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We study nonparametric density estimation in non-stationary drift settings. Given a sequence of independent samples taken from a distribution that gradually changes in time, the goal is to compute the best estimate for the current distribution. We prove tight minimax risk bounds for both discrete and continuous smooth densities, where the minimum is over all possible estimates and the maximum is over all possible distributions that satisfy the drift constraints. Our technique handles a broad class of drift models and generalizes previous results on agnostic learning under drift. 

Welfare measures overall utility across a population, whereas malfare measures overall disutility, and the social planner’s problem can be cast either as maximizing the former or minimizing the latter. We show novel bounds on the expectations and tail probabilities of estimators of welfare, malfare, and regret of per-group (dis)utility values, where estimates are made from a finite sample drawn from each group. In particular, we consider estimating these quantities for individual functions (e.g., allocations or classifiers) with standard probabilistic bounds, and optimizing and bounding generalization error over hypothesis classes (i.e., we quantify overfitting) using Rademacher averages. We then study algorithmic fairness through the lens of sample complexity, finding that because marginalized or minority groups are often understudied, and fewer data are therefore available, the social planner is more likely to overfit to these groups, thus even models that seem fair in training can be systematically biased against such groups. We argue that this effect can be mitigated by ensuring sufficient sample sizes for each group, and our sample complexity analysis characterizes these sample sizes. Motivated by these conclusions, we present progressive sampling algorithms to efficiently optimize various fairness objectives. 

We develop a rigorous mathematical analysis of zero-shot learning with attributes. In this setting, the goal is to label novel classes with no training data, only detectors for attributes and a description of how those attributes are correlated with the target classes, called the class-attribute matrix. We develop the first non-trivial lower bound on the worst-case error of the best map from attributes to classes for this setting, even with perfect attribute detectors. The lower bound characterizes the theoretical intrinsic difficulty of the zero-shot problem based on the available information---the class-attribute matrix---and the bound is practically computable from it. Our lower bound is tight, as we show that we can always find a randomized map from attributes to classes whose expected error is upper bounded by the value of the lower bound. We show that our analysis can be predictive of how standard zero-shot methods behave in practice, including which classes will likely be confused with others. 

We develop a rigorous approach for using a set of arbitrarily correlated weak supervision sources in order to solve a multiclass classification task when only a very small set of labeled data is available. Our learning algorithm provably converges to a model that has minimum empirical risk with respect to an adversarial choice over feasible labelings for a set of unlabeled data, where the feasibility of a labeling is computed through constraints defined by rigorously estimated statistics of the weak supervision sources. We show theoretical guarantees for this approach that depend on the information provided by the weak supervision sources. Notably, this method does not require the weak supervision sources to have the same labeling space as the multiclass classification task. We demonstrate the effectiveness of our approach with experiments on various image classification tasks. 

We introduce Tiered Sampling , a novel technique for estimating the count of sparse motifs in massive graphs whose edges are observed in a stream. Our technique requires only a single pass on the data and uses a memory of fixed size M , which can be magnitudes smaller than the number of edges. Our methods address the challenging task of counting sparse motifs—sub-graph patterns—that have a low probability of appearing in a sample of M edges in the graph, which is the maximum amount of data available to the algorithms in each step. To obtain an unbiased and low variance estimate of the count, we partition the available memory into tiers (layers) of reservoir samples. While the base layer is a standard reservoir sample of edges, other layers are reservoir samples of sub-structures of the desired motif. By storing more frequent sub-structures of the motif, we increase the probability of detecting an occurrence of the sparse motif we are counting, thus decreasing the variance and error of the estimate. While we focus on the designing and analysis of algorithms for counting 4-cliques, we present a method which allows generalizing Tiered Sampling to obtain high-quality estimates for the number of occurrence of any sub-graph of interest, while reducing the analysis effort due to specific properties of the pattern of interest. We present a complete analytical analysis and extensive experimental evaluation of our proposed method using both synthetic and real-world data. Our results demonstrate the advantage of our method in obtaining high-quality approximations for the number of 4 and 5-cliques for large graphs using a very limited amount of memory, significantly outperforming the single edge sample approach for counting sparse motifs in large scale graphs. 

We develop a novel method that provides theoretical guarantees for learning from weak labelers without the (mostly unrealistic) assumption that the errors of the weak labelers are independent or come from a particular family of distributions. We show a rigorous technique for efficiently selecting small subsets of the labelers so that a majority vote from such subsets has a provably low error rate. We explore several extensions of this method and provide experimental results over a range of labeled data set sizes on 45 image classification tasks. Our performance-guaranteed methods consistently match the best performing alternative, which varies based on problem difficulty. On tasks with accurate weak labelers, our methods are on average 3 percentage points more accurate than the state-of-the-art adversarial method. On tasks with inaccurate weak labelers, our methods are on average 15 percentage points more accurate than the semi-supervised Dawid-Skene model (which assumes independence). 

The topology of the hyperlink graph among pages expressing different opinions may influence the exposure of readers to diverse content. Structural bias may trap a reader in a 'polarized' bubble with no access to other opinions. We model readers' behavior as random walks. A node is in a 'polarized' bubble if the expected length of a random walk from it to a page of different opinion is large. The structural bias of a graph is the sum of the radii of highly-polarized bubbles. We study the problem of decreasing the structural bias through edge insertions. 'Healing' all nodes with high polarized bubble radius is hard to approximate within a logarithmic factor, so we focus on finding the best k edges to insert to maximally reduce the structural bias. We present RePBubLik, an algorithm that leverages a variant of the random walk closeness centrality to select the edges to insert. RePBubLik obtains, under mild conditions, a constant-factor approximation. It reduces the structural bias faster than existing edge-recommendation methods, including some designed to reduce the polarization of a graph. 

We introduce the use of empirical centralization to derive novel practical, probabilistic, sample-dependent bounds to the Supremum Deviation (SD) of empirical means of functions in a family from their expectations. Our bounds have optimal dependence on the maximum (i.e., wimpy) variance and the function ranges, and the same dependence on the number of samples as existing SD bounds. To compute the bounds in practice, we develop novel tightly-concentrated Monte-Carlo estimators of the empirical Rademacher average of the empirically-centralized family, and we show novel concentration results for the empirical wimpy variance. Our experimental evaluation shows that our bounds greatly outperform non-centralized bounds and are extremely practical even at small sample sizes. 

We present an algorithm for approximating the diameter of massive weighted undirected graphs on distributed platforms supporting a MapReduce-like abstraction. In order to be efficient in terms of both time and space, our algorithm is based on a decomposition strategy which partitions the graph into disjoint clusters of bounded radius. Theoretically, our algorithm uses linear space and yields a polylogarithmic approximation guarantee; most importantly, for a large family of graphs, it features a round complexity asymptotically smaller than the one exhibited by a natural approximation algorithm based on the state-of-the-art Δ-stepping SSSP algorithm, which is its only practical, linear-space competitor in the distributed setting. We complement our theoretical findings with a proof-of-concept experimental analysis on large benchmark graphs, which suggests that our algorithm may attain substantial improvements in terms of running time compared to the aforementioned competitor, while featuring, in practice, a similar approximation ratio.