Does generalization performance of lql^q regularization learning depend on qq? A negative example

$l^q$-regularization has been demonstrated to be an attractive technique in machine learning and statistical modeling. It attempts to improve the generalization (prediction) capability of a machine (model) through appropriately shrinking its coefficients. The shape of a $l^q$ estimator differs in varying choices of the regularization order $q$. In particular, $l^1$ leads to the LASSO estimate, while $l^{2}$ corresponds to the smooth ridge regression. This makes the order $q$ a potential tuning parameter in applications. To facilitate the use of $l^{q}$-regularization, we intend to seek for a modeling strategy where an elaborative selection on $q$ is avoidable. In this spirit, we place our investigation within a general framework of $l^{q}$-regularized kernel learning under a sample dependent hypothesis space (SDHS). For a designated class of kernel functions, we show that all $l^{q}$ estimators for $0< q < \infty$ attain similar generalization error bounds. These estimated bounds are almost optimal in the sense that up to a logarithmic factor, the upper and lower bounds are asymptotically identical. This finding tentatively reveals that, in some modeling contexts, the choice of $q$ might not have a strong impact in terms of the generalization capability. From this perspective, $q$ can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..Comment: 35 pages, 3 figure

Regularization and Compression of Deep Neural Networks

Deep neural networks (DNN) are the state-of-the-art machine learning models outperforming traditional machine learning methods in a number of domains from vision and speech to natural language understanding and autonomous control. With large amounts of data becoming available, the task performance of DNNs in these domains predictably scales with the size of the DNNs. However, in data-scarce scenarios, large DNNs overfit to the training dataset resulting in inferior performance. Additionally, in scenarios where enormous amounts of data is available, large DNNs incur large inference latencies and memory costs. Thus, while imperative for achieving state-of-the-art performances, large DNNs require large amounts of data for training and large computational resources during inference. These two problems could be mitigated by sparsely training large DNNs. Imposing sparsity constraints during training limits the capacity of the model to overfit to the training set while still being able to obtain good generalization. Sparse DNNs have most of their weights close to zero after training. Therefore, most of the weights could be removed resulting in smaller inference costs. To effectively train sparse DNNs, this thesis proposes two new sparse stochastic regularization techniques called Bridgeout and Sparseout. Furthermore, Bridgeout is used to prune convolutional neural networks for low-cost inference. Bridgeout randomly perturbs the weights of a parametric model such as a DNN. It is theoretically shown that Bridgeout constrains the weights of linear models to a sparse subspace. Empirically, Bridgeout has been shown to perform better, on image classification tasks, than state-of-the-art DNNs when the data is limited. Sparseout is an activations counter-part of Bridgeout, operating on the outputs of the neurons instead of the weights of the neurons. Theoretically, Sparseout has been shown to be a general case of the commonly used Dropout regularization method. Empirical evidence suggests that Sparseout is capable of controlling the level of activations sparsity in neural networks. This flexibility allows Sparseout to perform better than Dropout on image classification and language modelling tasks. Furthermore, using Sparseout, it is found that activation sparsity is beneficial to recurrent neural networks for language modeling but densification of activations favors convolutional neural networks for image classification. To address the problem of high computational cost during inference, this thesis evaluates Bridgeout for pruning convolutional neural networks (CNN). It is shown that recent CNN architectures such as VGG, ResNet and Wide-ResNet trained with Bridgeout are more robust to one-shot filter pruning compared to non-sparse stochastic regularization

Learning Theory and Approximation

Learning theory studies data structures from samples and aims at understanding unknown function relations behind them. This leads to interesting theoretical problems which can be often attacked with methods from Approximation Theory. This workshop - the second one of this type at the MFO - has concentrated on the following recent topics: Learning of manifolds and the geometry of data; sparsity and dimension reduction; error analysis and algorithmic aspects, including kernel based methods for regression and classification; application of multiscale aspects and of refinement algorithms to learning