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Building cost functions minimizing to some summary statistics.

M Saerens1

  • 1IRIDIA Laboratory, Université Libre de Bruxelles, B-1050 Bruxelles, Belgium. saerens@ulb.ac.be

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Summary

This study explores how cost functions impact machine learning model outputs. It uses calculus of variations to ensure models approximate conditional expectation, median, geometric mean, and variance.

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Area of Science:

  • Machine Learning
  • Statistical Modeling
  • Calculus of Variations

Background:

  • Machine learning models are trained by minimizing cost functions, which measure discrepancies between predicted and actual outputs.
  • The choice of cost function significantly influences the probabilistic interpretation of a trained model's output.
  • Existing methods often focus on specific statistical properties, limiting the model's interpretability.

Purpose of the Study:

  • To investigate the relationship between cost functions and the probabilistic interpretation of machine learning model outputs.
  • To derive conditions on cost functions that ensure approximation of various statistical measures.
  • To explore alternative cost functions for nonlinear regression and discuss regression quantiles.

Main Methods:

  • Utilizing the calculus of variations to derive necessary and sufficient conditions for cost functions.
  • Analyzing the impact of different cost functions on approximating conditional expectation, median, q-quantile, geometric mean, and variance.
  • Applying the derived conditions to nonlinear regression scenarios.

Main Results:

  • Established conditions for cost functions to ensure trained models approximate conditional expectation, median (q-quantile), geometric mean, and variance.
  • Demonstrated the applicability of the calculus of variations method for estimating various summary statistics.
  • Highlighted the potential of least absolute deviations as an alternative to ordinary least squares in nonlinear regression.

Conclusions:

  • The choice of cost function is critical for the probabilistic interpretation of machine learning models.
  • Calculus of variations provides a robust framework for designing cost functions that yield desired statistical properties.
  • This research offers a unified approach to cost function selection for diverse statistical estimations in machine learning.