Although quantile regression has emerged as a powerful tool for understanding various quantiles of a response variable conditioned on a set of covariates, the development of quantile regression for count responses has received far less attention. This paper proposes a new Bayesian approach to quantile regression for count data, which provides a more flexible and interpretable alternative to the existing approaches. The proposed approach associates the continuous latent variable with the discrete response and nonparametrically estimates the joint distribution of the latent variable and a set of covariates. Then, by regressing the estimated continuous conditional quantile on the covariates, the posterior distributions of the covariate effects on the conditional quantiles are obtained through general Bayesian updating via simple optimization. The simulation study and real data analysis demonstrate that the proposed method overcomes the existing limitations and enhances quantile estimation and interpretation of variable relationships, making it a valuable tool for practitioners handling count data.
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We examine the last-iterate convergence rate of Bregman proximal methods - from mirror descent to mirror-prox and its optimistic variants - as a function of the local geometry induced by the prox-mapping defining the method. For generality, we focus on local solutions of constrained, non-monotone variational inequalities, and we show that the convergence rate of a given method depends sharply on its associated Legendre exponent, a notion that measures the growth rate of the underlying Bregman function (Euclidean, entropic, or other) near a solution. In particular, we show that boundary solutions exhibit a stark separation of regimes between methods with a zero and non-zero Legendre exponent: the former converge at a linear rate, while the latter converge, in general, sublinearly. This dichotomy becomes even more pronounced in linearly constrained problems where methods with entropic regularization achieve a linear convergence rate along sharp directions, compared to convergence in a finite number of steps under Euclidean regularization.
The goal of uplift modeling is to recommend actions that optimize specific outcomes by determining which entities should receive treatment. One common approach involves two steps: first, an inference step that estimates conditional average treatment effects (CATEs), and second, an optimization step that ranks entities based on their CATE values and assigns treatment to the top k within a given budget. While uplift modeling typically focuses on binary treatments, many real-world applications are characterized by continuous-valued treatments, i.e., a treatment dose. This paper presents a predict-then-optimize framework to allow for continuous treatments in uplift modeling. First, in the inference step, conditional average dose responses (CADRs) are estimated from data using causal machine learning techniques. Second, in the optimization step, we frame the assignment task of continuous treatments as a dose-allocation problem and solve it using integer linear programming (ILP). This approach allows decision-makers to efficiently and effectively allocate treatment doses while balancing resource availability, with the possibility of adding extra constraints like fairness considerations or adapting the objective function to take into account instance-dependent costs and benefits to maximize utility. The experiments compare several CADR estimators and illustrate the trade-offs between policy value and fairness, as well as the impact of an adapted objective function. This showcases the framework's advantages and flexibility across diverse applications in healthcare, lending, and human resource management. All code is available on github.com/SimonDeVos/UMCT.
The maximal regularity property of discontinuous Galerkin methods for linear parabolic equations is used together with variational techniques to establish a priori and a posteriori error estimates of optimal order under optimal regularity assumptions. The analysis is set in the maximal regularity framework of UMD Banach spaces. Similar results were proved in an earlier work, based on the consistency analysis of Radau IIA methods. The present error analysis, which is based on variational techniques, is of independent interest, but the main motivation is that it extends to nonlinear parabolic equations; in contrast to the earlier work. Both autonomous and nonautonomous linear equations are considered.
Unlabeled sensing is a linear inverse problem with permuted measurements. We propose an alternating minimization (AltMin) algorithm with a suitable initialization for two widely considered permutation models: partially shuffled/$k$-sparse permutations and $r$-local/block diagonal permutations. Key to the performance of the AltMin algorithm is the initialization. For the exact unlabeled sensing problem, assuming either a Gaussian measurement matrix or a sub-Gaussian signal, we bound the initialization error in terms of the number of blocks $s$ and the number of shuffles $k$. Experimental results show that our algorithm is fast, applicable to both permutation models, and robust to choice of measurement matrix. We also test our algorithm on several real datasets for the linked linear regression problem and show superior performance compared to baseline methods.
Two sequential estimators are proposed for the odds p/(1-p) and log odds log(p/(1-p)) respectively, using independent Bernoulli random variables with parameter p as inputs. The estimators are unbiased, and guarantee that the variance of the estimation error divided by the true value of the odds, or the variance of the estimation error of the log odds, are less than a target value for any p in (0,1). The estimators are close to optimal in the sense of Wolfowitz's bound.
An extremely schematic model of the forces acting an a sailing yacht equipped with a system of foils is here presented and discussed. The role of the foils is to raise the hull from the water in order to reduce the total resistance and then increase the speed. Some CFD simulations are providing the total resistance of the bare hull at some values of speed and displacement, as well as the characteristics (drag and lift coefficients) of the 2D foil sections used for the appendages. A parametric study has been performed for the characterization of a foil of finite dimensions. The equilibrium of the vertical forces and longitudinal moments, as well as a reduced displacement, is obtained by controlling the pitch angle of the foils. The value of the total resistance of the yacht with foils is then compared with the case without foils, evidencing the speed regime where an advantage is obtained, if any.
This manuscript describes the notions of blocker and interdiction applied to well-known optimization problems. The main interest of these two concepts is the capability to analyze the existence of a combinatorial structure after some modifications. We focus on graph modification, like removing vertices or links in a network. In the interdiction version, we have a budget for modification to reduce as much as possible the size of a given combinatorial structure. Whereas, for the blocker version, we minimize the number of modifications such that the network does not contain a given combinatorial structure. Blocker and interdiction problems have some similarities and can be applied to well-known optimization problems. We consider matching, connectivity, shortest path, max flow, and clique problems. For these problems, we analyze either the blocker version or the interdiction one. Applying the concept of blocker or interdiction to well-known optimization problems can change their complexities. Some optimization problems become harder when one of these two notions is applied. For this reason, we propose some complexity analysis to show when an optimization problem, or the associated decision problem, becomes harder. Another fundamental aspect developed in the manuscript is the use of exact methods to tackle these optimization problems. The main way to solve these problems is to use integer linear programming to model them. An interesting aspect of integer linear programming is the possibility to analyze theoretically the strength of these models, using cutting planes. For most of the problems studied in this manuscript, a polyhedral analysis is performed to prove the strength of inequalities or describe new families of inequalities. The exact algorithms proposed are based on Branch-and-Cut or Branch-and-Price algorithm, where dedicated separation and pricing algorithms are proposed.
We consider the vorticity formulation of the Euler equations describing the flow of a two-dimensional incompressible ideal fluid on the sphere. Zeitlin's model provides a finite-dimensional approximation of the vorticity formulation that preserves the underlying geometric structure: it consists of an isospectral Lie--Poisson flow on the Lie algebra of skew-Hermitian matrices. We propose an approximation of Zeitlin's model based on a time-dependent low-rank factorization of the vorticity matrix and evolve a basis of eigenvectors according to the Euler equations. In particular, we show that the approximate flow remains isospectral and Lie--Poisson and that the error in the solution, in the approximation of the Hamiltonian and of the Casimir functions only depends on the approximation of the vorticity matrix at the initial time. The computational complexity of solving the approximate model is shown to scale quadratically with the order of the vorticity matrix and linearly if a further approximation of the stream function is introduced.
Weighting with the inverse probability of censoring is an approach to deal with censoring in regression analyses where the outcome may be missing due to right-censoring. In this paper, three separate approaches involving this idea in a setting where the Kaplan--Meier estimator is used for estimating the censoring probability are compared. In more detail, the three approaches involve weighted regression, regression with a weighted outcome, and regression of a jack-knife pseudo-observation based on a weighted estimator. Expressions of the asymptotic variances are given in each case and the expressions are compared to each other and to the uncensored case. In terms of low asymptotic variance, a clear winner cannot be found. Which approach will have the lowest asymptotic variance depends on the censoring distribution. Expressions of the limit of the standard sandwich variance estimator in the three cases are also provided, revealing an overestimation under the implied assumptions.
Randomly pivoted Cholesky (RPCholesky) is an algorithm for constructing a low-rank approximation of a positive-semidefinite matrix using a small number of columns. This paper develops an accelerated version of RPCholesky that employs block matrix computations and rejection sampling to efficiently simulate the execution of the original algorithm. For the task of approximating a kernel matrix, the accelerated algorithm can run over $40\times$ faster. The paper contains implementation details, theoretical guarantees, experiments on benchmark data sets, and an application to computational chemistry.
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