Machine Learning (stat.ML)

Abstract: This paper presents a comprehensive exploration of the theoretical properties inherent in the Adaptive Lasso and the Transfer Lasso. The Adaptive Lasso, a well-established method, employs regularization divided by initial estimators and is characterized by asymptotic normality and variable selection consistency. In contrast, the recently proposed Transfer Lasso employs regularization subtracted by initial estimators with the demonstrated capacity to curtail non-asymptotic estimation errors. A pivotal question thus emerges: Given the distinct ways the Adaptive Lasso and the Transfer Lasso employ initial estimators, what benefits or drawbacks does this disparity confer upon each method? This paper conducts a theoretical examination of the asymptotic properties of the Transfer Lasso, thereby elucidating its differentiation from the Adaptive Lasso. Informed by the findings of this analysis, we introduce a novel method, one that amalgamates the strengths and compensates for the weaknesses of both methods. The paper concludes with validations of our theory and comparisons of the methods via simulation experiments.

Abstract: Given observed data and a probabilistic generative model, Bayesian inference searches for the distribution of the model's parameters that could have yielded the data. Inference is challenging for large population studies where millions of measurements are performed over a cohort of hundreds of subjects, resulting in a massive parameter space. This large cardinality renders off-the-shelf Variational Inference (VI) computationally impractical. In this work, we design structured VI families that efficiently tackle large population studies. Our main idea is to share the parameterization and learning across the different i.i.d. variables in a generative model, symbolized by the model's \textit{plates}. We name this concept \textit{plate amortization}. Contrary to off-the-shelf stochastic VI, which slows down inference, plate amortization results in orders of magnitude faster to train variational distributions. Applied to large-scale hierarchical problems, PAVI yields expressive, parsimoniously parameterized VI with an affordable training time. This faster convergence effectively unlocks inference in those large regimes. We illustrate the practical utility of PAVI through a challenging Neuroimaging example featuring 400 million latent parameters, demonstrating a significant step towards scalable and expressive Variational Inference.

Abstract: A covariance matrix estimator using two bits per entry was recently developed by Dirksen, Maly and Rauhut [Annals of Statistics, 50(6), pp. 3538-3562]. The estimator achieves near minimax rate for general sub-Gaussian distributions, but also suffers from two downsides: theoretically, there is an essential gap on operator norm error between their estimator and sample covariance when the diagonal of the covariance matrix is dominated by only a few entries; practically, its performance heavily relies on the dithering scale, which needs to be tuned according to some unknown parameters. In this work, we propose a new 2-bit covariance matrix estimator that simultaneously addresses both issues. Unlike the sign quantizer associated with uniform dither in Dirksen et al., we adopt a triangular dither prior to a 2-bit quantizer inspired by the multi-bit uniform quantizer. By employing dithering scales varying across entries, our estimator enjoys an improved operator norm error rate that depends on the effective rank of the underlying covariance matrix rather than the ambient dimension, thus closing the theoretical gap. Moreover, our proposed method eliminates the need of any tuning parameter, as the dithering scales are entirely determined by the data. Experimental results under Gaussian samples are provided to showcase the impressive numerical performance of our estimator. Remarkably, by halving the dithering scales, our estimator oftentimes achieves operator norm errors less than twice of the errors of sample covariance.