Deep learning architectures with a huge number of parameters are often compressed using pruning techniques to ensure computational efficiency of inference during deployment. Despite multitude of empirical advances, there is no theoretical understanding of the effectiveness of different pruning methods. We address this issue by setting up the problem in the statistical mechanics formulation of a teacher-student framework and deriving generalization error (GE) bounds of specific pruning methods. This theoretical premise allows comparison between pruning methods and we use it to investigate compression of neural networks via diversity-based pruning methods. A recent work showed that Determinantal Point Process (DPP) based node pruning method is notably superior to competing approaches when tested on real datasets. Using GE bounds in the aforementioned setup we provide theoretical guarantees for their empirical observations. Another consistent finding in literature is that sparse neural networks (edge pruned) generalize better than dense neural networks (node pruned) for a fixed number of parameters. We use our theoretical setup to prove that baseline random edge pruning method performs better than DPP node pruning method. Finally, we draw motivation from our theoretical results to propose a DPP edge pruning technique for neural networks which empirically outperforms other competing pruning methods on real datasets.

We investigate the problem of regression where one is allowed to abstain from predicting. We refer to this framework as regression with reject option as an extension of classification with reject option. In this context, we focus on the case where the rejection rate is fixed and derive the optimal rule which relies on thresholding the conditional variance function. We provide a semi-supervised estimation procedure of the optimal rule involving two datasets: a first labeled dataset is used to estimate both regression function and conditional variance function while a second unlabeled dataset is exploited to calibrate the desired rejection rate. The resulting predictor with reject option is shown to be almost as good as the optimal predictor with reject option both in terms of risk and rejection rate. We additionally apply our methodology with kNN algorithm and establish rates of convergence for the resulting kNN predictor under mild conditions. Finally, a numerical study is performed to illustrate the benefit of using the proposed procedure.

Over the past years, the task of learning in the presence of outliers has become an increasingly important objective in both statistics and machine learning. Indeed, in many situations, training data can be contaminated by undesired samples, which may badly affect the resulting learning task, especially in adversarial settings. Building robust estimators and algorithms that are resilient to outliers is therefore becoming crucial in many learning procedures. In particular, the inference of a probability density function from a contaminated random sample is of major concerns. Density estimation methods are mostly divided into parametric and nonparametric techniques. Among the nonparametric family, the Kernel Density Estimator (KDE) is probably the most known and used for both univariate and multivariate densities [Parzen, 1962; Silverman, 1986; Scott, 2015], but it also known to be sensitive to dataset contaminated by outliers [Kim and Scott, 2011, 2012; Vandermeulen and Scott, 2014].

The subspace approximation problem with outliers, for given $n$ points in $d$ dimensions $x_{1},\ldots, x_{n} \in R^{d}$, an integer $1 \leq k \leq d$, and an outlier parameter $0 \leq \alpha \leq 1$, is to find a $k$-dimensional linear subspace of $R^{d}$ that minimizes the sum of squared distances to its nearest $(1-\alpha)n$ points. More generally, the $\ell_{p}$ subspace approximation problem with outliers minimizes the sum of $p$-th powers of distances instead of the sum of squared distances. Even the case of robust PCA is non-trivial, and previous work requires additional assumptions on the input. Any multiplicative approximation algorithm for the subspace approximation problem with outliers must solve the robust subspace recovery problem, a special case in which the $(1-\alpha)n$ inliers in the optimal solution are promised to lie exactly on a $k$-dimensional linear subspace. However, robust subspace recovery is Small Set Expansion (SSE)-hard. We show how to extend dimension reduction techniques and bi-criteria approximations based on sampling to the problem of subspace approximation with outliers. To get around the SSE-hardness of robust subspace recovery, we assume that the squared distance error of the optimal $k$-dimensional subspace summed over the optimal $(1-\alpha)n$ inliers is at least $\delta$ times its squared-error summed over all $n$ points, for some $0 < \delta \leq 1 - \alpha$. With this assumption, we give an efficient algorithm to find a subset of $poly(k/\epsilon) \log(1/\delta) \log\log(1/\delta)$ points whose span contains a $k$-dimensional subspace that gives a multiplicative $(1+\epsilon)$-approximation to the optimal solution. The running time of our algorithm is linear in $n$ and $d$. Interestingly, our results hold even when the fraction of outliers $\alpha$ is large, as long as the obvious condition $0 < \delta \leq 1 - \alpha$ is satisfied.

We consider a multi-armed bandit problem specified by a set of Gaussian or Bernoulli distributions endowed with a unimodal structure. Although this problem has been addressed in the literature (Combes and Proutiere, 2014), the state-of-the-art algorithms for such structure make appear a forced-exploration mechanism. We introduce IMED-UB, the first forced-exploration free strategy that exploits the unimodal-structure, by adapting to this setting the Indexed Minimum Empirical Divergence (IMED) strategy introduced by Honda and Takemura (2015). This strategy is proven optimal. We then derive KLUCB-UB, a KLUCB version of IMED-UB, which is also proven optimal. Owing to our proof technique, we are further able to provide a concise finite-time analysis of both strategies in an unified way. Numerical experiments show that both IMED-UB and KLUCB-UB perform similarly in practice and outperform the state-of-the-art algorithms.

Continual learning is the problem of sequentially learning new tasks or knowledge while protecting previously acquired knowledge. However, catastrophic forgetting poses a grand challenge for neural networks performing such learning process. Thus, neural networks that are deployed in the real world often struggle in scenarios where the data distribution is non-stationary (concept drift), imbalanced, or not always fully available, i.e., rare edge cases. We propose a Differentiable Hebbian Consolidation model which is composed of a Differentiable Hebbian Plasticity (DHP) Softmax layer that adds a rapid learning plastic component (compressed episodic memory) to the fixed (slow changing) parameters of the softmax output layer; enabling learned representations to be retained for a longer timescale. We demonstrate the flexibility of our method by integrating well-known task-specific synaptic consolidation methods to penalize changes in the slow weights that are important for each target task. We evaluate our approach on the Permuted MNIST, Split MNIST and Vision Datasets Mixture benchmarks, and introduce an imbalanced variant of Permuted MNIST -- a dataset that combines the challenges of class imbalance and concept drift. Our proposed model requires no additional hyperparameters and outperforms comparable baselines by reducing forgetting.

Automated equipment health monitoring from streaming multisensor time-series data can be used to enable condition-based maintenance, avoid sudden catastrophic failures, and ensure high operational availability. We note that most complex machinery has a well-documented and readily accessible underlying structure capturing the inter-dependencies between sub-systems or modules. Deep learning models such as those based on recurrent neural networks (RNNs) or convolutional neural networks (CNNs) fail to explicitly leverage this potentially rich source of domain-knowledge into the learning procedure. In this work, we propose to capture the structure of a complex equipment in the form of a graph, and use graph neural networks (GNNs) to model multi-sensor time-series data. Using remaining useful life estimation as an application task, we evaluate the advantage of incorporating the graph structure via GNNs on the publicly available turbofan engine benchmark dataset. We observe that the proposed GNN-based RUL estimation model compares favorably to several strong baselines from literature such as those based on RNNs and CNNs. Additionally, we observe that the learned network is able to focus on the module (node) with impending failure through a simple attention mechanism, potentially paving the way for actionable diagnosis.

Malware remains a big threat to cyber security, calling for machine learning based malware detection. While promising, such detectors are known to be vulnerable to evasion attacks. Ensemble learning typically facilitates countermeasures, while attackers can leverage this technique to improve attack effectiveness as well. This motivates us to investigate which kind of robustness the ensemble defense or effectiveness the ensemble attack can achieve, particularly when they combat with each other. We thus propose a new attack approach, named mixture of attacks, by rendering attackers capable of multiple generative methods and multiple manipulation sets, to perturb a malware example without ruining its malicious functionality. This naturally leads to a new instantiation of adversarial training, which is further geared to enhancing the ensemble of deep neural networks. We evaluate defenses using Android malware detectors against 26 different attacks upon two practical datasets. Experimental results show that the new adversarial training significantly enhances the robustness of deep neural networks against a wide range of attacks, ensemble methods promote the robustness when base classifiers are robust enough, and yet ensemble attacks can evade the enhanced malware detectors effectively, even notably downgrading the VirusTotal service.

In the context of stochastic gradient descent(SGD) and adaptive moment estimation (Adam),researchers have recently proposed optimization techniques that transition from Adam to SGD with the goal of improving both convergence and generalization performance. However, precisely how each approach trades off early progress and generalization is not well understood; thus, it is unclear when or even if, one should transition from one approach to the other. In this work, by first studying the convex setting, we identify potential contributors to observed differences in performance between SGD and Adam. In particular,we provide theoretical insights for when and why Adam outperforms SGD and vice versa. We ad-dress the performance gap by adapting a single global learning rate for SGD, which we refer to as AdaSGD. We justify this proposed approach with empirical analyses in non-convex settings. On several datasets that span three different domains,we demonstrate how AdaSGD combines the benefits of both SGD and Adam, eliminating the need for approaches that transition from Adam to SGD.

Kernelized Stein discrepancy (KSD), though being extensively used in goodness-of-fit tests and model learning, suffers from the curse-of-dimensionality. We address this issue by proposing the sliced Stein discrepancy and its scalable and kernelized variants, which employs kernel-based test functions defined on the optimal onedimensional projections instead of the full input in high dimensions. When applied to goodness-of-fit tests, extensive experiments show the proposed discrepancy significantly outperforms KSD and various baselines in high dimensions. For model learning, we show its advantages by training an independent component analysis when compared with existing Stein discrepancy baselines. We further propose a novel particle inference method called sliced Stein variational gradient descent (S-SVGD) which alleviates the mode-collapse issue of SVGD in training variational autoencoders.