1,340 research outputs found
Approximation Algorithms for Correlated Knapsacks and Non-Martingale Bandits
In the stochastic knapsack problem, we are given a knapsack of size B, and a
set of jobs whose sizes and rewards are drawn from a known probability
distribution. However, we know the actual size and reward only when the job
completes. How should we schedule jobs to maximize the expected total reward?
We know O(1)-approximations when we assume that (i) rewards and sizes are
independent random variables, and (ii) we cannot prematurely cancel jobs. What
can we say when either or both of these assumptions are changed?
The stochastic knapsack problem is of interest in its own right, but
techniques developed for it are applicable to other stochastic packing
problems. Indeed, ideas for this problem have been useful for budgeted learning
problems, where one is given several arms which evolve in a specified
stochastic fashion with each pull, and the goal is to pull the arms a total of
B times to maximize the reward obtained. Much recent work on this problem focus
on the case when the evolution of the arms follows a martingale, i.e., when the
expected reward from the future is the same as the reward at the current state.
What can we say when the rewards do not form a martingale?
In this paper, we give constant-factor approximation algorithms for the
stochastic knapsack problem with correlations and/or cancellations, and also
for budgeted learning problems where the martingale condition is not satisfied.
Indeed, we can show that previously proposed LP relaxations have large
integrality gaps. We propose new time-indexed LP relaxations, and convert the
fractional solutions into distributions over strategies, and then use the LP
values and the time ordering information from these strategies to devise a
randomized adaptive scheduling algorithm. We hope our LP formulation and
decomposition methods may provide a new way to address other correlated bandit
problems with more general contexts
Quenched lattice calculation of the B --> D l nu decay rate
We calculate, in the continuum limit of quenched lattice QCD, the form factor
that enters in the decay rate of the semileptonic decay B --> D l nu. Making
use of the step scaling method (SSM), previously introduced to handle two scale
problems in lattice QCD, and of flavour twisted boundary conditions we extract
G(w) at finite momentum transfer and at the physical values of the heavy quark
masses. Our results can be used in order to extract the CKM matrix element Vcb
by the experimental decay rate without model dependent extrapolations.Comment: 5 pages, 4 figures, accepted for publication on Phys. Lett. B,
corrected one typ
Colour and stellar population gradients in galaxies
We discuss the colour, age and metallicity gradients in a wide sample of
local SDSS early- and late-type galaxies. From the fitting of stellar
population models we find that metallicity is the main driver of colour
gradients and the age in the central regions is a dominant parameter which
rules the scatter in both metallicity and age gradients. We find a consistency
with independent observations and a set of simulations. From the comparison
with simulations and theoretical considerations we are able to depict a general
picture of a formation scenario.Comment: 4 pages, 4 figures. Proceedings of 54th Congresso Nazionale della
SAIt, Napoli 4-7 May 201
Geometry of Online Packing Linear Programs
We consider packing LP's with rows where all constraint coefficients are
normalized to be in the unit interval. The n columns arrive in random order and
the goal is to set the corresponding decision variables irrevocably when they
arrive so as to obtain a feasible solution maximizing the expected reward.
Previous (1 - \epsilon)-competitive algorithms require the right-hand side of
the LP to be Omega((m/\epsilon^2) log (n/\epsilon)), a bound that worsens with
the number of columns and rows. However, the dependence on the number of
columns is not required in the single-row case and known lower bounds for the
general case are also independent of n.
Our goal is to understand whether the dependence on n is required in the
multi-row case, making it fundamentally harder than the single-row version. We
refute this by exhibiting an algorithm which is (1 - \epsilon)-competitive as
long as the right-hand sides are Omega((m^2/\epsilon^2) log (m/\epsilon)). Our
techniques refine previous PAC-learning based approaches which interpret the
online decisions as linear classifications of the columns based on sampled dual
prices. The key ingredient of our improvement comes from a non-standard
covering argument together with the realization that only when the columns of
the LP belong to few 1-d subspaces we can obtain small such covers; bounding
the size of the cover constructed also relies on the geometry of linear
classifiers. General packing LP's are handled by perturbing the input columns,
which can be seen as making the learning problem more robust
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