Matteo Russo

Friedrich Eisenbrand, Thomas Rothvoss, Matteo Russo et al.

We investigate the convex-body containment problem $\max\{s >0 : s Z \subseteq Q\}$, where the outer body $Q \subseteq \mathbb R^d$ is described by a membership oracle and the inner body $Z \subseteq \mathbb R^d$ is a zonotope. Our main result is a sampling-based $O(\sqrt{d})$-approximation algorithm for this problem that almost matches the lower bound of $Ω(\sqrt{d/\log d})$ by Khot and Naor in the oracle model. Assuming zonotopes can be sparsified by a linear number of generators, which is referred to as Talagrand conjecture, our approach attains the optimal approximation factor of $Θ(\sqrt{d/\log d})$. Our second main result is a proof of Talagrand's conjecture for $Δ$-modular zonotopes whenever $Δ$ is constant. Those zonotopes are of the form $Z = \{ Wx \colon \| x\|_\infty \leq 1\}$ where the non-zero $d \times d$ sub-determinants of $W$ are between $1$ and $Δ$. This result establishes a connection between zonoid sparsification and spectral sparsification of Batson, Spielman and Srivastava. We complement these results with a universal $Ω(\sqrt{d/\log d})$ lower bound holding for all zonotopes. Finally, we consider containment problems $\max\{s >0 : s K \subseteq Q\}$, for general convex bodies $K \subseteq \mathbb R^d$. A result of Naszódi on approximating $K \subseteq \mathbb R^d$ by a polytope implies a $Θ(d/\log d)$ approximation algorithm in polynomial time. We show the tightness of this approximation factor in the oracle model via a reduction to the circumradius computation. Our lower bound holds for centrally symmetric convex sets, implying that Barvinok's optimal $O(\sqrt{d})$-approximation of a centrally symmetric convex body by a polytope with a polynomial number of vertices cannot be computed in polynomial time.

Romain Cosson, Federico Fusco, Anupam Gupta et al.

We study repeated bilateral trade when the valuations of the sellers and the buyers are contextual. More precisely, the agents' valuations are given by the inner product of a context vector with two unknown $d$-dimensional vectors -- one for the buyers and one for the sellers. At each time step $t$, the learner receives a context and posts two prices, one for the seller and one for the buyer, and the trade happens if both agents accept their price. We study two objectives for this problem, gain from trade and profit, proving no-regret with respect to a surprisingly strong benchmark: the best omniscient dynamic strategy. In the natural scenario where the learner observes \emph{separately} whether the agents accept their price -- the so-called \emph{two-bit} feedback -- we design algorithms that achieve $O(d\log d)$ regret for gain from trade, and $O(d \log\log T + d\log d)$ regret for profit maximization. Both results are tight, up to the $\log(d)$ factor, and implement per-step budget balance, meaning that the learner never incurs negative profit. In the less informative \emph{one-bit} feedback model, the learner only observes whether a trade happens or not. For this scenario, we show that the tight two-bit regret regimes are still attainable, at the cost of allowing the learner to possibly incur a small negative profit of order $O(d\log d)$, which is notably independent of the time horizon. As a final set of results, we investigate the combination of one-bit feedback and per-step budget balance. There, we design an algorithm for gain from trade that suffers regret independent of the time horizon, but \emph{exponential} in the dimension $d$. For profit maximization, we maintain this exponential dependence on the dimension, which gets multiplied by a $\log T$ factor.

Alexander Munteanu, Matteo Russo, David Saulpic et al.

Terminal embeddings have emerged as a powerful tool for dimension reduction. Given a set of points $P\subset \mathbb{R}^d$, a terminal embedding is a mapping $f:\mathbb{R}^d\rightarrow \mathbb{R}^t$ that preserves the pairwise distance between any pair of points $p\in P$ and $q\in \mathbb{R}^d$ up to small distortion under this mapping. Terminal embeddings have been particularly fruitful for constructing $k$-means and $k$-median coresets, where the objective is to find a typically weighted subset $Ω$ of $P$ such that for any candidate solution, the cost of the clustering objective on $Ω$ approximates the cost of the clustering objective on $P$ up to small distortion. Unfortunately, these techniques have not been extended to more complicated structures such as clustering time-series data under common straight-line interpolation between measurements. The main issue is that terminal embeddings, arguably the central technique in this line of research, cannot be linear and are thus not immediately suitable to preserve linear structures. In this work, we develop a generalization of terminal embeddings to affine line-segments that overcomes this issue. We showcase their applicability by using our lines-preserving terminal embeddings to obtain the first dimension-free coresets for clustering time-series under the Fréchet distance. The underlying dimension reduction uses Johnson-Lindenstrauss (JL) embeddings, and our experiments indicate that terminal embeddings perform similarly to JL and favorably against PCA for synthetic and real-world time-series, while only terminal embeddings extend pairwise distance preservation to the full ambient space.

Vincent Cohen-Addad, Andrew Draganov, Matteo Russo et al.

We consider coresets for $k$-clustering problems, where the goal is to assign points to centers minimizing powers of distances. A popular example is the $k$-median objective $\sum_{p}\min_{c\in C}dist(p,C)$. Given a point set $P$, a coreset $Ω$ is a small weighted subset that approximates the cost of $P$ for all candidate solutions $C$ up to a $(1\pm\varepsilon )$ multiplicative factor. In this paper, we give a sharp VC-dimension based analysis for coreset construction. As a consequence, we obtain improved $k$-median coreset bounds for the following metrics: Coresets of size $\tilde{O}\left(k\varepsilon^{-2}\right)$ for shortest path metrics in planar graphs, improving over the bounds $\tilde{O}\left(k\varepsilon^{-6}\right)$ by [Cohen-Addad, Saulpic, Schwiegelshohn, STOC'21] and $\tilde{O}\left(k^2\varepsilon^{-4}\right)$ by [Braverman, Jiang, Krauthgamer, Wu, SODA'21]. Coresets of size $\tilde{O}\left(kd\ell\varepsilon^{-2}\log m\right)$ for clustering $d$-dimensional polygonal curves of length at most $m$ with curves of length at most $\ell$ with respect to Frechet metrics, improving over the bounds $\tilde{O}\left(k^3d\ell\varepsilon^{-3}\log m\right)$ by [Braverman, Cohen-Addad, Jiang, Krauthgamer, Schwiegelshohn, Toftrup, and Wu, FOCS'22] and $\tilde{O}\left(k^2d\ell\varepsilon^{-2}\log m \log |P|\right)$ by [Conradi, Kolbe, Psarros, Rohde, SoCG'24].

Martino Bernasconi, Andrea Celli, Riccardo Colini-Baldeschi et al.

In the random-order model for online learning, the sequence of losses is chosen upfront by an adversary and presented to the learner after a random permutation. Any random-order input is \emph{asymptotically} equivalent to a stochastic i.i.d. one, but, for finite times, it may exhibit significant {\em non-stationarity}, which can hinder the performance of stochastic learning algorithms. While algorithms for adversarial inputs naturally maintain their regret guarantees in random order, simple no-regret algorithms exist for the stochastic model that fail against random-order instances. In this paper, we propose a general template to adapt stochastic learning algorithms to the random-order model without substantially affecting their regret guarantees. This allows us to recover improved regret bounds for prediction with delays, online learning with constraints, and bandits with switching costs. Finally, we investigate online classification and prove that, in random order, learnability is characterized by the VC dimension rather than the Littlestone dimension, thus providing a further separation from the general adversarial model.

Luis Muñoz-González, Bjarne Pfitzner, Matteo Russo et al.

Machine learning algorithms are vulnerable to poisoning attacks: An adversary can inject malicious points in the training dataset to influence the learning process and degrade the algorithm's performance. Optimal poisoning attacks have already been proposed to evaluate worst-case scenarios, modelling attacks as a bi-level optimization problem. Solving these problems is computationally demanding and has limited applicability for some models such as deep networks. In this paper we introduce a novel generative model to craft systematic poisoning attacks against machine learning classifiers generating adversarial training examples, i.e. samples that look like genuine data points but that degrade the classifier's accuracy when used for training. We propose a Generative Adversarial Net with three components: generator, discriminator, and the target classifier. This approach allows us to model naturally the detectability constrains that can be expected in realistic attacks and to identify the regions of the underlying data distribution that can be more vulnerable to data poisoning. Our experimental evaluation shows the effectiveness of our attack to compromise machine learning classifiers, including deep networks.