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Deep Neural Networks for Image-Based Dietary Assessment
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Gradient Descent Learning With Floats.

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    Summary
    This summary is machine-generated.

    Gradient descent, crucial for AI, is analyzed in floating-point domains. New convergence rates are established for smooth and PŁ conditions, accounting for machine precision limitations.

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    Area of Science:

    • Artificial Intelligence
    • Machine Learning
    • Numerical Analysis

    Background:

    • Gradient descent is fundamental to AI and machine learning.
    • Existing theory assumes continuous domains, not practical for computer float point arithmetic.
    • Current models are insufficient for low-precision computing environments.

    Purpose of the Study:

    • Analyze gradient descent performance in floating-point domains.
    • Investigate convergence rates for smooth and PŁ objective functions.
    • Determine iteration bounds considering machine epsilon.

    Main Methods:

    • Theoretical analysis of gradient descent algorithms.
    • Performance evaluation of three gradient descent variants in floating-point arithmetic.
    • Derivation of convergence bounds for deterministic and stochastic cases.

    Main Results:

    • Established iteration complexity of O(1/ϵ) for general smooth functions and O(ln(1/ϵ)) for PŁ functions.
    • Proved error bounds dependent on machine epsilon (δ(s)) for deterministic (ϵ ≥ Ω(δ(s))) and stochastic (ϵ ≥ Ω(√{δ(s)})) cases.
    • Showed floating stochastic and sign gradient descents achieve O(1/ϵ²) iterations.

    Conclusions:

    • The study provides a theoretical understanding of gradient descent with floats.
    • Convergence rates are improved under PŁ conditions, considering practical computational limits.
    • Findings are crucial for AI/ML applications using low-precision arithmetic.