25,426 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
Glassy dynamics in granular compaction
Two models are presented to study the influence of slow dynamics on granular
compaction. It is found in both cases that high values of packing fraction are
achieved only by the slow relaxation of cooperative structures. Ongoing work to
study the full implications of these results is discussed.Comment: 12 pages, 9 figures; accepted in J. Phys: Condensed Matter,
proceedings of the Trieste workshop on 'Unifying concepts in glass physics
A two-species model of a two-dimensional sandpile surface: a case of asymptotic roughening
We present and analyze a model of an evolving sandpile surface in (2 + 1)
dimensions where the dynamics of mobile grains ({\rho}(x, t)) and immobile
clusters (h(x, t)) are coupled. Our coupling models the situation where the
sandpile is flat on average, so that there is no bias due to gravity. We find
anomalous scaling: the expected logarithmic smoothing at short length and time
scales gives way to roughening in the asymptotic limit, where novel and
non-trivial exponents are found.Comment: 7 Pages, 6 Figures; Granular Matter, 2012 (Online
Competition and cooperation:aspects of dynamics in sandpiles
In this article, we review some of our approaches to granular dynamics, now
well known to consist of both fast and slow relaxational processes. In the
first case, grains typically compete with each other, while in the second, they
cooperate. A typical result of {\it cooperation} is the formation of stable
bridges, signatures of spatiotemporal inhomogeneities; we review their
geometrical characteristics and compare theoretical results with those of
independent simulations. {\it Cooperative} excitations due to local density
fluctuations are also responsible for relaxation at the angle of repose; the
{\it competition} between these fluctuations and external driving forces, can,
on the other hand, result in a (rare) collapse of the sandpile to the
horizontal. Both these features are present in a theory reviewed here. An arena
where the effects of cooperation versus competition are felt most keenly is
granular compaction; we review here a random graph model, where three-spin
interactions are used to model compaction under tapping. The compaction curve
shows distinct regions where 'fast' and 'slow' dynamics apply, separated by
what we have called the {\it single-particle relaxation threshold}. In the
final section of this paper, we explore the effect of shape -- jagged vs.
regular -- on the compaction of packings near their jamming limit. One of our
major results is an entropic landscape that, while microscopically rough,
manifests {\it Edwards' flatness} at a macroscopic level. Another major result
is that of surface intermittency under low-intensity shaking.Comment: 36 pages, 23 figures, minor correction
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
Crossover in the scaling of island size and capture zone distributions
Simulations of irreversible growth of extended (fractal and square) islands
with critical island sizes i=1 and 2 are performed in broad ranges of coverage
\theta and diffusion-to-deposition ratios R in order to investigate scaling of
island size and capture zone area distributions (ISD, CZD). Large \theta and
small R lead to a crossover from the CZD predicted by the theory of Pimpinelli
and Einstein (PE), with Gaussian right tail, to CZD with simple exponential
decays. The corresponding ISD also cross over from Gaussian or faster decays to
simple exponential ones. For fractal islands, these features are explained by
changes in the island growth kinetics, from a competition for capture of
diffusing adatoms (PE scaling) to aggregation of adatoms with effectively
irrelevant diffusion, which is characteristic of random sequential adsorption
(RSA) without surface diffusion. This interpretation is confirmed by studying
the crossover with similar CZ areas (of order 100 sites) in a model with
freezing of diffusing adatoms that corresponds to i=0. For square islands,
deviations from PE predictions appear for coverages near \theta=0.2 and are
mainly related to island coalescence. Our results show that the range of
applicability of the PE theory is narrow, thus observing the predicted Gaussian
tail of CZD may be difficult in real systems.Comment: 9 pages, 7 figure
Lax Operator for the Quantised Orthosymplectic Superalgebra U_q[osp(2|n)]
Each quantum superalgebra is a quasi-triangular Hopf superalgebra, so
contains a \textit{universal -matrix} in the tensor product algebra which
satisfies the Yang-Baxter equation. Applying the vector representation ,
which acts on the vector module , to one side of a universal -matrix
gives a Lax operator. In this paper a Lax operator is constructed for the
-type quantum superalgebras . This can in turn be used to
find a solution to the Yang-Baxter equation acting on
where is an arbitrary module. The case is included
here as an example.Comment: 15 page
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