Search

We study a stochastic first order primal-dual method for solving convex-concave saddle point problems over real reflexive Banach spaces using Bregman divergences and relative smoothness assumptions, in which we allow for stochastic error in the computation of gradient terms within the algorithm. We show ergodic convergence in expectation of the Lagrangian optimality gap with a rate of O (1/k) and that every almost sure weak cluster point of the ergodic sequence is a saddle point in expectation under mild assumptions. Under slightly stricter assumptions, we show almost sure weak convergence of the pointwise iterates to a saddle point. Under a relative strong convexity assumption on the objective functions and a total convexity assumption on the entropies of the Bregman divergences, we establish almost sure strong convergence of the pointwise iterates to a saddle point. Our framework is general and does not need strong convexity of the entropies inducing the Bregman divergences in the algorithm. Numerical applications are considered including entropically regularized Wasserstein barycenter problems and regularized inverse problems on the simplex.

L’articolo si focalizza sulla drammaturgia degli attori-autori della Commedia dell’Arte cercando di estrapolarne i tratti comuni. La produzione letteraria dei comici è imponente per quantità e varietà. Tuttavia è ovvio che la nostra attenzione si concentri soprattutto sugli scritti che hanno un rapporto diretto con il teatro, e che possiamo classificare grosso modo in tre categorie: la trattatistica, che può avere carattere apologetico (come La Supplica di Nicolò Barbieri) o teorico-normativo (come I Frutti delle moderne comedie di Pier Maria Cecchini); gli zibaldoni, o generici; e la drammaturgia, che può consistere di commedie distese come di scenari o soggetti. In questa saggio mi soffermo in particolare sulla drammaturgia a partire dagli scenari di Flaminio Scala. Aldilà delle distinzioni di genere (dramma romanzesco, commedia, farsa), sono individuabili alcuni scenari con azione complessa; altri dalla struttura più semplice e lineare; altri infine in cui ogni reale principio compositivo sembra saltare a favore di un susseguirsi casuale di situazioni diverse, ma canoniche. In questi ultimi testi prevale la comicità dell’azione, gli spazi per i lazzi e le scene topiche sono forse più ampi. Ma soprattutto il livello verbale viene subordinato a quello mimico-coreografico. Ciò detto, varrà la pena verificare da un lato se e in quale misura le opere drammatiche degli attori si inseriscano nella tradizione della drammaturgia ‘regolare’; e dall’altro se e fino a che punto utilizzino gli elementi specifici della commedia dell’arte e quindi, in particolare, le maschere.

In this paper we propose and analyze inexact and stochastic versions of the CGALP algorithm developed in [25], which we denote ICGALP , that allow for errors in the computation of several important quantities. In particular this allows one to compute some gradients, proximal terms, and/or linear minimization oracles in an inexact fashion that facilitates the practical application of the algorithm to computationally intensive settings, e.g., in high (or possibly infinite) dimensional Hilbert spaces commonly found in machine learning problems. The algorithm is able to solve composite minimization problems involving the sum of three convex proper lower-semicontinuous functions subject to an affine constraint of the form Ax = b for some bounded linear operator A. Only one of the functions in the objective is assumed to be differentiable, the other two are assumed to have an accessible proximal operator and a linear minimization oracle. As main results, we show convergence of the Lagrangian values (so-called convergence in the Bregman sense) and asymptotic feasibility of the affine constraint as well as strong convergence of the sequence of dual variables to a solution of the dual problem, in an almost sure sense. Almost sure convergence rates are given for the Lagrangian values and the feasibility gap for the ergodic primal variables. Rates in expectation are given for the Lagrangian values and the feasibility gap subsequentially in the pointwise sense. Numerical experiments verifying the predicted rates of convergence are shown as well.

In this paper we propose a splitting scheme which hybridizes generalized conditional gradient with a prox-imal step which we call CGALP algorithm, for minimizing the sum of three proper convex and lower-semicontinuous functions in real Hilbert spaces. The minimization is subject to an affine constraint, that allows in particular to deal with composite problems (sum of more than three functions) in a separate way by the usual product space technique. While classical conditional gradient methods require Lipschitz-continuity of the gradient of the differentiable part of the objective, CGALP needs only differentiability (on an appropriate subset), hence circumventing the intricate question of Lipschitz continuity of gradients. For the two remaining functions in the objective, we do not require any additional regularity assumption. The second function, possibly nonsmooth, is assumed simple, i.e., the associated proximal mapping is easily computable. For the third function, again nonsmooth, we just assume that its domain is weakly compact and that a linearly perturbed minimization oracle is accessible. In particular, this last function can be chosen to be the indicator of a nonempty bounded closed convex set, in order to deal with additional constraints. Finally, the affine constraint is addressed by the augmented Lagrangian approach. Our analysis is carried out for a wide choice of algorithm parameters satisfying so called "open loop" rules. As main results, under mild conditions, we show asymptotic feasibility with respect to the affine constraint, boundedness of the dual multipliers, and convergence of the Lagrangian values to the saddle-point optimal value. We also provide (subsequential) rates of convergence for both the feasibility gap and the Lagrangian values., Dans ce travail, nous proposons un schéma d’éclatement en optimisation non lisse, hybridant le gradient conditionnel avec une étapeproximale que nous appelons CGALP , pour minimiser la somme de fonctions propres fermées et convexes sur un compact de R n . La minimisationest de plus sujette à une contrainte affine, que nous prenons en compte par un Lagrangien augmenté, en qui permet en particulier de traiter desproblèmes composites à plusieurs fonctions par une technique d’espace produit. Certaines fonctions sont autorisées à être non lisses mais dontl’opérateur proximal est simple à calculer. Notre analyse et garanties de convergence sont assurées pour un large choix de paramètres "en boucleouverte". Comme résultats principaux, nous montrons la faisabilité asymptotique de la variable primale, la convergence de toute sous-suite versune solution du problème primal, la convergence de la variable duale à une solution du problème dual, et la convergence du Lagrangien. Des tauxde convergence sont aussi fournis. Les implications et illustrations de l’algorithme en traitement des données sont discutées.

Best Student Paper Award; International audience; In this paper we propose a splitting scheme which hybridizes generalized conditional gradient with a proximal step which we call CGALP algorithm, for minimizing the sum of closed, convex, and proper functions over a bounded subset of a Hilbert space. The minimization is subject to an affine constraint, which we address by the augmented Lagrangian approach, that allows in particular to deal with composite problems of sum of three or more functions by the usual product space technique. We allow for possibly nonsmooth functions which are simple, i.e., the associated proximal mapping is easily computable. Our analysis is carried out for a wide choice of algorithm parameters satisfying so called open loop rules. As main results, under mild conditions, we show asymptotic feasibility with respect to the affine constraint, weak convergence of the dual variable to a solution of the dual problem, and convergence of the Lagrangian values to the saddle-point optimal value. We also provide (subsequential) rates of convergence for both the feasibility gap and the Lagrangian values. Experimental results in signal processing are finally reported.

This paper deals with an optimal control problem of Bolza type for a class of parabolic equations. It consists in finding the initial datum that minimizes a cost functional, which comprises an energy term on the control and a regulation term given by a distributed cost on the state, and such that the final state lies within a prescribed distance to a given target. Beyond the particularities of the chosen problem, our aim here is to propose a new methodology based on a spectral decomposition of the operator governing the evolution of the system, describe its implementation for this kind of problem, and finally perform some preliminary numerical experiments to assess its behaviour.

In 1880, the first modern editor to publish a collection of commedia dell'arte scenarios noted, not without surprise, that ‘the earliest professional actors were almost all writers as well’. ‘Almost all’ is perhaps an exaggeration. Yet the literary production of comic players, and particularly that of the second and third generation of actors, is impressive for its quantity as well as its variety. It ranges from Giovan Paolo Fabbri's Capitoli alla Carlona and Adriano Valerini's historically important essay on the beauties of Verona, to the Rime of Isabella Andreini, in which she established a dialectic connection between the art of writing and that of the theatre:If you will read one daymy neglected verses, don't be taken in bythese feigned ardoursI am used to playing in dramatic love sceneswith such insincere passionswith lying and mock expressions.I unfold the Muses' high furoressometimes weeping in false sorrowsometimes singing with false pleasure.