48,994 research outputs found
Combining Expert Advice Efficiently
We show how models for prediction with expert advice can be defined concisely
and clearly using hidden Markov models (HMMs); standard HMM algorithms can then
be used to efficiently calculate, among other things, how the expert
predictions should be weighted according to the model. We cast many existing
models as HMMs and recover the best known running times in each case. We also
describe two new models: the switch distribution, which was recently developed
to improve Bayesian/Minimum Description Length model selection, and a new
generalisation of the fixed share algorithm based on run-length coding. We give
loss bounds for all models and shed new light on their relationships.Comment: 50 page
The Computational Power of Optimization in Online Learning
We consider the fundamental problem of prediction with expert advice where
the experts are "optimizable": there is a black-box optimization oracle that
can be used to compute, in constant time, the leading expert in retrospect at
any point in time. In this setting, we give a novel online algorithm that
attains vanishing regret with respect to experts in total
computation time. We also give a lower bound showing
that this running time cannot be improved (up to log factors) in the oracle
model, thereby exhibiting a quadratic speedup as compared to the standard,
oracle-free setting where the required time for vanishing regret is
. These results demonstrate an exponential gap between
the power of optimization in online learning and its power in statistical
learning: in the latter, an optimization oracle---i.e., an efficient empirical
risk minimizer---allows to learn a finite hypothesis class of size in time
. We also study the implications of our results to learning in
repeated zero-sum games, in a setting where the players have access to oracles
that compute, in constant time, their best-response to any mixed strategy of
their opponent. We show that the runtime required for approximating the minimax
value of the game in this setting is , yielding
again a quadratic improvement upon the oracle-free setting, where
is known to be tight
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