Majority-of-Three: The Simplest Optimal Learner?

Developing an optimal PAC learning algorithm in the realizable setting, where empirical risk minimization (ERM) is suboptimal, was a major open problem in learning theory for decades. The problem was finally resolved by Hanneke a few years ago. Unfortunately, Hanneke's algorithm is quite complex as it returns the majority vote of many ERM classifiers that are trained on carefully selected subsets of the data. It is thus a natural goal to determine the simplest algorithm that is optimal. In this work we study the arguably simplest algorithm that could be optimal: returning the majority vote of three ERM classifiers. We show that this algorithm achieves the optimal in-expectation bound on its error which is provably unattainable by a single ERM classifier. Furthermore, we prove a near-optimal high-probability bound on this algorithm's error. We conjecture that a better analysis will prove that this algorithm is in fact optimal in the high-probability regime.

High-Probability Risk Bounds via Sequential Predictors

Online learning methods yield sequential regret bounds under minimal assumptions and provide in-expectation risk bounds for statistical learning. However, despite the apparent advantage of online guarantees over their statistical counterparts, recent findings indicate that in many important cases, regret bounds may not guarantee tight high-probability risk bounds in the statistical setting. In this work we show that online to batch conversions applied to general online learning algorithms can bypass this limitation. Via a general second-order correction to the loss function defining the regret, we obtain nearly optimal high-probability risk bounds for several classical statistical estimation problems, such as discrete distribution estimation, linear regression, logistic regression, and conditional density estimation. Our analysis relies on the fact that many online learning algorithms are improper, as they are not restricted to use predictors from a given reference class. The improper nature of our estimators enables significant improvements in the dependencies on various problem parameters. Finally, we discuss some computational advantages of our sequential algorithms over their existing batch counterparts.

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 Lecu\'e and Rigollet for random design regression.

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.

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 $O( d / n + \log( 1 / \delta) / n )$ 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 $\delta$ is the confidence level. Our result is based on a unified geometric assumption on the gradient of losses and the notion of local norms.

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 $\mu$ and covariance $\Sigma$. 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 $\widehat \mu$ of the mean satisfying, with probability at least $1 - \delta$, a bound of the form \[ \|\widehat{\mu} - \mu\|_2 \le c\left(\sqrt{\frac{\operatorname{Tr}(\Sigma)}{N}} + \sqrt{\frac{\|\Sigma\|\log(1/\delta)}{N}} + \varepsilon\sqrt{\|\Sigma\|}\right), \] where $c > 0$ is an absolute constant and $\|\Sigma\|$ denotes the operator norm of $\Sigma$. In the same contaminated Gaussian setup, we construct an estimator $\widehat \Sigma$ of the covariance matrix $\Sigma$ that satisfies, with probability at least $1 - \delta$, \[ \left\|\widehat{\Sigma} - \Sigma\right\| \le c\left(\sqrt{\frac{\|\Sigma\|\operatorname{Tr}(\Sigma)}{N}} + \|\Sigma\|\sqrt{\frac{\log(1/\delta)}{N}} + \varepsilon\|\Sigma\|\right). \] 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 $\Sigma$ 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.

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.

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.

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.

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

The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondr\'{a}k, 2018, 2019), (Bousquet, Klochkov, Zhivotovskiy, 2020) contain a generally inevitable sampling error term of order $\Theta(1/\sqrt{n})$. 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 $\Theta(1/\sqrt{n})$ 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.