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Register a new userTowards Optimal Problem Dependent Generalization Error Bounds in Statistical Learning Theory
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We study problem-dependent rates, i.e., generalization errors that scale near-optimally with the variance, the effective loss, or the gradient norms evaluated at the best hypothesis. We introduce a principled framework dubbed uniform localized convergence, and characterize sharp problem-dependent rates for central statistical learning problems. From a methodological viewpoint, our framework resolves several fundamental limitations of existing uniform convergence and localization analysis approaches. It also provides improvements and some level of unification in the study of localized complexities, one-sided uniform inequalities, and sample-based iterative algorithms. In the so-called slow rate regime, we provides the first (moment-penalized) estimator that achieves the optimal variance-dependent rate for general rich classes; we also establish improved loss-dependent rate for standard empirical risk minimization. In the fast rate regime, we establish finite-sample problem-dependent bounds that are comparable to precise asymptotics. In addition, we show that iterative algorithms like gradient descent and first-order Expectation-Maximization can achieve optimal generalization error in several representative problems across the areas of non-convex learning, stochastic optimization, and learning with missing data.
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Machine learning models have traditionally been developed under the assumption that the training and test distributions match exactly. However, recent success in few-shot learning and related problems are encouraging signs that these models can be adapted to more realistic settings where train and test distributions differ. Unfortunately, there is severely limited theoretical support for these algorithms and little is known about the difficulty of these problems. In this work, we provide novel information-theoretic lower-bounds on minimax rates of convergence for algorithms that are trained on data from multiple sources and tested on novel data. Our bounds depend intuitively on the information shared between sources of data, and characterize the difficulty of learning in this setting for arbitrary algorithms. We demonstrate these bounds on a hierarchical Bayesian model of meta-learning, computing both upper and lower bounds on parameter estimation via maximum-a-posteriori inference.
Gaussian Processes (GPs) are a generic modelling tool for supervised learning. While they have been successfully applied on large datasets, their use in safety-critical applications is hindered by the lack of good performance guarantees. To this end, we propose a method to learn GPs and their sparse approximations by directly optimizing a PAC-Bayesian bound on their generalization performance, instead of maximizing the marginal likelihood. Besides its theoretical appeal, we find in our evaluation that our learning method is robust and yields significantly better generalization guarantees than other common GP approaches on several regression benchmark datasets.
Generalization in deep learning has been the topic of much recent theoretical and empirical research. Here we introduce desiderata for techniques that predict generalization errors for deep learning models in supervised learning. Such predictions should 1) scale correctly with data complexity; 2) scale correctly with training set size; 3) capture differences between architectures; 4) capture differences between optimization algorithms; 5) be quantitatively not too far from the true error (in particular, be non-vacuous); 6) be efficiently computable; and 7) be rigorous. We focus on generalization error upper bounds, and introduce a categorisation of bounds depending on assumptions on the algorithm and data. We review a wide range of existing approaches, from classical VC dimension to recent PAC-Bayesian bounds, commenting on how well they perform against the desiderata. We next use a function-based picture to derive a marginal-likelihood PAC-Bayesian bound. This bound is, by one definition, optimal up to a multiplicative constant in the asymptotic limit of large training sets, as long as the learning curve follows a power law, which is typically found in practice for deep learning problems. Extensive empirical analysis demonstrates that our marginal-likelihood PAC-Bayes bound fulfills desiderata 1-3 and 5. The results for 6 and 7 are promising, but not yet fully conclusive, while only desideratum 4 is currently beyond the scope of our bound. Finally, we comment on why this function-based bound performs significantly better than current parameter-based PAC-Bayes bounds.
The randomized-feature approach has been successfully employed in large-scale kernel approximation and supervised learning. The distribution from which the random features are drawn impacts the number of features required to efficiently perform a learning task. Recently, it has been shown that employing data-dependent randomization improves the performance in terms of the required number of random features. In this paper, we are concerned with the randomized-feature approach in supervised learning for good generalizability. We propose the Energy-based Exploration of Random Features (EERF) algorithm based on a data-dependent score function that explores the set of possible features and exploits the promising regions. We prove that the proposed score function with high probability recovers the spectrum of the best fit within the model class. Our empirical results on several benchmark datasets further verify that our method requires smaller number of random features to achieve a certain generalization error compared to the state-of-the-art while introducing negligible pre-processing overhead. EERF can be implemented in a few lines of code and requires no additional tuning parameters.
Large-margin classifiers are popular methods for classification. We derive the asymptotic expression for the generalization error of a family of large-margin classifiers in the limit of both sample size $n$ and dimension $p$ going to $infty$ with fixed ratio $alpha=n/p$. This family covers a broad range of commonly used classifiers including support vector machine, distance weighted discrimination, and penalized logistic regression. Our result can be used to establish the phase transition boundary for the separability of two classes. We assume that the data are generated from a single multivariate Gaussian distribution with arbitrary covariance structure. We explore two special choices for the covariance matrix: spiked population model and two layer neural networks with random first layer weights. The method we used for deriving the closed-form expression is from statistical physics known as the replica method. Our asymptotic results match simulations already when $n,p$ are of the order of a few hundreds. For two layer neural networks, we reproduce the recently developed `double descent phenomenology for several classification models. We also discuss some statistical insights that can be drawn from these analysis.
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