Sharp generalization error bounds for randomly-projected classifiers

We derive sharp bounds on the generalization error of a generic linear classifier trained by empirical risk minimization on randomly projected data. We make no restrictive assumptions (such as sparsity or separability) on the data: Instead we use the fact that, in a classification setting, the question of interest is really ‘what is the effect of random projection on the predicted class labels?’ and we therefore derive the exact probability of ‘label flipping’ under Gaussian random projection in order to quantify this effect precisely in our bounds

Generalization Error in Deep Learning

Deep learning models have lately shown great performance in various fields such as computer vision, speech recognition, speech translation, and natural language processing. However, alongside their state-of-the-art performance, it is still generally unclear what is the source of their generalization ability. Thus, an important question is what makes deep neural networks able to generalize well from the training set to new data. In this article, we provide an overview of the existing theory and bounds for the characterization of the generalization error of deep neural networks, combining both classical and more recent theoretical and empirical results

COMET: A Recipe for Learning and Using Large Ensembles on Massive Data

COMET is a single-pass MapReduce algorithm for learning on large-scale data. It builds multiple random forest ensembles on distributed blocks of data and merges them into a mega-ensemble. This approach is appropriate when learning from massive-scale data that is too large to fit on a single machine. To get the best accuracy, IVoting should be used instead of bagging to generate the training subset for each decision tree in the random forest. Experiments with two large datasets (5GB and 50GB compressed) show that COMET compares favorably (in both accuracy and training time) to learning on a subsample of data using a serial algorithm. Finally, we propose a new Gaussian approach for lazy ensemble evaluation which dynamically decides how many ensemble members to evaluate per data point; this can reduce evaluation cost by 100X or more

Investigating Randomised Sphere Covers in Supervised Learning

c©This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that no quotation from the thesis, nor any information derived therefrom, may be published without the author’s prior, written consent. In this thesis, we thoroughly investigate a simple Instance Based Learning (IBL) classifier known as Sphere Cover. We propose a simple Randomized Sphere Cover Classifier (αRSC) and use several datasets in order to evaluate the classification performance of the αRSC classifier. In addition, we analyse the generalization error of the proposed classifier using bias/variance decomposition. A Sphere Cover Classifier may be described from the compression scheme which stipulates data compression as the reason for high generalization performance. We investigate the compression capacity of αRSC using a sample compression bound. The Compression Scheme prompted us to search new compressibility methods for αRSC. As such, we used a Gaussian kernel to investigate further data compression

A PAC-Bayesian bound for Lifelong Learning

Transfer learning has received a lot of attention in the machine learning community over the last years, and several effective algorithms have been developed. However, relatively little is known about their theoretical properties, especially in the setting of lifelong learning, where the goal is to transfer information to tasks for which no data have been observed so far. In this work we study lifelong learning from a theoretical perspective. Our main result is a PAC-Bayesian generalization bound that offers a unified view on existing paradigms for transfer learning, such as the transfer of parameters or the transfer of low-dimensional representations. We also use the bound to derive two principled lifelong learning algorithms, and we show that these yield results comparable with existing methods.Comment: to appear at ICML 201

Efficient Learning with Partially Observed Attributes

We describe and analyze efficient algorithms for learning a linear predictor from examples when the learner can only view a few attributes of each training example. This is the case, for instance, in medical research, where each patient participating in the experiment is only willing to go through a small number of tests. Our analysis bounds the number of additional examples sufficient to compensate for the lack of full information on each training example. We demonstrate the efficiency of our algorithms by showing that when running on digit recognition data, they obtain a high prediction accuracy even when the learner gets to see only four pixels of each image.Comment: This is a full version of the paper appearing in The 27th International Conference on Machine Learning (ICML 2010

PAC-Bayesian Bounds on Rate-Efficient Classifiers

We derive analytic bounds on the noise invariance of majority vote classifiers operating on compressed inputs. Specifically, starting from recent bounds on the true risk of majority vote classifiers, we extend the applicability of PAC-Bayesian theory to quantify the resilience of majority votes to input noise stemming from compression. The derived bounds are intuitive in binary classification settings, where they can be measured as expressions of voter differentials and voter pair agreement. By combining measures of input distortion with analytic guarantees on noise invariance, we prescribe rate-efficient machines to compress inputs without affecting subsequent classification. Our validation shows how bounding noise invariance can inform the compression stage for any majority vote classifier such that worst-case implications of bad input reconstructions are known, and inputs can be compressed to the minimum amount of information needed prior to inference

Convergence of Online Mirror Descent

In this paper we consider online mirror descent (OMD) algorithms, a class of scalable online learning algorithms exploiting data geometric structures through mirror maps. Necessary and sufficient conditions are presented in terms of the step size sequence $\{\eta_t\}_{t}$ for the convergence of an OMD algorithm with respect to the expected Bregman distance induced by the mirror map. The condition is $\lim_{t\to\infty}\eta_t=0, \sum_{t=1}^{\infty}\eta_t=\infty$ in the case of positive variances. It is reduced to $\sum_{t=1}^{\infty}\eta_t=\infty$ in the case of zero variances for which the linear convergence may be achieved by taking a constant step size sequence. A sufficient condition on the almost sure convergence is also given. We establish tight error bounds under mild conditions on the mirror map, the loss function, and the regularizer. Our results are achieved by some novel analysis on the one-step progress of the OMD algorithm using smoothness and strong convexity of the mirror map and the loss function.Comment: Published in Applied and Computational Harmonic Analysis, 202