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Intuitive Analysis of the Quantization-based Optimization: From Stochastic and Quantum Mechanical Perspective

Jinwuk Seok, Changsik Cho

TL;DR

The paper addresses global optimization of non-convex functions by introducing a quantization-based optimization method and a physics-inspired analysis. It formulates an overdamped Langevin SDE driven by a time-varying quantization parameter $Q_p(t)$, and derives a corresponding Fokker-Planck equation and a Witten-Laplacian to connect thermodynamic and quantum-mechanical perspectives. The proposed approach models quantization error as $f^Q(x_t) = f(x_t) + Q_t \varepsilon_t$, yielding an annealing-like diffusion that facilitates escape from local minima, with a Boltzmann-Gibbs stationary distribution in the continuous limit. Experiments on standard benchmarks show competitive performance against simulated and quantum annealing, and the results motivate a tunneling-based interpretation of optimization dynamics. The work offers a physics-grounded framework that could inform future integration with learning dynamics in machine learning.

Abstract

In this paper, we present an intuitive analysis of the optimization technique based on the quantization of an objective function. Quantization of an objective function is an effective optimization methodology that decreases the measure of a level set containing several saddle points and local minima and finds the optimal point at the limit level set. To investigate the dynamics of quantization-based optimization, we derive an overdamped Langevin dynamics model from an intuitive analysis to minimize the level set by iterative quantization. We claim that quantization-based optimization involves the quantities of thermodynamical and quantum mechanical optimization as the core methodologies of global optimization. Furthermore, on the basis of the proposed SDE, we provide thermodynamic and quantum mechanical analysis with Witten-Laplacian. The simulation results with the benchmark functions, which compare the performance of the nonlinear optimization, demonstrate the validity of the quantization-based optimization.

Intuitive Analysis of the Quantization-based Optimization: From Stochastic and Quantum Mechanical Perspective

TL;DR

The paper addresses global optimization of non-convex functions by introducing a quantization-based optimization method and a physics-inspired analysis. It formulates an overdamped Langevin SDE driven by a time-varying quantization parameter , and derives a corresponding Fokker-Planck equation and a Witten-Laplacian to connect thermodynamic and quantum-mechanical perspectives. The proposed approach models quantization error as , yielding an annealing-like diffusion that facilitates escape from local minima, with a Boltzmann-Gibbs stationary distribution in the continuous limit. Experiments on standard benchmarks show competitive performance against simulated and quantum annealing, and the results motivate a tunneling-based interpretation of optimization dynamics. The work offers a physics-grounded framework that could inform future integration with learning dynamics in machine learning.

Abstract

In this paper, we present an intuitive analysis of the optimization technique based on the quantization of an objective function. Quantization of an objective function is an effective optimization methodology that decreases the measure of a level set containing several saddle points and local minima and finds the optimal point at the limit level set. To investigate the dynamics of quantization-based optimization, we derive an overdamped Langevin dynamics model from an intuitive analysis to minimize the level set by iterative quantization. We claim that quantization-based optimization involves the quantities of thermodynamical and quantum mechanical optimization as the core methodologies of global optimization. Furthermore, on the basis of the proposed SDE, we provide thermodynamic and quantum mechanical analysis with Witten-Laplacian. The simulation results with the benchmark functions, which compare the performance of the nonlinear optimization, demonstrate the validity of the quantization-based optimization.
Paper Structure (9 sections, 2 theorems, 13 equations, 2 tables)

This paper contains 9 sections, 2 theorems, 13 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition and Assumption
  4. Primitive Analysis of Quantization-based Optimization Algorithm
  5. Intuitive Analysis of the Objective Quantization
  6. Association to the Stochastic Differential Equation(SDE)
  7. Quantum Mechanical Quantities of the Quantization-Based Optimization
  8. Experimental Result
  9. Conclusion

Key Result

Lemma 1

Given the candidate of the search equation as sde-eq01, suppose that the virtual objective function satisfying Assumption assum_02 for the condition $f^Q(\boldsymbol{x}_{t+1}) = f^Q(\boldsymbol{x}_{t})$, before the $Q_p(t)$ is updated. Then, the norm of the random vector $\boldsymbol{r}_{t}$ satisfi

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Theorem 2