Mathematical Analysis of Gradient Descent Convergence
Abstract
We present a rigorous mathematical analysis of gradient descent convergence rates under various smoothness conditions. Our main theorem establishes that for $L$-smooth functions, the convergence rate is $\mathcal{O}(1/k)$ where $k$ is the iteration number.
Keywords
Details
Mathematical Analysis of Gradient Descent Convergence
Overview
This paper provides a comprehensive mathematical analysis of gradient descent algorithms, focusing on convergence guarantees under different smoothness assumptions.
Main Results
Theorem 1: Convergence Rate for Smooth Functions
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be an $L$-smooth function. Then the gradient descent algorithm with step size $\alpha = \frac{1}{L}$ satisfies:
\[f(x_k) - f(x^*) \leq \frac{L \|x_0 - x^*\|^2}{2k}\]where $x^*$ is the global minimum and $k$ is the iteration number.
Corollary 1.1: Linear Convergence for Strongly Convex Functions
If $f$ is additionally $\mu$-strongly convex, then:
\[\|x_k - x^*\|^2 \leq \left(1 - \frac{\mu}{L}\right)^k \|x_0 - x^*\|^2\]This gives us a linear convergence rate with condition number $\kappa = \frac{L}{\mu}$.
Key Technical Contributions
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Smoothness Analysis: We extend classical smoothness conditions to handle non-Euclidean norms, showing that convergence holds for general norm balls $B_p(x, r) = {y : |y - x|_p \leq r}$.
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Adaptive Step Sizes: Our analysis covers adaptive step size strategies where $\alpha_k = \frac{c}{\sqrt{k}}$ for some constant $c > 0$.
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Stochastic Extensions: The results extend to the stochastic setting where we observe $\nabla f(x_k) + \epsilon_k$ with noise $\epsilon_k$.
Mathematical Framework
Smoothness Definition
A function $f$ is $L$-smooth if for all $x, y \in \mathbb{R}^d$: \(\|\nabla f(x) - \nabla f(y)\| \leq L \|x - y\|\)
This is equivalent to the quadratic upper bound: \(f(y) \leq f(x) + \nabla f(x)^T (y - x) + \frac{L}{2} \|y - x\|^2\)
Proof Technique
Our proof uses the potential function approach. Define $\Phi_k = f(x_k) - f(x^*)$. Then:
\[\Phi_{k+1} \leq \Phi_k - \frac{\alpha}{2} \|\nabla f(x_k)\|^2 + \frac{\alpha^2 L}{2} \|\nabla f(x_k)\|^2\]Setting $\alpha = \frac{1}{L}$ gives: \(\Phi_{k+1} \leq \Phi_k - \frac{1}{2L} \|\nabla f(x_k)\|^2\)
Computational Complexity
The algorithm requires $\mathcal{O}(\epsilon^{-1})$ iterations to achieve $f(x_k) - f(x^*) \leq \epsilon$, where each iteration costs $\mathcal{O}(d)$ operations for gradient computation.
For strongly convex functions, this improves to $\mathcal{O}(\kappa \log(\epsilon^{-1}))$ iterations.
Impact and Applications
This theoretical framework has been applied to:
- Machine learning optimization
- Signal processing algorithms
- Control theory problems
- Numerical analysis of PDEs
The convergence guarantees provide theoretical backing for practical implementations across these domains.
Citation
BibTeX
@article{academic2024mathematical,
title={Mathematical Analysis of Gradient Descent Convergence},
author={Academic, Dr. and Expert, Prof. Mathematics},
journal={Journal of Mathematical Optimization},
volume={15},
number={3},
pages={234--251},
year={2024},
publisher={Optimization Society}
}