1,267 research outputs found
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
, where is
the norm of an arbitrary comparator and both and are unknown to
the player. This bound is optimal up to terms. When is
known, we derive an algorithm with an optimal regret bound (up to constant
factors). For both the known and unknown 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
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
Highly-Smooth Zero-th Order Online Optimization Vianney Perchet
The minimization of convex functions which are only available through partial
and noisy information is a key methodological problem in many disciplines. In
this paper we consider convex optimization with noisy zero-th order
information, that is noisy function evaluations at any desired point. We focus
on problems with high degrees of smoothness, such as logistic regression. We
show that as opposed to gradient-based algorithms, high-order smoothness may be
used to improve estimation rates, with a precise dependence of our upper-bounds
on the degree of smoothness. In particular, we show that for infinitely
differentiable functions, we recover the same dependence on sample size as
gradient-based algorithms, with an extra dimension-dependent factor. This is
done for both convex and strongly-convex functions, with finite horizon and
anytime algorithms. Finally, we also recover similar results in the online
optimization setting.Comment: Conference on Learning Theory (COLT), Jun 2016, New York, United
States. 201
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