Wai Ming Tai

Bryon Aragam, Wai Ming Tai

We study the problem of learning nonparametric distributions in a finite mixture, and establish tight bounds on the sample complexity for learning the component distributions in such models. Namely, we are given i.i.d. samples from a pdf $f$ where $$ f=w_1f_1+w_2f_2, \quad w_1+w_2=1, \quad w_1,w_2>0 $$ and we are interested in learning each component $f_i$. Without any assumptions on $f_i$, this problem is ill-posed. In order to identify the components $f_i$, we assume that each $f_i$ can be written as a convolution of a Gaussian and a compactly supported density $ν_i$ with $\text{supp}(ν_1)\cap \text{supp}(ν_2)=\emptyset$. Our main result shows that $(\frac{1}{\varepsilon})^{Ω(\log\log \frac{1}{\varepsilon})}$ samples are required for estimating each $f_i$. The proof relies on a quantitative Tauberian theorem that yields a fast rate of approximation with Gaussians, which may be of independent interest. To show this is tight, we also propose an algorithm that uses $(\frac{1}{\varepsilon})^{O(\log\log \frac{1}{\varepsilon})}$ samples to estimate each $f_i$. Unlike existing approaches to learning latent variable models based on moment-matching and tensor methods, our proof instead involves a delicate analysis of an ill-conditioned linear system via orthogonal functions. Combining these bounds, we conclude that the optimal sample complexity of this problem properly lies in between polynomial and exponential, which is not common in learning theory.

Benwei Shi, Aditya Bhaskara, Wai Ming Tai et al.

We show that a constant-size constant-error coreset for polytope distance is simple to maintain under merges of coresets. However, increasing the size cannot improve the error bound significantly beyond that constant.

Rafael Cabral, Tuan Manh Do, Xuejun Yu et al.

Proof autoformalization, the task of translating natural language theorems and proofs into machine-verifiable code, is a critical step for integrating large language models into rigorous mathematical workflows. Current approaches focus on producing executable code, but they frequently fail to preserve the semantic meaning and logical structure of the original human-written argument. To address this, we introduce ProofFlow, a novel pipeline that treats structural fidelity as a primary objective. ProofFlow first constructs a directed acyclic graph (DAG) to map the logical dependencies between proof steps. Then, it employs a novel lemma-based approach to systematically formalize each step as an intermediate lemma, preserving the logical structure of the original argument. To facilitate evaluation, we present a new benchmark of 184 undergraduate-level problems, manually annotated with step-by-step solutions and logical dependency graphs, and introduce ProofScore, a new composite metric to evaluate syntactic correctness, semantic faithfulness, and structural fidelity. Experimental results show our pipeline sets a new state-of-the-art for autoformalization, achieving a ProofScore of 0.545, substantially exceeding baselines like full-proof formalization (0.123), which processes the entire proof at once, and step-proof formalization (0.072), which handles each step independently. Our pipeline, benchmark, and score metric are open-sourced to encourage further progress at https://github.com/Huawei-AI4Math/ProofFlow.

Aarshvi Gajjar, Wai Ming Tai, Xingyu Xu et al.

We study active learning methods for single index models of the form $F({\mathbf x}) = f(\langle {\mathbf w}, {\mathbf x}\rangle)$, where $f:\mathbb{R} \to \mathbb{R}$ and ${\mathbf x,\mathbf w} \in \mathbb{R}^d$. In addition to their theoretical interest as simple examples of non-linear neural networks, single index models have received significant recent attention due to applications in scientific machine learning like surrogate modeling for partial differential equations (PDEs). Such applications require sample-efficient active learning methods that are robust to adversarial noise. I.e., that work even in the challenging agnostic learning setting. We provide two main results on agnostic active learning of single index models. First, when $f$ is known and Lipschitz, we show that $\tilde{O}(d)$ samples collected via {statistical leverage score sampling} are sufficient to learn a near-optimal single index model. Leverage score sampling is simple to implement, efficient, and already widely used for actively learning linear models. Our result requires no assumptions on the data distribution, is optimal up to log factors, and improves quadratically on a recent ${O}(d^{2})$ bound of \cite{gajjar2023active}. Second, we show that $\tilde{O}(d)$ samples suffice even in the more difficult setting when $f$ is \emph{unknown}. Our results leverage tools from high dimensional probability, including Dudley's inequality and dual Sudakov minoration, as well as a novel, distribution-aware discretization of the class of Lipschitz functions.

Yu Xuejun, Jianyuan Zhong, Zijin Feng et al.

Recent advances in large language models show strong promise for formal reasoning. However, most LLM-based theorem provers have long been constrained by the need for expert-written formal statements as inputs, limiting their applicability to real-world problems expressed in natural language. We tackle this gap with Mathesis, the first end-to-end theorem proving pipeline processing informal problem statements. It contributes Mathesis-Autoformalizer, the first autoformalizer using reinforcement learning to enhance the formalization ability of natural language problems, aided by our novel LeanScorer framework for nuanced formalization quality assessment. It also proposes a Mathesis-Prover, which generates formal proofs from the formalized statements. To evaluate the real-world applicability of end-to-end formal theorem proving, we introduce Gaokao-Formal, a benchmark of 488 complex problems from China's national college entrance exam. Our approach is carefully designed, with a thorough study of each component. Experiments demonstrate Mathesis's effectiveness, with the autoformalizer outperforming the best baseline by 22% in pass-rate on Gaokao-Formal. The full system surpasses other model combinations, achieving 64% accuracy on MiniF2F with pass@32 and a state-of-the-art 18% on Gaokao-Formal.

Robert A. Vandermeulen, Wai Ming Tai, Bryon Aragam

We show that deep neural networks achieve dimension-independent rates of convergence for learning structured densities such as those arising in image, audio, video, and text applications. More precisely, we demonstrate that neural networks with a simple $L^2$-minimizing loss achieve a rate of $n^{-1/(4+r)}$ in nonparametric density estimation when the underlying density is Markov to a graph whose maximum clique size is at most $r$, and we provide evidence that in the aforementioned applications, this size is typically constant, i.e., $r=O(1)$. We then establish that the optimal rate in $L^1$ is $n^{-1/(2+r)}$ which, compared to the standard nonparametric rate of $n^{-1/(2+d)}$, reveals that the effective dimension of such problems is the size of the largest clique in the Markov random field. These rates are independent of the data's ambient dimension, making them applicable to realistic models of image, sound, video, and text data. Our results provide a novel justification for deep learning's ability to circumvent the curse of dimensionality, demonstrating dimension-independent convergence rates in these contexts.

Yuhao Wang, Ming Gao, Wai Ming Tai et al.

We develop optimal algorithms for learning undirected Gaussian trees and directed Gaussian polytrees from data. We consider both problems of distribution learning (i.e. in KL distance) and structure learning (i.e. exact recovery). The first approach is based on the Chow-Liu algorithm, and learns an optimal tree-structured distribution efficiently. The second approach is a modification of the PC algorithm for polytrees that uses partial correlation as a conditional independence tester for constraint-based structure learning. We derive explicit finite-sample guarantees for both approaches, and show that both approaches are optimal by deriving matching lower bounds. Additionally, we conduct numerical experiments to compare the performance of various algorithms, providing further insights and empirical evidence.

Yi Li, Wai Ming Tai

The active regression problem of the single-index model is to solve $\min_x \lVert f(Ax)-b\rVert_p$, where $A$ is fully accessible and $b$ can only be accessed via entry queries, with the goal of minimizing the number of queries to the entries of $b$. When $f$ is Lipschitz, previous results only obtain constant-factor approximations. This work presents the first algorithm that provides a $(1+\varepsilon)$-approximation solution by querying $\tilde{O}(d^{\frac{p}{2}\vee 1}/\varepsilon^{p\vee 2})$ entries of $b$. This query complexity is also shown to be optimal up to logarithmic factors for $p\in [1,2]$ and the $\varepsilon$-dependence of $1/\varepsilon^p$ is shown to be optimal for $p>2$.

Zhao Lyu, Wai Ming Tai, Mladen Kolar et al.

Despite numerous years of research into the merits and trade-offs of various model selection criteria, obtaining robust results that elucidate the behavior of cross-validation remains a challenging endeavor. In this paper, we highlight the inherent limitations of cross-validation when employed to discern the structure of a Gaussian graphical model. We provide finite-sample bounds on the probability that the Lasso estimator for the neighborhood of a node within a Gaussian graphical model, optimized using a prediction oracle, misidentifies the neighborhood. Our results pertain to both undirected and directed acyclic graphs, encompassing general, sparse covariance structures. To support our theoretical findings, we conduct an empirical investigation of this inconsistency by contrasting our outcomes with other commonly used information criteria through an extensive simulation study. Given that many algorithms designed to learn the structure of graphical models require hyperparameter selection, the precise calibration of this hyperparameter is paramount for accurately estimating the inherent structure. Consequently, our observations shed light on this widely recognized practical challenge.

Wai Ming Tai, Bryon Aragam

We study the problem of learning mixtures of Gaussians with censored data. Statistical learning with censored data is a classical problem, with numerous practical applications, however, finite-sample guarantees for even simple latent variable models such as Gaussian mixtures are missing. Formally, we are given censored data from a mixture of univariate Gaussians $$ \sum_{i=1}^k w_i \mathcal{N}(μ_i,σ^2), $$ i.e. the sample is observed only if it lies inside a set $S$. The goal is to learn the weights $w_i$ and the means $μ_i$. We propose an algorithm that takes only $\frac{1}{\varepsilon^{O(k)}}$ samples to estimate the weights $w_i$ and the means $μ_i$ within $\varepsilon$ error.

Ming Gao, Wai Ming Tai, Bryon Aragam

We study the optimal sample complexity of learning a Gaussian directed acyclic graph (DAG) from observational data. Our main results establish the minimax optimal sample complexity for learning the structure of a linear Gaussian DAG model in two settings of interest: 1) Under equal variances without knowledge of the true ordering, and 2) For general linear models given knowledge of the ordering. In both cases the sample complexity is $n\asymp q\log(d/q)$, where $q$ is the maximum number of parents and $d$ is the number of nodes. We further make comparisons with the classical problem of learning (undirected) Gaussian graphical models, showing that under the equal variance assumption, these two problems share the same optimal sample complexity. In other words, at least for Gaussian models with equal error variances, learning a directed graphical model is statistically no more difficult than learning an undirected graphical model. Our results also extend to more general identification assumptions as well as subgaussian errors.

Wai Ming Tai

Given a point set $P\subset \mathbb{R}^d$, the kernel density estimate of $P$ is defined as \[ \overline{\mathcal{G}}_P(x) = \frac{1}{\left|P\right|}\sum_{p\in P}e^{-\left\lVert x-p \right\rVert^2} \] for any $x\in\mathbb{R}^d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimate of $P$ is approximated by the kernel density estimate of $Q$. This subset $Q$ is called a coreset. The main technique in this work is constructing a $\pm 1$ coloring on the point set $P$ by discrepancy theory and we leverage Banaszczyk's Theorem. When $d>1$ is a constant, our construction gives a coreset of size $O\left(\frac{1}{\varepsilon}\right)$ as opposed to the best-known result of $O\left(\frac{1}{\varepsilon}\sqrt{\log\frac{1}{\varepsilon}}\right)$. It is the first result to give a breakthrough on the barrier of $\sqrt{\log}$ factor even when $d=2$.

Aditya Bhaskara, Wai Ming Tai

In the dictionary learning (or sparse coding) problem, we are given a collection of signals (vectors in $\mathbb{R}^d$), and the goal is to find a "basis" in which the signals have a sparse (approximate) representation. The problem has received a lot of attention in signal processing, learning, and theoretical computer science. The problem is formalized as factorizing a matrix $X (d \times n)$ (whose columns are the signals) as $X = AY$, where $A$ has a prescribed number $m$ of columns (typically $m \ll n$), and $Y$ has columns that are $k$-sparse (typically $k \ll d$). Most of the known theoretical results involve assuming that the columns of the unknown $A$ have certain incoherence properties, and that the coefficient matrix $Y$ has random (or partly random) structure. The goal of our work is to understand what can be said in the absence of such assumptions. Can we still find $A$ and $Y$ such that $X \approx AY$? We show that this is possible, if we allow violating the bounds on $m$ and $k$ by appropriate factors that depend on $k$ and the desired approximation. Our results rely on an algorithm for what we call the threshold correlation problem, which turns out to be related to hypercontractive norms of matrices. We also show that our algorithmic ideas apply to a setting in which some of the columns of $X$ are outliers, thus giving similar guarantees even in this challenging setting.

Jeff M. Phillips, Wai Ming Tai

We introduce two versions of a new sketch for approximately embedding the Gaussian kernel into Euclidean inner product space. These work by truncating infinite expansions of the Gaussian kernel, and carefully invoking the RecursiveTensorSketch [Ahle et al. SODA 2020]. After providing concentration and approximation properties of these sketches, we use them to approximate the kernel distance between points sets. These sketches yield almost $(1+\varepsilon)$-relative error, but with a small additive $α$ term. In the first variants the dependence on $1/α$ is poly-logarithmic, but has higher degree of polynomial dependence on the original dimension $d$. In the second variant, the dependence on $1/α$ is still poly-logarithmic, but the dependence on $d$ is linear.

Jeff M. Phillips, Wai Ming Tai

We construct near-optimal coresets for kernel density estimates for points in $\mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(\sqrt{d}/\varepsilon\cdot \sqrt{\log 1/\varepsilon} )$, and we show a near-matching lower bound of size $Ω(\min\{\sqrt{d}/\varepsilon, 1/\varepsilon^2\})$. When $d\geq 1/\varepsilon^2$, it is known that the size of coreset can be $O(1/\varepsilon^2)$. The upper bound is a polynomial-in-$(1/\varepsilon)$ improvement when $d \in [3,1/\varepsilon^2)$ and the lower bound is the first known lower bound to depend on $d$ for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.

Jeff M. Phillips, Wai Ming Tai

We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation preserves the $L_\infty$ error between kernel density estimates within error $ε$, with coreset size $2/ε^2$, but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is unrestricted, we show this bound is tight for these kernels as well as a much broader set. This work provides a careful analysis of the iterative Frank-Wolfe algorithm adapted to this context, an algorithm called \emph{kernel herding}. This analysis unites a broad line of work that spans statistics, machine learning, and geometry. When the dimension $d$ is constant, we demonstrate much tighter bounds on the size of the coreset specifically for Gaussian kernels, showing that it is bounded by the size of the coreset for axis-aligned rectangles. Currently the best known constructive bound is $O(\frac{1}ε \log^d \frac{1}ε)$, and non-constructively, this can be improved by $\sqrt{\log \frac{1}ε}$. This improves the best constant dimension bounds polynomially for $d \geq 3$.