1,230 research outputs found
A Combinatorial Analog of a Theorem of F.J.Dyson
Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the
case n=2 was proposed by Tucker in 1945. Numerous generalizations and
applications of the Lemma have appeared since then. In 2006 Meunier proved the
Lemma in its full generality in his Ph.D. thesis. There are generalizations and
extensions of the Borsuk-Ulam theorem that do not yet have combinatorial
analogs. In this note, we give a combinatorial analog of a result of Freeman J.
Dyson and show that our result is equivalent to Dyson's theorem. As with
Tucker's Lemma, we hope that this will lead to generalizations and applications
and ultimately a combinatorial analog of Yang's theorem of which both
Borsuk-Ulam and Dyson are special cases.Comment: Original version: 7 pages, 2 figures. Revised version: 12 pages, 4
figures, revised proofs. Final revised version: 9 pages, 2 figures, revised
proof
Second-order Quantile Methods for Experts and Combinatorial Games
We aim to design strategies for sequential decision making that adjust to the
difficulty of the learning problem. We study this question both in the setting
of prediction with expert advice, and for more general combinatorial decision
tasks. We are not satisfied with just guaranteeing minimax regret rates, but we
want our algorithms to perform significantly better on easy data. Two popular
ways to formalize such adaptivity are second-order regret bounds and quantile
bounds. The underlying notions of 'easy data', which may be paraphrased as "the
learning problem has small variance" and "multiple decisions are useful", are
synergetic. But even though there are sophisticated algorithms that exploit one
of the two, no existing algorithm is able to adapt to both.
In this paper we outline a new method for obtaining such adaptive algorithms,
based on a potential function that aggregates a range of learning rates (which
are essential tuning parameters). By choosing the right prior we construct
efficient algorithms and show that they reap both benefits by proving the first
bounds that are both second-order and incorporate quantiles
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