Global Convergence in Training Large-Scale Transformers

Despite the widespread success of Transformers across various domains, their optimization guarantees in large-scale model settings are not well-understood. This paper rigorously analyzes the convergence properties of gradient flow in training Transformers with weight decay regularization. First, we construct the mean-field limit of large-scale Transformers, showing that as the model width and depth go to infinity, gradient flow converges to the Wasserstein gradient flow, which is represented by a partial differential equation. Then, we demonstrate that the gradient flow reaches a global minimum consistent with the PDE solution when the weight decay regularization parameter is sufficiently small. Our analysis is based on a series of novel mean-field techniques that adapt to Transformers. Compared with existing tools for deep networks (Lu et al., 2020) that demand homogeneity and global Lipschitz smoothness, we utilize a refined analysis assuming only $\textit{partial homogeneity}$ and $\textit{local Lipschitz smoothness}$. These new techniques may be of independent interest.

Analysis of the ICML 2023 Ranking Data: Can Authors' Opinions of Their Own Papers Assist Peer Review in Machine Learning?

We conducted an experiment during the review process of the 2023 International Conference on Machine Learning (ICML) that requested authors with multiple submissions to rank their own papers based on perceived quality. We received 1,342 rankings, each from a distinct author, pertaining to 2,592 submissions. In this paper, we present an empirical analysis of how author-provided rankings could be leveraged to improve peer review processes at machine learning conferences. We focus on the Isotonic Mechanism, which calibrates raw review scores using author-provided rankings. Our analysis demonstrates that the ranking-calibrated scores outperform raw scores in estimating the ground truth ``expected review scores'' in both squared and absolute error metrics. Moreover, we propose several cautious, low-risk approaches to using the Isotonic Mechanism and author-provided rankings in peer review processes, including assisting senior area chairs' oversight of area chairs' recommendations, supporting the selection of paper awards, and guiding the recruitment of emergency reviewers. We conclude the paper by addressing the study's limitations and proposing future research directions.

Factor Adjusted Spectral Clustering for Mixture Models

This paper studies a factor modeling-based approach for clustering high-dimensional data generated from a mixture of strongly correlated variables. Statistical modeling with correlated structures pervades modern applications in economics, finance, genomics, wireless sensing, etc., with factor modeling being one of the popular techniques for explaining the common dependence. Standard techniques for clustering high-dimensional data, e.g., naive spectral clustering, often fail to yield insightful results as their performances heavily depend on the mixture components having a weakly correlated structure. To address the clustering problem in the presence of a latent factor model, we propose the Factor Adjusted Spectral Clustering (FASC) algorithm, which uses an additional data denoising step via eliminating the factor component to cope with the data dependency. We prove this method achieves an exponentially low mislabeling rate, with respect to the signal to noise ratio under a general set of assumptions. Our assumption bridges many classical factor models in the literature, such as the pervasive factor model, the weak factor model, and the sparse factor model. The FASC algorithm is also computationally efficient, requiring only near-linear sample complexity with respect to the data dimension. We also show the applicability of the FASC algorithm with real data experiments and numerical studies, and establish that FASC provides significant results in many cases where traditional spectral clustering fails.

Covariate Assisted Entity Ranking with Sparse Intrinsic Scores

This paper addresses the item ranking problem with associate covariates, focusing on scenarios where the preference scores can not be fully explained by covariates, and the remaining intrinsic scores, are sparse. Specifically, we extend the pioneering Bradley-Terry-Luce (BTL) model by incorporating covariate information and considering sparse individual intrinsic scores. Our work introduces novel model identification conditions and examines the regularized penalized Maximum Likelihood Estimator (MLE) statistical rates. We then construct a debiased estimator for the penalized MLE and analyze its distributional properties. Additionally, we apply our method to the goodness-of-fit test for models with no latent intrinsic scores, namely, the covariates fully explaining the preference scores of individual items. We also offer confidence intervals for ranks. Our numerical studies lend further support to our theoretical findings, demonstrating validation for our proposed method

When can weak latent factors be statistically inferred?

This article establishes a new and comprehensive estimation and inference theory for principal component analysis (PCA) under the weak factor model that allow for cross-sectional dependent idiosyncratic components under nearly minimal the factor strength relative to the noise level or signal-to-noise ratio. Our theory is applicable regardless of the relative growth rate between the cross-sectional dimension $N$ and temporal dimension $T$. This more realistic assumption and noticeable result requires completely new technical device, as the commonly-used leave-one-out trick is no longer applicable to the case with cross-sectional dependence. Another notable advancement of our theory is on PCA inference $ - $ for example, under the regime where $N\asymp T$, we show that the asymptotic normality for the PCA-based estimator holds as long as the signal-to-noise ratio (SNR) grows faster than a polynomial rate of $\log N$. This finding significantly surpasses prior work that required a polynomial rate of $N$. Our theory is entirely non-asymptotic, offering finite-sample characterizations for both the estimation error and the uncertainty level of statistical inference. A notable technical innovation is our closed-form first-order approximation of PCA-based estimator, which paves the way for various statistical tests. Furthermore, we apply our theories to design easy-to-implement statistics for validating whether given factors fall in the linear spans of unknown latent factors, testing structural breaks in the factor loadings for an individual unit, checking whether two units have the same risk exposures, and constructing confidence intervals for systematic risks. Our empirical studies uncover insightful correlations between our test results and economic cycles.

Optimal Aggregation of Prediction Intervals under Unsupervised Domain Shift

As machine learning models are increasingly deployed in dynamic environments, it becomes paramount to assess and quantify uncertainties associated with distribution shifts. A distribution shift occurs when the underlying data-generating process changes, leading to a deviation in the model's performance. The prediction interval, which captures the range of likely outcomes for a given prediction, serves as a crucial tool for characterizing uncertainties induced by their underlying distribution. In this paper, we propose methodologies for aggregating prediction intervals to obtain one with minimal width and adequate coverage on the target domain under unsupervised domain shift, under which we have labeled samples from a related source domain and unlabeled covariates from the target domain. Our analysis encompasses scenarios where the source and the target domain are related via i) a bounded density ratio, and ii) a measure-preserving transformation. Our proposed methodologies are computationally efficient and easy to implement. Beyond illustrating the performance of our method through a real-world dataset, we also delve into the theoretical details. This includes establishing rigorous theoretical guarantees, coupled with finite sample bounds, regarding the coverage and width of our prediction intervals. Our approach excels in practical applications and is underpinned by a solid theoretical framework, ensuring its reliability and effectiveness across diverse contexts.

Causality Pursuit from Heterogeneous Environments via Neural Adversarial Invariance Learning

Statistics suffers from a fundamental problem, "the curse of endogeneity" -- the regression function, or more broadly the prediction risk minimizer with infinite data, may not be the target we wish to pursue. This is because when complex data are collected from multiple sources, the biases deviated from the interested (causal) association inherited in individuals or sub-populations are not expected to be canceled. Traditional remedies are of hindsight and restrictive in being tailored to prior knowledge like untestable cause-effect structures, resulting in methods that risk model misspecification and lack scalable applicability. This paper seeks to offer a purely data-driven and universally applicable method that only uses the heterogeneity of the biases in the data rather than following pre-offered commandments. Such an idea is formulated as a nonparametric invariance pursuit problem, whose goal is to unveil the invariant conditional expectation $m^\star(x)\equiv \mathbb{E}[Y^{(e)}|X_{S^\star}^{(e)}=x_{S^\star}]$ with unknown important variable set $S^\star$ across heterogeneous environments $e\in \mathcal{E}$. Under the structural causal model framework, $m^\star$ can be interpreted as certain data-driven causality in general. The paper contributes to proposing a novel framework, called Focused Adversarial Invariance Regularization (FAIR), formulated as a single minimax optimization program that can solve the general invariance pursuit problem. As illustrated by the unified non-asymptotic analysis, our adversarial estimation framework can attain provable sample-efficient estimation akin to standard regression under a minimal identification condition for various tasks and models. As an application, the FAIR-NN estimator realized by two Neural Network classes is highlighted as the first approach to attain statistically efficient estimation in general nonparametric invariance learning.

An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization

Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible high-dimensional data modeling, and act as a sampler for generating new samples under active guidance towards task-desired properties. Despite the significant empirical success, theory of diffusion models is very limited, potentially slowing down principled methodological innovations for further harnessing and improving diffusion models. In this paper, we review emerging applications of diffusion models, understanding their sample generation under various controls. Next, we overview the existing theories of diffusion models, covering their statistical properties and sampling capabilities. We adopt a progressive routine, beginning with unconditional diffusion models and connecting to conditional counterparts. Further, we review a new avenue in high-dimensional structured optimization through conditional diffusion models, where searching for solutions is reformulated as a conditional sampling problem and solved by diffusion models. Lastly, we discuss future directions about diffusion models. The purpose of this paper is to provide a well-rounded theoretical exposure for stimulating forward-looking theories and methods of diffusion models.

A general theory for robust clustering via trimmed mean

Clustering is a fundamental tool in statistical machine learning in the presence of heterogeneous data. Many recent results focus primarily on optimal mislabeling guarantees, when data are distributed around centroids with sub-Gaussian errors. Yet, the restrictive sub-Gaussian model is often invalid in practice, since various real-world applications exhibit heavy tail distributions around the centroids or suffer from possible adversarial attacks that call for robust clustering with a robust data-driven initialization. In this paper, we introduce a hybrid clustering technique with a novel multivariate trimmed mean type centroid estimate to produce mislabeling guarantees under a weak initialization condition for general error distributions around the centroids. A matching lower bound is derived, up to factors depending on the number of clusters. In addition, our approach also produces the optimal mislabeling even in the presence of adversarial outliers. Our results reduce to the sub-Gaussian case when errors follow sub-Gaussian distributions. To solve the problem thoroughly, we also present novel data-driven robust initialization techniques and show that, with probabilities approaching one, these initial centroid estimates are sufficiently good for the subsequent clustering algorithm to achieve the optimal mislabeling rates. Furthermore, we demonstrate that the Lloyd algorithm is suboptimal for more than two clusters even when errors are Gaussian, and for two clusters when errors distributions have heavy tails. Both simulated data and real data examples lend further support to both of our robust initialization procedure and clustering algorithm.

Structured Matrix Learning under Arbitrary Entrywise Dependence and Estimation of Markov Transition Kernel

The problem of structured matrix estimation has been studied mostly under strong noise dependence assumptions. This paper considers a general framework of noisy low-rank-plus-sparse matrix recovery, where the noise matrix may come from any joint distribution with arbitrary dependence across entries. We propose an incoherent-constrained least-square estimator and prove its tightness both in the sense of deterministic lower bound and matching minimax risks under various noise distributions. To attain this, we establish a novel result asserting that the difference between two arbitrary low-rank incoherent matrices must spread energy out across its entries, in other words cannot be too sparse, which sheds light on the structure of incoherent low-rank matrices and may be of independent interest. We then showcase the applications of our framework to several important statistical machine learning problems. In the problem of estimating a structured Markov transition kernel, the proposed method achieves the minimax optimality and the result can be extended to estimating the conditional mean operator, a crucial component in reinforcement learning. The applications to multitask regression and structured covariance estimation are also presented. We propose an alternating minimization algorithm to approximately solve the potentially hard optimization problem. Numerical results corroborate the effectiveness of our method which typically converges in a few steps.