1,785 research outputs found
Towards Minimax Online Learning with Unknown Time Horizon
We consider online learning when the time horizon is unknown. We apply a
minimax analysis, beginning with the fixed horizon case, and then moving on to
two unknown-horizon settings, one that assumes the horizon is chosen randomly
according to some known distribution, and the other which allows the adversary
full control over the horizon. For the random horizon setting with restricted
losses, we derive a fully optimal minimax algorithm. And for the adversarial
horizon setting, we prove a nontrivial lower bound which shows that the
adversary obtains strictly more power than when the horizon is fixed and known.
Based on the minimax solution of the random horizon setting, we then propose a
new adaptive algorithm which "pretends" that the horizon is drawn from a
distribution from a special family, but no matter how the actual horizon is
chosen, the worst-case regret is of the optimal rate. Furthermore, our
algorithm can be combined and applied in many ways, for instance, to online
convex optimization, follow the perturbed leader, exponential weights algorithm
and first order bounds. Experiments show that our algorithm outperforms many
other existing algorithms in an online linear optimization setting
Tight Lower Bounds for Multiplicative Weights Algorithmic Families
We study the fundamental problem of prediction with expert advice and develop
regret lower bounds for a large family of algorithms for this problem. We
develop simple adversarial primitives, that lend themselves to various
combinations leading to sharp lower bounds for many algorithmic families. We
use these primitives to show that the classic Multiplicative Weights Algorithm
(MWA) has a regret of , there by completely closing
the gap between upper and lower bounds. We further show a regret lower bound of
for a much more general family of
algorithms than MWA, where the learning rate can be arbitrarily varied over
time, or even picked from arbitrary distributions over time. We also use our
primitives to construct adversaries in the geometric horizon setting for MWA to
precisely characterize the regret at for the case
of experts and a lower bound of
for the case of arbitrary number of experts
A Quasi-Bayesian Perspective to Online Clustering
When faced with high frequency streams of data, clustering raises theoretical
and algorithmic pitfalls. We introduce a new and adaptive online clustering
algorithm relying on a quasi-Bayesian approach, with a dynamic (i.e.,
time-dependent) estimation of the (unknown and changing) number of clusters. We
prove that our approach is supported by minimax regret bounds. We also provide
an RJMCMC-flavored implementation (called PACBO, see
https://cran.r-project.org/web/packages/PACBO/index.html) for which we give a
convergence guarantee. Finally, numerical experiments illustrate the potential
of our procedure
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