Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

The notion of relative universality with respect to a {\sigma}-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of this result in the existing literature. The existing definition of relative universality seems to be too strong for the proof to be valid. In this note we modify the definition of relative universality using the concept of \k{o}-measurability, and rigorously establish the mentioned unbiasedness and Fisher consistency. The significance of this result is beyond its original context of sufficient dimension reduction, because relative universality allows us to use the regression operator to fully characterize conditional independence, a crucially important statistical relation that sits at the core of many areas and methodologies in statistics and machine learning, such as dimension reduction, graphical models, probability embedding, causal inference, and Bayesian estimation.

Privacy preservation has become a critical concern in high-dimensional data analysis due to the growing prevalence of data-driven applications. Proposed by Li (1991), sliced inverse regression has emerged as a widely utilized statistical technique for reducing covariate dimensionality while maintaining sufficient statistical information. In this paper, we propose optimally differentially private algorithms specifically designed to address privacy concerns in the context of sufficient dimension reduction. We proceed to establish lower bounds for differentially private sliced inverse regression in both the low and high-dimensional settings. Moreover, we develop differentially private algorithms that achieve the minimax lower bounds up to logarithmic factors. Through a combination of simulations and real data analysis, we illustrate the efficacy of these differentially private algorithms in safeguarding privacy while preserving vital information within the reduced dimension space. As a natural extension, we can readily offer analogous lower and upper bounds for differentially private sparse principal component analysis, a topic that may also be of potential interest to the statistical and machine learning community.

Supervised dimension reduction (SDR) has been a topic of growing interest in data science, as it enables the reduction of high-dimensional covariates while preserving the functional relation with certain response variables of interest. However, existing SDR methods are not suitable for analyzing datasets collected from case-control studies. In this setting, the goal is to learn and exploit the low-dimensional structure unique to or enriched by the case group, also known as the foreground group. While some unsupervised techniques such as the contrastive latent variable model and its variants have been developed for this purpose, they fail to preserve the functional relationship between the dimension-reduced covariates and the response variable. In this paper, we propose a supervised dimension reduction method called contrastive inverse regression (CIR) specifically designed for the contrastive setting. CIR introduces an optimization problem defined on the Stiefel manifold with a non-standard loss function. We prove the convergence of CIR to a local optimum using a gradient descent-based algorithm, and our numerical study empirically demonstrates the improved performance over competing methods for high-dimensional data.

Federated learning is a distributed machine learning paradigm that collaboratively trains a model with data on many clients. Unlike traditional distributed machine learning methods, which partition data into different clients to improve the efficiency of the learning algorithm, the goal of federated learning is to solve the learning problem without requiring the clients to reveal too much local information. With the increasing demand for data security and privacy protection, federated learning has received significant attention in both industry and academia. For example, banks want to collaboratively train a credit card scoring model without disclosing information about their customers, or hospitals want to carry out researches on a rare disease with each other due to the small number of sample cases, but they can't expose their patients' identity. For more on the progress of federated learning, see [1, 2]. The term federated learning was introduced by McMahan et al. [3], they also proposed the Federated Averaging(FedAvg) algorithm. FedAvg composes multiple rounds of local stochastic gradient descent updates and server-side averaging aggregation to train a centralized model.

We review sufficient dimension reduction (SDR) estimators with multivariate response in this paper. A wide range of SDR methods are characterized as inverse regression SDR estimators or forward regression SDR estimators. The inverse regression family include pooled marginal estimators, projective resampling estimators, and distance-based estimators. Ordinary least squares, partial least squares, and semiparametric SDR estimators, on the other hand, are discussed as estimators from the forward regression family.

The application of standard sufficient dimension reduction methods for reducing the dimension space of predictors without losing regression information requires inverting the covariance matrix of the predictors. This has posed a number of challenges especially when analyzing high-dimensional data sets in which the number of predictors $\mathit{p}$ is much larger than number of samples $n,~(n\ll p)$. A new covariance estimator, called the \textit{Maximum Entropy Covariance} (MEC) that addresses loss of covariance information when similar covariance matrices are linearly combined using \textit{Maximum Entropy} (ME) principle is proposed in this work. By benefitting naturally from slicing or discretizing range of the response variable, y into \textit{H} non-overlapping categories, $\mathit{h_{1},\ldots ,h_{H}}$, MEC first combines covariance matrices arising from samples in each y slice $\mathit{h\in H}$ and then select the one that maximizes entropy under the principle of maximum uncertainty. The MEC estimator is then formed from convex mixture of such entropy-maximizing sample covariance $S_{\mbox{mec}}$ estimate and pooled sample covariance $\mathbf{S}_{\mathit{p}}$ estimate across the $\mathit{H}$ slices without requiring time-consuming covariance optimization procedures. MEC deals directly with singularity and instability of sample group covariance estimate in both regression and classification problems. The efficiency of the MEC estimator is studied with the existing sufficient dimension reduction methods such as \textit{Sliced Inverse Regression} (SIR) and \textit{Sliced Average Variance Estimator} (SAVE) as demonstrated on both classification and regression problems using real life Leukemia cancer data and customers' electricity load profiles from smart meter data sets respectively.