Applying statistical learning theory to deep learning (2311.15404v2)

Abstract: Although statistical learning theory provides a robust framework to understand supervised learning, many theoretical aspects of deep learning remain unclear, in particular how different architectures may lead to inductive bias when trained using gradient based methods. The goal of these lectures is to provide an overview of some of the main questions that arise when attempting to understand deep learning from a learning theory perspective. After a brief reminder on statistical learning theory and stochastic optimization, we discuss implicit bias in the context of benign overfitting. We then move to a general description of the mirror descent algorithm, showing how we may go back and forth between a parameter space and the corresponding function space for a given learning problem, as well as how the geometry of the learning problem may be represented by a metric tensor. Building on this framework, we provide a detailed study of the implicit bias of gradient descent on linear diagonal networks for various regression tasks, showing how the loss function, scale of parameters at initialization and depth of the network may lead to various forms of implicit bias, in particular transitioning between kernel or feature learning.

• The paper demonstrates that gradient descent induces an implicit bias toward low-norm solutions in over-parameterized deep models.
• The authors explore mirror descent and Riemannian geometry to explain optimization in non-convex learning landscapes.
• The study contrasts deep linear and convolutional architectures, showing how design choices shape regularization and sparsity.

Insights into the Application of Statistical Learning Theory to Deep Learning

The paper by Cedric Gerbelot, Avetik Karagulyan, Stefani Karp, Kavya Ravichandran, Nathan Srebro, and Menachem Stern investigates how statistical learning theory, a robust framework for understanding supervised learning, applies to deep learning. Despite deep learning models' extensive empirical success, their theoretical underpinnings remain under-explored, especially how different architectures may induce particular inductive biases when trained with gradient-based methods. The paper navigates through implicit bias in the context of benign overfitting, the specific case of mirror descent, and an exploration of learning problem geometry via the use of metric tensors.

The authors ground their work by revisiting classical statistical learning theory, focusing on how inductive biases motivate learning rules like empirical risk minimization (ERM) and structural risk minimization (SRM). They reflect on the conundrum posed by the 'no free lunch' theorem, which mandates that knowledge about the population distribution is essential for minimizing generalization error. To circumvent this, students of machine learning must design models with an inductive bias towards hypotheses that render certain model classes more feasible than others given empirical data.

One critical discussion within the paper is the dual nature of deep neural networks. On one hand, they are powerful enough to become universal approximators, thus equally able to fit random label data perfectly. On the other hand, in practice, these networks generalize well from real data, a process often explained through the lens of implicit regularization. It suggests that, although over-parameterized models can learn complex patterns, the inherent inductive bias of gradient descent methods steers the solution towards those with minimal norm in parameter space.

By analyzing deep linear networks and convolutional networks, the authors elucidate how the architecture impacts learning biases. For instance, linear networks exhibit biases equivalent to minimizing the Euclidean norm in parameters, supporting the classical view of kernel methods. Convolutional networks introduce an additional layer of complexity. In particular scenarios, they can lean towards sparsity, especially when interpreted within Fourier space. This implicit sparsity regularization propels us beyond classical kernel learning theory into the contemporary rich regime of feature learning.

Further, the authors investigate mirror descent, a generalization of gradient descent that incorporates a more nuanced form of regularization via Bregman divergences. This framework becomes not only a more flexible approach than typical gradient descent but also offers an abstract connection between parameter and function space through metric tensors — or rather, Riemannian manifolds. It establishes an optimization approach that encapsulates the geometric intrinsic geometry of problems, importantly linking to implicit bias.

Another meaningful contribution of the paper is the contrast between the convex and non-convex landscapes in deep model training. Convex landscapes allow more predictable behavior in optimization and generalization, whereas non-convex landscapes — characteristic of deep networks — are less understood in theoretical terms. The authors propose understanding this behavior through implicit regularization, showing how stochastic optimization using gradient descent drives solutions to particular regions in parameter space.

Among the theoretical challenges, benign overfitting remains an issue beyond current understanding. Unlike overfitting, benign overfitting does not harm generalization, suggesting that certain noise-fitting is innocuous. The authors suggest this phenomenon may widen the range of acceptable regularization parameters during training, further complicating how one might approach model adequacy assessments.

In conclusion, this paper examines multiple theoretical facets of deep learning, highlighting its biases and implications from a statistical learning perspective. It indirectly questions whether more extensive theories can capture these intricacies, leading to a future where theories encompass architectural and optimization nuances. The insights furnish not only a basis for designing architectures with desired properties but also provide pathways to translate these practices into actionable insights in AI research and development.