19,991 research outputs found
Boundary Crossing Probabilities for General Exponential Families
We consider parametric exponential families of dimension on the real
line. We study a variant of \textit{boundary crossing probabilities} coming
from the multi-armed bandit literature, in the case when the real-valued
distributions form an exponential family of dimension . Formally, our result
is a concentration inequality that bounds the probability that
, where
is the parameter of an unknown target distribution, is the empirical parameter estimate built from observations,
is the log-partition function of the exponential family and
is the corresponding Bregman divergence. From the
perspective of stochastic multi-armed bandits, we pay special attention to the
case when the boundary function is logarithmic, as it is enables to analyze
the regret of the state-of-the-art \KLUCB\ and \KLUCBp\ strategies, whose
analysis was left open in such generality. Indeed, previous results only hold
for the case when , while we provide results for arbitrary finite
dimension , thus considerably extending the existing results. Perhaps
surprisingly, we highlight that the proof techniques to achieve these strong
results already existed three decades ago in the work of T.L. Lai, and were
apparently forgotten in the bandit community. We provide a modern rewriting of
these beautiful techniques that we believe are useful beyond the application to
stochastic multi-armed bandits
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
Dynamic programming for graphs on surfaces
We provide a framework for the design and analysis of dynamic
programming algorithms for surface-embedded graphs on n vertices
and branchwidth at most k. Our technique applies to general families
of problems where standard dynamic programming runs in 2O(k·log k).
Our approach combines tools from topological graph theory and
analytic combinatorics.Postprint (updated version
Asymptotic enumeration of non-crossing partitions on surfaces
We generalize the notion of non-crossing partition on a disk to general surfaces
with boundary. For this, we consider a surface S and introduce the number CS(n) of noncrossing partitions of a set of n points laying on the boundary of SPostprint (author's final draft
Neptune's resonances in the Scattered Disk
The Scattered Disk Objects (SDOs) are thought to be a small fraction of the
ancient population of leftover planetesimals in the outer solar system that
were gravitationally scattered by the giant planets and have managed to survive
primarily by capture and sticking in Neptune's exterior mean motion resonances
(MMRs). In order to advance understanding of the role of MMRs in the dynamics
of the SDOs, we investigate the phase space structure of a large number of
Neptune's MMRs in the semi-major axis range 33--140~au by use of Poincar\'e
sections of the circular planar restricted three body model for the full range
of particle eccentricity pertinent to SDOs. We find that, for eccentricities
corresponding to perihelion distances near Neptune's orbit, distant MMRs have
stable regions with widths that are surprisingly large and of similar size to
those of the closer-in MMRs. We identify a phase-shifted second resonance zone
that exists in the phase space at planet-crossing eccentricities but not at
lower eccentricities; this second resonance zone plays an important role in the
dynamics of SDOs in lengthening their dynamical lifetimes. Our non-perturbative
measurements of the sizes of the stable resonance zones confirm previous
results and provide an additional explanation for the prominence of the :1
sequence of MMRs over the :2, :3 sequences and other MMRs in the
population statistics of SDOs; our results also provide a tool to more easily
identify resonant objects.Comment: 20 pages, 15 figures, 1 table. Some re-organization and minor
revisions; to be published in CMD
A Finite-Volume Version of Aizenman-Higuchi Theorem for the 2d Ising Model
In the late 1970s, in two celebrated papers, Aizenman and Higuchi
independently established that all infinite-volume Gibbs measures of the
two-dimensional ferromagnetic nearest-neighbor Ising model are convex
combinations of the two pure phases. We present here a new approach to this
result, with a number of advantages: (i) We obtain an optimal finite-volume,
quantitative analogue (implying the classical claim); (ii) the scheme of our
proof seems more natural and provides a better picture of the underlying
phenomenon; (iii) this new approach might be applicable to systems for which
the classical method fails.Comment: A couple of typos corrected. To appear in Probab. Theory Relat.
Field
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