scholarly journals Box-constrained optimization for minimax supervised learning

2021 ◽  
Vol 71 ◽  
pp. 101-113
Author(s):  
Cyprien Gilet ◽  
Susana Barbosa ◽  
Lionel Fillatre
Keyword(s):  
Constrained Optimization ◽  
Empirical Bayes ◽  
Optimization Procedure ◽  
Affine Function ◽  
Bayes Risk ◽  
Piecewise Affine ◽  
Empirical Risk ◽  
Polyhedral Domain ◽  
Subgradient Algorithm ◽  
Constrained Minimax

In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.

Analytical expression of explicit MPC solution via lattice piecewise-affine function

Automatica ◽  
2009 ◽  
Vol 45 (4) ◽  
pp. 910-917 ◽  
Author(s):  
Chengtao Wen ◽  
Xiaoyan Ma ◽  
B. Erik Ydstie
Keyword(s):  
Affine Function ◽  
Analytical Expression ◽  
Piecewise Affine

An Adaptive Primal-Dual Subgradient Algorithm for Online Distributed Constrained Optimization

2018 ◽  
Vol 48 (11) ◽  
pp. 3045-3055 ◽  
Author(s):  
Deming Yuan ◽  
Daniel W. C. Ho ◽  
Guo-Ping Jiang
Keyword(s):  
Constrained Optimization ◽  
Primal Dual ◽  
Subgradient Algorithm

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

scholarly journals Aiding Dictionary Learning Through Multi-Parametric Sparse Representation

Algorithms ◽  
2019 ◽  
Vol 12 (7) ◽  
pp. 131 ◽  
Author(s):  
Florin Stoican ◽  
Paul Irofti
Keyword(s):  
Constrained Optimization ◽  
Sparse Representation ◽  
Dictionary Learning ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Parameter Vector ◽  
Constrained Optimization Problems ◽  
Piecewise Affine ◽  
Current Vector ◽  
Learning Procedure

The ℓ 1 relaxations of the sparse and cosparse representation problems which appear in the dictionary learning procedure are usually solved repeatedly (varying only the parameter vector), thus making them well-suited to a multi-parametric interpretation. The associated constrained optimization problems differ only through an affine term from one iteration to the next (i.e., the problem’s structure remains the same while only the current vector, which is to be (co)sparsely represented, changes). We exploit this fact by providing an explicit, piecewise affine with a polyhedral support, representation of the solution. Consequently, at runtime, the optimal solution (the (co)sparse representation) is obtained through a simple enumeration throughout the non-overlapping regions of the polyhedral partition and the application of an affine law. We show that, for a suitably large number of parameter instances, the explicit approach outperforms the classical implementation.

STRETCHING THE NET: MULTIDIMENSIONAL REGULARIZATION

2021 ◽  
pp. 1-30
Author(s):  
Jaume Vives-i-Bastida
Keyword(s):  
Empirical Bayes ◽  
Cross Validation ◽  
Elastic Net ◽  
Shrinkage Estimators ◽  
Expected Loss ◽  
Bayes Risk ◽  
Asymptotic Risk ◽  
Regularization Parameters ◽  
Unbiased Risk ◽  
Sparsity Level

This paper derives asymptotic risk (expected loss) results for shrinkage estimators with multidimensional regularization in high-dimensional settings. We introduce a class of multidimensional shrinkage estimators (MuSEs), which includes the elastic net, and show that—as the number of parameters to estimate grows—the empirical loss converges to the oracle-optimal risk. This result holds when the regularization parameters are estimated empirically via cross-validation or Stein’s unbiased risk estimate. To help guide applied researchers in their choice of estimator, we compare the empirical Bayes risk of the lasso, ridge, and elastic net in a spike and normal setting. Of the three estimators, we find that the elastic net performs best when the data are moderately sparse and the lasso performs best when the data are highly sparse. Our analysis suggests that applied researchers who are unsure about the level of sparsity in their data might benefit from using MuSEs such as the elastic net. We exploit these insights to propose a new estimator, the cubic net, and demonstrate through simulations that it outperforms the three other estimators for any sparsity level.

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