Statistical Mechanics of Optimal Convex Inference in High Dimensions

By Madhu S. Advani, Surya Ganguli. 2016

57 citations recorded by the source corpus.

To model modern large-scale datasets, we need efficient algorithms to infer a set of $P$ unknown model parameters from $N$ noisy measurements. What are fundamental limits on the accuracy of parameter inference, given finite signal-to-noise ratios, limited measurements, prior information, and computational tractability requirements? How can we combine prior information with measurements to achieve these limits? Classical statistics gives incisive answers to these questions as the measurement density $\alpha = \frac{N}{P}\rightarrow \infty$. However, these classical results are not relevant to modern high-dimensional inference problems, which instead occur at finite $\alpha$. We formulate and analyze high-dimensional inference as a problem in the statistical physics of quenched disorder. Our analysis uncovers fundamental limits on the accuracy of inference in high dimensions, and reveals that widely cherished inference algorithms like maximum likelihood (ML) and maximum-a posteriori (MAP) inference cannot achieve these limits. We further find optimal, computationally tractable algorithms that can achieve these limits. Intriguingly, in high dimensions, these optimal algorithms become computationally simpler than MAP and ML, while still outperforming them. For example, such optimal algorithms can lead to as much as a 20% reduction in the amount of data to achieve the same performance relative to MAP. Moreover, our analysis reveals simple relations between optimal high dimensional inference and low dimensional scalar Bayesian inference, insights into the nature of generalization and predictive power in high dimensions, information theoretic limits on compressed sensing, phase transitions in quadratic inference, and connections to central mathematical objects in convex optimization theory and random matrix theory.

Subfields: Artificial Intelligence, Statistics and Probability, Scaling & Learning Theory

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Citations in the corpus

High-dimensional dynamics of generalization error in neural networks (2017, methodology): “In fact, under the assumptions we make of Gaussian noise and parameter distributions, no algorithm can outperform optimal L2 regularization in terms of generalization error performance (as shown in e.g. Advani and Ganguli, 2016a; Advani and Ganguli, 2016c).”
Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks (2020, methodology): “The generalization of regularized linear regression with isotropic Gaussian features was analyzed with the replica method in [28] and diagrammatic methods in [29, 30, 31].”
An equivalence between high dimensional Bayes optimal inference and M-estimation (2016, methodology): “Following this, the performance of regularized M-estimators in the linear additive setting was characterized in [7], using non-rigorous statistical physics methods based on replica theory, and in [8], using rigorous methods different from [4, 5].”