The Bregman chord divergence

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} distances, one rather prefers to consider parametric classes of distances that are exhaustively characterized by axioms derived from first principles. Bregman divergences are such a class. However fine-tuning a Bregman divergence is delicate since it requires to smoothly adjust a functional generator. In this work, we propose an extension of Bregman divergences called the Bregman chord divergences. This new class of distances does not require gradient calculations, uses two scalar parameters that can be easily tailored in applications, and generalizes asymptotically Bregman divergences.Comment: 10 page

Total Jensen divergences: Definition, Properties and k-Means++ Clustering

We present a novel class of divergences induced by a smooth convex function called total Jensen divergences. Those total Jensen divergences are invariant by construction to rotations, a feature yielding regularization of ordinary Jensen divergences by a conformal factor. We analyze the relationships between this novel class of total Jensen divergences and the recently introduced total Bregman divergences. We then proceed by defining the total Jensen centroids as average distortion minimizers, and study their robustness performance to outliers. Finally, we prove that the k-means++ initialization that bypasses explicit centroid computations is good enough in practice to guarantee probabilistically a constant approximation factor to the optimal k-means clustering.Comment: 27 page

Defending Elections Against Malicious Spread of Misinformation

The integrity of democratic elections depends on voters' access to accurate information. However, modern media environments, which are dominated by social media, provide malicious actors with unprecedented ability to manipulate elections via misinformation, such as fake news. We study a zero-sum game between an attacker, who attempts to subvert an election by propagating a fake new story or other misinformation over a set of advertising channels, and a defender who attempts to limit the attacker's impact. Computing an equilibrium in this game is challenging as even the pure strategy sets of players are exponential. Nevertheless, we give provable polynomial-time approximation algorithms for computing the defender's minimax optimal strategy across a range of settings, encompassing different population structures as well as models of the information available to each player. Experimental results confirm that our algorithms provide near-optimal defender strategies and showcase variations in the difficulty of defending elections depending on the resources and knowledge available to the defender.Comment: Full version of paper accepted to AAAI 201

The Last-Iterate Convergence Rate of Optimistic Mirror Descent in Stochastic Variational Inequalities

In this paper, we analyze the local convergence rate of optimistic mirror descent methods in stochastic variational inequalities, a class of optimization problems with important applications to learning theory and machine learning. Our analysis reveals an intricate relation between the algorithm's rate of convergence and the local geometry induced by the method's underlying Bregman function. We quantify this relation by means of the Legendre exponent, a notion that we introduce to measure the growth rate of the Bregman divergence relative to the ambient norm near a solution. We show that this exponent determines both the optimal step-size policy of the algorithm and the optimal rates attained, explaining in this way the differences observed for some popular Bregman functions (Euclidean projection, negative entropy, fractional power, etc.).Comment: 31 pages, 3 figures, 1 table; to be presented at the 34th Annual Conference on Learning Theory (COLT 2021

TUTTI! - Music Composition as Dialogue

As an engineer, when I could not comprehend a physical phenomenon, I turned to mathematics. As a mathematician, when I could not link sciences to humanity, I turned to music. As a music composer, I no longer see things, I see others. The novel method of music composition presented herein is a first comprehensive framework, system and architectonic template relying on the ideologies of Mikhail Bakhtin's dialogism as well as on research in auditory perception and cognition to create music dialogue as a means of including and engaging participants in musical communication. Beyond immediate artistic intent, I strive to compose music that fosters inclusiveness and collaboration as a relational social gesture in hope that it might incite people and society to embrace their differences and collaborate with the 'others' around them. After probing aesthetics, communication studies and sociology, I argue that dialogism reveals itself well-suited to the aims of the current research. With dialogism as a guiding philosophy, the chapters then look at the relationship between music and language, perception as authorship, intertextuality, the interplay of imagination and understanding, means of arousal in music, mimesis, motion in music and rhythmic entrainment. Employing findings from Gestalt psychology, psychoacoustics, auditory scene analysis, cognition and psychology of expectation, the remaining chapters propose a cognitively informed polyphonic music composition method capable of reproducing the different constituents of dialogic communication by creating and organizing melodic, harmonic, rhythmic and structural elements. Music theory and principles of orchestration then move to music composition as examples demonstrate how dialogue scored between voice-parts provides opportunities for performers to interact with each other and, consequently, engage listeners experiencing the collaboration. As dialogue can be identified in various works, I postulate that the presented Dialogical Music Composition Method can also serve as a method of music analysis. This personal method of composition also supplies tools that other musicians can opt to employ when endeavouring to build balanced dialogue in music. If visibility is key to identity, then composing music that potentially enters into dialogue which each and every voice promotes 'humanity' through inclusivity, yielding a united Tutti