Statistical Mechanics of High-Dimensional Inference

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.Comment: See http://ganguli-gang.stanford.edu/pdf/HighDimInf.Supp.pdf for supplementary materia

Statistical Mechanics of Learning: A Variational Approach for Real Data

Using a variational technique, we generalize the statistical physics approach of learning from random examples to make it applicable to real data. We demonstrate the validity and relevance of our method by computing approximate estimators for generalization errors that are based on training data alone.Comment: 4 pages, 2 figure

Learning curves for Gaussian process regression: Approximations and bounds

We consider the problem of calculating learning curves (i.e., average generalization performance) of Gaussian processes used for regression. On the basis of a simple expression for the generalization error, in terms of the eigenvalue decomposition of the covariance function, we derive a number of approximation schemes. We identify where these become exact, and compare with existing bounds on learning curves; the new approximations, which can be used for any input space dimension, generally get substantially closer to the truth. We also study possible improvements to our approximations. Finally, we use a simple exactly solvable learning scenario to show that there are limits of principle on the quality of approximations and bounds expressible solely in terms of the eigenvalue spectrum of the covariance function.Comment: 25 pages, 10 figure

The Information Complexity of Learning Tasks, their Structure and their Distance

We introduce an asymmetric distance in the space of learning tasks, and a framework to compute their complexity. These concepts are foundational for the practice of transfer learning, whereby a parametric model is pre-trained for a task, and then fine-tuned for another. The framework we develop is non-asymptotic, captures the finite nature of the training dataset, and allows distinguishing learning from memorization. It encompasses, as special cases, classical notions from Kolmogorov complexity, Shannon, and Fisher Information. However, unlike some of those frameworks, it can be applied to large-scale models and real-world datasets. Our framework is the first to measure complexity in a way that accounts for the effect of the optimization scheme, which is critical in Deep Learning

Combining predictions from linear models when training and test inputs differ

Methods for combining predictions from different models in a supervised learning setting must somehow estimate/predict the quality of a model's predictions at unknown future inputs. Many of these methods (often implicitly) make the assumption that the test inputs are identical to the training inputs, which is seldom reasonable. By failing to take into account that prediction will generally be harder for test inputs that did not occur in the training set, this leads to the selection of too complex models. Based on a novel, unbiased expression for KL divergence, we propose XAIC and its special case FAIC as versions of AIC intended for prediction that use different degrees of knowledge of the test inputs. Both methods substantially differ from and may outperform all the known versions of AIC even when the training and test inputs are iid, and are especially useful for deterministic inputs and under covariate shift. Our experiments on linear models suggest that if the test and training inputs differ substantially, then XAIC and FAIC predictively outperform AIC, BIC and several other methods including Bayesian model averaging.Comment: 12 pages, 2 figures. To appear in Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence (UAI2014). This version includes the supplementary material (regularity assumptions, proofs