High-dimensional learning of narrow neural networks (2409.13904v2)

Abstract: Recent years have been marked with the fast-pace diversification and increasing ubiquity of machine learning applications. Yet, a firm theoretical understanding of the surprising efficiency of neural networks to learn from high-dimensional data still proves largely elusive. In this endeavour, analyses inspired by statistical physics have proven instrumental, enabling the tight asymptotic characterization of the learning of neural networks in high dimensions, for a broad class of solvable models. This manuscript reviews the tools and ideas underlying recent progress in this line of work. We introduce a generic model -- the sequence multi-index model -- which encompasses numerous previously studied models as special instances. This unified framework covers a broad class of machine learning architectures with a finite number of hidden units, including multi-layer perceptrons, autoencoders, attention mechanisms; and tasks, including (un)supervised learning, denoising, contrastive learning, in the limit of large data dimension, and comparably large number of samples. We explicate in full detail the analysis of the learning of sequence multi-index models, using statistical physics techniques such as the replica method and approximate message-passing algorithms. This manuscript thus provides a unified presentation of analyses reported in several previous works, and a detailed overview of central techniques in the field of statistical physics of machine learning. This review should be a useful primer for machine learning theoreticians curious of statistical physics approaches; it should also be of value to statistical physicists interested in the transfer of such ideas to the study of neural networks.

• The paper introduces the sequence multi-index model to characterize narrow neural networks' learning dynamics in high dimensions.
• It employs the replica method for precise asymptotic analysis, reducing high-dimensional integration to a finite order parameter optimization.
• It maps saddle-point equations to GAMP state evolution, offering practical algorithms for estimating network weights and training dynamics.

High-Dimensional Learning of Narrow Neural Networks in Sequence Multi-Index Models

The paper "High-dimensional learning of narrow neural networks" explores the intricate theoretical examination of neural networks operating within high-dimensional spaces, particularly focusing on architectures with a finite number of hidden units. The author builds upon the foundations laid by statistical physics to present a coherent and unified framework known as the sequence multi-index model, which encompasses and extends a variety of previously studied architectures and tasks.

Overview

The sequence multi-index model proposed in this work is designed to capture the learning dynamics of neural networks with a finite number of hidden units when trained on high-dimensional sequential data. The model's versatility allows it to adapt to different neural network architectures such as multi-layer perceptrons (MLPs), autoencoders, and attention mechanisms. It can also accommodate various learning tasks, from supervised learning to denoising and contrastive learning.

Data Model and Empirical Risk Minimization

The data samples in the sequence multi-index model are structured as sequences, with each token drawn from a Gaussian mixture distribution:

x_\ell, c_\ell \sim \sum_{k=1}^{K_\ell} \rho_{\ell,k} \delta_{c_\ell,k} \mathcal{N}(\mu_{\ell,k}, \Sigma_{\ell,k}).

The sequence multi-index model employs empirical risk minimization (ERM) defined by:

\hat{w} = \underset{w \in \mathbb{R}^{d \times r}}{\mathrm{argmin}} \; \mathcal{R}(w), <h2 class='paper-heading' id='high-dimensional-learning-of-narrow-neural-networks-in-sequence-multi-index-models'>High-Dimensional Learning of Narrow Neural Networks in Sequence Multi-Index Models</h2> <p>The paper "High-dimensional learning of narrow neural networks" explores the intricate theoretical examination of neural networks operating within high-dimensional spaces, particularly focusing on architectures with a finite number of hidden units. The author builds upon the foundations laid by statistical physics to present a coherent and unified framework known as the sequence multi-index model, which encompasses and extends a variety of previously studied architectures and tasks.</p> <h3 class='paper-heading' id='overview'>Overview</h3> <p>The sequence multi-index model proposed in this work is designed to capture the learning dynamics of neural networks with a finite number of hidden units when trained on high-dimensional sequential data. The model's versatility allows it to adapt to different neural network architectures such as multi-layer perceptrons (MLPs), autoencoders, and attention mechanisms. It can also accommodate various learning tasks, from supervised learning to denoising and <a href="https://www.emergentmind.com/topics/contrastive-learning-cl" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">contrastive learning</a>.</p> <h3 class='paper-heading' id='data-model-and-empirical-risk-minimization'>Data Model and Empirical Risk Minimization</h3> <p>The data samples in the sequence multi-index model are structured as sequences, with each token drawn from a Gaussian mixture distribution:</p> <p><pre><code class="plaintext">x_\ell, c_\ell \sim \sum_{k=1}^{K_\ell} \rho_{\ell,k} \delta_{c_\ell,k} \mathcal{N}(\mu_{\ell,k}, \Sigma_{\ell,k}).

The sequence multi-index model employs empirical risk minimization (ERM) defined by:

`` with the empirical risk: \mathcal{R}(w) = \sum_{\mu=1}^n \ell\left(\frac{x^\mu w}{\sqrt{d}}, \frac{x^\mu w}{\sqrt{d}}, \frac{w^\top w}{d}, c^\mu \right) + \frac{\lambda}{2} | w |^2.

### Asymptotic Analysis and the Replica Method A key contribution of the paper is the rigorous asymptotic analysis using the replica method from statistical physics. This method allows for a precise characterization of the learning properties by evaluating the free energy of the system. The free energy is computed via the cumulant-generating function, leveraging the replica trick:

\mathbb{E}{\mathcal{D}} \ln Z = \lim{s \to 0} \frac{\mathbb{E}_{\mathcal{D}} Z^s - 1}{s}, ```

where $Z$ is the partition function. The high-dimensional integration over the network weights $w$ is reduced to an optimization problem over a finite number of order parameters, significantly simplifying the analysis.

State Evolution and Generalized Approximate Message Passing (GAMP)

The paper extends its analysis by mapping the saddle-point equations derived from the replica method to the state evolution of a GAMP algorithm. This provides an algorithmic perspective that not only aligns with the theoretical findings but also offers practical computation methods for estimating the learned weights and their properties. The fixed points of GAMP are shown to be critical points of the ERM landscape, bridging the gap between theoretical analyses and practical learning dynamics.

Implications and Future Developments

The theoretical insights garnered from the sequence multi-index model offer a robust foundation for understanding the learning dynamics of narrow neural networks in high-dimensional spaces. This framework informs the optimal design and regularization of neural networks, shedding light on their efficiency and generalization abilities.

Future research directions may explore:

• Extending to Broader Architectures: Investigating other neural network structures beyond those covered, such as convolutional neural networks and recurrent neural networks.
• Real Data Distributions: Bridging the gap between the idealized data models used in analysis and the complex, real-world data distributions encountered in practice.
• Algorithmic Developments: Developing more efficient algorithms for training these networks in high dimensions, potentially leveraging insights from the GAMP framework.

Conclusion

The paper "High-dimensional learning of narrow neural networks" makes significant strides in the theoretical understanding of neural networks with a finite number of hidden units. By unifying various neural network architectures and tasks under the sequence multi-index model and employing the replica method and GAMP algorithm, the author provides a comprehensive framework that not only elucidates the learning dynamics of these networks but also offers actionable insights for their practical application. This work stands as a foundational piece in the continued exploration and optimization of neural network learning in high-dimensional settings.