Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models (1708.03395v3)

Abstract: Generalized linear models (GLMs) arise in high-dimensional machine learning, statistics, communications and signal processing. In this paper we analyze GLMs when the data matrix is random, as relevant in problems such as compressed sensing, error-correcting codes or benchmark models in neural networks. We evaluate the mutual information (or "free entropy") from which we deduce the Bayes-optimal estimation and generalization errors. Our analysis applies to the high-dimensional limit where both the number of samples and the dimension are large and their ratio is fixed. Non-rigorous predictions for the optimal errors existed for special cases of GLMs, e.g. for the perceptron, in the field of statistical physics based on the so-called replica method. Our present paper rigorously establishes those decades old conjectures and brings forward their algorithmic interpretation in terms of performance of the generalized approximate message-passing algorithm. Furthermore, we tightly characterize, for many learning problems, regions of parameters for which this algorithm achieves the optimal performance, and locate the associated sharp phase transitions separating learnable and non-learnable regions. We believe that this random version of GLMs can serve as a challenging benchmark for multi-purpose algorithms. This paper is divided in two parts that can be read independently: The first part (main part) presents the model and main results, discusses some applications and sketches the main ideas of the proof. The second part (supplementary informations) is much more detailed and provides more examples as well as all the proofs.

• The paper derives precise asymptotic expressions for free entropy and mutual information to calculate Bayes-optimal estimation errors in GLMs.
• It rigorously validates the GAMP algorithm against theoretical error limits using state evolution analysis in the high-dimensional regime.
• The study identifies phase transitions that delineate learnable from non-learnable regimes, informing optimal design of learning algorithms.

Overview of Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models

In "Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models," the authors present a comprehensive paper of generalized linear models (GLMs) in the context of high-dimensional statistics, focusing on cases where the data matrix is random. This paper is particularly relevant to problems in fields such as compressed sensing, machine learning, and neural networks. The paper rigorously establishes the decades-old theoretical predictions associated with these models using statistical physics methods, thereby offering new insights and extending previous non-rigorous results.

The authors employ the replica method from statistical physics, a sophisticated analytical tool traditionally used for systems with disorder, to calculate the mutual information necessary for determining Bayes-optimal estimation and generalization errors. They proceed to demonstrate that the generalized approximate message-passing (GAMP) algorithm is capable of achieving these optimal errors under defined conditions, thus bridging the gap between theoretical insights and practical algorithmic applications.

Main Results

The paper's main findings are derived in the asymptotic regime where both the number of samples (m) and the dimensionality (n) are large, with their ratio (m/n) fixed. Several key results are highlighted:

1. Free Entropy and Mutual Information: The authors derive a precise expression for the asymptotic free entropy, establishing its relation to the mutual information in GLMs. This is a critical step for computing accurate estimation and prediction errors.
2. Optimal Estimation and Generalization Error: They rigorously show the limits of optimal estimation errors, affirming predictions made in prior non-rigorous work. Notably, the generalization error results from an extension of the I-MMSE relationship, adapting it to the setting of high-dimensional GLMs.
3. GAMP's Performance: The GAMP algorithm's asymptotic behavior is analyzed through state evolution, which allows for tracking the progression of its estimation accuracy against the Bayes-optimal benchmark.
4. Phase Transitions: Several parameter regions are identified, demarcating learnable and non-learnable areas, with sharp phase transitions observed. These transitions are indicative of shifts between attainable and unattainable performance for learning tasks depending on sampling rates and noise levels.

Implications and Future Directions

The implications of this paper are far-reaching. Practically, the clear characterization of phase transitions and error bounds guides the design of learning algorithms, providing criteria for their deployment in specific contexts. Theoretically, the results validate the use of statistical physics-based methods in the rigorous analysis of GLMs, opening pathways for similar applications in other complex systems faced in machine learning and signal processing.

The paper also suggests that these high-dimensional GLMs serve as valuable benchmarks for testing the efficacy of general-purpose algorithms. This facilitates a deeper understanding of sample complexity and guides the development of algorithms aiming to achieve performance levels close to the theoretical limits.

Future Research: The authors propose exploring the adaptation of their methods to additional inference models, extending both the theory and practical implications. Moreover, addressing computational complexity in regions identified as "hard" in parameter space remains an open question, warranting further investigation into potentially new or hybrid algorithms that could approach Bayes-optimal performance efficiently.

This paper provides a rigorous, structured framework for understanding estimation limits and algorithmic performance in GLMs, marking a prominent contribution to the field of high-dimensional statistics and its intersection with signal processing and machine learning.