Ata Kabán

Andrew J. Turner, Ata Kabán

Approximate learning machines have become popular in the era of small devices, including quantised, factorised, hashed, or otherwise compressed predictors, and the quest to explain and guarantee good generalisation abilities for such methods has just begun. In this paper we study the role of approximability in learning, both in the full precision and the approximated settings of the predictor that is learned from the data, through a notion of sensitivity of predictors to the action of the approximation operator at hand. We prove upper bounds on the generalisation of such predictors, yielding the following main findings, for any PAC-learnable class and any given approximation operator. 1) We show that under mild conditions, approximable target concepts are learnable from a smaller labelled sample, provided sufficient unlabelled data. 2) We give algorithms that guarantee a good predictor whose approximation also enjoys the same generalisation guarantees. 3) We highlight natural examples of structure in the class of sensitivities, which reduce, and possibly even eliminate the otherwise abundant requirement of additional unlabelled data, and henceforth shed new light onto what makes one problem instance easier to learn than another. These results embed the scope of modern model compression approaches into the general goal of statistical learning theory, which in return suggests appropriate algorithms through minimising uniform bounds.

Efstratios Palias, Ata Kabán

Metric learning aims at finding a suitable distance metric over the input space, to improve the performance of distance-based learning algorithms. In high-dimensional settings, it can also serve as dimensionality reduction by imposing a low-rank restriction to the learnt metric. In this paper, we consider the problem of learning a Mahalanobis metric, and instead of training a low-rank metric on high-dimensional data, we use a randomly compressed version of the data to train a full-rank metric in this reduced feature space. We give theoretical guarantees on the error for Mahalanobis metric learning, which depend on the stable dimension of the data support, but not on the ambient dimension. Our bounds make no assumptions aside from i.i.d. data sampling from a bounded support, and automatically tighten when benign geometrical structures are present. An important ingredient is an extension of Gordon's theorem, which may be of independent interest. We also corroborate our findings by numerical experiments.

Sijia Zhou, Yunwen Lei, Ata Kabán

We study stochastic optimization with data-adaptive sampling schemes to train pairwise learning models. Pairwise learning is ubiquitous, and it covers several popular learning tasks such as ranking, metric learning and AUC maximization. A notable difference of pairwise learning from pointwise learning is the statistical dependencies among input pairs, for which existing analyses have not been able to handle in the general setting considered in this paper. To this end, we extend recent results that blend together two algorithm-dependent frameworks of analysis -- algorithmic stability and PAC-Bayes -- which allow us to deal with any data-adaptive sampling scheme in the optimizer. We instantiate this framework to analyze (1) pairwise stochastic gradient descent, which is a default workhorse in many machine learning problems, and (2) pairwise stochastic gradient descent ascent, which is a method used in adversarial training. All of these algorithms make use of a stochastic sampling from a discrete distribution (sample indices) before each update. Non-uniform sampling of these indices has been already suggested in the recent literature, to which our work provides generalization guarantees in both smooth and non-smooth convex problems.