20 research outputs found
Prediction with Expert Advice under Discounted Loss
We study prediction with expert advice in the setting where the losses are
accumulated with some discounting---the impact of old losses may gradually
vanish. We generalize the Aggregating Algorithm and the Aggregating Algorithm
for Regression to this case, propose a suitable new variant of exponential
weights algorithm, and prove respective loss bounds.Comment: 26 pages; expanded (2 remarks -> theorems), some misprints correcte
Competing with Markov prediction strategies
Assuming that the loss function is convex in the prediction, we construct a
prediction strategy universal for the class of Markov prediction strategies,
not necessarily continuous. Allowing randomization, we remove the requirement
of convexity.Comment: 11 page
Generalised Mixability, Constant Regret, and Bayesian Updating
Mixability of a loss is known to characterise when constant regret bounds are
achievable in games of prediction with expert advice through the use of Vovk's
aggregating algorithm. We provide a new interpretation of mixability via convex
analysis that highlights the role of the Kullback-Leibler divergence in its
definition. This naturally generalises to what we call -mixability where
the Bregman divergence replaces the KL divergence. We prove that
losses that are -mixable also enjoy constant regret bounds via a
generalised aggregating algorithm that is similar to mirror descent.Comment: 12 page
Competing with stationary prediction strategies
In this paper we introduce the class of stationary prediction strategies and
construct a prediction algorithm that asymptotically performs as well as the
best continuous stationary strategy. We make mild compactness assumptions but
no stochastic assumptions about the environment. In particular, no assumption
of stationarity is made about the environment, and the stationarity of the
considered strategies only means that they do not depend explicitly on time; we
argue that it is natural to consider only stationary strategies even for highly
non-stationary environments.Comment: 20 page
Generalized Mixability via Entropic Duality
Mixability is a property of a loss which characterizes when fast convergence
is possible in the game of prediction with expert advice. We show that a key
property of mixability generalizes, and the exp and log operations present in
the usual theory are not as special as one might have thought. In doing this we
introduce a more general notion of -mixability where is a general
entropy (\ie, any convex function on probabilities). We show how a property
shared by the convex dual of any such entropy yields a natural algorithm (the
minimizer of a regret bound) which, analogous to the classical aggregating
algorithm, is guaranteed a constant regret when used with -mixable
losses. We characterize precisely which have -mixable losses and
put forward a number of conjectures about the optimality and relationships
between different choices of entropy.Comment: 20 pages, 1 figure. Supersedes the work in arXiv:1403.2433 [cs.LG
Competitive on-line learning with a convex loss function
We consider the problem of sequential decision making under uncertainty in
which the loss caused by a decision depends on the following binary
observation. In competitive on-line learning, the goal is to design decision
algorithms that are almost as good as the best decision rules in a wide
benchmark class, without making any assumptions about the way the observations
are generated. However, standard algorithms in this area can only deal with
finite-dimensional (often countable) benchmark classes. In this paper we give
similar results for decision rules ranging over an arbitrary reproducing kernel
Hilbert space. For example, it is shown that for a wide class of loss functions
(including the standard square, absolute, and log loss functions) the average
loss of the master algorithm, over the first observations, does not exceed
the average loss of the best decision rule with a bounded norm plus
. Our proof technique is very different from the standard ones and
is based on recent results about defensive forecasting. Given the probabilities
produced by a defensive forecasting algorithm, which are known to be well
calibrated and to have good resolution in the long run, we use the expected
loss minimization principle to find a suitable decision.Comment: 26 page
Prediction with expert advice for the Brier game
We show that the Brier game of prediction is mixable and find the optimal
learning rate and substitution function for it. The resulting prediction
algorithm is applied to predict results of football and tennis matches. The
theoretical performance guarantee turns out to be rather tight on these data
sets, especially in the case of the more extensive tennis data.Comment: 34 pages, 22 figures, 2 tables. The conference version (8 pages) is
published in the ICML 2008 Proceeding
From Stochastic Mixability to Fast Rates
Empirical risk minimization (ERM) is a fundamental algorithm for statistical learning problems where the data is generated according to some unknown distribution P and returns a hypothesis f chosen from a fixed class F with small loss `. In the parametric setting, depending upon (`,F,P) ERM can have slow (1/ n) or fast (1/n) rates of convergence of the excess risk as a function of the sample size n. There exist several results that give sufficient conditions for fast rates in terms of joint properties of `, F, and P, such as the margin condition and the Bernstein condition. In the non-statistical prediction with experts setting, there is an analogous slow and fast rate phenomenon, and it is entirely characterized in terms of the mixability of the loss ` (there being no role there for F or P). The notion of stochastic mixability builds a bridge between these two models of learning, reducing to classical mixability in a special case. The present paper presents a direct proof of fast rates for ERM in terms of stochastic mixability of (`,F,P), and in so doing provides new insight into the fast-rates phenomenon. The proof exploits an old result of Kemperman on the solution to the generalized moment problem. We also show a partial converse that suggests a characterization of fast rates for ERM in terms of stochastic mixability is possible.
From Stochastic Mixability to Fast Rates
Empirical risk minimization (ERM) is a fundamental learning rule for
statistical learning problems where the data is generated according to some
unknown distribution and returns a hypothesis chosen from a
fixed class with small loss . In the parametric setting,
depending upon ERM can have slow
or fast rates of convergence of the excess risk as a
function of the sample size . There exist several results that give
sufficient conditions for fast rates in terms of joint properties of ,
, and , such as the margin condition and the Bernstein
condition. In the non-statistical prediction with expert advice setting, there
is an analogous slow and fast rate phenomenon, and it is entirely characterized
in terms of the mixability of the loss (there being no role there for
or ). The notion of stochastic mixability builds a
bridge between these two models of learning, reducing to classical mixability
in a special case. The present paper presents a direct proof of fast rates for
ERM in terms of stochastic mixability of , and
in so doing provides new insight into the fast-rates phenomenon. The proof
exploits an old result of Kemperman on the solution to the general moment
problem. We also show a partial converse that suggests a characterization of
fast rates for ERM in terms of stochastic mixability is possible.Comment: 21 pages, accepted to NIPS 201