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Causality

Feng Xie, Zhen Yao, Lin Xie, Yan Zeng, Zhi Geng

We consider the challenging problem of estimating causal effects from purely observational data in the bi-directional Mendelian randomization (MR), where some invalid instruments, as well as unmeasured confounding, usually exist. To address this problem, most existing methods attempt to find proper valid instrumental variables (IVs) for the target causal effect by expert knowledge or by assuming that the causal model is a one-directional MR model. As such, in this paper, we first theoretically investigate the identification of the bi-directional MR from observational data. In particular, we provide necessary and sufficient conditions under which valid IV sets are correctly identified such that the bi-directional MR model is identifiable, including the causal directions of a pair of phenotypes (i.e., the treatment and outcome).Moreover, based on the identification theory, we develop a cluster fusion-like method to discover valid IV sets and estimate the causal effects of interest.We theoretically demonstrate the correctness of the proposed algorithm.Experimental results show the effectiveness of our method for estimating causal effects in both one-directional and bi-directional MR models.

Miscellaneous Aspects of Machine Learning · Kernel methods

Anna Korba, Francis Bach, Clémentine CHAZAL

In this paper, we study the statistical and geometrical properties of the Kullback-Leibler divergence with kernel covariance operators (KKL) introduced by [Bach, 2022, Information Theory with Kernel Methods]. Unlike the classical Kullback-Leibler (KL) divergence that involves density ratios, the KKL compares probability distributions through covariance operators (embeddings) in a reproducible kernel Hilbert space (RKHS), and compute the Kullback-Leibler quantum divergence. This novel divergence hence shares parallel but different aspects with both the standard Kullback-Leibler between probability distributions and kernel embeddings metrics such as the maximum mean discrepancy. A limitation faced with the original KKL divergence is its inability to be defined for distributions with disjoint supports. To solve this problem, we propose in this paper a regularised variant that guarantees that divergence is well defined for all distributions. We derive bounds that quantify the deviation of the regularised KKL to the original one, as well as concentration bounds. In addition, we provide a closed-form expression for the regularised KKL, specifically applicable when the distributions consist of finite sets of points, which makes it implementable. Furthermore, we derive a Wasserstein gradient descent scheme of the KKL divergence in the case of discrete distributions, and study empirically its properties to transport a set of points to a target distribution.

Generative Models · New Approaches

Lujun Li, Peijie Dong, Zhenheng Tang, Xiang Liu, Qiang Wang, Wenhan Luo, Wei Xue, Qifeng Liu, Xiaowen Chu, Yike Guo

In this paper, we present DSA, the first automated framework for discovering sparsity allocation schemes for layer-wise pruning in Large Language Models (LLMs). LLMs have become increasingly powerful, but their large parameter counts make them computationally expensive. Existing pruning methods for compressing LLMs primarily focus on evaluating redundancies and removing element-wise weights. However, these methods fail to allocate adaptive layer-wise sparsities, leading to performance degradation in challenging tasks. We observe that per-layer importance statistics can serve as allocation indications, but their effectiveness depends on the allocation function between layers. To address this issue, we develop an expression discovery framework to explore potential allocation strategies. Our allocation functions involve two steps: reducing element-wise metrics to per-layer importance scores, and modelling layer importance to sparsity ratios. To search for the most effective allocation function, we construct a search space consisting of pre-process, reduction, transform, and post-process operations. We leverage an evolutionary algorithm to perform crossover and mutation on superior candidates within the population, guided by performance evaluation. Finally, we seamlessly integrate our discovered functions into various uniform methods, resulting in significant performance improvements. We conduct extensive experiments on multiple challenging tasks such as arithmetic, knowledge reasoning, and multimodal benchmarks spanning GSM8K, MMLU, SQA, and VQA, demonstrating that our DSA method achieves significant performance gains on the LLaMA-1|2|3, Mistral, and OPT models. Notably, the LLaMA-1|2|3 model pruned by our DSA reaches 4.73\%|6.18\%|10.65\% gain over the state-of-the-art techniques (e.g., Wanda and SparseGPT).

Theory · Statistics

Yihan Zhang, Marco Mondelli

We study the matrix denoising problem of estimating the singular vectors of a rank-$1$ signal corrupted by noise with both column and row correlations. Existing works are either unable to pinpoint the exact asymptotic estimation error or, when they do so, the resulting approaches (e.g., based on whitening or singular value shrinkage) remain vastly suboptimal. On top of this, most of the literature has focused on the special case of estimating the left singular vector of the signal when the noise only possesses row correlation (one-sided heteroscedasticity). In contrast, our work establishes the information-theoretic and algorithmic limits of matrix denoising with doubly heteroscedastic noise. We characterize the exact asymptotic minimum mean square error, and design a novel spectral estimator with rigorous optimality guarantees: under a technical condition, it attains positive correlation with the signals whenever information-theoretically possible and, for one-sided heteroscedasticity, it also achieves the Bayes-optimal error. Numerical experiments demonstrate the significant advantage of our theoretically principled method with the state of the art. The proofs draw connections with statistical physics and approximate message passing, departing drastically from standard random matrix theory techniques.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Ayush Jain, Andrea Montanari, Eren Sasoglu

Collecting large quantities of high-quality data can be prohibitively expensive or impractical, and a bottleneck in machine learning. One may instead augment a small set of $n$ data points from the target distribution with data from more accessible sources, e.g. data collected under different circumstances or synthesized by generative models. We refer to such data as `surrogate data'. We study a weighted empirical risk minimization (ERM) approach for integrating surrogate data into training. We analyze mathematically this method under several classical statistical models, and validate our findings empirically on datasets from different domains. Our main findings are: $(i)$ Integrating surrogate data can significantly reduce the test error on the original distribution. Surprisingly, this can happen even when the surrogate data is unrelated to the original ones. We trace back this behavior to the classical Stein's paradox. $(ii)$ In order to reap the benefit of surrogate data, it is crucial to use optimally weighted ERM. $(iii)$ The test error of models trained on mixtures of real and surrogate data is approximately described by a scaling law. This scaling law can be used to predict the optimal weighting scheme, and to choose the amount of surrogate data to add.

Optimization · Large Scale, Parallel and Distributed

Tongle Wu, Ying Sun

We consider learning a sparse model from linear measurements taken by a network of agents. Different from existing decentralized methods designed based on the LASSO regression with explicit $\ell_1$ norm regularization, we exploit the implicit regularization of decentralized optimization method applied to an over-parameterized nonconvex least squares formulation without penalization. Our first result shows that despite nonconvexity, if the network connectivity is good, the well-known decentralized gradient descent algorithm (DGD) with small initialization and early stopping can compute the statistically optimal solution. Sufficient conditions on the initialization scale, choice of step size, network connectivity, and stopping time are further provided to achieve convergence. Our result recovers the convergence rate of gradient descent in the centralized setting, showing its tightness. Based on the analysis of DGD, we further propose a communication-efficient version, termed T-DGD, by truncating the iterates before transmission. In the high signal-to-noise ratio (SNR) regime, we show that T-DGD achieves comparable statistical accuracy to DGD, while the communication cost is logarithmic in the number of parameters. Numerical results are provided to validate the effectiveness of DGD and T-DGD for sparse learning through implicit regularization.

Statistical Methods

Weichao Yang, Hongwei Shi, Xu Guo, Changliang Zou

The high-dimensional single index model (SIM), which assumes that the response is independent of the predictors given a linear combination of predictors, has drawn attention due to its flexibility and interpretability, but its efficiency is adversely affected by outlying observations and heavy-tailed distributions. This paper introduces a robust procedure by recasting the SIM into a pseudo-linear model with transformed responses. It relaxes the distributional conditions on random errors from sub-Gaussian to more general distributions and thus it is robust with substantial efficiency gain for heavy-tailed random errors. Under this paradigm, we provide asymptotically honest group inference procedures based on the idea of orthogonalization, which enjoys the feature that it does not require the zero and nonzero coefficients to be well-separated. Asymptotic null distribution and bootstrap implementation are both established. Moreover, we develop a multiple testing procedure for determining if the individual coefficients are relevant simultaneously, and show that it is able to control the false discovery rate asymptotically. Numerical results indicate that the new procedures can be highly competitive among existing methods, especially for heavy-tailed errors.

Privacy

Bonwoo Lee, Jeongyoun Ahn, Cheolwoo Park

As the volume of data invested in statistical learning increases and concerns regarding privacy grow, the privacy leakage issue has drawn significant attention. Differential privacy has emerged as a widely accepted concept capable of mitigating privacy concerns, and numerous differentially private (DP) versions of machine learning algorithms have been developed. However, existing works on DP kernel learning algorithms have exhibited practical limitations, including scalability, restricted choice of kernels, or dependence on test data availability. We propose DP scalable kernel empirical risk minimization (ERM) algorithms and a DP kernel mean embedding (KME) release algorithm suitable for general kernels. Our approaches address the shortcomings of previous algorithms by employing Nyström methods, classical techniques in non-private scalable kernel learning. These methods provide data-dependent low-rank approximations of the kernel matrix for general kernels in a DP manner. We present excess empirical risk bounds and computational complexities for the scalable kernel DP ERM, KME algorithms, contrasting them with established methodologies. Furthermore, we develop a private data-generating algorithm capable of learning diverse kernel models. We conduct experiments to demonstrate the performance of our algorithms, comparing them with existing methods to highlight their superiority.

Miscellaneous Aspects of Machine Learning · Kernel methods

Sejun Park, Kihun Hong, Ganguk Hwang

Over the past decade, there is a growing interest in collaborative learning that can enhance AI models of multiple parties.However, it is still challenging to enhance performance them without sharing private data and models from individual parties.One recent promising approach is to develop distillation-based algorithms that exploit unlabeled public data but the results are still unsatisfactory in both theory and practice.To tackle this problem, we rigorously analyze a representative distillation-based algorithm in the view of kernel regression.This work provides the first theoretical results to prove the (nearly) minimax optimality of the nonparametric collaborative learning algorithm that does not directly share local data or models in massively distributed statistically heterogeneous environments.Inspired by our theoretical results, we also propose a practical distillation-based collaborative learning algorithm based on neural network architecture.Our algorithm successfully bridges the gap between our theoretical assumptions and practical settings with neural networks through feature kernel matching.We simulate various regression tasks to verify our theory and demonstrate the practical feasibility of our proposed algorithm.

Miscellaneous Aspects of Machine Learning · Kernel methods

Florian Kalinke, Zoltan Szabo

Kernel techniques are among the most influential approaches in data science and statistics. Under mild conditions, the reproducing kernel Hilbert space associated to a kernel is capable of encoding the independence of $M\ge2$ random variables. Probably the most widespread independence measure relying on kernels is the so-called Hilbert-Schmidt independence criterion (HSIC; also referred to as distance covariance in the statistics literature). Despite various existing HSIC estimators designed since its introduction close to two decades ago, the fundamental question of the rate at which HSIC can be estimated is still open. In this work, we prove that the minimax optimal rate of HSIC estimation on $\mathbb{R}^d$ for Borel measures containing the Gaussians with continuous bounded translation-invariant characteristic kernels is $\mathcal{O}\left(n^{-1/2}\right)$. Specifically, our result implies the optimality in the minimax sense of many of the most-frequently used estimators (including the U-statistic, the V-statistic, and the Nyström-based one) on $\mathbb{R}^d$.

Probabilistic Methods

Daniela de Albuquerque, John Pearson

Beyond estimating parameters of interest from data, one of the key goals of statistical inference is to properly quantify uncertainty in these estimates. In Bayesian inference, this uncertainty is provided by the posterior distribution, the computation of which typically involves an intractable high-dimensional integral. Among available approximation methods, sampling-based approaches come with strong theoretical guarantees but scale poorly to large problems, while variational approaches scale well but offer few theoretical guarantees. In particular, variational methods are known to produce overconfident estimates of posterior uncertainty and are typically non-identifiable, with many latent variable configurations generating equivalent predictions. Here, we address these challenges by showing how diffusion-based models (DBMs), which have recently produced state-of-the-art performance in generative modeling tasks, can be repurposed for performing calibrated, identifiable Bayesian inference. By exploiting a previously established connection between the stochastic and probability flow ordinary differential equations (pfODEs) underlying DBMs, we derive a class of models, \emph{inflationary flows,} that uniquely and deterministically map high-dimensional data to a lower-dimensional Gaussian distribution via ODE integration. This map is both invertible and neighborhood-preserving, with controllable numerical error, with the result that uncertainties in the data are correctly propagated to the latent space. We demonstrate how such maps can be learned via standard DBM training using a novel noise schedule and are effective at both preserving and reducing intrinsic data dimensionality. The result is a class of highly expressive generative models, uniquely defined on a low-dimensional latent space, that afford principled Bayesian inference.

Miscellaneous Aspects of Machine Learning

Xiang Liu, Liangxi Liu, Feiyang Ye, Yunheng Shen, Xia Li, Linshan Jiang, Jialin Li

Efficiently aggregating trained neural networks from local clients into a global model on a server is a widely researched topic in federated learning. Recently, motivated by diminishing privacy concerns, mitigating potential attacks, and reducing communication overhead, one-shot federated learning (i.e., limiting client-server communication into a single round) has gained popularity among researchers. However, the one-shot aggregation performances are sensitively affected by the non-identical training data distribution, which exhibits high statistical heterogeneity in some real-world scenarios. To address this issue, we propose a novel one-shot aggregation method with layer-wise posterior aggregation, named FedLPA. FedLPA aggregates local models to obtain a more accurate global model without requiring extra auxiliary datasets or exposing any private label information, e.g., label distributions. To effectively capture the statistics maintained in the biased local datasets in the practical non-IID scenario, we efficiently infer the posteriors of each layer in each local model using layer-wise Laplace approximation and aggregate them to train the global parameters. Extensive experimental results demonstrate that FedLPA significantly improves learning performance over state-of-the-art methods across several metrics.

Trustworthy Machine Learning

Huancheng Chen, Haris Vikalo

Statistical heterogeneity of data present at client devices in a federated learning (FL) system renders the training of a global model in such systems difficult. Particularly challenging are the settings where due to communication resource constraints only a small fraction of clients can participate in any given round of FL. Recent approaches to training a global model in FL systems with non-IID data have focused on developing client selection methods that aim to sample clients with more informative updates of the model. However, existing client selection techniques either introduce significant computation overhead or perform well only in the scenarios where clients have data with similar heterogeneity profiles. In this paper, we propose HiCS-FL (Federated Learning via Hierarchical Clustered Sampling), a novel client selection method in which the server estimates statistical heterogeneity of a client's data using the client’s update of the network’s output layer and relies on this information to cluster and sample the clients. We analyze the ability of the proposed techniques to compare heterogeneity of different datasets, and characterize convergence of the training process that deploys the introduced client selection method. Extensive experimental results demonstrate that in non-IID settings HiCS-FL achieves faster convergence than state-of-the-art FL client selection schemes. Notably, HiCS-FL drastically reduces computation cost compared to existing selection schemes and is adaptable to different heterogeneity scenarios.

Optimization

Paul Mangold, Sergey Samsonov, Safwan Labbi, Ilya Levin, REDA ALAMI, Alexey Naumov, Eric Moulines

In this paper, we analyze the sample and communication complexity of the federated linear stochastic approximation (FedLSA) algorithm. We explicitly quantify the effects of local training with agent heterogeneity. We show that the communication complexity of FedLSA scales polynomially with the inverse of the desired accuracy ϵ. To overcome this, we propose SCAFFLSA a new variant of FedLSA that uses control variates to correct for client drift, and establish its sample and communication complexities. We show that for statistically heterogeneous agents, its communication complexity scales logarithmically with the desired accuracy, similar to Scaffnew. An important finding is that, compared to the existing results for Scaffnew, the sample complexity scales with the inverse of the number of agents, a property referred to as linear speed-up. Achieving this linear speed-up requires completely new theoretical arguments. We apply the proposed method to federated temporal difference learning with linear function approximation and analyze the corresponding complexity improvements.

Statistical Methods

Haoqun Cao, Zizhuo Meng, Tianjun Ke, Feng Zhou

Score matching estimators for point processes have gained widespread attention in recent years because they do not require the calculation of intensity integrals, thereby effectively addressing the computational challenges in maximum likelihood estimation (MLE). Some existing works have proposed score matching estimators for point processes. However, this work demonstrates that the incompleteness of the estimators proposed in those works renders them applicable only to specific problems, and they fail for more general point processes. To address this issue, this work introduces the weighted score matching estimator to point processes. Theoretically, we prove the consistency of the estimator we propose. Experimental results indicate that our estimator accurately estimates model parameters on synthetic data and yields results consistent with MLE on real data. In contrast, existing score matching estimators fail to perform effectively. Codes are publicly available at \url{https://github.com/KenCao2007/WSM_TPP}.

Computational Biology

Gokul Gowri, Xiaokang Lun, Allon Klein, Peng Yin

Mutual information (MI) is a general measure of statistical dependence with widespread application across the sciences. However, estimating MI between multi-dimensional variables is challenging because the number of samples necessary to converge to an accurate estimate scales unfavorably with dimensionality. In practice, existing techniques can reliably estimate MI in up to tens of dimensions, but fail in higher dimensions, where sufficient sample sizes are infeasible. Here, we explore the idea that underlying low-dimensional structure in high-dimensional data can be exploited to faithfully approximate MI in high-dimensional settings with realistic sample sizes. We develop a method that we call latent MI (LMI) approximation, which applies a nonparametric MI estimator to low-dimensional representations learned by a simple, theoretically-motivated model architecture. Using several benchmarks, we show that unlike existing techniques, LMI can approximate MI well for variables with $> 10^3$ dimensions if their dependence structure is captured by low-dimensional representations. Finally, we showcase LMI on two open problems in biology. First, we approximate MI between protein language model (pLM) representations of interacting proteins, and find that pLMs encode non-trivial information about protein-protein interactions. Second, we quantify cell fate information contained in single-cell RNA-seq (scRNA-seq) measurements of hematopoietic stem cells, and find a sharp transition during neutrophil differentiation when fate information captured by scRNA-seq increases dramatically. An implementation of LMI is available at *latentmi.readthedocs.io.*

Fairness

Muhammad Faaiz Taufiq, Jean-Francois Ton, Yang Liu

In machine learning fairness, training models that minimize disparity across different sensitive groups often leads to diminished accuracy, a phenomenon known as the fairness-accuracy trade-off. The severity of this trade-off inherently depends on dataset characteristics such as dataset imbalances or biases and therefore, using a uniform fairness requirement across diverse datasets remains questionable. To address this, we present a computationally efficient approach to approximate the fairness-accuracy trade-off curve tailored to individual datasets, backed by rigorous statistical guarantees. By utilizing the You-Only-Train-Once (YOTO) framework, our approach mitigates the computational burden of having to train multiple models when approximating the trade-off curve. Crucially, we introduce a novel methodology for quantifying uncertainty in our estimates, thereby providing practitioners with a robust framework for auditing model fairness while avoiding false conclusions due to estimation errors. Our experiments spanning tabular (e.g., Adult), image (CelebA), and language (Jigsaw) datasets underscore that our approach not only reliably quantifies the optimum achievable trade-offs across various data modalities but also helps detect suboptimality in SOTA fairness methods.

Data-centric AI · Data augmentation

Andrei Margeloiu, Xiangjian Jiang, Nikola Simidjievski, Mateja Jamnik

Data collection is often difficult in critical fields such as medicine, physics, and chemistry, yielding typically only small tabular datasets. However, classification methods tend to struggle with these small datasets, leading to poor predictive performance. Increasing the training set with additional synthetic data, similar to data augmentation in images, is commonly believed to improve downstream tabular classification performance. However, current tabular generative methods that learn either the joint distribution $ p(\mathbf{x}, y) $ or the class-conditional distribution $ p(\mathbf{x} \mid y) $ often overfit on small datasets, resulting in poor-quality synthetic data, usually worsening classification performance compared to using real data alone. To solve these challenges, we introduce TabEBM, a novel class-conditional generative method using Energy-Based Models (EBMs). Unlike existing tabular methods that use a shared model to approximate all class-conditional densities, our key innovation is to create distinct EBM generative models for each class, each modelling its class-specific data distribution individually. This approach creates robust energy landscapes, even in ambiguous class distributions. Our experiments show that TabEBM generates synthetic data with higher quality and better statistical fidelity than existing methods. When used for data augmentation, our synthetic data consistently leads to improved classification performance across diverse datasets of various sizes, especially small ones. Code is available at https://github.com/andreimargeloiu/TabEBM.

Yiwen Kou, Zixiang Chen, Quanquan Gu, Sham Kakade

The $k$-sparse parity problem is a classical problem in computational complexity and algorithmic theory, serving as a key benchmark for understanding computational classes. In this paper, we solve the $k$-sparse parity problem with sign stochastic gradient descent, a variant of stochastic gradient descent (SGD) on two-layer fully-connected neural networks. We demonstrate that this approach can efficiently solve the $k$-sparse parity problem on a $d$-dimensional hypercube ($k\le O(\sqrt{d})$) with a sample complexity of $\tilde{O}(d^{k-1})$ using $2^{\Theta(k)}$ neurons, matching the established $\Omega(d^{k})$ lower bounds of Statistical Query (SQ) models. Our theoretical analysis begins by constructing a good neural network capable of correctly solving the $k$-parity problem. We then demonstrate how a trained neural network with sign SGD can effectively approximate this good network, solving the $k$-parity problem with small statistical errors. To the best of our knowledge, this is the first result that matches the SQ lower bound for solving $k$-sparse parity problem using gradient-based methods.

Data-centric AI · Data-centric AI methods and tools

Jonathan Hayase, Alisa Liu, Yejin Choi, Sewoong Oh, Noah Smith

The pretraining data of today's strongest language models remains opaque, even when their parameters are open-sourced.In particular, little is known about the proportions of different domains, languages, or code represented in the data. While a long line of membership inference attacks aim to identify training examples on an instance level, they do not extend easily to *global* statistics about the corpus. In this work, we tackle a task which we call *data mixture inference*, which aims to uncover the distributional make-up of the pretraining data. We introduce a novel attack based on a previously overlooked source of information — byte-pair encoding (BPE) tokenizers, used by the vast majority of modern language models. Our key insight is that the ordered vocabulary learned by a BPE tokenizer naturally reveals information about the token frequencies in its training data: the first token is the most common byte pair, the second is the most common pair after merging the first token, and so on. Given a tokenizer's merge list along with data samples for each category of interest (e.g., different natural languages), we formulate a linear program that solves for the relative proportion of each category in the tokenizer's training set. Importantly, to the extent to which tokenizer training data is representative of the pretraining data, we indirectly learn about the pretraining data. In controlled experiments, we show that our attack can recover mixture ratios with high precision for tokenizers trained on known mixtures of natural languages, programming languages, and data sources. We then apply our approach to off-the-shelf tokenizers released alongside recent LMs. We confirm much publicly disclosed information about these models, and also make several new inferences: GPT-4o is much more multilingual than its predecessors, training on 10x more non-English data than GPT-3.5, Llama 3 and Claude are trained on predominantly code, and many recent models are trained on 7-16% books. We hope our work sheds light on current design practices for pretraining data, and inspires continued research into data mixture inference for LMs.