Learning Smooth Distance Functions via Queries

In this work, we investigate the problem of learning distance functions within the query-based learning framework, where a learner is able to pose triplet queries of the form: ``Is $x_i$ closer to $x_j$ or $x_k$?'' We establish formal guarantees on the query complexity required to learn smooth, but otherwise general, distance functions under two notions of approximation: $\omega$-additive approximation and $(1 + \omega)$-multiplicative approximation. For the additive approximation, we propose a global method whose query complexity is quadratic in the size of a finite cover of the sample space. For the (stronger) multiplicative approximation, we introduce a method that combines global and local approaches, utilizing multiple Mahalanobis distance functions to capture local geometry. This method has a query complexity that scales quadratically with both the size of the cover and the ambient space dimension of the sample space.

Online Consistency of the Nearest Neighbor Rule

In the realizable online setting, a learner is tasked with making predictions for a stream of instances, where the correct answer is revealed after each prediction. A learning rule is online consistent if its mistake rate eventually vanishes. The nearest neighbor rule (Fix and Hodges, 1951) is a fundamental prediction strategy, but it is only known to be consistent under strong statistical or geometric assumptions: the instances come i.i.d. or the label classes are well-separated. We prove online consistency for all measurable functions in doubling metric spaces under the mild assumption that the instances are generated by a process that is uniformly absolutely continuous with respect to a finite, upper doubling measure.

Convergence Behavior of an Adversarial Weak Supervision Method

Labeling data via rules-of-thumb and minimal label supervision is central to Weak Supervision, a paradigm subsuming subareas of machine learning such as crowdsourced learning and semi-supervised ensemble learning. By using this labeled data to train modern machine learning methods, the cost of acquiring large amounts of hand labeled data can be ameliorated. Approaches to combining the rules-of-thumb falls into two camps, reflecting different ideologies of statistical estimation. The most common approach, exemplified by the Dawid-Skene model, is based on probabilistic modeling. The other, developed in the work of Balsubramani-Freund and others, is adversarial and game-theoretic. We provide a variety of statistical results for the adversarial approach under log-loss: we characterize the form of the solution, relate it to logistic regression, demonstrate consistency, and give rates of convergence. On the other hand, we find that probabilistic approaches for the same model class can fail to be consistent. Experimental results are provided to corroborate the theoretical results.

New bounds on the cohesion of complete-link and other linkage methods for agglomeration clustering

Linkage methods are among the most popular algorithms for hierarchical clustering. Despite their relevance the current knowledge regarding the quality of the clustering produced by these methods is limited. Here, we improve the currently available bounds on the maximum diameter of the clustering obtained by complete-link for metric spaces. One of our new bounds, in contrast to the existing ones, allows us to separate complete-link from single-link in terms of approximation for the diameter, which corroborates the common perception that the former is more suitable than the latter when the goal is producing compact clusters. We also show that our techniques can be employed to derive upper bounds on the cohesion of a class of linkage methods that includes the quite popular average-link.

Online nearest neighbor classification

We study an instance of online non-parametric classification in the realizable setting. In particular, we consider the classical 1-nearest neighbor algorithm, and show that it achieves sublinear regret - that is, a vanishing mistake rate - against dominated or smoothed adversaries in the realizable setting.

Active learning using region-based sampling

We present a general-purpose active learning scheme for data in metric spaces. The algorithm maintains a collection of neighborhoods of different sizes and uses label queries to identify those that have a strong bias towards one particular label; when two such neighborhoods intersect and have different labels, the region of overlap is treated as a ``known unknown'' and is a target of future active queries. We give label complexity bounds for this method that do not rely on assumptions about the data and we instantiate them in several cases of interest.

Data-Copying in Generative Models: A Formal Framework

There has been some recent interest in detecting and addressing memorization of training data by deep neural networks. A formal framework for memorization in generative models, called "data-copying," was proposed by Meehan et. al. (2020). We build upon their work to show that their framework may fail to detect certain kinds of blatant memorization. Motivated by this and the theory of non-parametric methods, we provide an alternative definition of data-copying that applies more locally. We provide a method to detect data-copying, and provably show that it works with high probability when enough data is available. We also provide lower bounds that characterize the sample requirement for reliable detection.

Streaming Encoding Algorithms for Scalable Hyperdimensional Computing

Hyperdimensional computing (HDC) is a paradigm for data representation and learning originating in computational neuroscience. HDC represents data as high-dimensional, low-precision vectors which can be used for a variety of information processing tasks like learning or recall. The mapping to high-dimensional space is a fundamental problem in HDC, and existing methods encounter scalability issues when the input data itself is high-dimensional. In this work, we explore a family of streaming encoding techniques based on hashing. We show formally that these methods enjoy comparable guarantees on performance for learning applications while being substantially more efficient than existing alternatives. We validate these results experimentally on a popular high-dimensional classification problem and show that our approach easily scales to very large data sets.

Convergence of online $k$-means

We prove asymptotic convergence for a general class of $k$-means algorithms performed over streaming data from a distribution: the centers asymptotically converge to the set of stationary points of the $k$-means cost function. To do so, we show that online $k$-means over a distribution can be interpreted as stochastic gradient descent with a stochastic learning rate schedule. Then, we prove convergence by extending techniques used in optimization literature to handle settings where center-specific learning rates may depend on the past trajectory of the centers.

Framework for Evaluating Faithfulness of Local Explanations

We study the faithfulness of an explanation system to the underlying prediction model. We show that this can be captured by two properties, consistency and sufficiency, and introduce quantitative measures of the extent to which these hold. Interestingly, these measures depend on the test-time data distribution. For a variety of existing explanation systems, such as anchors, we analytically study these quantities. We also provide estimators and sample complexity bounds for empirically determining the faithfulness of black-box explanation systems. Finally, we experimentally validate the new properties and estimators.