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Flavio Chierichetti

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Data Structures and Algorithms Computer Science and Game Theory Machine Learning Social and Information Networks Discrete Mathematics Information Retrieval math.PR math.ST physics.soc-ph Statistics Theory

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preprint2026arXiv

Learning Multinomial Logits in $O(n \log n)$ time

A Multinomial Logit (MNL) model is composed of a finite universe of items $[n]=\{1,..., n\}$, each assigned a positive weight. A query specifies an admissible subset -- called a slate -- and the model chooses one item from that slate with probability proportional to its weight. This query model is also known as the Plackett-Luce model or conditional sampling oracle in the literature. Although MNLs have been studied extensively, a basic computational question remains open: given query access to slates, how efficiently can we learn weights so that, for every slate, the induced choice distribution is within total variation distance $\varepsilon$ of the ground truth? This question is central to MNL learning and has direct implications for modern recommender system interfaces. We provide two algorithms for this task, one with adaptive queries and one with non-adaptive queries. Each algorithm outputs an MNL $M'$ that induces, for each slate $S$, a distribution $M'_S$ on $S$ that is within $\varepsilon$ total variation distance of the true distribution. Our adaptive algorithm makes $O\left(\frac{n}{\varepsilon^{3}}\log n\right)$ queries, while our non-adaptive algorithm makes $O\left(\frac{n^{2}}{\varepsilon^{3}}\log n \log\frac{n}{\varepsilon}\right)$ queries. Both algorithms query only slates of size two and run in time proportional to their query complexity. We complement these upper bounds with lower bounds of $Ω\left(\frac{n}{\varepsilon^{2}}\log n\right)$ for adaptive queries and $Ω\left(\frac{n^{2}}{\varepsilon^{2}}\log n\right)$ for non-adaptive queries, thus proving that our adaptive algorithm is optimal in its dependence on the support size $n$, while the non-adaptive one is tight within a $\log n$ factor.

preprint2026arXiv

On the LSH Distortion of Ulam and Cayley Similarities

Locality-sensitive hashing (LSH) has found widespread use as a fundamental primitive, particularly to accelerate nearest neighbor search. An LSH scheme for a similarity function $S:\mathcal{X} \times \mathcal{X} \to [0,1]$ is a distribution over hash functions on $\mathcal{X}$ with the property that the probability of collision of any two elements $x,y\in \mathcal{X}$ is exactly equal to $S(x,y)$. However, not all similarity functions admit exact LSH schemes. The notion of LSH distortion measures how multiplicatively close a similarity function is to having an LSH scheme. In this work, we study the LSH distortion of the Ulam and Cayley similarities, which are popular similarity measures on permutations of $n$ elements. We show that the Ulam similarity admits a sublinear LSH distortion of $O(n / \sqrt{\log n})$; we also prove a lower bound of $Ω(n^{0.12})$ on the best LSH distortion achievable. On the other hand, we show that the LSH distortion of the Cayley similarity is $Θ(n)$.

preprint2022arXiv

Spectral Robustness for Correlation Clustering Reconstruction in Semi-Adversarial Models

Correlation Clustering is an important clustering problem with many applications. We study the reconstruction version of this problem in which one is seeking to reconstruct a latent clustering that has been corrupted by random noise and adversarial modifications. Concerning the latter, there is a standard "post-adversarial" model in the literature, in which adversarial modifications come after the noise. Here, we introduce and analyse a "pre-adversarial" model in which adversarial modifications come before the noise. Given an input coming from such a semi-adversarial generative model, the goal is to reconstruct almost perfectly and with high probability the latent clustering. We focus on the case where the hidden clusters have nearly equal size and show the following. In the pre-adversarial setting, spectral algorithms are optimal, in the sense that they reconstruct all the way to the information-theoretic threshold beyond which no reconstruction is possible. This is in contrast to the post-adversarial setting, in which their ability to restore the hidden clusters stops before the threshold, but the gap is optimally filled by SDP-based algorithms. These results highlight a heretofore unknown robustness of spectral algorithms, showing them less brittle than previously thought.

preprint2013arXiv

On Discrete Preferences and Coordination

An active line of research has considered games played on networks in which payoffs depend on both a player's individual decision and also the decisions of her neighbors. Such games have been used to model issues including the formation of opinions and the adoption of technology. A basic question that has remained largely open in this area is to consider games where the strategies available to the players come from a fixed, discrete set, and where players may have different intrinsic preferences among the possible strategies. It is natural to model the tension among these different preferences by positing a distance function on the strategy set that determines a notion of "similarity" among strategies; a player's payoff is determined by the distance from her chosen strategy to her preferred strategy and to the strategies chosen by her network neighbors. Even when there are only two strategies available, this framework already leads to natural open questions about a version of the classical Battle of the Sexes problem played on a graph. We develop a set of techniques for analyzing this class of games, which we refer to as discrete preference games. We parametrize the games by the relative extent to which a player takes into account the effect of her preferred strategy and the effect of her neighbors' strategies, allowing us to interpolate between network coordination games and unilateral decision-making. When these two effects are balanced, we show that the price of stability is equal to 1 for any discrete preference game in which the distance function on the strategies is a tree metric; as a special case, this includes the Battle of the Sexes on a graph. We also show that trees form the maximal family of metrics for which the price of stability is 1, and produce a collection of metrics on which the price of stability converges to a tight bound of 2.

preprint2013arXiv

Trace Complexity of Network Inference

The network inference problem consists of reconstructing the edge set of a network given traces representing the chronology of infection times as epidemics spread through the network. This problem is a paradigmatic representative of prediction tasks in machine learning that require deducing a latent structure from observed patterns of activity in a network, which often require an unrealistically large number of resources (e.g., amount of available data, or computational time). A fundamental question is to understand which properties we can predict with a reasonable degree of accuracy with the available resources, and which we cannot. We define the trace complexity as the number of distinct traces required to achieve high fidelity in reconstructing the topology of the unobserved network or, more generally, some of its properties. We give algorithms that are competitive with, while being simpler and more efficient than, existing network inference approaches. Moreover, we prove that our algorithms are nearly optimal, by proving an information-theoretic lower bound on the number of traces that an optimal inference algorithm requires for performing this task in the general case. Given these strong lower bounds, we turn our attention to special cases, such as trees and bounded-degree graphs, and to property recovery tasks, such as reconstructing the degree distribution without inferring the network. We show that these problems require a much smaller (and more realistic) number of traces, making them potentially solvable in practice.

preprint2011arXiv

Voting with Limited Information and Many Alternatives

The traditional axiomatic approach to voting is motivated by the problem of reconciling differences in subjective preferences. In contrast, a dominant line of work in the theory of voting over the past 15 years has considered a different kind of scenario, also fundamental to voting, in which there is a genuinely "best" outcome that voters would agree on if they only had enough information. This type of scenario has its roots in the classical Condorcet Jury Theorem; it includes cases such as jurors in a criminal trial who all want to reach the correct verdict but disagree in their inferences from the available evidence, or a corporate board of directors who all want to improve the company's revenue, but who have different information that favors different options. This style of voting leads to a natural set of questions: each voter has a {\em private signal} that provides probabilistic information about which option is best, and a central question is whether a simple plurality voting system, which tabulates votes for different options, can cause the group decision to arrive at the correct option. We show that plurality voting is powerful enough to achieve this: there is a way for voters to map their signals into votes for options in such a way that --- with sufficiently many voters --- the correct option receives the greatest number of votes with high probability. We show further, however, that any process for achieving this is inherently expensive in the number of voters it requires: succeeding in identifying the correct option with probability at least $1 - η$ requires $Ω(n^3 ε^{-2} \log η^{-1})$ voters, where $n$ is the number of options and $ε$ is a distributional measure of the minimum difference between the options.