Scaling and renormalization in high-dimensional regression (2405.00592v4)

Abstract: From benign overfitting in overparameterized models to rich power-law scalings in performance, simple ridge regression displays surprising behaviors sometimes thought to be limited to deep neural networks. This balance of phenomenological richness with analytical tractability makes ridge regression the model system of choice in high-dimensional machine learning. In this paper, we present a unifying perspective on recent results on ridge regression using the basic tools of random matrix theory and free probability, aimed at readers with backgrounds in physics and deep learning. We highlight the fact that statistical fluctuations in empirical covariance matrices can be absorbed into a renormalization of the ridge parameter. This `deterministic equivalence' allows us to obtain analytic formulas for the training and generalization errors in a few lines of algebra by leveraging the properties of the $S$-transform of free probability. From these precise asymptotics, we can easily identify sources of power-law scaling in model performance. In all models, the $S$-transform corresponds to the train-test generalization gap, and yields an analogue of the generalized-cross-validation estimator. Using these techniques, we derive fine-grained bias-variance decompositions for a very general class of random feature models with structured covariates. This allows us to discover a scaling regime for random feature models where the variance due to the features limits performance in the overparameterized setting. We also demonstrate how anisotropic weight structure in random feature models can limit performance and lead to nontrivial exponents for finite-width corrections in the overparameterized setting. Our results extend and provide a unifying perspective on earlier models of neural scaling laws.

• The paper derives deterministic equivalence using S-transforms and subordination formulas to simplify complex random matrices in high-dimensional regression.
• It characterizes training and generalization errors in linear and kernel ridge regression models through detailed spectral analysis.
• It uncovers scaling laws and variance-dominated regimes that provide actionable insights for designing robust high-dimensional learning systems.

Understanding High-Dimensional Regression Through the Lens of Random Matrix Theory

Linear Regression and Kernel Methods in High-Dimensional Spaces

When faced with the challenge of high-dimensional data analysis, both classical linear regression and modern machine learning methodologies like kernel regression are immensely valuable. But as the dimensionality of the data (number of predictors) increases, especially relative to the number of observations, traditional analysis techniques face difficulty due to high variance among other issues.

Focusing on a scenario where data points are drawn from a high-dimensional Gaussian distribution, this paper journeys through an analytical framework using tools from random matrix theory and free probability. This kind of analysis is critical because it helps us understand the behavior of estimators in high-dimensional settings objectively, by connecting abstract mathematical concepts with practical regression problems.

Utilizing $S$-transforms in Random Matrix Theory

A key mathematical tool employed in this paper is the $S$-transform from free probability theory, which helps manage and simplify the complexity associated with products of random matrices. This is particularly useful when analyzing properties like the covariance of the datasets in high dimensions, which often appear in the form of random matrices in practical problems.

Subordination Formulas and Deterministic Equivalence:

These concepts allow us to replace complex random matrices (like those we encounter in data covariance matrices) with simpler, deterministic equivalents under certain conditions. This simplification is incredibly beneficial for theoretical analyses, making a range of calculations more tractable.

Bridging Theory with Practical Learning Models

Linear and Kernel Ridge Regression:

The paper explores models that add a regularization term (ridge regression) to address issues of overfitting in high-dimensional settings. By applying deterministic equivalence and subordination formulas, the authors derive expressions for training error, generalization error, and provide a sharp characterization of these errors in terms of the data's spectral properties (distribution of eigenvalues of the covariance matrix).

Linear Random Features Model:

Expanding the analysis to random feature models introduces another layer of stochasticity and complexity. Here, the paper scrutinizes how random projections (features) affect learning outcomes, again harnessing the power of $S$-transforms to untangle the effects of randomness introduced by such features.

Generalization Error and Scaling Laws:

A particularly practical aspect of the paper revolves around understanding how generalization error scales with parameters like the number of features and samples. This is crucial for designing machine learning systems that are both efficient and robust.

Challenges and Opportunities

One of the takeaways is the nuanced understanding of "variance-dominated" regimes in learning, which occur when parameters or features have nonlinear effects on the learning outcomes. Detecting and understanding these regimes can lead to better model design and parameter adjustment.

Future Paths and Theoretical Implications

The analytical methods highlighted in the paper open pathways for exploring more complex models, including deep learning networks, under the rigorous mathematical framework offered by random matrix theory. This could lead to a deeper understanding of why certain deep learning models perform exceptionally well and how to systematically improve models that underperform.

As we continue to push the boundaries of what's achievable with high-dimensional statistical models, the blend of theoretical rigor with practical application as demonstrated in this paper will be indispensable. Echoing through this paper is a powerful narrative about how abstract mathematical concepts can provide profound insights into the data structures and learning algorithms that drive modern AI systems.