253 research outputs found
Quantitative estimates for the flux of TASEP with dilute site disorder
We prove that the flux function of the totally asymmetric simple exclusion
process (TASEP) with site disorder exhibits a flat segment for sufficiently
dilute disorder. For high dilution, we obtain an accurate description of the
flux. The result is established undera decay assumption of the maximum current
in finite boxes, which is implied in particular by a sufficiently slow power
tail assumption on the disorder distribution near its minimum. To circumvent
the absence of explicit invariant measures, we use an original renormalization
procedure and some ideas inspired by homogenization
Multilinear Superhedging of Lookback Options
In a pathbreaking paper, Cover and Ordentlich (1998) solved a max-min
portfolio game between a trader (who picks an entire trading algorithm,
) and "nature," who picks the matrix of gross-returns of all
stocks in all periods. Their (zero-sum) game has the payoff kernel
, where is the trader's final wealth and
is the final wealth that would have accrued to a deposit into the best
constant-rebalanced portfolio (or fixed-fraction betting scheme) determined in
hindsight. The resulting "universal portfolio" compounds its money at the same
asymptotic rate as the best rebalancing rule in hindsight, thereby beating the
market asymptotically under extremely general conditions. Smitten with this
(1998) result, the present paper solves the most general tractable version of
Cover and Ordentlich's (1998) max-min game. This obtains for performance
benchmarks (read: derivatives) that are separately convex and homogeneous in
each period's gross-return vector. For completely arbitrary (even
non-measurable) performance benchmarks, we show how the axiom of choice can be
used to "find" an exact maximin strategy for the trader.Comment: 41 pages, 3 figure
Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas
We investigate the approximability of several classes of real-valued
functions by functions of a small number of variables ({\em juntas}). Our main
results are tight bounds on the number of variables required to approximate a
function within -error over
the uniform distribution: 1. If is submodular, then it is -close
to a function of variables.
This is an exponential improvement over previously known results. We note that
variables are necessary even for linear
functions. 2. If is fractionally subadditive (XOS) it is -close
to a function of variables. This result holds for all
functions with low total -influence and is a real-valued analogue of
Friedgut's theorem for boolean functions. We show that
variables are necessary even for XOS functions.
As applications of these results, we provide learning algorithms over the
uniform distribution. For XOS functions, we give a PAC learning algorithm that
runs in time . For submodular functions we give
an algorithm in the more demanding PMAC learning model (Balcan and Harvey,
2011) which requires a multiplicative factor approximation with
probability at least over the target distribution. Our uniform
distribution algorithm runs in time .
This is the first algorithm in the PMAC model that over the uniform
distribution can achieve a constant approximation factor arbitrarily close to 1
for all submodular functions. As follows from the lower bounds in (Feldman et
al., 2013) both of these algorithms are close to optimal. We also give
applications for proper learning, testing and agnostic learning with value
queries of these classes.Comment: Extended abstract appears in proceedings of FOCS 201
Hierarchical testing designs for pattern recognition
We explore the theoretical foundations of a ``twenty questions'' approach to
pattern recognition. The object of the analysis is the computational process
itself rather than probability distributions (Bayesian inference) or decision
boundaries (statistical learning). Our formulation is motivated by applications
to scene interpretation in which there are a great many possible explanations
for the data, one (``background'') is statistically dominant, and it is
imperative to restrict intensive computation to genuinely ambiguous regions.
The focus here is then on pattern filtering: Given a large set Y of possible
patterns or explanations, narrow down the true one Y to a small (random) subset
\hat Y\subsetY of ``detected'' patterns to be subjected to further, more
intense, processing. To this end, we consider a family of hypothesis tests for
Y\in A versus the nonspecific alternatives Y\in A^c. Each test has null type I
error and the candidate sets A\subsetY are arranged in a hierarchy of nested
partitions. These tests are then characterized by scope (|A|), power (or type
II error) and algorithmic cost. We consider sequential testing strategies in
which decisions are made iteratively, based on past outcomes, about which test
to perform next and when to stop testing. The set \hat Y is then taken to be
the set of patterns that have not been ruled out by the tests performed. The
total cost of a strategy is the sum of the ``testing cost'' and the
``postprocessing cost'' (proportional to |\hat Y|) and the corresponding
optimization problem is analyzed.Comment: Published at http://dx.doi.org/10.1214/009053605000000174 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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