Existing methods for reducing disparate performance of a classifier across different demographic groups assume that one has access to a large data set, thereby focusing on the algorithmic aspect of optimizing overall performance subject to additional constraints. However, poor data collection and imbalanced data sets can severely affect the quality of these methods. In this work, we consider a setting where data collection and optimization are performed simultaneously. In such a scenario, a natural strategy to mitigate the performance difference of the classifier is to provide additional training data drawn from the demographic groups that are worse off. In this paper, we propose to consistently follow this strategy throughout the whole training process and to guide the resulting classifier towards equal performance on the different groups by adaptively sampling each data point from the group that is currently disadvantaged. We provide a rigorous theoretical analysis of our approach in a simplified one-dimensional setting and an extensive experimental evaluation on numerous real-world data sets, including a case study on the data collected during the Flint water crisis.

A common distinction in fair machine learning, in particular in fair classification, is between group fairness and individual fairness. In the context of clustering, group fairness has been studied extensively in recent years; however, individual fairness for clustering has hardly been explored. In this paper, we propose a natural notion of individual fairness for clustering. Our notion asks that every data point, on average, is closer to the points in its own cluster than to the points in any other cluster. We study several questions related to our proposed notion of individual fairness. On the negative side, we show that deciding whether a given data set allows for such an individually fair clustering in general is NP-hard. On the positive side, for the special case of a data set lying on the real line, we propose an efficient dynamic programming approach to find an individually fair clustering. For general data sets, we investigate heuristics aimed at minimizing the number of individual fairness violations and compare them to standard clustering approaches on real data sets.

Robustness is a key requirement for widespread deployment of machine learning algorithms, and has received much attention in both statistics and computer science. We study a natural model of robustness for high-dimensional statistical estimation problems that we call the adversarial perturbation model. An adversary can perturb every sample arbitrarily up to a specified magnitude $\delta$ measured in some $\ell_q$ norm, say $\ell_\infty$. Our model is motivated by emerging paradigms such as low precision machine learning and adversarial training. We study the classical problem of estimating the top-$r$ principal subspace of the Gaussian covariance matrix in high dimensions, under the adversarial perturbation model. We design a computationally efficient algorithm that given corrupted data, recovers an estimate of the top-$r$ principal subspace with error that depends on a robustness parameter $\kappa$ that we identify. This parameter corresponds to the $q \to 2$ operator norm of the projector onto the principal subspace, and generalizes well-studied analytic notions of sparsity. Additionally, in the absence of corruptions, our algorithmic guarantees recover existing bounds for problems such as sparse PCA and its higher rank analogs. We also prove that the above dependence on the parameter $\kappa$ is almost optimal asymptotically, not just in a minimax sense, but remarkably for every instance of the problem. This instance-optimal guarantee shows that the $q \to 2$ operator norm of the subspace essentially characterizes the estimation error under adversarial perturbations.

Despite the ubiquity of clustering as a tool in unsupervised learning, there is not yet a consensus on a formal theory, and the vast majority of work in this direction has focused on unsupervised clustering. We study a recently proposed framework for supervised clustering where there is access to a teacher. We give an improved generic algorithm to cluster any concept class in that model. Our algorithm is query-efficient in the sense that it involves only a small amount of interaction with the teacher. The model assumes that the teacher response to the algorithm is perfect.

Most approaches for ensuring or improving a model's fairness with respect to a protected attribute (such as race or gender) assume access to the true value of the protected attribute for every data point. In many scenarios, however, perfect knowledge of the protected attribute is unrealistic. In this paper, we ask to what extent fairness interventions can be effective even with imperfect information about the protected attribute. In particular, we study this question in the context of the prominent equalized odds method of Hardt et al. (2016). We claim that as long as the perturbation of the protected attribute is somewhat moderate, one should still run equalized odds if one would run it knowing the true protected attribute: the bias of the classifier that we obtain using the perturbed attribute is smaller than the bias of the original classifier, and its error is not larger than the error of the equalized odds classifier obtained when working with the true protected attribute.

In data summarization we want to choose k prototypes in order to summarize a data set. We study a setting where the data set comprises several demographic groups and we are restricted to choose k_i prototypes belonging to group i. A common approach to the problem without the fairness constraint is to optimize a centroid-based clustering objective such as k-center. A natural extension then is to incorporate the fairness constraint into the clustering objective. Existing algorithms for doing so run in time super-quadratic in the size of the data set. This is in contrast to the standard k-center objective that can be approximately optimized in linear time. In this paper, we resolve this gap by providing a simple approximation algorithm for the k-center problem under the fairness constraint with running time linear in the size of the data set and k. If the number of demographic groups is small, the approximation guarantee of our algorithm only incurs a constant-factor overhead. We demonstrate the applicability of our algorithm on both synthetic and real data sets.

Given the widespread popularity of spectral clustering (SC) for partitioning graph data, we study a version of constrained SC in which we try to incorporate the fairness notion proposed by Chierichetti et al. (2017). According to this notion, a clustering is fair if every demographic group is approximately proportionally represented in each cluster. To this end, we develop variants of both normalized and unnormalized constrained SC and show that they help find fairer clusterings on both synthetic and real data. We also provide a rigorous theoretical analysis of our algorithms. While there have been efforts to incorporate various constraints into the SC framework, theoretically analyzing them is a challenging problem. We overcome this by proposing a natural variant of the stochastic block model where h groups have strong inter-group connectivity, but also exhibit a "natural" clustering structure which is fair. We prove that our algorithms can recover this fair clustering with high probability.

Dictionary learning is a popular approach for inferring a hidden basis or dictionary in which data has a sparse representation. Data generated from the dictionary A (an n by m matrix, with m > n in the over-complete setting) is given by Y = AX where X is a matrix whose columns have supports chosen from a distribution over k-sparse vectors, and the non-zero values chosen from a symmetric distribution. Given Y, the goal is to recover A and X in polynomial time. Existing algorithms give polytime guarantees for recovering incoherent dictionaries, under strong distributional assumptions both on the supports of the columns of X, and on the values of the non-zero entries. In this work, we study the following question: Can we design efficient algorithms for recovering dictionaries when the supports of the columns of X are arbitrary? To address this question while circumventing the issue of non-identifiability, we study a natural semirandom model for dictionary learning where there are a large number of samples $y=Ax$ with arbitrary k-sparse supports for x, along with a few samples where the sparse supports are chosen uniformly at random. While the few samples with random supports ensures identifiability, the support distribution can look almost arbitrary in aggregate. Hence existing algorithmic techniques seem to break down as they make strong assumptions on the supports. Our main contribution is a new polynomial time algorithm for learning incoherent over-complete dictionaries that works under the semirandom model. Additionally the same algorithm provides polynomial time guarantees in new parameter regimes when the supports are fully random. Finally using these techniques, we also identify a minimal set of conditions on the supports under which the dictionary can be (information theoretically) recovered from polynomial samples for almost linear sparsity, i.e., $k=\tilde{O}(n)$.

We introduce a new approach for designing computationally efficient learning algorithms that are tolerant to noise, and demonstrate its effectiveness by designing algorithms with improved noise tolerance guarantees for learning linear separators. We consider both the malicious noise model and the adversarial label noise model. For malicious noise, where the adversary can corrupt both the label and the features, we provide a polynomial-time algorithm for learning linear separators in $\Re^d$ under isotropic log-concave distributions that can tolerate a nearly information-theoretically optimal noise rate of $\eta = \Omega(\epsilon)$. For the adversarial label noise model, where the distribution over the feature vectors is unchanged, and the overall probability of a noisy label is constrained to be at most $\eta$, we also give a polynomial-time algorithm for learning linear separators in $\Re^d$ under isotropic log-concave distributions that can handle a noise rate of $\eta = \Omega\left(\epsilon\right)$. We show that, in the active learning model, our algorithms achieve a label complexity whose dependence on the error parameter $\epsilon$ is polylogarithmic. This provides the first polynomial-time active learning algorithm for learning linear separators in the presence of malicious noise or adversarial label noise.

Variational inference is an efficient, popular heuristic used in the context of latent variable models. We provide the first analysis of instances where variational inference algorithms converge to the global optimum, in the setting of topic models. Our initializations are natural, one of them being used in LDA-c, the mostpopular implementation of variational inference.In addition to providing intuition into why this heuristic might work in practice, the multiplicative, rather than additive nature of the variational inference updates forces us to usenon-standard proof arguments, which we believe might be of general theoretical interest.