Paper Details

Name: Manju Dhand
Country: India

Year: 2014
Volume: Volume-1, Issue-1 (January-June)
Page Number: 44-59

ABSTRACT
Modern deep neural networks operate in the overparameterized regime, where the number of
parameters vastly exceeds the number of training samples. Classical statistical learning theory
suggests such models should severely overfit, yet they generalize remarkably well in practice.
This paper provides a comprehensive mathematical analysis of implicit regularization—the
phenomenon where optimization algorithms introduce inductive biases that favor certain
solutions over others without explicit regularization terms. We examine how gradient descent
and its variants implicitly regularize neural networks through the lens of optimization
geometry, kernel methods, and dynamical systems theory. Our analysis reveals that the
trajectory of gradient-based optimization in overparameterized networks converges to
solutions with specific geometric and spectral properties that promote generalization. We
present theoretical results on the implicit bias toward minimum norm solutions, characterize
the role of initialization and learning rate, and discuss connections to classical regularization
methods. Our findings provide mathematical justification for the success of deep learning and
offer insights for designing more effective training procedures.
Keywords: Implicit regularization, overparameterization, gradient descent, neural networks,
generalization theory, optimization geometry