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Gradient Descent Efficiency Index

Aviral Dhingra

TL;DR

A new efficiency metric, Ek, designed to quantify the effectiveness of each iteration of gradient descent is introduced, which has the potential to guide more informed decisions in the selection and tuning of optimization algorithms in machine learning applications.

Abstract

Gradient descent is a widely used iterative algorithm for finding local minima in multivariate functions. However, the final iterations often either overshoot the minima or make minimal progress, making it challenging to determine an optimal stopping point. This study introduces a new efficiency metric, Ek, designed to quantify the effectiveness of each iteration. The proposed metric accounts for both the relative change in error and the stability of the loss function across iterations. This measure is particularly valuable in resource-constrained environments, where costs are closely tied to training time. Experimental validation across multiple datasets and models demonstrates that Ek provides valuable insights into the convergence behavior of gradient descent, complementing traditional performance metrics. The index has the potential to guide more informed decisions in the selection and tuning of optimization algorithms in machine learning applications and be used to compare the "effectiveness" of models relative to each other.

Gradient Descent Efficiency Index

TL;DR

A new efficiency metric, Ek, designed to quantify the effectiveness of each iteration of gradient descent is introduced, which has the potential to guide more informed decisions in the selection and tuning of optimization algorithms in machine learning applications.

Abstract

Gradient descent is a widely used iterative algorithm for finding local minima in multivariate functions. However, the final iterations often either overshoot the minima or make minimal progress, making it challenging to determine an optimal stopping point. This study introduces a new efficiency metric, Ek, designed to quantify the effectiveness of each iteration. The proposed metric accounts for both the relative change in error and the stability of the loss function across iterations. This measure is particularly valuable in resource-constrained environments, where costs are closely tied to training time. Experimental validation across multiple datasets and models demonstrates that Ek provides valuable insights into the convergence behavior of gradient descent, complementing traditional performance metrics. The index has the potential to guide more informed decisions in the selection and tuning of optimization algorithms in machine learning applications and be used to compare the "effectiveness" of models relative to each other.

Paper Structure

This paper contains 27 sections, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Momentum
  4. AdaGrad
  5. RMSProp
  6. Adam
  7. Nesterov Accelerated Gradient (NAG)
  8. Other Algorithms and Variants
  9. Derivation of GDEI
  10. External Parameters
  11. Mean Squared Error (MSE)
  12. Proportion of Initial Loss Reduced ($P_k$)
  13. Absolute Change in Loss Function ($\Delta_k$)
  14. Formula & Theoretical Explanation
  15. Logarithmic Term $\log\left(1 + \Delta_k^2\right)$
  16. ...and 12 more sections

Figures (3)

  • Figure 1: Contour plot of cost b,w with path of gradient descent $J(w,b)$.
  • Figure 2: Comparison of cost with different domain restrictions.
  • Figure 3: Line graph of efficiency with respect to iterations in a sample dataset.