Loss landscapes and optimization in over-parameterized non-linear systems and neural networks (2003.00307v2)

Abstract: The success of deep learning is due, to a large extent, to the remarkable effectiveness of gradient-based optimization methods applied to large neural networks. The purpose of this work is to propose a modern view and a general mathematical framework for loss landscapes and efficient optimization in over-parameterized machine learning models and systems of non-linear equations, a setting that includes over-parameterized deep neural networks. Our starting observation is that optimization problems corresponding to such systems are generally not convex, even locally. We argue that instead they satisfy PL$^*$, a variant of the Polyak-Lojasiewicz condition on most (but not all) of the parameter space, which guarantees both the existence of solutions and efficient optimization by (stochastic) gradient descent (SGD/GD). The PL$^*$ condition of these systems is closely related to the condition number of the tangent kernel associated to a non-linear system showing how a PL$^*$-based non-linear theory parallels classical analyses of over-parameterized linear equations. We show that wide neural networks satisfy the PL$^*$ condition, which explains the (S)GD convergence to a global minimum. Finally we propose a relaxation of the PL$^*$ condition applicable to "almost" over-parameterized systems.

• The paper introduces the PL* condition, linking the tangent kernel's condition number to guarantee efficient GD/SGD convergence in non-convex landscapes.
• It demonstrates that wide neural networks satisfy the PL* condition, explaining the effectiveness of gradient-based optimization methods.
• The analysis shows that over-parameterized systems form solution manifolds, achieving exponential convergence despite inherent non-convexity.

Overview of "Loss landscapes and optimization in over-parameterized non-linear systems and neural networks"

The paper "Loss landscapes and optimization in over-parameterized non-linear systems and neural networks" proposes a mathematical framework for understanding loss landscapes in over-parameterized machine learning models, such as deep neural networks. It focuses on explaining why gradient-based optimization methods perform effectively for these complex, non-convex systems.

The authors introduce the concept of the PL$^*$ condition, a variant of the Polyak-{\L}ojasiewicz condition, which captures the optimization dynamics in over-parameterized settings. The key assertion is that while these landscapes are non-convex, they satisfy the PL$^*$ condition in most of the parameter space. This condition ensures the existence of solutions and guarantees efficient convergence of gradient descent (GD) and stochastic gradient descent (SGD) to a global minimum.

Key Contributions

1. PL$^*$ Condition: The paper argues that the PL$^*$ condition, which relates closely to the condition number of the tangent kernel associated with the non-linear system, provides the right framework for analyzing the optimization landscapes of over-parameterized systems. Satisfying this condition implies both the existence of solutions and efficient optimization.
2. Wide Neural Networks: The authors show that wide neural networks satisfy the PL$^*$ condition, providing an explanation for the success of SGD in these models. The paper explores the mathematical underpinnings based on the spectrum of the tangent kernel and examines the impact of over-parameterization on these landscapes.
3. Essential Non-convexity: The paper discusses that while the landscapes of over-parameterized systems are non-convex, they are fundamentally different from those of under-parameterized systems where local convexity around minima is often observed. The over-parameterized systems form solution manifolds, indicating non-convexity even locally around global minima.
4. Convergence Analysis: By establishing the PL$^*$ condition in a bounded region, the authors provide strong theoretical results on the exponential convergence rate of GD and SGD for these loss landscapes. The paper also introduces a relaxed PL$^*$ condition, termed PL$^*_\epsilon$, for scenarios where models may not be fully over-parameterized throughout the optimization trajectory.

Implications and Future Directions

The work presents significant theoretical advancements in understanding the mechanisms driving the success of gradient-based optimization in modern machine learning models. The implications are substantial for designing new optimization methods and improving existing algorithms for over-parameterized systems. Questions remain about the broader applicability of these analyses across various architectures and datasets, suggesting future work might explore adaptive methods that better exploit PL$^*$ properties.

Future developments could involve extending the PL$^*$ framework to other classes of non-linear systems and exploring its relationship with generalization and regularization within the scope of extremely large models. Moreover, insights into how practical architectures like CNNs and ResNets behave under these conditions could illuminate further directions for model design and training strategies.

This paper provides a comprehensive mathematical approach for tackling the challenges posed by non-convex optimization in deep learning, contributing valuable insights to the field.