Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates.Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems.Comment: Neural Information Processing Systems (2011

Minimizing Finite Sums with the Stochastic Average Gradient

We propose the stochastic average gradient (SAG) method for optimizing the sum of a finite number of smooth convex functions. Like stochastic gradient (SG) methods, the SAG method's iteration cost is independent of the number of terms in the sum. However, by incorporating a memory of previous gradient values the SAG method achieves a faster convergence rate than black-box SG methods. The convergence rate is improved from O(1/k^{1/2}) to O(1/k) in general, and when the sum is strongly-convex the convergence rate is improved from the sub-linear O(1/k) to a linear convergence rate of the form O(p^k) for p \textless{} 1. Further, in many cases the convergence rate of the new method is also faster than black-box deterministic gradient methods, in terms of the number of gradient evaluations. Numerical experiments indicate that the new algorithm often dramatically outperforms existing SG and deterministic gradient methods, and that the performance may be further improved through the use of non-uniform sampling strategies.Comment: Revision from January 2015 submission. Major changes: updated literature follow and discussion of subsequent work, additional Lemma showing the validity of one of the formulas, somewhat simplified presentation of Lyapunov bound, included code needed for checking proofs rather than the polynomials generated by the code, added error regions to the numerical experiment

State aid and tacit collusion

Both literature and policy debate on State aid or government subsidies have focused on the trade-off between the potential ine¢ ciencies caused by state intervention (inefficient allocation of resources, moral hazard) and the potential gains from intervention (whether related to the resolution of market failures or to the achievement of some dimension of social equity). The debate however has ignored another important negative e¤ect of State aid: governments, by setting up aid schemes to ailing firms, may increase the likelihood of (tacit) collusion in an industry characterised by idiosyncratic shocks. Indeed, in a repeated-game setting, a systematic bailout regime increases the expected profits of a firm from cooperation and simultaneously raises the probability that competitors will still be in business to carry out punishment against cheaters. Despite the generality of the model and of its key insight, we study this problem through an application to the banking sector, as it has recently been subject of much attention within the context of the ongoing economic crisis.Subsidies, dynamic oligopoly, government policy, banking

Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma

We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking configurations. This bijection preserves their equivalence classes with respect to the sandpile group. The study of these operators in the special case of the complete bipartite graph ${K}_{m,n}$ naturally leads to a generalization of the well known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic bi-infinite paths in the plane having slightly different slopes. We achieve our results by interpreting the action of these operators as an action on a point in the grid $\mathbb{Z}^2$ which is pointed to by one of these pairs of paths. Our Cyclic lemma allows us to enumerate several classes of polyominoes, and therefore builds on the work of Irving and Rattan (2009), Chapman et al. (2009), and Bonin et al. (2003).Comment: 28 page

Internet e-ethics in confrontation with an activists' agenda: Yahoo! on trial

A prolonged confrontation between Yahoo! Inc. and French anti-racism activists who ask for the removal of Nazi items from auction sites as well as restricted access to neo-Nazis sites is analyzed. We present the case and its development up to the decision of Yahoo! Inc. to remove the items from yahoo.com following a French court’s verdict against the firm. Using a business ethics approach, we distinguish the legal, technical, philosophical and managerial issues involved in the case and their management by Yahoo! We conclude on the difficulty of governing relations with society from corporate and legal affairs departments at the headquarters level, and on the clash of two visions over the regulation of social freedom.E-business, e-ethics, auctions, legal compliance, internet regulation, freedom of expression, media crisis