A simple uniformly optimal method without line search for convex optimization

Line search (or backtracking) procedures have been widely employed into first-order methods for solving convex optimization problems, especially those with unknown problem parameters (e.g., Lipschitz constant). In this paper, we show that line search is superfluous in attaining the optimal rate of convergence for solving a convex optimization problem whose parameters are not given a priori. In particular, we present a novel accelerated gradient descent type algorithm called auto-conditioned fast gradient method (AC-FGM) that can achieve an optimal $\mathcal{O}(1/k^2)$ rate of convergence for smooth convex optimization without requiring the estimate of a global Lipschitz constant or the employment of line search procedures. We then extend AC-FGM to solve convex optimization problems with H\"{o}lder continuous gradients and show that it automatically achieves the optimal rates of convergence uniformly for all problem classes with the desired accuracy of the solution as the only input. Finally, we report some encouraging numerical results that demonstrate the advantages of AC-FGM over the previously developed parameter-free methods for convex optimization

On controllability of neuronal networks with constraints on the average of control gains

Control gains play an important role in the control of a natural or a technical system since they reflect how much resource is required to optimize a certain control objective. This paper is concerned with the controllability of neuronal networks with constraints on the average value of the control gains injected in driver nodes, which are in accordance with engineering and biological backgrounds. In order to deal with the constraints on control gains, the controllability problem is transformed into a constrained optimization problem (COP). The introduction of the constraints on the control gains unavoidably leads to substantial difficulty in finding feasible as well as refining solutions. As such, a modified dynamic hybrid framework (MDyHF) is developed to solve this COP, based on an adaptive differential evolution and the concept of Pareto dominance. By comparing with statistical methods and several recently reported constrained optimization evolutionary algorithms (COEAs), we show that our proposed MDyHF is competitive and promising in studying the controllability of neuronal networks. Based on the MDyHF, we proceed to show the controlling regions under different levels of constraints. It is revealed that we should allocate the control gains economically when strong constraints are considered. In addition, it is found that as the constraints become more restrictive, the driver nodes are more likely to be selected from the nodes with a large degree. The results and methods presented in this paper will provide useful insights into developing new techniques to control a realistic complex network efficiently

Acceleration and new analysis of convex optimization algorithms

Ces dernières années ont vu une résurgence de l’algorithme de Frank-Wolfe (FW) (également connu sous le nom de méthodes de gradient conditionnel) dans l’optimisation clairsemée et les problèmes d’apprentissage automatique à grande échelle avec des objectifs convexes lisses. Par rapport aux méthodes de gradient projeté ou proximal, une telle méthode sans projection permet d’économiser le coût de calcul des projections orthogonales sur l’ensemble de contraintes. Parallèlement, FW propose également des solutions à structure clairsemée. Malgré ces propriétés prometteuses, FW ne bénéficie pas des taux de convergence optimaux obtenus par les méthodes accélérées basées sur la projection. Nous menons une enquête dé- taillée sur les essais récents pour accélérer FW dans différents contextes et soulignons où se situe la difficulté lorsque l’on vise des taux linéaires globaux en théorie. En outre, nous fournissons une direction prometteuse pour accélérer FW sur des ensembles fortement convexes en utilisant des techniques d’intervalle de dualité et une nouvelle notion de régularité. D’autre part, l’algorithme FW est une covariante affine et bénéficie de taux de convergence accélérés lorsque l’ensemble de contraintes est fortement convexe. Cependant, ces résultats reposent sur des hypothèses dépendantes de la norme, entraînant généralement des bornes invariantes non affines, en contradiction avec la propriété de covariante affine de FW. Dans ce travail, nous introduisons de nouvelles hypothèses structurelles sur le problème (comme la régularité directionnelle) et dérivons une analyse affine invariante et indépendante de la norme de Frank-Wolfe. Sur la base de notre analyse, nous proposons une recherche par ligne affine invariante. Fait intéressant, nous montrons que les recherches en ligne classiques utilisant la régularité de la fonction objectif convergent étonnamment vers une taille de pas invariante affine, malgré l’utilisation de normes dépendantes de l’affine dans le calcul des tailles de pas. Cela indique que nous n’avons pas nécessairement besoin de connaître à l’avance la structure des ensembles pour profiter du taux accéléré affine-invariant. Dans un autre axe de recherche, nous étudions les algorithmes au-delà des méthodes du premier ordre. Les techniques Quasi-Newton approchent le pas de Newton en estimant le Hessien en utilisant les équations dites sécantes. Certaines de ces méthodes calculent le Hessien en utilisant plusieurs équations sécantes mais produisent des mises à jour non symétriques. D’autres schémas quasi-Newton, tels que BFGS, imposent la symétrie mais ne peuvent pas satisfaire plus d’une équation sécante. Nous proposons un nouveau type de mise à jour symétrique quasi-Newton utilisant plusieurs équations sécantes au sens des moindres carrés. Notre approche généralise et unifie la conception de mises à jour quasi-Newton et satisfait des garanties de robustesse prouvables.Recent years have witnessed a resurgence of the Frank-Wolfe (FW) algorithm, also known as conditional gradient methods, in sparse optimization and large-scale machine learning problems with smooth convex objectives. Compared to projected or proximal gradient methods, such projection-free method saves the computational cost of orthogonal projections onto the constraint set. Meanwhile, FW also gives solutions with sparse structure. Despite of these promising properties, FW does not enjoy the optimal convergence rates achieved by projection-based accelerated methods. On the other hand, FW algorithm is affine-covariant, and enjoys accelerated convergence rates when the constraint set is strongly convex. However, these results rely on norm-dependent assumptions, usually incurring non-affine invariant bounds, in contradiction with FW’s affine-covariant property. In this work, we introduce new structural assumptions on the problem (such as the directional smoothness) and derive an affine in- variant, norm-independent analysis of Frank-Wolfe. Based on our analysis, we pro- pose an affine invariant backtracking line-search. Interestingly, we show that typical back-tracking line-search techniques using smoothness of the objective function surprisingly converge to an affine invariant stepsize, despite using affine-dependent norms in the computation of stepsizes. This indicates that we do not necessarily need to know the structure of sets in advance to enjoy the affine-invariant accelerated rate. Additionally, we provide a promising direction to accelerate FW over strongly convex sets using duality gap techniques and a new version of smoothness. In another line of research, we study algorithms beyond first-order methods. Quasi-Newton techniques approximate the Newton step by estimating the Hessian using the so-called secant equations. Some of these methods compute the Hessian using several secant equations but produce non-symmetric updates. Other quasi- Newton schemes, such as BFGS, enforce symmetry but cannot satisfy more than one secant equation. We propose a new type of quasi-Newton symmetric update using several secant equations in a least-squares sense. Our approach generalizes and unifies the design of quasi-Newton updates and satisfies provable robustness guarantees