Finite Sample Analysis and Bounds on Generalization Error of Gradient Descent in Linear Regression

Core Concepts

The core message of this work is to provide a comprehensive analysis of the generalization properties of a single step of gradient descent in the context of linear regression with well-specified models. The analysis derives analytical expressions for the statistical properties of generalization error in a non-asymptotic (finite sample) setting, and contrasts the results with classical least squares regression.

Abstract

This work investigates the generalization properties of a single step of gradient descent in the context of linear regression with well-specified models. A random design setting is considered, and analytical expressions are derived for the statistical properties of generalization error in a non-asymptotic (finite sample) setting. These expressions avoid arbitrary constants, providing robust quantitative information and scaling relationships.
The key highlights and insights are:

The expected generalization error of in-context gradient descent is derived and compared to classical least squares regression. The breakdown of the systematic and noise components is provided, along with an expression for the optimal step size.

Probabilistic bounds are derived for the generalization error of gradient descent and least squares regression. These bounds are verified using extensive empirical evaluations.

Several identities involving high-order products of Gaussian random matrices are presented as a byproduct of the analysis, which may have broader applications.

The results demonstrate that a single step of gradient descent can provide comparable performance to least squares regression, especially in high noise settings. This has implications for reducing computational complexity in one-shot scenarios and resource-constrained environments.

The work addresses gaps in the literature by providing finite sample, non-asymptotic results that do not rely on arbitrary constants, which are rare even in the case of linear regression.

Stats

The expected generalization error of gradient descent is given by: E[ℓ] = ||W1 - W0||^2 (1 - η)^2 + η^2 (n + 1)/N + σ^2 (1 + η^2 n/N).
The optimal step size for gradient descent is: η_opt = N / (N + n + 1 + σ^2 / ||W1||^2 n).
The generalization error of least squares regression is given by: E[ℓ] = ||W1 - W0||^2 (1 - N/n) + σ^2 (1 + N/(n - N - 1)) for 2 ≤ N ≤ n - 1, and E[ℓ] = σ^2 (1 + n/(N - n - 1)) for n + 1 ≤ N ≤ ∞.

Quotes

"Recent studies show that transformer-based architectures emulate gradient descent during a forward pass, contributing to in-context learning capabilities— an ability where the model adapts to new tasks based on a sequence of prompt examples without being explicitly trained or fine tuned to do so."
"The connections between in-context learning and gradient descent have been widely studied over the past two years."

Key Insights Distilled From

Finite Sample Analysis and Bounds of Generalization Error of Gradient Descent in In-Context Linear Regression

by Karthik Dura... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02462.pdf

Deeper Inquiries

How can the insights from this work on gradient descent in linear regression be extended to more complex regression tasks, including non-linear and incomplete parametrizations?

The insights gained from the study on gradient descent in linear regression can be extended to more complex regression tasks by considering non-linear and incomplete parametrizations. In the context of non-linear regression, the principles of gradient descent can still be applied, but the optimization process may involve more complex surfaces with multiple local minima. Techniques such as stochastic gradient descent, adaptive learning rates, and regularization methods can help navigate these challenges. Additionally, techniques like feature engineering and kernel methods can be employed to transform the input data into higher-dimensional spaces where linear models may perform better.
For incomplete parametrizations, where not all parameters are known or specified, gradient descent can still be utilized by treating the missing parameters as variables to be optimized. This approach, known as data imputation, can help fill in missing values and improve the overall performance of the regression model. By incorporating these strategies and adapting the principles of gradient descent, the insights from this work can be effectively applied to more complex regression tasks.

What are the implications of these results on the design and understanding of transformer-based architectures and other machine learning algorithms in practical applications?

The results of this study have significant implications for the design and understanding of transformer-based architectures and other machine learning algorithms in practical applications. By demonstrating the effectiveness of a single step of gradient descent in linear regression tasks, the study highlights the potential for gradient-based optimization methods to generalize effectively in in-context learning scenarios. This finding can be leveraged in the development of transformer-based models, where the ability to adapt to new tasks based on prompt examples is crucial.
In practical applications, the insights from this work can inform the design of more efficient and adaptive machine learning algorithms. By understanding the statistical properties of generalization error and optimal step sizes in gradient descent, researchers and practitioners can fine-tune their models for better performance on new tasks. This can lead to improved accuracy, faster convergence, and reduced computational complexity in real-world applications of machine learning.

Can the identities involving high-order products of Gaussian random matrices derived in this work find applications beyond the context of regression tasks?

The identities involving high-order products of Gaussian random matrices derived in this work have the potential to find applications beyond the context of regression tasks. These identities provide valuable insights into the statistical properties of random matrices and their products, which can be applicable in various fields of mathematics, statistics, and machine learning.
One potential application of these identities is in the analysis of complex systems with random components, such as in signal processing, image recognition, and natural language processing. By understanding the properties of high-order products of Gaussian random matrices, researchers can gain a deeper insight into the underlying structures of these systems and develop more efficient algorithms for processing and analyzing them.
Furthermore, these identities can also be useful in the study of random matrix theory, where the properties of large matrices with random entries are analyzed. By extending the applications of these identities to diverse fields, researchers can uncover new connections and insights that can advance the understanding of complex systems and improve the performance of machine learning algorithms.