However, the communication complexity for VFL is typically very high.

Here, we show that marginal variance tends to increase along the causal order for generically sampled additive noise models.

Here, we show that marginal variance tends to increase along the causal order for generically sampled additive noise models.

We study the problem of inferring heterogeneous treatment effects from time-to-event data.

Optimization problems over discrete or quantized variables are very challenging in general due to the combinatorial nature of their search space. Piecewise-affine regularization (PAR) provides a flexible modeling and computational framework for quantization based on continuous optimization. In this work, we focus on the setting of supervised learning and investigate the theoretical foundations of PAR from optimization and statistical perspectives. First, we show that in the overparameterized regime, where the number of parameters exceeds the number of samples, every critical point of the PAR-regularized loss function exhibits a high degree of quantization. Second, we derive closed-form proximal mappings for various (convex, quasi-convex, and non-convex) PARs and show how to solve PAR-regularized problems using the proximal gradient method, its accelerated variant, and the Alternating Direction Method of Multipliers. Third, we study statistical guarantees of PAR-regularized linear regression problems; specifically, we can approximate classical formulations of $\ell_1$-, squared $\ell_2$-, and nonconvex regularizations using PAR and obtain similar statistical guarantees with quantized solutions.

Predictive coding, the term used in the legal industry for document classification using machine learning, presents additional challenges when the dataset comprises instant messages, due to their informal nature and smaller sizes. In this paper, we exploit a data management workflow to group messages into day chats, followed by feature selection and a logistic regression classifier to provide an economically feasible predictive coding solution. We also improve the solution's baseline model performance by dimensionality reduction, with focus on quantitative features. We test our methodology on an Instant Bloomberg dataset, rich in quantitative information. In parallel, we provide an example of the cost savings of our approach.

We propose a new decentralized robust kernel-based learning algorithm within the framework of reproducing kernel Hilbert spaces (RKHSs) by utilizing a networked system that can be represented as a connected graph. The robust loss function $\huaL_σ$ induced by a windowing function $W$ and a robustness scaling parameter $σ>0$ can encompass a broad spectrum of robust losses. Consequently, the proposed algorithm effectively provides a unified decentralized learning framework for robust regression, which fundamentally differs from the existing distributed robust kernel-based learning schemes, all of which are divide-and-conquer based. We rigorously establish a learning theory and offer comprehensive convergence analysis for the algorithm. We show each local robust estimator generated from the decentralized algorithm can be utilized to approximate the regression function. Based on kernel-based integral operator techniques, we derive general high confidence convergence bounds for the local approximating sequence in terms of the mean square distance, RKHS norm, and generalization error, respectively. Moreover, we provide rigorous selection rules for local sample size and show that, under properly selected step size and scaling parameter $σ$, the decentralized robust algorithm can achieve optimal learning rates (up to logarithmic factors) in both norms. The parameter $σ$ is shown to be essential for enhancing robustness and ensuring favorable convergence behavior. The intrinsic connection among decentralization, sample selection, robustness of the algorithm, and its convergence is clearly reflected.

Supplementary material is at the end of this document.

V ariable selection is often of fundamental interest in many data science applications, providing benefits in prediction error, interpretability, and computation by excluding unnecessary variables.