8,699 research outputs found
An Active Learning Algorithm for Ranking from Pairwise Preferences with an Almost Optimal Query Complexity
We study the problem of learning to rank from pairwise preferences, and solve
a long-standing open problem that has led to development of many heuristics but
no provable results for our particular problem. Given a set of
elements, we wish to linearly order them given pairwise preference labels. A
pairwise preference label is obtained as a response, typically from a human, to
the question "which if preferred, u or v?u,v\in V{n\choose 2}$ possibilities only. We present an active learning algorithm for
this problem, with query bounds significantly beating general (non active)
bounds for the same error guarantee, while almost achieving the information
theoretical lower bound. Our main construct is a decomposition of the input
s.t. (i) each block incurs high loss at optimum, and (ii) the optimal solution
respecting the decomposition is not much worse than the true opt. The
decomposition is done by adapting a recent result by Kenyon and Schudy for a
related combinatorial optimization problem to the query efficient setting. We
thus settle an open problem posed by learning-to-rank theoreticians and
practitioners: What is a provably correct way to sample preference labels? To
further show the power and practicality of our solution, we show how to use it
in concert with an SVM relaxation.Comment: Fixed a tiny error in theorem 3.1 statemen
A Comparison of Aggregation Methods for Probabilistic Forecasts of COVID-19 Mortality in the United States
The COVID-19 pandemic has placed forecasting models at the forefront of
health policy making. Predictions of mortality and hospitalization help
governments meet planning and resource allocation challenges. In this paper, we
consider the weekly forecasting of the cumulative mortality due to COVID-19 at
the national and state level in the U.S. Optimal decision-making requires a
forecast of a probability distribution, rather than just a single point
forecast. Interval forecasts are also important, as they can support decision
making and provide situational awareness. We consider the case where
probabilistic forecasts have been provided by multiple forecasting teams, and
we aggregate the forecasts to extract the wisdom of the crowd. With only
limited information available regarding the historical accuracy of the
forecasting teams, we consider aggregation (i.e. combining) methods that do not
rely on a record of past accuracy. In this empirical paper, we evaluate the
accuracy of aggregation methods that have been previously proposed for interval
forecasts and predictions of probability distributions. These include the use
of the simple average, the median, and trimming methods, which enable robust
estimation and allow the aggregate forecast to reduce the impact of a tendency
for the forecasting teams to be under- or overconfident. We use data that has
been made publicly available from the COVID-19 Forecast Hub. While the simple
average performed well for the high mortality series, we obtained greater
accuracy using the median and certain trimming methods for the low and medium
mortality series. It will be interesting to see if this remains the case as the
pandemic evolves.Comment: 32 pages, 11 figures, 5 table
Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors
Compressed Sensing suggests that the required number of samples for
reconstructing a signal can be greatly reduced if it is sparse in a known
discrete basis, yet many real-world signals are sparse in a continuous
dictionary. One example is the spectrally-sparse signal, which is composed of a
small number of spectral atoms with arbitrary frequencies on the unit interval.
In this paper we study the problem of line spectrum denoising and estimation
with an ensemble of spectrally-sparse signals composed of the same set of
continuous-valued frequencies from their partial and noisy observations. Two
approaches are developed based on atomic norm minimization and structured
covariance estimation, both of which can be solved efficiently via semidefinite
programming. The first approach aims to estimate and denoise the set of signals
from their partial and noisy observations via atomic norm minimization, and
recover the frequencies via examining the dual polynomial of the convex
program. We characterize the optimality condition of the proposed algorithm and
derive the expected convergence rate for denoising, demonstrating the benefit
of including multiple measurement vectors. The second approach aims to recover
the population covariance matrix from the partially observed sample covariance
matrix by motivating its low-rank Toeplitz structure without recovering the
signal ensemble. Performance guarantee is derived with a finite number of
measurement vectors. The frequencies can be recovered via conventional spectrum
estimation methods such as MUSIC from the estimated covariance matrix. Finally,
numerical examples are provided to validate the favorable performance of the
proposed algorithms, with comparisons against several existing approaches.Comment: 14 pages, 10 figure
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