Behind the Scenes of Gradient Descent: A Trajectory Analysis via Basis Function Decomposition

TL;DR

This paper introduces a basis function decomposition to analyze gradient descent trajectories, revealing monotonic behavior in the coefficients and improving convergence results for matrix and tensor factorizations, with empirical validation on deep neural networks.

Contribution

It proposes a novel basis function decomposition framework for analyzing gradient descent trajectories, leading to new convergence results and insights into neural network training.

Findings

Gradient descent trajectories are nearly monotonic when projected onto an appropriate basis.

The framework improves convergence analysis for symmetric matrix factorization.

Empirical results show monotonic learning of basis coefficients in deep neural networks.

Abstract

This work analyzes the solution trajectory of gradient-based algorithms via a novel basis function decomposition. We show that, although solution trajectories of gradient-based algorithms may vary depending on the learning task, they behave almost monotonically when projected onto an appropriate orthonormal function basis. Such projection gives rise to a basis function decomposition of the solution trajectory. Theoretically, we use our proposed basis function decomposition to establish the convergence of gradient descent (GD) on several representative learning tasks. In particular, we improve the convergence of GD on symmetric matrix factorization and provide a completely new convergence result for the orthogonal symmetric tensor decomposition. Empirically, we illustrate the promise of our proposed framework on realistic deep neural networks (DNNs) across different architectures, gradient-based solvers, and datasets. Our key finding is that gradient-based algorithms monotonically learn the coefficients of a particular orthonormal function basis of DNNs defined as the eigenvectors of the conjugate kernel after training. Our code is available at https://github.com/jianhaoma/function-basis-decomposition.