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Bounds on the Excess Minimum Risk via Generalized Information Divergence Measures.

Ananya Omanwar1, Fady Alajaji1, Tamás Linder1

  • 1Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada.

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Summary
This summary is machine-generated.

Researchers developed new upper bounds for estimating target vectors using generalized information divergence measures. These bounds improve upon existing methods by not requiring constant sub-Gaussian parameters, broadening applicability in machine learning and information theory.

Keywords:
Rényi divergenceSibson mutual informationexcess minimum riskinformation divergencesstatistical inferencesub-Gaussianityvariational characterizationsα-Jensen–Shannon divergence

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Area of Science:

  • Information Theory
  • Machine Learning
  • Statistical Inference

Background:

  • Estimating a target vector Y from observed data X or its degraded version Z is crucial in various fields.
  • The excess minimum risk quantifies the performance loss due to data degradation (Z vs. X).
  • Existing bounds often rely on mutual information and specific distribution assumptions.

Purpose of the Study:

  • To derive generalized upper bounds on the excess minimum risk for estimating Y from X or Z.
  • To introduce novel bounds using Rényi and α-Jensen-Shannon divergences, generalizing prior work.
  • To extend the applicability of these bounds to a wider range of joint distributions by relaxing constant sub-Gaussianity assumptions.

Main Methods:

  • Utilizing generalized information divergence measures, including Rényi and α-Jensen-Shannon divergences.
  • Analyzing a Markov chain Y→X→Z where Y is the target, X is the observed feature, and Z is the degraded version.
  • Developing theoretical bounds that do not assume a constant sub-Gaussian parameter.

Main Results:

  • A family of generalized upper bounds on the excess minimum risk was derived.
  • The new bounds are shown to be applicable to broader classes of joint distributions compared to previous methods.
  • Numerical examples demonstrate that the generalized divergence-based bounds can be tighter than mutual information-based bounds.

Conclusions:

  • The developed generalized information divergence measures provide tighter and more broadly applicable bounds for excess minimum risk.
  • These findings advance the understanding of information loss in data degradation and estimation tasks.
  • The relaxed assumptions on sub-Gaussianity enhance the utility of these bounds in practical machine learning scenarios.