Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent

First-order methods play a central role in large-scale machine learning. Even though many variations exist, each suited to a particular problem, almost all such methods fundamentally rely on two types of algorithmic steps: gradient descent, which yields primal progress, and mirror descent, which yields dual progress. We observe that the performances of gradient and mirror descent are complementary, so that faster algorithms can be designed by LINEARLY COUPLING the two. We show how to reconstruct Nesterov's accelerated gradient methods using linear coupling, which gives a cleaner interpretation than Nesterov's original proofs. We also discuss the power of linear coupling by extending it to many other settings that Nesterov's methods cannot apply to.Comment: A new section added; polished writin

A Modern Introduction to Online Learning

In this monograph, I introduce the basic concepts of Online Learning through a modern view of Online Convex Optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I present first-order and second-order algorithms for online learning with convex losses, in Euclidean and non-Euclidean settings. All the algorithms are clearly presented as instantiation of Online Mirror Descent or Follow-The-Regularized-Leader and their variants. Particular attention is given to the issue of tuning the parameters of the algorithms and learning in unbounded domains, through adaptive and parameter-free online learning algorithms. Non-convex losses are dealt through convex surrogate losses and through randomization. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. These notes do not require prior knowledge of convex analysis and all the required mathematical tools are rigorously explained. Moreover, all the proofs have been carefully chosen to be as simple and as short as possible.Comment: Fixed more typos, added more history bits, added local norms bounds for OMD and FTR

Unconstrained Online Linear Learning in Hilbert Spaces: Minimax Algorithms and Normal Approximations

We study algorithms for online linear optimization in Hilbert spaces, focusing on the case where the player is unconstrained. We develop a novel characterization of a large class of minimax algorithms, recovering, and even improving, several previous results as immediate corollaries. Moreover, using our tools, we develop an algorithm that provides a regret bound of $\mathcal{O}\Big(U \sqrt{T \log(U \sqrt{T} \log^2 T +1)}\Big)$, where $U$ is the $L_2$ norm of an arbitrary comparator and both $T$ and $U$ are unknown to the player. This bound is optimal up to $\sqrt{\log \log T}$ terms. When $T$ is known, we derive an algorithm with an optimal regret bound (up to constant factors). For both the known and unknown $T$ case, a Normal approximation to the conditional value of the game proves to be the key analysis tool.Comment: Proceedings of the 27th Annual Conference on Learning Theory (COLT 2014