4,569 research outputs found
Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression
In high dimensional settings, sparse structures are crucial for efficiency,
both in term of memory, computation and performance. It is customary to
consider penalty to enforce sparsity in such scenarios. Sparsity
enforcing methods, the Lasso being a canonical example, are popular candidates
to address high dimension. For efficiency, they rely on tuning a parameter
trading data fitting versus sparsity. For the Lasso theory to hold this tuning
parameter should be proportional to the noise level, yet the latter is often
unknown in practice. A possible remedy is to jointly optimize over the
regression parameter as well as over the noise level. This has been considered
under several names in the literature: Scaled-Lasso, Square-root Lasso,
Concomitant Lasso estimation for instance, and could be of interest for
confidence sets or uncertainty quantification. In this work, after illustrating
numerical difficulties for the Smoothed Concomitant Lasso formulation, we
propose a modification we coined Smoothed Concomitant Lasso, aimed at
increasing numerical stability. We propose an efficient and accurate solver
leading to a computational cost no more expansive than the one for the Lasso.
We leverage on standard ingredients behind the success of fast Lasso solvers: a
coordinate descent algorithm, combined with safe screening rules to achieve
speed efficiency, by eliminating early irrelevant features
A Scalable Algorithm For Sparse Portfolio Selection
The sparse portfolio selection problem is one of the most famous and
frequently-studied problems in the optimization and financial economics
literatures. In a universe of risky assets, the goal is to construct a
portfolio with maximal expected return and minimum variance, subject to an
upper bound on the number of positions, linear inequalities and minimum
investment constraints. Existing certifiably optimal approaches to this problem
do not converge within a practical amount of time at real world problem sizes
with more than 400 securities. In this paper, we propose a more scalable
approach. By imposing a ridge regularization term, we reformulate the problem
as a convex binary optimization problem, which is solvable via an efficient
outer-approximation procedure. We propose various techniques for improving the
performance of the procedure, including a heuristic which supplies high-quality
warm-starts, a preprocessing technique for decreasing the gap at the root node,
and an analytic technique for strengthening our cuts. We also study the
problem's Boolean relaxation, establish that it is second-order-cone
representable, and supply a sufficient condition for its tightness. In
numerical experiments, we establish that the outer-approximation procedure
gives rise to dramatic speedups for sparse portfolio selection problems.Comment: Submitted to INFORMS Journal on Computin
Comparing Experiments to the Fault-Tolerance Threshold
Achieving error rates that meet or exceed the fault-tolerance threshold is a
central goal for quantum computing experiments, and measuring these error rates
using randomized benchmarking is now routine. However, direct comparison
between measured error rates and thresholds is complicated by the fact that
benchmarking estimates average error rates while thresholds reflect worst-case
behavior when a gate is used as part of a large computation. These two measures
of error can differ by orders of magnitude in the regime of interest. Here we
facilitate comparison between the experimentally accessible average error rates
and the worst-case quantities that arise in current threshold theorems by
deriving relations between the two for a variety of physical noise sources. Our
results indicate that it is coherent errors that lead to an enormous mismatch
between average and worst case, and we quantify how well these errors must be
controlled to ensure fair comparison between average error probabilities and
fault-tolerance thresholds.Comment: 5 pages, 2 figures, 13 page appendi
Designing Optimal Quantum Detectors Via Semidefinite Programming
We consider the problem of designing an optimal quantum detector to minimize
the probability of a detection error when distinguishing between a collection
of quantum states, represented by a set of density operators. We show that the
design of the optimal detector can be formulated as a semidefinite programming
problem. Based on this formulation, we derive a set of necessary and sufficient
conditions for an optimal quantum measurement. We then show that the optimal
measurement can be found by solving a standard (convex) semidefinite program
followed by the solution of a set of linear equations or, at worst, a standard
linear programming problem. By exploiting the many well-known algorithms for
solving semidefinite programs, which are guaranteed to converge to the global
optimum, the optimal measurement can be computed very efficiently in polynomial
time.
Using the semidefinite programming formulation, we also show that the rank of
each optimal measurement operator is no larger than the rank of the
corresponding density operator. In particular, if the quantum state ensemble is
a pure-state ensemble consisting of (not necessarily independent) rank-one
density operators, then we show that the optimal measurement is a pure-state
measurement consisting of rank-one measurement operators.Comment: Submitted to IEEE Transactions on Information Theor
Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming
We propose a pivotal method for estimating high-dimensional sparse linear
regression models, where the overall number of regressors is large,
possibly much larger than , but only regressors are significant. The
method is a modification of the lasso, called the square-root lasso. The method
is pivotal in that it neither relies on the knowledge of the standard deviation
or nor does it need to pre-estimate . Moreover, the method
does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle
performance, attaining the convergence rate in
the prediction norm, and thus matching the performance of the lasso with known
. These performance results are valid for both Gaussian and
non-Gaussian errors, under some mild moment restrictions. We formulate the
square-root lasso as a solution to a convex conic programming problem, which
allows us to implement the estimator using efficient algorithmic methods, such
as interior-point and first-order methods
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