arXiv subjects

Nikita Zhivotovskiy

Publications and source records attributed to Nikita Zhivotovskiy.

At least 37 records · Page 2Linked to original sources

arXiv2023

Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails

We provide an estimator of the covariance matrix that achieves the optimal rate of convergence (up to constant factors) in the operator norm under two standard notions of data contamination: We allow the adversary to corrupt an $η$-fraction of the sample arbitrarily, while the distribution of the remaining data points only satisfies that the $L_{p}$-marginal moment with some $p \ge 4$ is equivalent to the corresponding $L_2$-marginal moment. Despite requiring the existence of only a few moments, our estimator achieves the same tail estimates as if the underlying distribution were Gaussian. As a part of our analysis, we prove a dimension-free Bai-Yin type theorem in the regime $p > 4$.

math.ST↗

arXiv2023

Exploring Local Norms in Exp-concave Statistical Learning

We consider the problem of stochastic convex optimization with exp-concave losses using Empirical Risk Minimization in a convex class. Answering a question raised in several prior works, we provide a $O( d / n + \log( 1 / δ) / n )$ excess risk bound valid for a wide class of bounded exp-concave losses, where $d$ is the dimension of the convex reference set, $n$ is the sample size, and $δ$ is the confidence level. Our result is based on a unified geometric assumption on the gradient of losses and the notion of local norms.

cs.LG↗

arXiv2023

Local Risk Bounds for Statistical Aggregation

In the problem of aggregation, the aim is to combine a given class of base predictors to achieve predictions nearly as accurate as the best one. In this flexible framework, no assumption is made on the structure of the class or the nature of the target. Aggregation has been studied in both sequential and statistical contexts. Despite some important differences between the two problems, the classical results in both cases feature the same global complexity measure. In this paper, we revisit and tighten classical results in the theory of aggregation in the statistical setting by replacing the global complexity with a smaller, local one. Some of our proofs build on the PAC-Bayes localization technique introduced by Catoni. Among other results, we prove localized versions of the classical bound for the exponential weights estimator due to Leung and Barron and deviation-optimal bounds for the Q-aggregation estimator. These bounds improve over the results of Dai, Rigollet and Zhang for fixed design regression and the results of Lecué and Rigollet for random design regression.

math.ST↗

arXiv2023

Optimal PAC Bounds Without Uniform Convergence

In statistical learning theory, determining the sample complexity of realizable binary classification for VC classes was a long-standing open problem. The results of Simon and Hanneke established sharp upper bounds in this setting. However, the reliance of their argument on the uniform convergence principle limits its applicability to more general learning settings such as multiclass classification. In this paper, we address this issue by providing optimal high probability risk bounds through a framework that surpasses the limitations of uniform convergence arguments. Our framework converts the leave-one-out error of permutation invariant predictors into high probability risk bounds. As an application, by adapting the one-inclusion graph algorithm of Haussler, Littlestone, and Warmuth, we propose an algorithm that achieves an optimal PAC bound for binary classification. Specifically, our result shows that certain aggregations of one-inclusion graph algorithms are optimal, addressing a variant of a classic question posed by Warmuth. We further instantiate our framework in three settings where uniform convergence is provably suboptimal. For multiclass classification, we prove an optimal risk bound that scales with the one-inclusion hypergraph density of the class, addressing the suboptimality of the analysis of Daniely and Shalev-Shwartz. For partial hypothesis classification, we determine the optimal sample complexity bound, resolving a question posed by Alon, Hanneke, Holzman, and Moran. For realizable bounded regression with absolute loss, we derive an optimal risk bound that relies on a modified version of the scale-sensitive dimension, refining the results of Bartlett and Long. Our rates surpass standard uniform convergence-based results due to the smaller complexity measure in our risk bound.

cs.LG↗

arXiv2023

Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians

Assume that $X_{1}, \ldots, X_{N}$ is an $\varepsilon$-contaminated sample of $N$ independent Gaussian vectors in $\mathbb{R}^d$ with mean $μ$ and covariance $Σ$. In the strong $\varepsilon$-contamination model we assume that the adversary replaced an $\varepsilon$ fraction of vectors in the original Gaussian sample by any other vectors. We show that there is an estimator $\widehat μ$ of the mean satisfying, with probability at least $1 - δ$, a bound of the form \[ \|\widehatμ - μ\|_2 \le c\left(\sqrt{\frac{\operatorname{Tr}(Σ)}{N}} + \sqrt{\frac{\|Σ\|\log(1/δ)}{N}} + \varepsilon\sqrt{\|Σ\|}\right), \] where $c > 0$ is an absolute constant and $\|Σ\|$ denotes the operator norm of $Σ$. In the same contaminated Gaussian setup, we construct an estimator $\widehat Σ$ of the covariance matrix $Σ$ that satisfies, with probability at least $1 - δ$, \[ \left\|\widehatΣ - Σ\right\| \le c\left(\sqrt{\frac{\|Σ\|\operatorname{Tr}(Σ)}{N}} + \|Σ\|\sqrt{\frac{\log(1/δ)}{N}} + \varepsilon\|Σ\|\right). \] Both results are optimal up to multiplicative constant factors. Despite the recent significant interest in robust statistics, achieving both dimension-free bounds in the canonical Gaussian case remained open. In fact, several previously known results were either dimension-dependent and required $Σ$ to be close to identity, or had a sub-optimal dependence on the contamination level $\varepsilon$. As a part of the analysis, we derive sharp concentration inequalities for central order statistics of Gaussian, folded normal, and chi-squared distributions.

math.ST↗

arXiv2022

Robustifying Markowitz

Markowitz mean-variance portfolios with sample mean and covariance as input parameters feature numerous issues in practice. They perform poorly out of sample due to estimation error, they experience extreme weights together with high sensitivity to change in input parameters. The heavy-tail characteristics of financial time series are in fact the cause for these erratic fluctuations of weights that consequently create substantial transaction costs. In robustifying the weights we present a toolbox for stabilizing costs and weights for global minimum Markowitz portfolios. Utilizing a projected gradient descent (PGD) technique, we avoid the estimation and inversion of the covariance operator as a whole and concentrate on robust estimation of the gradient descent increment. Using modern tools of robust statistics we construct a computationally efficient estimator with almost Gaussian properties based on median-of-means uniformly over weights. This robustified Markowitz approach is confirmed by empirical studies on equity markets. We demonstrate that robustified portfolios reach the lowest turnover compared to shrinkage-based and constrained portfolios while preserving or slightly improving out-of-sample performance.

econ.EM↗

arXiv2022

The One-Inclusion Graph Algorithm is not Always Optimal

The one-inclusion graph algorithm of Haussler, Littlestone, and Warmuth achieves an optimal in-expectation risk bound in the standard PAC classification setup. In one of the first COLT open problems, Warmuth conjectured that this prediction strategy always implies an optimal high probability bound on the risk, and hence is also an optimal PAC algorithm. We refute this conjecture in the strongest sense: for any practically interesting Vapnik-Chervonenkis class, we provide an in-expectation optimal one-inclusion graph algorithm whose high probability risk bound cannot go beyond that implied by Markov's inequality. Our construction of these poorly performing one-inclusion graph algorithms uses Varshamov-Tenengolts error correcting codes. Our negative result has several implications. First, it shows that the same poor high-probability performance is inherited by several recent prediction strategies based on generalizations of the one-inclusion graph algorithm. Second, our analysis shows yet another statistical problem that enjoys an estimator that is provably optimal in expectation via a leave-one-out argument, but fails in the high-probability regime. This discrepancy occurs despite the boundedness of the binary loss for which arguments based on concentration inequalities often provide sharp high probability risk bounds.

cs.LG↗

arXiv2022

A remark on Kashin's discrepancy argument and partial coloring in the Komlós conjecture

In this expository note, we discuss an early partial coloring result of B. Kashin [C. R. Acad. Bulgare Sci., 1985]. Although this result only implies Spencer's six standard deviations [Trans. Amer. Math. Soc., 1985] up to a $\log\log n$ factor, Kashin's argument gives a simple proof of the existence of a constant discrepancy partial coloring in the setup of Komlós conjecture.

math.CO↗

arXiv2022

A Regret-Variance Trade-Off in Online Learning

We consider prediction with expert advice for strongly convex and bounded losses, and investigate trade-offs between regret and "variance" (i.e., squared difference of learner's predictions and best expert predictions). With $K$ experts, the Exponentially Weighted Average (EWA) algorithm is known to achieve $O(\log K)$ regret. We prove that a variant of EWA either achieves a negative regret (i.e., the algorithm outperforms the best expert), or guarantees a $O(\log K)$ bound on both variance and regret. Building on this result, we show several examples of how variance of predictions can be exploited in learning. In the online to batch analysis, we show that a large empirical variance allows to stop the online to batch conversion early and outperform the risk of the best predictor in the class. We also recover the optimal rate of model selection aggregation when we do not consider early stopping. In online prediction with corrupted losses, we show that the effect of corruption on the regret can be compensated by a large variance. In online selective sampling, we design an algorithm that samples less when the variance is large, while guaranteeing the optimal regret bound in expectation. In online learning with abstention, we use a similar term as the variance to derive the first high-probability $O(\log K)$ regret bound in this setting. Finally, we extend our results to the setting of online linear regression.

cs.LG↗

arXiv2022

Dimension-free Bounds for Sums of Independent Matrices and Simple Tensors via the Variational Principle

We consider the deviation inequalities for the sums of independent $d$ by $d$ random matrices, as well as rank one random tensors. Our focus is on the non-isotropic case and the bounds that do not depend explicitly on the dimension $d$, but rather on the effective rank. In an elementary and unified manner, we show the following results: 1) A deviation bound for the sums of independent positive-semi-definite matrices. This result complements the dimension-free bound of Koltchinskii and Lounici [Bernoulli, 2017] on the sample covariance matrix in the sub-Gaussian case. 2) A new bound for truncated covariance matrices that is used to prove a dimension-free version of the bound of Adamczak, Litvak, Pajor and Tomczak-Jaegermann [Journal Of Amer. Math. Soc., 2010] on the sample covariance matrix in the log-concave case. 3) Dimension-free bounds for the operator norm of the sums of random tensors of rank one formed either by sub-Gaussian or by log-concave random vectors. This complements the result of Guédon and Rudelson [Adv. in Math., 2007]. 4) A non-isotropic version of the result of Alesker [Geom. Asp. of Funct. Anal., 1995] on the deviation of the norm of sub-exponential random vectors. 5) A dimension-free lower tail bound for sums of positive semi-definite matrices with heavy-tailed entries, sharpening the bound of Oliveira [Prob. Th. and Rel. Fields, 2016]. Our approach is based on the duality formula between entropy and moment generating functions. In contrast to the known proofs of dimension-free bounds, we avoid Talagrand's majorizing measure theorem, as well as generic chaining bounds for empirical processes. Some of our tools were pioneered by O. Catoni and co-authors in the context of robust statistical estimation.

math.PR↗

arXiv2022

Exponential Savings in Agnostic Active Learning through Abstention

We show that in pool-based active classification without assumptions on the underlying distribution, if the learner is given the power to abstain from some predictions by paying the price marginally smaller than the average loss $1/2$ of a random guess, exponential savings in the number of label requests are possible whenever they are possible in the corresponding realizable problem. We extend this result to provide a necessary and sufficient condition for exponential savings in pool-based active classification under the model misspecification.

cs.LG↗

arXiv2021

Stability and Deviation Optimal Risk Bounds with Convergence Rate $O(1/n)$

The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondrák, 2018, 2019), (Bousquet, Klochkov, Zhivotovskiy, 2020) contain a generally inevitable sampling error term of order $Θ(1/\sqrt{n})$. When applied to excess risk bounds, this leads to suboptimal results in several standard stochastic convex optimization problems. We show that if the so-called Bernstein condition is satisfied, the term $Θ(1/\sqrt{n})$ can be avoided, and high probability excess risk bounds of order up to $O(1/n)$ are possible via uniform stability. Using this result, we show a high probability excess risk bound with the rate $O(\log n/n)$ for strongly convex and Lipschitz losses valid for \emph{any} empirical risk minimization method. This resolves a question of Shalev-Shwartz, Shamir, Srebro, and Sridharan (2009). We discuss how $O(\log n/n)$ high probability excess risk bounds are possible for projected gradient descent in the case of strongly convex and Lipschitz losses without the usual smoothness assumption.

cs.LG↗

arXiv2021

Distribution-Free Robust Linear Regression

We study random design linear regression with no assumptions on the distribution of the covariates and with a heavy-tailed response variable. In this distribution-free regression setting, we show that boundedness of the conditional second moment of the response given the covariates is a necessary and sufficient condition for achieving nontrivial guarantees. As a starting point, we prove an optimal version of the classical in-expectation bound for the truncated least squares estimator due to Györfi, Kohler, Krzyżak, and Walk. However, we show that this procedure fails with constant probability for some distributions despite its optimal in-expectation performance. Then, combining the ideas of truncated least squares, median-of-means procedures, and aggregation theory, we construct a non-linear estimator achieving excess risk of order $d/n$ with an optimal sub-exponential tail. While existing approaches to linear regression for heavy-tailed distributions focus on proper estimators that return linear functions, we highlight that the improperness of our procedure is necessary for attaining nontrivial guarantees in the distribution-free setting.

math.ST↗

arXiv2021

Suboptimality of Constrained Least Squares and Improvements via Non-Linear Predictors

We study the problem of predicting as well as the best linear predictor in a bounded Euclidean ball with respect to the squared loss. When only boundedness of the data generating distribution is assumed, we establish that the least squares estimator constrained to a bounded Euclidean ball does not attain the classical $O(d/n)$ excess risk rate, where $d$ is the dimension of the covariates and $n$ is the number of samples. In particular, we construct a bounded distribution such that the constrained least squares estimator incurs an excess risk of order $Ω(d^{3/2}/n)$ hence refuting a recent conjecture of Ohad Shamir [JMLR 2015]. In contrast, we observe that non-linear predictors can achieve the optimal rate $O(d/n)$ with no assumptions on the distribution of the covariates. We discuss additional distributional assumptions sufficient to guarantee an $O(d/n)$ excess risk rate for the least squares estimator. Among them are certain moment equivalence assumptions often used in the robust statistics literature. While such assumptions are central in the analysis of unbounded and heavy-tailed settings, our work indicates that in some cases, they also rule out unfavorable bounded distributions.

math.ST↗

arXiv2020

Robust $k$-means Clustering for Distributions with Two Moments

We consider the robust algorithms for the $k$-means clustering problem where a quantizer is constructed based on $N$ independent observations. Our main results are median of means based non-asymptotic excess distortion bounds that hold under the two bounded moments assumption in a general separable Hilbert space. In particular, our results extend the renowned asymptotic result of Pollard, 1981 who showed that the existence of two moments is sufficient for strong consistency of an empirically optimal quantizer in $\mathbb{R}^d$. In a special case of clustering in $\mathbb{R}^d$, under two bounded moments, we prove matching (up to constant factors) non-asymptotic upper and lower bounds on the excess distortion, which depend on the probability mass of the lightest cluster of an optimal quantizer. Our bounds have the sub-Gaussian form, and the proofs are based on the versions of uniform bounds for robust mean estimators.

math.ST↗

arXiv2020

Fast classification rates without standard margin assumptions

We consider the classical problem of learning rates for classes with finite VC dimension. It is well known that fast learning rates up to $O\left(\frac{d}{n}\right)$ are achievable by the empirical risk minimization algorithm (ERM) if low noise or margin assumptions are satisfied. These usually require the optimal Bayes classifier to be in the class, and it has been shown that when this is not the case, the fast rates cannot be achieved even in the noise free case. In this paper, we further investigate the question of the fast rates under the misspecification, when the Bayes classifier is not in the class (also called the agnostic setting). First, we consider classification with a reject option, namely Chow's reject option model, and show that by slightly lowering the impact of hard instances, a learning rate of order $O\left(\frac{d}{n}\log \frac{n}{d}\right)$ is always achievable in the agnostic setting by a specific learning algorithm. Similar results were only known under special versions of margin assumptions. We also show that the performance of the proposed algorithm is never worse than the performance of ERM. Based on those results, we derive the necessary and sufficient conditions for classification (without a reject option) with fast rates in the agnostic setting achievable by improper learners. This simultaneously extends the work of Massart and Nédélec (Ann. of Statistics, 2006), which studied this question in the case where the Bayesian optimal rule belongs to the class, and the work of Ben-David and Urner (COLT, 2014), which allows the misspecification but is limited to the no noise setting. Our result also provides the first general setup in statistical learning theory in which an improper learning algorithm may significantly improve the learning rate for non-convex losses.

cs.LG↗

arXiv2020

On Mean Estimation for Heteroscedastic Random Variables

We study the problem of estimating the common mean $μ$ of $n$ independent symmetric random variables with different and unknown standard deviations $σ_1 \le σ_2 \le \cdots \leσ_n$. We show that, under some mild regularity assumptions on the distribution, there is a fully adaptive estimator $\widehatμ$ such that it is invariant to permutations of the elements of the sample and satisfies that, up to logarithmic factors, with high probability, \[ |\widehatμ - μ| \lesssim \min\left\{σ_{m^*}, \frac{\sqrt{n}}{\sum_{i = \sqrt{n}}^n σ_i^{-1}} \right\}~, \] where the index $m^* \lesssim \sqrt{n}$ satisfies $m^* \approx \sqrt{σ_{m^*}\sum_{i = m^*}^nσ_i^{-1}}$.

math.ST↗

arXiv2020

Fast Rates for Online Prediction with Abstention

In the setting of sequential prediction of individual $\{0, 1\}$-sequences with expert advice, we show that by allowing the learner to abstain from the prediction by paying a cost marginally smaller than $\frac 12$ (say, $0.49$), it is possible to achieve expected regret bounds that are independent of the time horizon $T$. We exactly characterize the dependence on the abstention cost $c$ and the number of experts $N$ by providing matching upper and lower bounds of order $\frac{\log N}{1-2c}$, which is to be contrasted with the best possible rate of $\sqrt{T\log N}$ that is available without the option to abstain. We also discuss various extensions of our model, including a setting where the sequence of abstention costs can change arbitrarily over time, where we show regret bounds interpolating between the slow and the fast rates mentioned above, under some natural assumptions on the sequence of abstention costs.

cs.LG↗