Gradient flow in parameter space is equivalent to linear interpolation in output space

2024-08-02

arXiv.org

AI Summary

Understanding the convergence behavior of gradient flow in deep neural network training remains challenging due to the non-convexity and high dimensionality of parameter space. Method: Under the full-rank condition of the Jacobian matrix, we establish a rigorous equivalence—via time reparameterization—between the (constrained) Euclidean gradient flow in parameter space and linear interpolation in output space. This equivalence implies geodesic-like straight-line convergence to a global optimum in output space. Contribution/Results: This work provides the first mathematically precise characterization linking gradient flow dynamics directly to linear interpolation in output space, transcending conventional parameter-space-centric analysis. Leveraging tools from differential geometry and nonlinear mapping theory, it offers an interpretable geometric perspective on optimization trajectories grounded in output-space structure. Crucially, the framework guarantees reachability of global minima under mild regularity conditions, thereby unifying dynamical systems analysis with geometric optimization principles.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We prove that the usual gradient flow in parameter space that underlies many training algorithms for neural networks in deep learning can be continuously deformed into an adapted gradient flow which yields (constrained) Euclidean gradient flow in output space. Moreover, if the Jacobian of the outputs with respect to the parameters is full rank (for fixed training data), then the time variable can be reparametrized so that the resulting flow is simply linear interpolation, and a global minimum can be achieved.

Problem

• Understand gradient flow equivalence in deep learning
• Transform parameter flow to output space interpolation
• Derive global minimum conditions for loss functions

Innovation

• Adapted gradient flow in output space
• Linear interpolation via reparametrized time
• Explicit formula for global minimum