52 research outputs found
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
Mathematics Is Physics
In this essay, I argue that mathematics is a natural science---just like
physics, chemistry, or biology---and that this can explain the alleged
"unreasonable" effectiveness of mathematics in the physical sciences. The main
challenge for this view is to explain how mathematical theories can become
increasingly abstract and develop their own internal structure, whilst still
maintaining an appropriate empirical tether that can explain their later use in
physics. In order to address this, I offer a theory of mathematical
theory-building based on the idea that human knowledge has the structure of a
scale-free network and that abstract mathematical theories arise from a
repeated process of replacing strong analogies with new hubs in this network.
This allows mathematics to be seen as the study of regularities, within
regularities, within ..., within regularities of the natural world. Since
mathematical theories are derived from the natural world, albeit at a much
higher level of abstraction than most other scientific theories, it should come
as no surprise that they so often show up in physics.
This version of the essay contains an addendum responding to Slyvia
Wenmackers' essay and comments that were made on the FQXi website.Comment: 15 pages, LaTeX. Second prize winner in 2015 FQXi Essay Contest (see
http://fqxi.org/community/forum/topic/2364
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
The No-Free-Lunch Theorems of Supervised Learning
The no-free-lunch theorems promote a skeptical conclusion that all possible machine learning algorithms equally lack justification. But how could this leave room for a learning theory, that shows that some algorithms are better than others? Drawing parallels to the philosophy of induction, we point out that the no-free-lunch results presuppose a conception of learning algorithms as purely data-driven. On this conception, every algorithm must have an inherent inductive bias, that wants justification. We argue that many standard learning algorithms should rather be understood as model-dependent: in each application they also require for input a model, representing a bias. Generic algorithms themselves, they can be given a model-relative justification
The FLUXNET2015 dataset and the ONEFlux processing pipeline for eddy covariance data
The FLUXNET2015 dataset provides ecosystem-scale data on CO2, water, and energy exchange between the biosphere and the atmosphere, and other meteorological and biological measurements, from 212 sites around the globe (over 1500 site-years, up to and including year 2014). These sites, independently managed and operated, voluntarily contributed their data to create global datasets. Data were quality controlled and processed using uniform methods, to improve consistency and intercomparability across sites. The dataset is already being used in a number of applications, including ecophysiology studies, remote sensing studies, and development of ecosystem and Earth system models. FLUXNET2015 includes derived-data products, such as gap-filled time series, ecosystem respiration and photosynthetic uptake estimates, estimation of uncertainties, and metadata about the measurements, presented for the first time in this paper. In addition, 206 of these sites are for the first time distributed under a Creative Commons (CC-BY 4.0) license. This paper details this enhanced dataset and the processing methods, now made available as open-source codes, making the dataset more accessible, transparent, and reproducible.Peer reviewe
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