8,839 research outputs found
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
Hypergraph conditions for the solvability of the ergodic equation for zero-sum games
The ergodic equation is a basic tool in the study of mean-payoff stochastic
games. Its solvability entails that the mean payoff is independent of the
initial state. Moreover, optimal stationary strategies are readily obtained
from its solution. In this paper, we give a general sufficient condition for
the solvability of the ergodic equation, for a game with finite state space but
arbitrary action spaces. This condition involves a pair of directed hypergraphs
depending only on the ``growth at infinity'' of the Shapley operator of the
game. This refines a recent result of the authors which only applied to games
with bounded payments, as well as earlier nonlinear fixed point results for
order preserving maps, involving graph conditions.Comment: 6 pages, 1 figure, to appear in Proc. 54th IEEE Conference on
Decision and Control (CDC 2015
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