arXiv subjects

Jiaze Qiu

Publications and source records attributed to Jiaze Qiu.

4 recordsLinked to original sources

arXiv2024

On Naive Mean-Field Approximation for high-dimensional canonical GLMs

We study the validity of the Naive Mean Field (NMF) approximation for canonical GLMs with product priors. This setting is challenging due to the non-conjugacy of the likelihood and the prior. Using the theory of non-linear large deviations (Austin 2019, Chatterjee, Dembo 2016, Eldan 2018), we derive sufficient conditions for the tightness of the NMF approximation to the log-normalizing constant of the posterior distribution. As a second contribution, we establish that under minor conditions on the design, any NMF optimizer is a product distribution where each component is a quadratic tilt of the prior. In turn, this suggests novel iterative algorithms for fitting the NMF optimizer to the target posterior. Finally, we establish that if the NMF optimization problem has a "well-separated maximizer", then this optimizer governs the probabilistic properties of the posterior. Specifically, we derive credible intervals with average coverage guarantees, and characterize the prediction performance on an out-of-sample datapoint in terms of this dominant optimizer.

math.ST↗

arXiv2023

Sub-optimality of the Naive Mean Field approximation for proportional high-dimensional Linear Regression

The Naïve Mean Field (NMF) approximation is widely employed in modern Machine Learning due to the huge computational gains it bestows on the statistician. Despite its popularity in practice, theoretical guarantees for high-dimensional problems are only available under strong structural assumptions (e.g., sparsity). Moreover, existing theory often does not explain empirical observations noted in the existing literature. In this paper, we take a step towards addressing these problems by deriving sharp asymptotic characterizations for the NMF approximation in high-dimensional linear regression. Our results apply to a wide class of natural priors and allow for model mismatch (i.e., the underlying statistical model can be different from the fitted model). We work under an \textit{iid} Gaussian design and the proportional asymptotic regime, where the number of features and the number of observations grow at a proportional rate. As a consequence of our asymptotic characterization, we establish two concrete corollaries: (a) we establish the inaccuracy of the NMF approximation for the log-normalizing constant in this regime, and (b) we provide theoretical results backing the empirical observation that the NMF approximation can be overconfident in terms of uncertainty quantification. Our results utilize recent advances in the theory of Gaussian comparison inequalities. To the best of our knowledge, this is the first application of these ideas to the analysis of Bayesian variational inference problems. Our theoretical results are corroborated by numerical experiments. Lastly, we believe our results can be generalized to non-Gaussian designs and provide empirical evidence to support it.

math.ST↗

arXiv2022

Hypothesis Testing in Sequentially Sampled Data: AdapRT to Maximize Power Beyond iid Sampling

Testing whether a variable of interest affects the outcome is one of the most fundamental problem in statistics and is often the main scientific question of interest. To tackle this problem, the conditional randomization test (CRT) is widely used to test the independence of variable(s) of interest (X) with an outcome (Y) holding other variable(s) (Z) fixed. The CRT uses randomization or design-based inference that relies solely on the iid sampling of (X,Z) to produce exact finite-sample p-values that are constructed using any test statistic. We propose a new method, the adaptive randomization test (ART), that tackles the independence problem while allowing the data to be adaptively sampled. We first showcase the ART in a particular multi-arm bandit problem known as the normal-mean model. Under this setting, we theoretically characterize the powers of both the iid sampling procedure and the adaptive sampling procedure and empirically find that the ART can uniformly outperform the CRT that pulls all arms independently with equal probability. We also surprisingly find that the ART can be more powerful than even the CRT that uses an oracle iid sampling procedure when the signal is relatively strong. We believe that the proposed adaptive procedure is successful because it takes arms that may initially look like "fake" signals due to random chance and stabilizes them closer to "null" signals. We additionally showcase the ART to a popular factorial survey design setting known as conjoint analysis. We find similar results through simulations and a recent application concerning the role of gender discrimination in political candidate evaluation.

stat.ME↗

arXiv2022

The TAP free energy for high-dimensional linear regression

We derive a variational representation for the log-normalizing constant of the posterior distribution in Bayesian linear regression with a uniform spherical prior and an i.i.d. Gaussian design. We work under the "proportional" asymptotic regime, where the number of observations and the number of features grow at a proportional rate. This rigorously establishes the Thouless-Anderson-Palmer (TAP) approximation arising from spin glass theory, and proves a conjecture of Krzakala et. al. (2014) in the special case of the spherical prior.

math.PR↗