied mathematics.
  • Existing methods may lack efficiency for high-order derivatives.

Purpose of the Study:

  • To develop and analyze tensor methods for finding epsilon-approximate stationary points of convex functions.
  • To establish improved iteration complexity bounds for these methods, considering accelerated and non-accelerated schemes.
  • To investigate the impact of knowing or not knowing the Hölder parameter nu.

Main Methods:

  • Development of non-accelerated tensor methods.
  • Development of accelerated tensor schemes.
  • Analysis of iteration complexity bounds based on function properties (p-times differentiability, nu-Hölder continuity).

Main Results:

  • Non-accelerated schemes achieve a gradient norm reduction below epsilon in at most O(1/epsilon^(2/p)) iterations.
  • Accelerated tensor schemes achieve improved complexity bounds of O(1/epsilon^(1/p)) when nu is known.
  • A universal accelerated scheme achieves O(1/epsilon^(2/(2p-1))) complexity when nu is unknown, and a lower bound of O(1/epsilon^(1/p)) is established.

Conclusions:

  • The proposed tensor methods offer efficient solutions for finding approximate stationary points of convex functions.
  • The established complexity bounds demonstrate significant improvements, especially for accelerated schemes.
  • The research provides theoretical guarantees for the performance of these optimization algorithms.