Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path? (1812.10004v1)

Abstract: Many modern learning tasks involve fitting nonlinear models to data which are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Due to this overparameterization, the training loss may have infinitely many global minima and it is critical to understand the properties of the solutions found by first-order optimization schemes such as (stochastic) gradient descent starting from different initializations. In this paper we demonstrate that when the loss has certain properties over a minimally small neighborhood of the initial point, first order methods such as (stochastic) gradient descent have a few intriguing properties: (1) the iterates converge at a geometric rate to a global optima even when the loss is nonconvex, (2) among all global optima of the loss the iterates converge to one with a near minimal distance to the initial point, (3) the iterates take a near direct route from the initial point to this global optima. As part of our proof technique, we introduce a new potential function which captures the precise tradeoff between the loss function and the distance to the initial point as the iterations progress. For Stochastic Gradient Descent (SGD), we develop novel martingale techniques that guarantee SGD never leaves a small neighborhood of the initialization, even with rather large learning rates. We demonstrate the utility of our general theory for a variety of problem domains spanning low-rank matrix recovery to neural network training. Underlying our analysis are novel insights that may have implications for training and generalization of more sophisticated learning problems including those involving deep neural network architectures.

• The paper demonstrates that gradient descent in overparameterized nonlinear settings converges geometrically to a global optimum along the path closest to the initial parameters.
• It introduces a novel potential function that links the loss function and distance from initialization, providing rigorous guarantees even for nonconvex objectives.
• The study develops martingale techniques for SGD, ensuring stability with large learning rates and validating the theory with applications in neural networks and low-rank regression.

Understanding Overparameterized Nonlinear Learning and Gradient Descent Dynamics

The paper "Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?" explores the intricacies of optimizing nonlinear models in an overparameterized regime. This paradigm occurs when the number of model parameters exceeds the size of the training dataset, leading to infinitely many global minima for the training loss. Understanding the properties and convergence behavior of first-order optimization methods, such as (stochastic) gradient descent, within this overparameterized context is crucial.

The authors provide three main insights regarding the behavior of these methods: (1) Even in the presence of nonconvex loss, gradient descent exhibits geometric convergence to a global optimum, (2) among the multiple global optima, the iterates converge to one nearest to the initial point, and (3) the path taken from the initial point to the global optimum is nearly direct. This behavior is explained through the introduction of a new potential function that captures the interplay between the loss function and the distance from the initial parameter.

To safeguard the convergence of Stochastic Gradient Descent (SGD), the researchers develop novel martingale techniques, ensuring the SGD iterates remain within a small neighborhood of the initialization despite the use of relatively large learning rates. These techniques expand the theoretical understanding by providing guarantees not only for simple gradient descent but also for its stochastic variant.

The general theoretical framework is demonstrated across various domains, including low-rank matrix recovery and training neural networks, enhancing its applicability. The analysis reveals novel insights that are potentially significant for understanding training dynamics and the generalization in more sophisticated learning architectures, such as deep neural networks.

The paper further provides a crucial observation on the optimization landscape in overparameterized learning circumstances, outlining that the condition number of the Jacobian matrix increases with more samples, making optimization progressively challenging. Yet, the utility of these theoretical results hinges on the Jacobian matrix being well-behaved over sufficiently small neighborhoods around the initialization, as established through both upper and lower bounds.

Theoretical implications aside, the paper ventures into case studies for generalized linear models, low-rank matrix regression, and neural network training. These cases not only corroborate the general theory but also provide specific insights tailored to these paradigms, illustrating how theoretical results play out in practical contexts.

Potential future research directions could explore these theoretical findings' implications for generalization in overparameterized regimes, especially as they pertain to the phenomenon of good generalization from finite data despite vast parameter spaces. Understanding this connection more deeply could clarify why first-order methods generalize well in practical overparameterized learning scenarios.

Overall, the paper provides a robust theoretical foundation for understanding the dynamics of gradient-based optimization in overparameterized nonlinear learning tasks, revealing a critical pathway for both theoretical advancement and practical application in AI and machine learning landscapes.