Deep learning: a statistical viewpoint (2103.09177v1)

Abstract: The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

• The paper highlights the pivotal role of overparameterization in simplifying the optimization landscape and enabling effective training with gradient descent.
• The paper shows how implicit regularization, arising from the optimization process, facilitates robust generalization even in overparameterized models.
• The paper explores benign overfitting in both linear regression and neural networks, demonstrating that minimum-norm solutions can achieve near-optimal test performance.

Deep Learning: A Statistical Viewpoint

The paper "Deep Learning: A Statistical Viewpoint" provides a comprehensive analysis of the surprising success of deep learning from a statistical learning theory perspective. The authors, Peter L. Bartlett, Andrea Montanari, and Alexander Rakhlin, delve into the phenomena of overparameterization, implicit regularization, and benign overfitting, which have emerged with the advancement of deep learning methodologies.

Core Hypotheses

The authors hypothesize two primary mechanisms underlying the empirical success of deep learning:

1. Tractability via Overparameterization: Contrary to classical learning theory, deep learning benefits from models that are sufficiently overparameterized. This overparameterization simplifies the optimization landscape, making it feasible to find global minima using simple, local optimization techniques like gradient descent.
2. Generalization via Implicit Regularization: Although overparameterized models are capable of perfectly fitting the training data, they still manage to generalize well. The conjecture is that the optimization process introduces an implicit form of regularization, thereby favoring solutions that generalize better despite the absence of explicit regularization techniques.

Statistical Learning Theory: Breakdown and Extensions

Traditional statistical learning theory relies on uniform convergence to explain the generalization ability of models. The authors review uniform laws of large numbers and the concept of Rademacher complexity, which provide upper bounds on the estimation error of empirical risk minimizers. However, these classical tools fall short in explaining the behavior of deep learning models, particularly due to their inability to manage the trade-off between model complexity and empirical fit in the interpolating regime.

The paper presents several results where empirical minimizers align with implicit regularization frameworks, such as minimum-norm interpolation solutions arising from using gradient descent methods in overparameterized linear models. This provides a glimpse into how implicit biases induced by optimization algorithms can act as regularizers in high-dimensional settings.

Overfitting Is Not Always Harmful

A significant focus is placed on the phenomenon of benign overfitting, where models that interpolate the training noise still manage to achieve low test error. The authors analyze linear regression under this lens, illustrating that for certain data distributions, minimum-norm interpolants can lead to nearly optimal generalization performance. They introduce the self-induced regularization phenomenon, which behaves similarly to explicit regularization methods, such as ridge regression, especially in overparameterized regimes.

Neural Networks and the Linear Regime

The paper also tackles the surprising tractability of neural network training via the so-called linear regime, where the network behaves like a linear model around the initial parameters. This regime is key in explaining why simple gradient methods can effectively optimize highly non-convex networks. Theoretical results support that in specific configurations, neural networks have empirical risks that converge exponentially to zero, and approximations can be made to simplify generalization analyses through neural tangent kernels or randomized features approaches.

Implications and Future Directions

The insights presented have significant implications for both the theoretical understanding and practical development of machine learning models. The decomposition of neural network predictions into smooth and spiky parts creates avenues for reformulating learning problems that adaptively balance complexity and fit. Further research can focus on extending these insights to fully characterize realistic non-linear deep learning scenarios and to explore alternate regimes that offer beneficial inductive biases beyond the linear tangent kernel approaches.

The paper underscores the idea that overparameterization and a careful understanding of implicit regularization mechanisms hold the key to unlocking the full capabilities of deep learning technologies in data-rich applications. As the field progresses, there is a vast terrain of unexplored methodologies inspired by the successes outlined by the authors, promising novel techniques that harness these properties for improved data modeling and inference.