arXiv subjects

Samuel Rosa

Publications and source records attributed to Samuel Rosa.

14 recordsLinked to original sources

arXiv2025

Laplacian Network Optimization via Information Functions

Designing networks to optimize robustness and other performance metrics is a well-established problem with applications ranging from electrical engineering to communication networks. Many such performance measures rely on the Laplacian spectrum; notable examples include total effective resistance, the number of spanning trees, and algebraic connectivity. This paper advances the study of Laplacian-based network optimization by drawing on ideas from experimental design in statistics. We present a theoretical framework for analyzing performance measures by introducing the notion of information functions, which captures a set of their desirable properties. Then, we formulate a new parametric family of information functions, Kiefer's measures, which encompasses the three most common spectral objectives. We provide a regular reformulation of the Laplacian optimization problem, and we use this reformulation to compute directional derivatives of Kiefer's measures. The directional derivatives provide a unified treatment of quantities recurring in Laplacian optimization, such as gradients and subgradients, and we show that they are connected to Laplacian-based measures of node distance, which we call node dissimilarities. We apply the node dissimilarities to derive efficient rank-one update formulas for Kiefer's criteria, and to devise a new edge-exchange method for network optimization. These update formulas enable greedy and exchange algorithms with reduced asymptotic time complexity.

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arXiv2025

Removal of Redundant Candidate Points for the Exact D-Optimal Design Problem

One of the most common problems in statistical experimentation is computing D-optimal designs on large finite candidate sets. While optimal approximate (i.e., infinite-sample) designs can be efficiently computed using convex methods, constructing optimal exact (i.e., finite-sample) designs is a substantially more difficult integer-optimization problem. In this paper, we propose necessary conditions, based on approximate designs, that must be satisfied by any support point of a D-optimal exact design. These conditions enable rapid elimination of redundant candidate points without loss of optimality, thereby reducing memory requirements and runtime of subsequent exact design algorithms. In addition, we prove that for sufficiently large sample sizes, the supports of D-optimal exact designs are contained in a typically small maximum-variance set. We demonstrate the approach on randomly generated benchmark models with candidate sets up to 100 million points, and on commonly used constrained mixture models with up to one million points. The proposed approach reduces the initial candidate sets by several orders of magnitude, thereby making it possible to compute exact D-optimal designs for these problems via mixed-integer second-order cone programming, which provides optimality guarantees.

stat.CO↗

arXiv2024

A Randomized Exchange Algorithm for Optimal Design of Multi-Response Experiments

Despite the increasing prevalence of vector observations, computation of optimal experimental design for multi-response models has received limited attention. To address this problem within the framework of approximate designs, we introduce mREX, an algorithm that generalizes the randomized exchange algorithm REX (J Am Stat Assoc 115:529, 2020), originally specialized for single-response models. The mREX algorithm incorporates several improvements: a novel method for computing efficient sparse initial designs, an extension to all differentiable Kiefer's optimality criteria, and an efficient method for performing optimal exchanges of weights. For the most commonly used D-optimality criterion, we propose a technique for optimal weight exchanges based on the characteristic matrix polynomial. The mREX algorithm is applicable to linear, nonlinear, and generalized linear models, and scales well to large problems. It typically converges to optimal designs faster than available alternative methods, although it does not require advanced mathematical programming solvers. We demonstrate the usefulness of mREX to bivariate dose-response Emax models for clinical trials, both without and with the inclusion of covariates.

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arXiv2023

The Polytope of Optimal Approximate Designs: Extending the Selection of Informative Experiments

Consider the problem of constructing an experimental design, optimal for estimating parameters of a given statistical model with respect to a chosen criterion. To address this problem, the literature usually provides a single solution. Often, however, there exists a rich set of optimal designs, and the knowledge of this set can lead to substantially greater freedom to select an appropriate experiment. In this paper, we demonstrate that the set of all optimal approximate designs generally corresponds to a polytope. Particularly important elements of the polytope are its vertices, which we call vertex optimal designs. We prove that the vertex optimal designs possess unique properties, such as small supports, and outline strategies for how they can facilitate the construction of suitable experiments. Moreover, we show that for a variety of situations it is possible to construct the vertex optimal designs with the assistance of a computer, by employing error-free rational-arithmetic calculations. In such cases the vertex optimal designs are exact, often closely related to known combinatorial designs. Using this approach, we were able to determine the polytope of optimal designs for some of the most common multifactor regression models, thereby extending the choice of informative experiments for a large variety of applications.

stat.CO↗

arXiv2023

Mixed-integer linear programming for computing optimal experimental designs

The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

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arXiv2021

Optimal Design of Multifactor Experiments via Grid Exploration

For computing efficient approximate designs of multifactor experiments, we propose a simple algorithm based on adaptive exploration of the grid of all combinations of factor levels. We demonstrate that the algorithm significantly outperforms several state-of-the-art competitors for problems with discrete, continuous, as well as mixed factors. Importantly, we provide a free R code that permits direct verification of the numerical results and allows the researchers to easily compute optimal or nearly-optimal experimental designs for their own statistical models.

stat.CO↗

arXiv2019

Optimal experimental designs for treatment contrasts in heteroscedastic models with covariates

In clinical trials, the response of a given subject often depends on the selected treatment as well as on some covariates. We study optimal approximate designs of experiments in the models with treatment and covariate effects. We allow for the variances of the responses to depend on the chosen treatments, which introduces heteroscedasticity into the models. For estimating systems of treatment contrasts and linear functions of the covariates, we extend known results on D-optimality of product designs by providing product designs that are optimal with respect to general eigenvalue-based criteria. In particular, A- and E-optimal product designs are obtained. We then formulate a method based on linear programming for constructing optimal designs with smaller supports from the optimal product designs. The sparser designs can be more easily converted to practically applicable exact designs. The provided results and the proposed sparsification method are demonstrated on some examples.

math.ST↗

arXiv2019

On greedy heuristics for computing D-efficient saturated subsets

Let $\mathcal{F}$ be a set consisting of $n$ real vectors of dimension $m \leq n$. For any saturated, i.e., $m$-element, subset $\mathcal{S}$ of $\mathcal{F}$, let $\mathrm{vol}(\mathcal{S})$ be the volume of the parallelotope formed by the vectors of $\mathcal{S}$. A set $\mathcal{S}^*$ is called a $D$-optimal saturated subset of $\mathcal{F}$, if it maximizes $\mathrm{vol}(\mathcal{S})$ among all saturated subsets of $\mathcal{F}$. In this paper, we propose two greedy heuristics for the construction of saturated subsets performing well with respect to the criterion of $D$-optimality: an improvement of the method suggested by Galil and Kiefer for the initiation of $D$-optimal experimental design algorithms, and a modification of the Kumar-Yildirim method, the original version of which was proposed for the initiation of the minimum-volume enclosing ellipsoid algorithms. We provide geometric and analytic insights into the two methods, and compare them to the commonly used random and regularized greedy heuristics. We also suggest variants of the greedy methods for a large set $\mathcal{F}$, for the construction of $D$-efficient non-saturated subsets, and for alternative optimality criteria.

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arXiv2018

Removal of the points that do not support an E-optimal experimental design

We propose a method of removal of design points that cannot support any E-optimal experimental design of a linear regression model with uncorrelated observations. The proposed method can be used to reduce the size of some large E-optimal design problems such that they can be efficiently solved by semidefinite programming. This paper complements the results of Pronzato [Pronzato, L., 2013. A delimitation of the support of optimal designs for Kiefer's $ϕ_p$-class of criteria. Statistics & Probability Letters 83, 2721--2728], who studied the same problem for analytically simpler criteria of design optimality.

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arXiv2018

E- and R-optimality of block designs for treatment-control comparisons

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs characterized by simple linear constraints. Employing this characterization, we obtain a class of E-optimal exact designs for treatment-control comparisons for unequal block sizes. In the studied model, we justify the use of E-optimality by providing a statistical interpretation for all E-optimal approximate designs and for the known classes of E-optimal exact designs. Moreover, we consider the R-optimality criterion, which minimizes the volume of the rectangular confidence region based on the Bonferroni confidence intervals. We show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This further reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.

math.ST↗

arXiv2016

Optimal designs for treatment comparisons represented by graphs

Consider an experiment consisting of a set of independent trials for comparing a set of treatments. In each trial, one treatment is chosen and the mean response of the trial is equal to the effect of the chosen treatment. We examine the optimal approximate designs for the estimation of a system of treatment contrasts under such model. These approximate treatment designs can be used to provide optimal treatment proportions for designs in more general models with nuisance effects (e.g., time trend, effects of blocks). For any system of pairwise treatment comparisons, we propose to represent such system by a graph. In particular, we represent the treatment designs for these sets of contrasts by the inverses of the vertex weights in the corresponding graph G. We show that then the positive eigenvalues of the information matrix of a treatment design are inverse to the positive eigenvalues of the vertex-weighted Laplacian of G. Note that such representation of treatment designs differs from the well known graph representation of block designs, which are represented by edges. We provide a graph-theoretic interpretation of the D-, A- and E-optimality for estimating sets of pairwise comparisons; as well as some optimality results for both the systems of pairwise comparisons and the general systems of treatment contrasts. Moreover, we provide a class of 'symmetric' systems of treatment contrasts for which the uniform treatment design is optimal with respect to a wide range of optimality criteria.

math.ST↗

arXiv2016

On weighted optimality of experimental designs

When the experimental objective is expressed by a set of estimable functions, and any eigenvalue-based optimality criterion is selected, we prove the equivalence of the recently introduced weighted optimality and the 'standard' optimality criteria for estimating this set of functions of interest. Also, given a weighted eigenvalue-based criterion, we construct a system of estimable functions, so that the optimality for estimating this system of functions is equivalent to the weighted optimality. This allows one to use the large body of existing theoretical and computational results for the standard optimality criteria for estimating a system of interest to derive theorems and numerical algorithms for the weighted optimality of experimental designs. Moreover, we extend the theory of weighted optimality so that it captures the experimental objective consisting of any system of estimable functions, which was not the case in the literature on weighted optimality so far. For any set of estimable functions, we propose a corresponding weight matrix of a simple form, and with a straightforward interpretation. Given a set of estimable functions with their corresponding weights, we show that it is useful to distinguish between the primary weights selected by the experimenters and the secondary weights implied by the weight matrix.

math.ST↗

arXiv2016

Optimal Approximate Designs for Comparison with Control in Dose-Escalation Studies

Consider an experiment, where a new drug is tested for the first time on human subjects - healthy volunteers. Such experiments are often performed as dose-escalation studies: a set of increasing doses is pre-selected, individuals are grouped into cohorts, and in each cohort, the dose number $i$ can be administered only if the dose number $i-1$ has already been tested in the previous cohort. If an adverse effect of a dose is observed, the experiment stops and thus no subjects are exposed to higher doses. In this paper, we assume that the response is affected both by the dose or placebo effects as well as by the cohort effects. We provide optimal approximate designs for selected optimality criteria ($E$-, $MV$- and $LV$-optimality) for estimating the effects of the drug doses compared with the placebo. In particular, we obtain the optimality of Senn designs and extended Senn designs with respect to multiple criteria.

math.ST↗

arXiv2015

Optimal approximate designs for estimating treatment contrasts resistant to nuisance effects

Suppose that we intend to perform an experiment consisting of a set of independent trials. The mean value of the response of each trial is assumed to be equal to the sum of the effect of the treatment selected for the trial, and some nuisance effects, e.g., the effect of a time trend, or blocking. In this model, we examine optimal approximate designs for the estimation of a system of treatment contrasts, with respect to a wide range of optimality criteria. We show that it is necessary for any optimal design to attain the optimal treatment proportions, which may be obtained from the marginal model that excludes the nuisance effects. Moreover, we prove that for a design to be optimal, it is sufficient that it attains the optimal treatment proportions and satisfies conditions of resistance to nuisance effects. For selected natural choices of treatment contrasts and optimality criteria, we calculate the optimal treatment proportions and give an explicit form of optimal designs. In particular, we obtain optimal treatment proportions for comparison of a set of new treatments with a set of controls. The results allow us to construct a method of calculating optimal approximate designs with a small support by means of linear programming. As a consequence, we can construct efficient exact designs by a simple heuristic.

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