208 research outputs found
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
Fast rates in statistical and online learning
The speed with which a learning algorithm converges as it is presented with
more data is a central problem in machine learning --- a fast rate of
convergence means less data is needed for the same level of performance. The
pursuit of fast rates in online and statistical learning has led to the
discovery of many conditions in learning theory under which fast learning is
possible. We show that most of these conditions are special cases of a single,
unifying condition, that comes in two forms: the central condition for 'proper'
learning algorithms that always output a hypothesis in the given model, and
stochastic mixability for online algorithms that may make predictions outside
of the model. We show that under surprisingly weak assumptions both conditions
are, in a certain sense, equivalent. The central condition has a
re-interpretation in terms of convexity of a set of pseudoprobabilities,
linking it to density estimation under misspecification. For bounded losses, we
show how the central condition enables a direct proof of fast rates and we
prove its equivalence to the Bernstein condition, itself a generalization of
the Tsybakov margin condition, both of which have played a central role in
obtaining fast rates in statistical learning. Yet, while the Bernstein
condition is two-sided, the central condition is one-sided, making it more
suitable to deal with unbounded losses. In its stochastic mixability form, our
condition generalizes both a stochastic exp-concavity condition identified by
Juditsky, Rigollet and Tsybakov and Vovk's notion of mixability. Our unifying
conditions thus provide a substantial step towards a characterization of fast
rates in statistical learning, similar to how classical mixability
characterizes constant regret in the sequential prediction with expert advice
setting.Comment: 69 pages, 3 figure
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.
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
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
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