Several recent works have shown that state-of-the-art classifiers are vulnerable to worst-case (i.e., adversarial) perturbations of the datapoints. On the other hand, it has been empirically observed that these same classifiers are relatively robust to random noise. In this paper, we propose to study a \textit{semi-random} noise regime that generalizes both the random and worst-case noise regimes. We propose the first quantitative analysis of the robustness of nonlinear classifiers in this general noise regime. We establish precise theoretical bounds on the robustness of classifiers in this general regime, which depend on the curvature of the classifier's decision boundary. Our bounds confirm and quantify the empirical observations that classifiers satisfying curvature constraints are robust to random noise. Moreover, we quantify the robustness of classifiers in terms of the subspace dimension in the semi-random noise regime, and show that our bounds remarkably interpolate between the worst-case and random noise regimes. We perform experiments and show that the derived bounds provide very accurate estimates when applied to various state-of-the-art deep neural networks and datasets. This result suggests bounds on the curvature of the classifiers' decision boundaries that we support experimentally, and more generally offers important insights onto the geometry of high dimensional classification problems.

We summarize the potential impact that the European Union's new General Data Protection Regulation will have on the routine use of machine learning algorithms. Slated to take effect as law across the EU in 2018, it will restrict automated individual decision-making (that is, algorithms that make decisions based on user-level predictors) which "significantly affect" users. The law will also effectively create a "right to explanation," whereby a user can ask for an explanation of an algorithmic decision that was made about them. We argue that while this law will pose large challenges for industry, it highlights opportunities for computer scientists to take the lead in designing algorithms and evaluation frameworks which avoid discrimination and enable explanation.

Distance-based hierarchical clustering (HC) methods are widely used in unsupervised data analysis but few authors take account of uncertainty in the distance data. We incorporate a statistical model of the uncertainty through corruption or noise in the pairwise distances and investigate the problem of estimating the HC as unknown parameters from measurements. Specifically, we focus on single linkage hierarchical clustering (SLHC) and study its geometry. We prove that under fairly reasonable conditions on the probability distribution governing measurements, SLHC is equivalent to maximum partial profile likelihood estimation (MPPLE) with some of the information contained in the data ignored. At the same time, we show that direct evaluation of SLHC on maximum likelihood estimation (MLE) of pairwise distances yields a consistent estimator. Consequently, a full MLE is expected to perform better than SLHC in getting the correct HC results for the ground truth metric.

In this paper, we study the challenge of feature selection based on a relatively small collection of sample pairs $\{(x_i, y_i)\}_{1 \leq i \leq m}$. The observations $y_i \in \mathbb{R}$ are thereby supposed to follow a noisy single-index model, depending on a certain set of signal variables. A major difficulty is that these variables usually cannot be observed directly, but rather arise as hidden factors in the actual data vectors $x_i \in \mathbb{R}^d$ (feature variables). We will prove that a successful variable selection is still possible in this setup, even when the applied estimator does not have any knowledge of the underlying model parameters and only takes the 'raw' samples $\{(x_i, y_i)\}_{1 \leq i \leq m}$ as input. The model assumptions of our results will be fairly general, allowing for non-linear observations, arbitrary convex signal structures as well as strictly convex loss functions. This is particularly appealing for practical purposes, since in many applications, already standard methods, e.g., the Lasso or logistic regression, yield surprisingly good outcomes. Apart from a general discussion of the practical scope of our theoretical findings, we will also derive a rigorous guarantee for a specific real-world problem, namely sparse feature extraction from (proteomics-based) mass spectrometry data.

A benefit of using ensembles of decision tree methods like gradient boosting is that they can automatically provide estimates of feature importance from a trained predictive model. In this post you will discover how you can estimate the importance of features for a predictive modeling problem using the XGBoost library in Python. Feature Importance and Feature Selection With XGBoost in Python Photo by Keith Roper, some rights reserved. XGBoost is the high performance implementation of gradient boosting that you can now access directly in Python. A benefit of using gradient boosting is that after the boosted trees are constructed, it is relatively straightforward to retrieve importance scores for each attribute.

As the election season rampages on, we categorize voters into broad demographics -- soccer moms, NASCAR dads, blacks, whites, ALICEs, yuppies -- in an attempt to understand and discuss this complex, churning electorate. In doing so we're tapping into something fundamental about how we perceive the world: not as a sequence of singular individuals, but rather as a massive set of overlapping taxonomies that, taken together, comprise an impressively structured human experience. With fewer than 20 yes/no queries on category membership we can often identify a single object amidst a staggering breadth of possibilities. We've grouped everything that we know to exist and the groupings themselves are the primary subject of our thoughts. We can go the other direction as well -- taking an object and placing it in its many groups.

K-means' goal is to reveal patterns within the data. Let's imagine that you have a database with millions of rows representing your customers' orders. You might want to use the K-means algorithm to gather your customers into different groups based on key characteristics. K-means is easy to implement; you just need to specify how many clusters you want. Here you should look at two measures.

Epilepsy is a chronic neurological condition that affects approximately 70 million people worldwide. Characterised by sudden bursts of excess electricity in the brain, manifesting as seizures, epilepsy is still not well understood when compared with other neurological disorders. Seizures often happen unexpectedly and attempting to predict them has been a research topic for the last 30 years. Electroencephalograms have been integral to these studies, as the recordings that they produce can capture the brain's electrical signals. The diagnosis of epilepsy is usually made by a neurologist, but can be difficult to make in the early stages.

Getting into machine learning (ml) can seem like an unachievable task from the outside. However, after dedicating one week to learning the basics of the subject, I found it to be much more accessible than I anticipated. This article is intended to give others who're interested in getting into ml a roadmap of how to get started, drawing from the experiences I made in my intro week. Before my machine learning week, I had been reading about the subject for a while, and had gone through half of Andrew Ng's course on Coursera and a few other theoretical courses. So I had a tiny bit of conceptual understanding of ml, though I was completely unable to transfer any of my knowledge into code.

In this dissertation we propose alternative analysis of distributed stochastic gradient descent (SGD) algorithms that rely on spectral properties of the data covariance. As a consequence we can relate questions pertaining to speedups and convergence rates for distributed SGD to the data distribution instead of the regularity properties of the objective functions. More precisely we show that this rate depends on the spectral norm of the sample covariance matrix. An estimate of this norm can provide practitioners with guidance towards a potential gain in algorithm performance. For example many sparse datasets with low spectral norm prove to be amenable to gains in distributed settings. Towards establishing this data dependence we first study a distributed consensus-based SGD algorithm and show that the rate of convergence involves the spectral norm of the sample covariance matrix when the underlying data is assumed to be independent and identically distributed (homogenous). This dependence allows us to identify network regimes that prove to be beneficial for datasets with low sample covariance spectral norm. Existing consensus based analyses prove to be sub-optimal in the homogenous setting. Our analysis method also allows us to find data-dependent convergence rates as we limit the amount of communication. Spreading a fixed amount of data across more nodes slows convergence; in the asymptotic regime we show that adding more machines can help when minimizing twice-differentiable losses. Since the mini-batch results don't follow from the consensus results we propose a different data dependent analysis thereby providing theoretical validation for why certain datasets are more amenable to mini-batching. We also provide empirical evidence for results in this thesis.