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Aryeh Kontorovich

Aryeh Kontorovich appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
Machine Learningmath.STmath.PRStatistics TheoryData Structures and AlgorithmsInformation TheoryFormal Languages and Automata Theorymath.ITArtificial IntelligenceComputational ComplexityComputer VisionDiscrete Mathematicsmath.COmath.FAmath.MG

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Published work

31 published item(s)

preprint2026arXiv

A Fine-Grained Understanding of Uniform Convergence for Halfspaces

We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in $\mathbb{R}^d$ with $d\ge 2$, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error $Θ(d\ln(n/d)/n)$, and in the agnostic setting the deviation scales as $\sqrt{τ\ln(1/τ)}$ at true error $τ$. In contrast, homogeneous halfspaces in $\mathbb{R}^2$ exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error $O(1/n)$. In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a $\ln\ln n$ overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds.

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preprint2026arXiv

Realizable Bayes-Consistency for General Metric Losses

We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification (Bousquet et al., 2020; Hanneke et al., 2021) and real-valued regression (Attias et al., 2024). Given an instance space $(X,ρ)$, a label space $(Y,\ell)$ with possibly unbounded loss, and a hypothesis class $H \subseteq Y^{X}$, we resolve the realizable case of an open problem presented in Tsir Cohen and Kontorovich (2022). Specifically, we find the necessary and sufficient conditions on the hypothesis class $H$ under which there exists a distribution-free learning rule whose risk converges almost surely to the best-in-class risk (which is zero) for every realizable data-generating distribution. Our main contribution is this sharp characterization in terms of a combinatorial obstruction: Similarly to Attias et al. (2024), we introduce the notion of an infinite non-decreasing $(γ_k)$-Littlestone tree, where $γ_k \to \infty$. This extends the Littlestone tree structure used in Bousquet et al. (2020) to the metric loss setting.

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preprint2022arXiv

Adaptive Data Analysis with Correlated Observations

The vast majority of the work on adaptive data analysis focuses on the case where the samples in the dataset are independent. Several approaches and tools have been successfully applied in this context, such as differential privacy, max-information, compression arguments, and more. The situation is far less well-understood without the independence assumption. We embark on a systematic study of the possibilities of adaptive data analysis with correlated observations. First, we show that, in some cases, differential privacy guarantees generalization even when there are dependencies within the sample, which we quantify using a notion we call Gibbs-dependence. We complement this result with a tight negative example. Second, we show that the connection between transcript-compression and adaptive data analysis can be extended to the non-iid setting.

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preprint2022arXiv

Domain Invariant Adversarial Learning

The phenomenon of adversarial examples illustrates one of the most basic vulnerabilities of deep neural networks. Among the variety of techniques introduced to surmount this inherent weakness, adversarial training has emerged as the most effective strategy for learning robust models. Typically, this is achieved by balancing robust and natural objectives. In this work, we aim to further optimize the trade-off between robust and standard accuracy by enforcing a domain-invariant feature representation. We present a new adversarial training method, Domain Invariant Adversarial Learning (DIAL), which learns a feature representation that is both robust and domain invariant. DIAL uses a variant of Domain Adversarial Neural Network (DANN) on the natural domain and its corresponding adversarial domain. In the case where the source domain consists of natural examples and the target domain is the adversarially perturbed examples, our method learns a feature representation constrained not to discriminate between the natural and adversarial examples, and can therefore achieve a more robust representation. DIAL is a generic and modular technique that can be easily incorporated into any adversarial training method. Our experiments indicate that incorporating DIAL in the adversarial training process improves both robustness and standard accuracy.

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preprint2022arXiv

Efficient Kirszbraun Extension with Applications to Regression

We introduce a framework for performing regression between two Hilbert spaces. This is done based on Kirszbraun's extension theorem, to the best of our knowledge, the first application of this technique to supervised learning. We analyze the statistical and computational aspects of this method. We decompose this task into two stages: training (which corresponds operationally to smoothing/regularization) and prediction (which is achieved via Kirszbraun extension). Both are solved algorithmically via a novel multiplicative weight updates (MWU) scheme, which, for our problem formulation, achieves a quadratic runtime improvement over the state of the art. Our empirical results indicate a dramatic improvement over standard off-the-shelf solvers in our setting.

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preprint2022arXiv

Estimating the Mixing Time of Ergodic Markov Chains

We address the problem of estimating the mixing time $t_{\mathsf{mix}}$ of an arbitrary ergodic finite-state Markov chain from a single trajectory of length $m$. The reversible case was addressed by Hsu et al. [2019], who left the general case as an open problem. In the reversible case, the analysis is greatly facilitated by the fact that the Markov operator is self-adjoint, and Weyl's inequality allows for a dimension-free perturbation analysis of the empirical eigenvalues. As Hsu et al. point out, in the absence of reversibility (which induces asymmetric pair probabilities matrices), the existing perturbation analysis has a worst-case exponential dependence on the number of states $d$. Furthermore, even if an eigenvalue perturbation analysis with better dependence on $d$ were available, in the non-reversible case the connection between the spectral gap and the mixing time is not nearly as straightforward as in the reversible case. Our key insight is to estimate the pseudo-spectral gap $γ_{\mathsf{ps}}$ instead, which allows us to overcome the loss of symmetry and to achieve a polynomial dependence on the minimal stationary probability $π_\star$ and $γ_{\mathsf{ps}}$. Additionally, in the reversible case, we obtain simultaneous nearly (up to logarithmic factors) minimax rates in $t_{\mathsf{mix}}$ and precision $\varepsilon$, closing a gap in Hsu et al., who treated $\varepsilon$ as constant in the lower bounds. Finally, we construct fully empirical confidence intervals for $γ_{\mathsf{ps}}$, which shrink to zero at a rate of roughly $1/\sqrt{m}$, and improve the state of the art in even the reversible case.

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preprint2022arXiv

Improved Generalization Bounds for Adversarially Robust Learning

We consider a model of robust learning in an adversarial environment. The learner gets uncorrupted training data with access to possible corruptions that may be affected by the adversary during testing. The learner's goal is to build a robust classifier, which will be tested on future adversarial examples. The adversary is limited to $k$ possible corruptions for each input. We model the learner-adversary interaction as a zero-sum game. This model is closely related to the adversarial examples model of Schmidt et al. (2018); Madry et al. (2017). Our main results consist of generalization bounds for the binary and multiclass classification, as well as the real-valued case (regression). For the binary classification setting, we both tighten the generalization bound of Feige et al. (2015), and are also able to handle infinite hypothesis classes. The sample complexity is improved from $O(\frac{1}{ε^4}\log(\frac{|H|}δ))$ to $O\big(\frac{1}{ε^2}(kVC(H)\log^{\frac{3}{2}+α}(kVC(H))+\log(\frac{1}δ)\big)$ for any $α> 0$. Additionally, we extend the algorithm and generalization bound from the binary to the multiclass and real-valued cases. Along the way, we obtain results on fat-shattering dimension and Rademacher complexity of $k$-fold maxima over function classes; these may be of independent interest. For binary classification, the algorithm of Feige et al. (2015) uses a regret minimization algorithm and an ERM oracle as a black box; we adapt it for the multiclass and regression settings. The algorithm provides us with near-optimal policies for the players on a given training sample.

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preprint2020arXiv

Apportioned Margin Approach for Cost Sensitive Large Margin Classifiers

We consider the problem of cost sensitive multiclass classification, where we would like to increase the sensitivity of an important class at the expense of a less important one. We adopt an {\em apportioned margin} framework to address this problem, which enables an efficient margin shift between classes that share the same boundary. The decision boundary between all pairs of classes divides the margin between them in accordance to a given prioritization vector, which yields a tighter error bound for the important classes while also reducing the overall out-of-sample error. In addition to demonstrating an efficient implementation of our framework, we derive generalization bounds, demonstrate Fisher consistency, adapt the framework to Mercer's kernel and to neural networks, and report promising empirical results on all accounts.

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preprint2020arXiv

Fast and Bayes-consistent nearest neighbors

Research on nearest-neighbor methods tends to focus somewhat dichotomously either on the statistical or the computational aspects -- either on, say, Bayes consistency and rates of convergence or on techniques for speeding up the proximity search. This paper aims at bridging these realms: to reap the advantages of fast evaluation time while maintaining Bayes consistency, and further without sacrificing too much in the risk decay rate. We combine the locality-sensitive hashing (LSH) technique with a novel missing-mass argument to obtain a fast and Bayes-consistent classifier. Our algorithm's prediction runtime compares favorably against state of the art approximate NN methods, while maintaining Bayes-consistency and attaining rates comparable to minimax. On samples of size $n$ in $\R^d$, our pre-processing phase has runtime $O(d n \log n)$, while the evaluation phase has runtime $O(d\log n)$ per query point.

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preprint2020arXiv

Nested Barycentric Coordinate System as an Explicit Feature Map

We propose a new embedding method which is particularly well-suited for settings where the sample size greatly exceeds the ambient dimension. Our technique consists of partitioning the space into simplices and then embedding the data points into features corresponding to the simplices' barycentric coordinates. We then train a linear classifier in the rich feature space obtained from the simplices. The decision boundary may be highly non-linear, though it is linear within each simplex (and hence piecewise-linear overall). Further, our method can approximate any convex body. We give generalization bounds based on empirical margin and a novel hybrid sample compression technique. An extensive empirical evaluation shows that our method consistently outperforms a range of popular kernel embedding methods.

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preprint2020arXiv

Non-uniform packings

We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {\em non-uniform packing number}. We show that the non-uniform packing number can be upper-bounded in terms of the {\em average} radius of the balls, resulting in bounds of the familiar classical form.

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preprint2020arXiv

On Biased Random Walks, Corrupted Intervals, and Learning Under Adversarial Design

We tackle some fundamental problems in probability theory on corrupted random processes on the integer line. We analyze when a biased random walk is expected to reach its bottommost point and when intervals of integer points can be detected under a natural model of noise. We apply these results to problems in learning thresholds and intervals under a new model for learning under adversarial design.

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preprint2020arXiv

Statistical Estimation of Ergodic Markov Chain Kernel over Discrete State Space

We investigate the statistical complexity of estimating the parameters of a discrete-state Markov chain kernel from a single long sequence of state observations. In the finite case, we characterize (modulo logarithmic factors) the minimax sample complexity of estimation with respect to the operator infinity norm, while in the countably infinite case, we analyze the problem with respect to a natural entry-wise norm derived from total variation. We show that in both cases, the sample complexity is governed by the mixing properties of the unknown chain, for which, in the finite-state case, there are known finite-sample estimators with fully empirical confidence intervals.

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preprint2016arXiv

Concentration of measure without independence: a unified approach via the martingale method

The concentration of measure phenomenon may be summarized as follows: a function of many weakly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several powerful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less well understood. This chapter focuses on the martingale method for deriving concentration inequalities without independence assumptions. In particular, we use the machinery of so-called Wasserstein matrices to show that the Azuma-Hoeffding concentration inequality for martingales with almost surely bounded differences, when applied in a sufficiently abstract setting, is powerful enough to recover and sharpen several known concentration results for nonproduct measures. Wasserstein matrices provide a natural formalism for capturing the interplay between the metric and the probabilistic structures, which is fundamental to the concentration phenomenon.

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preprint2015arXiv

Adaptive Metric Dimensionality Reduction

We study adaptive data-dependent dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are doubling, or nearly doubling. On the algorithmic front, we describe an analogue of PCA for metric spaces: namely an efficient procedure that approximates the data's intrinsic dimension, which is often much lower than the ambient dimension. Our approach thus leverages the dual benefits of low dimensionality: (1) more efficient algorithms, e.g., for proximity search, and (2) more optimistic generalization bounds.

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preprint2015arXiv

Mixing Time Estimation in Reversible Markov Chains from a Single Sample Path

This article provides the first procedure for computing a fully data-dependent interval that traps the mixing time $t_{\text{mix}}$ of a finite reversible ergodic Markov chain at a prescribed confidence level. The interval is computed from a single finite-length sample path from the Markov chain, and does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time $t_{\text{relax}}$, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a $\sqrt{n}$ rate, where $n$ is the length of the sample path. Upper and lower bounds are given on the number of samples required to achieve constant-factor multiplicative accuracy. The lower bounds indicate that, unless further restrictions are placed on the chain, no procedure can achieve this accuracy level before seeing each state at least $Ω(t_{\text{relax}})$ times on the average. Finally, future directions of research are identified.

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preprint2015arXiv

Nearly optimal classification for semimetrics

We initiate the rigorous study of classification in semimetric spaces, which are point sets with a distance function that is non-negative and symmetric, but need not satisfy the triangle inequality. For metric spaces, the doubling dimension essentially characterizes both the runtime and sample complexity of classification algorithms --- yet we show that this is not the case for semimetrics. Instead, we define the {\em density dimension} and discover that it plays a central role in the statistical and algorithmic feasibility of learning in semimetric spaces. We present nearly optimal sample compression algorithms and use these to obtain generalization guarantees, including fast rates. The latter hold for general sample compression schemes and may be of independent interest.

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preprint2014arXiv

Consistency of weighted majority votes

We revisit the classical decision-theoretic problem of weighted expert voting from a statistical learning perspective. In particular, we examine the consistency (both asymptotic and finitary) of the optimal Nitzan-Paroush weighted majority and related rules. In the case of known expert competence levels, we give sharp error estimates for the optimal rule. When the competence levels are unknown, they must be empirically estimated. We provide frequentist and Bayesian analyses for this situation. Some of our proof techniques are non-standard and may be of independent interest. The bounds we derive are nearly optimal, and several challenging open problems are posed. Experimental results are provided to illustrate the theory.

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preprint2014arXiv

Efficient Classification for Metric Data

Recent advances in large-margin classification of data residing in general metric spaces (rather than Hilbert spaces) enable classification under various natural metrics, such as string edit and earthmover distance. A general framework developed for this purpose by von Luxburg and Bousquet [JMLR, 2004] left open the questions of computational efficiency and of providing direct bounds on generalization error. We design a new algorithm for classification in general metric spaces, whose runtime and accuracy depend on the doubling dimension of the data points, and can thus achieve superior classification performance in many common scenarios. The algorithmic core of our approach is an approximate (rather than exact) solution to the classical problems of Lipschitz extension and of Nearest Neighbor Search. The algorithm's generalization performance is guaranteed via the fat-shattering dimension of Lipschitz classifiers, and we present experimental evidence of its superiority to some common kernel methods. As a by-product, we offer a new perspective on the nearest neighbor classifier, which yields significantly sharper risk asymptotics than the classic analysis of Cover and Hart [IEEE Trans. Info. Theory, 1967].

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preprint2014arXiv

Maximum Margin Multiclass Nearest Neighbors

We develop a general framework for margin-based multicategory classification in metric spaces. The basic work-horse is a margin-regularized version of the nearest-neighbor classifier. We prove generalization bounds that match the state of the art in sample size $n$ and significantly improve the dependence on the number of classes $k$. Our point of departure is a nearly Bayes-optimal finite-sample risk bound independent of $k$. Although $k$-free, this bound is unregularized and non-adaptive, which motivates our main result: Rademacher and scale-sensitive margin bounds with a logarithmic dependence on $k$. As the best previous risk estimates in this setting were of order $\sqrt k$, our bound is exponentially sharper. From the algorithmic standpoint, in doubling metric spaces our classifier may be trained on $n$ examples in $O(n^2\log n)$ time and evaluated on new points in $O(\log n)$ time.

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preprint2014arXiv

Minimum KL-divergence on complements of $L_1$ balls

Pinsker's widely used inequality upper-bounds the total variation distance $||P-Q||_1$ in terms of the Kullback-Leibler divergence $D(P||Q)$. Although in general a bound in the reverse direction is impossible, in many applications the quantity of interest is actually $D^*(P,\eps)$ --- defined, for an arbitrary fixed $P$, as the infimum of $D(P||Q)$ over all distributions $Q$ that are $\eps$-far away from $P$ in total variation. We show that $D^*(P,\eps)\le C\eps^2 + O(\eps^3)$, where $C=C(P)=1/2$ for "balanced" distributions, thereby providing a kind of reverse Pinsker inequality. An application to large deviations is given, and some of the structural results may be of independent interest. Keywords: Pinsker inequality, Sanov's theorem, large deviations

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preprint2013arXiv

Concentration in unbounded metric spaces and algorithmic stability

We prove an extension of McDiarmid's inequality for metric spaces with unbounded diameter. To this end, we introduce the notion of the {\em subgaussian diameter}, which is a distribution-dependent refinement of the metric diameter. Our technique provides an alternative approach to that of Kutin and Niyogi's method of weakly difference-bounded functions, and yields nontrivial, dimension-free results in some interesting cases where the former does not. As an application, we give apparently the first generalization bound in the algorithmic stability setting that holds for unbounded loss functions. We furthermore extend our concentration inequality to strongly mixing processes.

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preprint2013arXiv

On learning parametric-output HMMs

We present a novel approach for learning an HMM whose outputs are distributed according to a parametric family. This is done by {\em decoupling} the learning task into two steps: first estimating the output parameters, and then estimating the hidden states transition probabilities. The first step is accomplished by fitting a mixture model to the output stationary distribution. Given the parameters of this mixture model, the second step is formulated as the solution of an easily solvable convex quadratic program. We provide an error analysis for the estimated transition probabilities and show they are robust to small perturbations in the estimates of the mixture parameters. Finally, we support our analysis with some encouraging empirical results.

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preprint2013arXiv

Predictive PAC Learning and Process Decompositions

We informally call a stochastic process learnable if it admits a generalization error approaching zero in probability for any concept class with finite VC-dimension (IID processes are the simplest example). A mixture of learnable processes need not be learnable itself, and certainly its generalization error need not decay at the same rate. In this paper, we argue that it is natural in predictive PAC to condition not on the past observations but on the mixture component of the sample path. This definition not only matches what a realistic learner might demand, but also allows us to sidestep several otherwise grave problems in learning from dependent data. In particular, we give a novel PAC generalization bound for mixtures of learnable processes with a generalization error that is not worse than that of each mixture component. We also provide a characterization of mixtures of absolutely regular ($β$-mixing) processes, of independent probability-theoretic interest.

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preprint2013arXiv

The state complexity of random DFAs

The state complexity of a Deterministic Finite-state automaton (DFA) is the number of states in its minimal equivalent DFA. We study the state complexity of random $n$-state DFAs over a $k$-symbol alphabet, drawn uniformly from the set $[n]^{[n]\times[k]}\times2^{[n]}$ of all such automata. We show that, with high probability, the latter is $α_k n + O(\sqrt n\log n)$ for a certain explicit constant $α_k$.

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preprint2013arXiv

Uniform Chernoff and Dvoretzky-Kiefer-Wolfowitz-type inequalities for Markov chains and related processes

We observe that the technique of Markov contraction can be used to establish measure concentration for a broad class of non-contracting chains. In particular, geometric ergodicity provides a simple and versatile framework. This leads to a short, elementary proof of a general concentration inequality for Markov and hidden Markov chains (HMM), which supercedes some of the known results and easily extends to other processes such as Markov trees. As applications, we give a Dvoretzky-Kiefer-Wolfowitz-type inequality and a uniform Chernoff bound. All of our bounds are dimension-free and hold for countably infinite state spaces.

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preprint2012arXiv

Efficiently decoding strings from their shingles

Determining whether an unordered collection of overlapping substrings (called shingles) can be uniquely decoded into a consistent string is a problem that lies within the foundation of a broad assortment of disciplines ranging from networking and information theory through cryptography and even genetic engineering and linguistics. We present three perspectives on this problem: a graph theoretic framework due to Pevzner, an automata theoretic approach from our previous work, and a new insight that yields a time-optimal streaming algorithm for determining whether a string of $n$ characters over the alphabet $Σ$ can be uniquely decoded from its two-character shingles. Our algorithm achieves an overall time complexity $Θ(n)$ and space complexity $O(|Σ|)$. As an application, we demonstrate how this algorithm can be extended to larger shingles for efficient string reconciliation.

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preprint2012arXiv

On the Convergence of the Empirical Distribution

We develop a general technique for bounding the tail of the total variation distance between the empirical and the true distributions over countable sets. Our methods sharpen a deviation bound of Devroye (1983) for distributions over finite sets, and also hold for the broader class of distributions with countable support. We also provide some lower bounds of possible independent interest.

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preprint2011arXiv

Unique decodability of bigram counts by finite automata

We revisit the problem of deciding whether a given string is uniquely decodable from its bigram counts by means of a finite automaton. An efficient algorithm for constructing a polynomial-size nondeterministic finite automaton that decides unique decodability is given. Conversely, we show that the minimum deterministic finite automaton for deciding unique decodability has at least exponentially many states in alphabet size.

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