49 research outputs found
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
Combining Adversarial Guarantees and Stochastic Fast Rates in Online Learning
We consider online learning algorithms that guarantee worst-case regret rates
in adversarial environments (so they can be deployed safely and will perform
robustly), yet adapt optimally to favorable stochastic environments (so they
will perform well in a variety of settings of practical importance). We
quantify the friendliness of stochastic environments by means of the well-known
Bernstein (a.k.a. generalized Tsybakov margin) condition. For two recent
algorithms (Squint for the Hedge setting and MetaGrad for online convex
optimization) we show that the particular form of their data-dependent
individual-sequence regret guarantees implies that they adapt automatically to
the Bernstein parameters of the stochastic environment. We prove that these
algorithms attain fast rates in their respective settings both in expectation
and with high probability
A Tight Excess Risk Bound via a Unified PAC-Bayesian-Rademacher-Shtarkov-MDL Complexity
We present a novel notion of complexity that interpolates between and
generalizes some classic existing complexity notions in learning theory: for
estimators like empirical risk minimization (ERM) with arbitrary bounded
losses, it is upper bounded in terms of data-independent Rademacher complexity;
for generalized Bayesian estimators, it is upper bounded by the data-dependent
information complexity (also known as stochastic or PAC-Bayesian,
complexity. For
(penalized) ERM, the new complexity reduces to (generalized) normalized maximum
likelihood (NML) complexity, i.e. a minimax log-loss individual-sequence
regret. Our first main result bounds excess risk in terms of the new
complexity. Our second main result links the new complexity via Rademacher
complexity to entropy, thereby generalizing earlier results of Opper,
Haussler, Lugosi, and Cesa-Bianchi who did the log-loss case with .
Together, these results recover optimal bounds for VC- and large (polynomial
entropy) classes, replacing localized Rademacher complexity by a simpler
analysis which almost completely separates the two aspects that determine the
achievable rates: 'easiness' (Bernstein) conditions and model complexity.Comment: 38 page