General Tail Bounds for Non-Smooth Stochastic Mirror Descent

In this paper, we provide novel tail bounds on the optimization error of Stochastic Mirror Descent for convex and Lipschitz objectives. Our analysis extends the existing tail bounds from the classical light-tailed Sub-Gaussian noise case to heavier-tailed noise regimes. We study the optimization error of the last iterate as well as the average of the iterates. We instantiate our results in two important cases: a class of noise with exponential tails and one with polynomial tails. A remarkable feature of our results is that they do not require an upper bound on the diameter of the domain. Finally, we support our theory with illustrative experiments that compare the behavior of the average of the iterates with that of the last iterate in heavy-tailed noise regimes.

An Improved Uniform Convergence Bound with Fat-Shattering Dimension

The fat-shattering dimension characterizes the uniform convergence property of real-valued functions. The state-of-the-art upper bounds feature a multiplicative squared logarithmic factor on the sample complexity, leaving an open gap with the existing lower bound. We provide an improved uniform convergence bound that closes this gap.

Active Learning of Classifiers with Label and Seed Queries

We study exact active learning of binary and multiclass classifiers with margin. Given an $n$-point set $X \subset \mathbb{R}^m$, we want to learn any unknown classifier on $X$ whose classes have finite strong convex hull margin, a new notion extending the SVM margin. In the standard active learning setting, where only label queries are allowed, learning a classifier with strong convex hull margin $\gamma$ requires in the worst case $\Omega\big(1+\frac{1}{\gamma}\big)^{(m-1)/2}$ queries. On the other hand, using the more powerful seed queries (a variant of equivalence queries), the target classifier could be learned in $O(m \log n)$ queries via Littlestone's Halving algorithm; however, Halving is computationally inefficient. In this work we show that, by carefully combining the two types of queries, a binary classifier can be learned in time $\operatorname{poly}(n+m)$ using only $O(m^2 \log n)$ label queries and $O\big(m \log \frac{m}{\gamma}\big)$ seed queries; the result extends to $k$-class classifiers at the price of a $k!k^2$ multiplicative overhead. Similar results hold when the input points have bounded bit complexity, or when only one class has strong convex hull margin against the rest. We complement the upper bounds by showing that in the worst case any algorithm needs $\Omega\big(k m \log \frac{1}{\gamma}\big)$ seed and label queries to learn a $k$-class classifier with strong convex hull margin $\gamma$.

Regret Analysis of Dyadic Search

We analyze the cumulative regret of the Dyadic Search algorithm of Bachoc et al. [2022].

High Probability Bounds for Stochastic Subgradient Schemes with Heavy Tailed Noise

In this work we study high probability bounds for stochastic subgradient methods under heavy tailed noise. In this case the noise is only assumed to have finite variance as opposed to a sub-Gaussian distribution for which it is known that standard subgradient methods enjoys high probability bounds. We analyzed a clipped version of the projected stochastic subgradient method, where subgradient estimates are truncated whenever they have large norms. We show that this clipping strategy leads both to near optimal any-time and finite horizon bounds for many classical averaging schemes. Preliminary experiments are shown to support the validity of the method.

A Near-Optimal Algorithm for Univariate Zeroth-Order Budget Convex Optimization

This paper studies a natural generalization of the problem of minimizing a univariate convex function $f$ by querying its values sequentially. At each time-step $t$, the optimizer can invest a budget $b_t$ in a query point $X_t$ of their choice to obtain a fuzzy evaluation of $f$ at $X_t$ whose accuracy depends on the amount of budget invested in $X_t$ across times. This setting is motivated by the minimization of objectives whose values can only be determined approximately through lengthy or expensive computations. We design an any-time parameter-free algorithm called Dyadic Search, for which we prove near-optimal optimization error guarantees. As a byproduct of our analysis, we show that the classical dependence on the global Lipschitz constant in the error bounds is an artifact of the granularity of the budget. Finally, we illustrate our theoretical findings with numerical simulations.

On Margin-Based Cluster Recovery with Oracle Queries

We study an active cluster recovery problem where, given a set of $n$ points and an oracle answering queries like "are these two points in the same cluster?", the task is to recover exactly all clusters using as few queries as possible. We begin by introducing a simple but general notion of margin between clusters that captures, as special cases, the margins used in previous work, the classic SVM margin, and standard notions of stability for center-based clusterings. Then, under our margin assumptions we design algorithms that, in a variety of settings, recover all clusters exactly using only $O(\log n)$ queries. For the Euclidean case, $\mathbb{R}^m$, we give an algorithm that recovers arbitrary convex clusters, in polynomial time, and with a number of queries that is lower than the best existing algorithm by $\Theta(m^m)$ factors. For general pseudometric spaces, where clusters might not be convex or might not have any notion of shape, we give an algorithm that achieves the $O(\log n)$ query bound, and is provably near-optimal as a function of the packing number of the space. Finally, for clusterings realized by binary concept classes, we give a combinatorial characterization of recoverability with $O(\log n)$ queries, and we show that, for many concept classes in Euclidean spaces, this characterization is equivalent to our margin condition. Our results show a deep connection between cluster margins and active cluster recoverability.

Multitask Online Mirror Descent

We introduce and analyze MT-OMD, a multitask generalization of Online Mirror Descent (OMD) which operates by sharing updates between tasks. We prove that the regret of MT-OMD is of order $\sqrt{1 + \sigma^2(N-1)}\sqrt{T}$, where $\sigma^2$ is the task variance according to the geometry induced by the regularizer, $N$ is the number of tasks, and $T$ is the time horizon. Whenever tasks are similar, that is, $\sigma^2 \le 1$, this improves upon the $\sqrt{NT}$ bound obtained by running independent OMDs on each task. Our multitask extensions of Online Gradient Descent and Exponentiated Gradient, two important instances of OMD, are shown to enjoy closed-form updates, making them easy to use in practice. Finally, we provide numerical experiments on four real-world datasets which support our theoretical findings.

Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries

We investigate the problem of exact cluster recovery using oracle queries. Previous results show that clusters in Euclidean spaces that are convex and separated with a margin can be reconstructed exactly using only $O(\log n)$ same-cluster queries, where $n$ is the number of input points. In this work, we study this problem in the more challenging non-convex setting. We introduce a structural characterization of clusters, called $(\beta,\gamma)$-convexity, that can be applied to any finite set of points equipped with a metric (or even a semimetric, as the triangle inequality is not needed). Using $(\beta,\gamma)$-convexity, we can translate natural density properties of clusters (which include, for instance, clusters that are strongly non-convex in $R^d$) into a graph-theoretic notion of convexity. By exploiting this convexity notion, we design a deterministic algorithm that recovers $(\beta,\gamma)$-convex clusters using $O(k^2 \log n + k^2 (\frac{6}{\beta\gamma})^{dens(X)})$ same-cluster queries, where $k$ is the number of clusters and $dens(X)$ is the density dimension of the semimetric. We show that an exponential dependence on the density dimension is necessary, and we also show that, if we are allowed to make $O(k^2 + k \log n)$ additional queries to a "cluster separation" oracle, then we can recover clusters that have different and arbitrary scales, even when the scale of each cluster is unknown.

A Perturbation Resilient Framework for Unsupervised Learning

Designing learning algorithms that are resistant to perturbations of the underlying data distribution is a problem of wide practical and theoretical importance. We present a general approach to this problem focusing on unsupervised learning. The key assumption is that the perturbing distribution is characterized by larger losses relative to a given class of admissible models. This is exploited by a general descent algorithm which minimizes an $L$-statistic criterion over the model class, weighting more small losses. We characterize the robustness of the method in terms of bounds on the reconstruction error for the assumed unperturbed distribution. Numerical experiments with \textsc{kmeans} clustering and principal subspace analysis demonstrate the effectiveness of our method.