论文检索

Probabilistic Methods · Gaussian Processes

David Dalton, Dirk Husmeier, Hao Gao

Physics-informed machine learning (PIML) has established itself as a new scientific paradigm which enables the seamless integration of observational data with partial differential equation (PDE) based physics models. A powerful tool for the analysis, reduction and solution of PDEs is the Lie symmetry method. Nevertheless, only recently has the integration of such symmetries into PIML frameworks begun to be explored. The present work adds to this growing literature by introducing an approach for incorporating a Lie symmetry into a physics-informed Gaussian process (GP) model. The symmetry is introduced as a constraint on the GP; either in a soft manner via virtual observations of an induced PDE called the invariant surface condition, or explicitly through the design of the kernel. Experimental results demonstrate that the use of symmetry constraints improves the performance of the GP for both forward and inverse problems, and that our approach offers competitive performance with neural networks in the low-data environment.

Miscellaneous Aspects of Machine Learning · Kernel methods

Vladimir Kostic, Karim Lounici, Prune Inzerilli, Pietro Novelli, Massimiliano Pontil

We study the problem of forecasting the evolution of a function of the state (observable) of a discrete ergodic dynamical system over multiple time steps. The elegant theory of Koopman and transfer operators can be used to evolve any such function forward in time. However, their estimators are usually unreliable in long-term forecasting. We show how classical techniques of eigenvalue deflation from operator theory and feature centering from statistics can be exploited to enhance standard estimators. We develop a novel technique to derive high probability bounds on powers of empirical estimators. Our approach, rooted in the stability _theory of non-normal operators_, allows us to establish uniform in time bounds for the forecasting error, which hold even on _infinite time horizons_. We further show that our approach can be seamlessly employed to forecast future state distributions from an initial one, with provably uniform error bounds. Numerical experiments illustrate the advantages of our approach in practice.

Deep Learning · Theory

Shokichi Takakura, Taiji Suzuki

In this paper, we study the feature learning ability of two-layer neural networks in the mean-field regime through the lens of kernel methods. To focus on the dynamics of the kernel induced by the first layer, we utilize a two-timescale limit, where the second layer moves much faster than the first layer. In this limit, the learning problem is reduced to the minimization problem over the intrinsic kernel. Then, we show the global convergence of the mean-field Langevin dynamics and derive time and particle discretization error. We also demonstrate that two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than any kernel methods, and neural networks aquire data-dependent kernel which aligns with the target function. In addition, we develop a label noise procedure, which converges to the global optimum and show that the degrees of freedom appears as an implicit reguralization.

Optimization

Sam Reifenstein, Timothee Leleu, Yoshihisa Yamamoto

We propose a novel algorithm that extends the methods of ball smoothing and Gaussian smoothing for noisy derivative-free optimization by accounting for the heterogeneous curvature of the objective function. The algorithm dynamically adapts the shape of the smoothing kernel to approximate the Hessian of the objective function around a local optimum. This approach significantly reduces the error in estimating the gradient from noisy evaluations through sampling. We demonstrate the efficacy of our method through numerical experiments on artificial problems. Additionally, we show improved performance when tuning NP-hard combinatorial optimization solvers compared to existing state-ofthe-art heuristic derivative-free and Bayesian optimization methods.

Miscellaneous Aspects of Machine Learning · Kernel methods

Lukas Gruber, Markus Holzleitner, Johannes Lehner, Sepp Hochreiter, Werner Zellinger

Estimating the ratio of two probability densities from finitely many samples, is a central task in machine learning and statistics. In this work, we show that a large class of kernel methods for density ratio estimation suffers from error saturation, which prevents algorithms from achieving fast error convergence rates on highly regular learning problems. To resolve saturation, we introduce iterated regularization in density ratio estimation to achieve fast error rates. Our methods outperform its non-iteratively regularized versions on benchmarks for density ratio estimation as well as on large-scale evaluations for importance-weighted ensembling of deep unsupervised domain adaptation models.

Deep Learning · Algorithms

Jiaxing Cui, Ruochen Wang, Yuanhao Xiong, Cho-Jui Hsieh

Dataset Condensation has emerged as a technique for compressing large datasets into smaller synthetic counterparts, facilitating downstream training tasks. In this paper, we study the impact of bias inside the original dataset on the performance of dataset condensation. With a comprehensive empirical evaluation on canonical datasets with color, corruption and background biases, we found that color and background biases in the original dataset will be amplified through the condensation process, resulting in a notable decline in the performance of models trained on the condensed dataset, while corruption bias is suppressed through the condensation process. To reduce bias amplification in dataset condensation, we introduce a simple yet highly effective approach based on a sample reweighting scheme utilizing kernel density estimation. Empirical results on multiple real-world and synthetic datasets demonstrate the effectiveness of the proposed method. Notably, on CMNIST with 5% bias-conflict ratio and IPC 50, our method achieves 91.5% test accuracy compared to 23.8% from vanilla DM, boosting the performance by 67.7%, whereas applying state-of-the-art debiasing method on the same dataset only achieves 53.7% accuracy. Our findings highlight the importance of addressing biases in dataset condensation and provide a promising avenue to address bias amplification in the process.

Reinforcement Learning

Uri Gadot, Kaixin Wang, Navdeep Kumar, Kfir Levy, Shie Mannor

Robust Markov Decision Processes (RMDPs) provide a framework for sequential decision-making that is robust to perturbations on the transition kernel. However, current RMDP methods are often limited to small-scale problems, hindering their use in high-dimensional domains. To bridge this gap, we present EWoK, a novel online approach to solve RMDP that Estimates the Worst transition Kernel to learn robust policies. Unlike previous works that regularize the policy or value updates, EWoK achieves robustness by simulating the worst scenarios for the agent while retaining complete flexibility in the learning process. Notably, EWoK can be applied on top of any off-the-shelf non-robust RL algorithm, enabling easy scaling to high-dimensional domains. Our experiments, spanning from simple Cartpole to high-dimensional DeepMind Control Suite environments, demonstrate the effectiveness and applicability of the EWoK paradigm as a practical method for learning robust policies.

Deep Learning

Hoyong Choi, Nohyun Ki, Hye Won Chung

Data subset selection aims to find a smaller yet informative subset of a large dataset that can approximate the full-dataset training, addressing challenges associated with training neural networks on large-scale datasets. However, existing methods tend to specialize in either high or low selection ratio regimes, lacking a universal approach that consistently achieves competitive performance across a broad range of selection ratios. We introduce a universal and efficient data subset selection method, Best Window Selection (BWS), by proposing a method to choose the best window subset from samples ordered based on their difficulty scores. This approach offers flexibility by allowing the choice of window intervals that span from easy to difficult samples. Furthermore, we provide an efficient mechanism for selecting the best window subset by evaluating its quality using kernel ridge regression. Our experimental results demonstrate the superior performance of BWS compared to other baselines across a broad range of selection ratios over datasets, including CIFAR-10/100 and ImageNet, and the scenarios involving training from random initialization or fine-tuning of pre-trained models.

Deep Learning

Maksim Zhdanov, David Ruhe, Maurice Weiler, Ana Lucic, Johannes Brandstetter, Patrick Forré

We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a novel class of ${\operatorname{E}}(p, q)$-equivariant CNNs. CS-CNNs process multivector fields on pseudo-Euclidean spaces $\mathbb{R}^{p,q}$. They specialize, for instance, to ${\operatorname{E}}(3)$-equivariance on $\mathbb{R}^3$ and Poincaré-equivariance on Minkowski spacetime $\mathbb{R}^{1,3}$. Our approach is based on an implicit parametrization of ${\operatorname{O}}(p,q)$-steerable kernels via Clifford group equivariant neural networks. We significantly and consistently outperform baseline methods on fluid dynamics as well as relativistic electrodynamics forecasting tasks.

Probabilistic Methods · Monte Carlo and Sampling Methods

Aimee Maurais, Youssef Marzouk

We introduce a new mean-field ODE and corresponding interacting particle systems (IPS) for sampling from an unnormalized target density. The IPS are gradient-free, available in closed form, and only require the ability to sample from a reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, $\pi_0^{1-t} \pi_1^t$, which is the path of a particular Fisher-Rao gradient flow. We employ a RKHS ansatz for the velocity field, which makes the Poisson equation tractable and enables discretization of the resulting mean-field ODE over finite samples. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Ampère equations within a framework known as sample-driven optimal transport. We introduce a stochastic variant of our approach and demonstrate empirically that our IPS can produce high-quality samples from varied target distributions, outperforming comparable gradient-free particle systems and competitive with gradient-based alternatives.

Probabilistic Methods · Gaussian Processes

Alan Matias, César Lincoln Mattos, Joao Paulo Gomes, Diego Mesquita

Deep kernel learning (DKL) marries the uncertainty quantification of Gaussian processes (GPs) and the representational power of deep neural networks. However, training DKL is challenging and often leads to overfitting. Most notably, DKL often learns “non-local” kernels — incurring spurious correlations. To remedy this issue, we propose using amortized inducing points and a parameter-sharing scheme, which ties together the amortization and DKL networks. This design imposes an explicit dependency between the ELBO’s model fit and capacity terms. In turn, this prevents the former from dominating the optimization procedure and incurring the aforementioned spurious correlations. Extensive experiments show that our resulting method, *amortized varitional* DKL (AVDKL), i) consistently outperforms DKL and standard GPs for tabular data; ii) achieves significantly higher accuracy than DKL in node classification tasks; and iii) leads to substantially better accuracy and negative log-likelihood than DKL on CIFAR100.

Optimization · Sampling and Optimization

Qiang Fu, Ashia Wilson

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

Weixuan Liang, En Zhu, Shengju Yu, Huiying Xu, Xinzhong Zhu, Xinwang Liu

In this paper, we derive an upper bound of the difference between a kernel matrix and its expectation under a mild assumption. Specifically, we assume that the true distribution of the training data is an unknown isotropic Gaussian distribution. When the kernel function is a Gaussian kernel, and the mean of each cluster is sufficiently separated, we find that the expectation of a kernel matrix can be close to a rank-$k$ matrix, where $k$ is the cluster number. Moreover, we prove that the normalized kernel matrix of the training set deviates (w.r.t. Frobenius norm) from its expectation in the order of $\widetilde{\mathcal{O}}(1/\sqrt{d})$, where $d$ is the dimension of samples. Based on the above theoretical results, we propose a novel multiple kernel clustering framework which attempts to learn the information of the expectation kernel matrices. First, we aim to minimize the distance between each base kernel and a rank-$k$ matrix, which is a proxy of the expectation kernel. Then, we fuse these rank-$k$ matrices into a consensus rank-$k$ matrix to find the clustering structure. Using an anchor-based method, the proposed framework is flexible with the sizes of input kernel matrices and able to handle large-scale datasets. We also provide the approximation guarantee by deriving two non-asymptotic bounds for the consensus kernel and clustering indicator matrices. Finally, we conduct extensive experiments to verify the clustering performance of the proposed method and the correctness of the proposed theoretical results.

Probabilistic Methods · Gaussian Processes

Nicolas Alder, Ralf Herbrich

The widespread use of artificial intelligence requires finding energy-efficient paradigms for the field. We propose to reduce the energy consumption of Gaussian process regression using low-precision floating-point representations. We explore how low-precision representations impact the results of Gaussian process regression and how data set properties, implementation approach, model performance, and energy consumption interact. Our findings show that a well-conditioned kernel matrix allows reducing the energy consumption by up to 89.01% for 98.08% of arithmetic operations with little to no impact on model performance. Our findings are relevant whenever one needs to invert a symmetric full-rank matrix.

Applications · Time Series

Shikai Fang, Qingsong Wen, Yingtao Luo, Shandian Zhe, Liang Sun

In real-world scenarios such as traffic and energy management, we frequently encounter large volumes of time-series data characterized by missing values, noise, and irregular sampling patterns. While numerous imputation methods have been proposed, the majority tend to operate within a local horizon, which involves dividing long sequences into batches of fixed-length segments for model training. This local horizon often leads to the overlooking of global trends and periodic patterns. More importantly, most methods assume the observations are sampled at regular timestamps, and fail to handle complex irregular sampled time series in various applications. Additionally, most existing methods are learned in an offline manner. Thus, it is not suitable for applications with rapidly arriving streaming data. To address these challenges, we propose BayOTIDE: Bayesian Online Multivariate Time series Imputation with functional decomposition. Our method conceptualizes multivariate time series as the weighted combination of groups of low-rank temporal factors with different patterns. We employ a suite of Gaussian Processes (GPs),each with a unique kernel, as functional priors to model these factors. For computational efficiency, we further convert the GPs into a state-space prior by constructing an equivalent stochastic differential equation (SDE), and developing a scalable algorithm for online inference. The proposed method can not only handle imputation over arbitrary timestamps, but also offer uncertainty quantification and interpretability for the downstream application. We evaluate our method on both synthetic and real-world datasets. We release the code at https://github.com/xuangu-fang/BayOTIDE.

Deep Learning · Generative Models and Autoencoders

Jaehyeong Jo, Sung Ju Hwang

Learning the distribution of data on Riemannian manifolds is crucial for modeling data from non-Euclidean space, which is required by many applications in diverse scientific fields. Yet, existing generative models on manifolds suffer from expensive divergence computation or rely on approximations of heat kernel. These limitations restrict their applicability to simple geometries and hinder scalability to high dimensions. In this work, we introduce the Riemannian Diffusion Mixture, a principled framework for building a generative diffusion process on manifolds. Instead of following the denoising approach of previous diffusion models, we construct a diffusion process using a mixture of bridge processes derived on general manifolds without requiring heat kernel estimations. We develop a geometric understanding of the mixture process, deriving the drift as a weighted mean of tangent directions to the data points that guides the process toward the data distribution. We further propose a scalable training objective for learning the mixture process that readily applies to general manifolds. Our method achieves superior performance on diverse manifolds with dramatically reduced number of in-training simulation steps for general manifolds.

Deep Learning · Generative Models and Autoencoders

Jingwei Zhang, Cheuk Ting Li, Farzan Farnia

The massive developments of generative model frameworks require principled methods for the evaluation of a model's novelty compared to a reference dataset. While the literature has extensively studied the evaluation of the quality, diversity, and generalizability of generative models, the assessment of a model's novelty compared to a reference model has not been adequately explored in the machine learning community. In this work, we focus on the novelty assessment for multi-modal distributions and attempt to address the following differential clustering task: Given samples of a generative model $P_\mathcal{G}$ and a reference model $P_\mathrm{ref}$, how can we discover the sample types expressed by $P_\mathcal{G}$ more frequently than in $P_\mathrm{ref}$? We introduce a spectral approach to the differential clustering task and propose the Kernel-based Entropic Novelty (KEN) score to quantify the mode-based novelty of $P_\mathcal{G}$ with respect to $P_\mathrm{ref}$. We analyze the KEN score for mixture distributions with well-separable components and develop a kernel-based method to compute the KEN score from empirical data. We support the KEN framework by presenting numerical results on synthetic and real image datasets, indicating the framework's effectiveness in detecting novel modes and comparing generative models. The paper's code is available at: github.com/buyeah1109/KEN.

Deep Learning · Large Language Models

Ruisi Cai, Yuandong Tian, Zhangyang “Atlas” Wang, Beidi Chen

This paper tackles the memory hurdle of of processing long context sequences in Large Language Models (LLMs), by presenting a novel approach, Dropping In Convolutions for **Lo**ng **Co**ntext **Co**mpression (**LoCoCo**). LoCoCo employs only a fixed-size Key-Value (KV) cache, and can enhance efficiency in both inference and fine-tuning stages. Diverging from prior methods that selectively drop KV pairs based on heuristics, LoCoCo leverages a data-driven adaptive fusion technique, blending previous KV pairs with incoming tokens to minimize the loss of contextual information and ensure accurate attention modeling. This token integration is achieved through injecting one-dimensional convolutional kernels that dynamically calculate mixing weights for each KV cache slot. Designed for broad compatibility with existing LLM frameworks, LoCoCo allows for straightforward "drop-in" integration without needing architectural modifications, while incurring minimal tuning overhead. Experiments demonstrate that LoCoCo maintains consistently outstanding performance across various context lengths and can achieve a high context compression rate during both inference and fine-tuning phases. During inference, we successfully compressed up to $3482$ tokens into a $128$-size KV cache, while retaining comparable performance to the full sequence - an accuracy improvement of up to $0.2791$ compared to baselines at the same cache size. During post-training tuning, we also effectively extended the context length from 4K to 32K using a KV cache of fixed size 512, achieving performance similar to fine-tuning with entire sequences.

Yulun Zhang, Kai Zhang, Luc Van Gool, Martin Danelljan, Fisher Yu

Lightweight image super-resolution (SR) methods have obtained promising results with moderate model complexity. These approaches primarily focus on a lightweight architecture design, but neglect to further reduce network redundancy. While some model compression techniques try to achieve more lightweight SR models with neural architecture search, knowledge distillation, or channel pruning, they typically require considerable extra computational resources or neglect to prune weights. To address these issues, we propose a flexible meta pruning (FMP) for lightweight image SR, where the network channels and weights are pruned simultaneously. Specifically, we control the network sparsity via channel vectors and weight indicators. We feed them into a hypernetwork, whose parameters act as meta-data for the parameters of the SR backbone. Consequently, for each network layer, we conduct structured pruning with channel vectors, which control the output and input channels. Besides, we conduct unstructured pruning with weight indicators to influence the sparsity of kernel weights, resulting in flexible pruning. During pruning, the sparsity of both channel vectors and weight indicators are regularized. We optimize the channel vectors and weight indicators with proximal gradient and SGD. We conduct extensive experiments to investigate critical factors in the flexible channel and weight pruning for image SR, demonstrating the superiority of our FMP when applied to baseline image SR architectures.

Deep Learning · Large Language Models

Jingtan Wang, Xiaoqiang Lin, Rui Qiao, Chuan-Sheng Foo, Bryan Kian Hsiang Low

The increasing complexity of foundational models underscores the necessity for explainability, particularly for fine-tuning, the most widely used training method for adapting models to downstream tasks. Instance attribution, one type of explanation, attributes the model prediction to each training example by an instance score. However, the robustness of instance scores, specifically towards dataset resampling, has been overlooked. To bridge this gap, we propose a notion of robustness on the sign of the instance score. We theoretically and empirically demonstrate that the popular leave-one-out-based methods lack robustness, while the Shapley value behaves significantly better, but at a higher computational cost. Accordingly, we introduce an efficient fine-tuning-free approximation of the Shapley value (FreeShap) for instance attribution based on the neural tangent kernel. We empirically demonstrate that FreeShap outperforms other methods for instance attribution and other data-centric applications such as data removal, data selection, and wrong label detection, and further generalize our scale to large language models (LLMs). Our code is available at https://github.com/JTWang2000/FreeShap.