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 $\ell_1$ 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 $p$ is large, possibly much larger than $n$, but only $s$ 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 $\sigma$ or nor does it need to pre-estimate $\sigma$. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate $\sigma \{(s/n)\log p\}^{1/2}$ in the prediction norm, and thus matching the performance of the lasso with known $\sigma$. 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