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Deep Learning · Graph Neural Networks

Qianlong Wen, Mingxuan Ju, Zhongyu Ouyang, Chuxu Zhang, Yanfang Ye

Self-supervised learning (SSL) has gained increasing attention in the graph learning community, owing to its capability of enabling powerful models pre-trained on large unlabeled graphs for general purposes, facilitating quick adaptation to specific domains. Though promising, existing graph SSL frameworks often struggle to capture both high-level abstract features and fine-grained features simultaneously, leading to sub-optimal generalization abilities across different downstream tasks. To bridge this gap, we present Multi-granularity Graph Semantic Ensemble via Knowledge Distillation, namely MGSE, a plug-and-play graph knowledge distillation framework that can be applied to any existing graph SSL framework to enhance its performance by incorporating the concept of multi-granularity. Specifically, MGSE captures multi-granular knowledge by employing multiple student models to learn from a single teacher model, conditioned by probability distributions with different granularities. We apply it to six state-of-the-art graph SSL frameworks and evaluate their performances over multiple graph datasets across different domains, the experimental results show that MGSE can consistently boost the performance of these existing graph SSL frameworks with up to 9.2% improvement.

Miscellaneous Aspects of Machine Learning · General Machine Learning Techniques

Li Sun, Zhenhao Huang, Hao Peng, YuJie Wang, Chunyang Liu, Philip Yu

Graph clustering is a fundamental problem in machine learning. Deep learning methods achieve the state-of-the-art results in recent years, but they still cannot work without predefined cluster numbers. Such limitation motivates us to pose a more challenging problem of graph clustering with unknown cluster number. We propose to address this problem from a fresh perspective of graph information theory (i.e., structural information). In the literature, structural information has not yet been introduced to deep clustering, and its classic definition falls short of discrete formulation and modeling node features. In this work, we first formulate a differentiable structural information (DSI) in the continuous realm, accompanied by several theoretical results. By minimizing DSI, we construct the optimal partitioning tree where densely connected nodes in the graph tend to have the same assignment, revealing the cluster struc- ture. DSI is also theoretically presented as a new graph clustering objective, not requiring the pre-defined cluster number. Furthermore, we design a neural LSEnet in the Lorentz model of hyperbolic space, where we integrate node features to structural information via manifold-valued graph convolution. Extensive empirical results on real graphs show the superiority of our approach.

Deep Learning

Pratik Rathore, Weimu Lei, Zachary Frangella, Lu Lu, Madeleine Udell

This paper explores challenges in training Physics-Informed Neural Networks (PINNs), emphasizing the role of the loss landscape in the training process. We examine difficulties in minimizing the PINN loss function, particularly due to ill-conditioning caused by differential operators in the residual term. We compare gradient-based optimizers Adam, L-BFGS, and their combination Adam+L-BFGS, showing the superiority of Adam+L-BFGS, and introduce a novel second-order optimizer, NysNewton-CG (NNCG), which significantly improves PINN performance. Theoretically, our work elucidates the connection between ill-conditioned differential operators and ill-conditioning in the PINN loss and shows the benefits of combining first- and second-order optimization methods. Our work presents valuable insights and more powerful optimization strategies for training PINNs, which could improve the utility of PINNs for solving difficult partial differential equations.

Woojin Cho, Minju Jo, Haksoo Lim, Kookjin Lee, Dongeun Lee, Sanghyun Hong, Noseong Park

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Raynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P$^2$INNs). P$^2$INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P$^2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known “failure modes”.

Siqi Miao, Zhiyuan Lu, Mia Liu, Javier Duarte, Pan Li

This study introduces a novel transformer model optimized for large-scale point cloud processing in scientific domains such as high-energy physics (HEP) and astrophysics. Addressing the limitations of graph neural networks and standard transformers, our model integrates local inductive bias and achieves near-linear complexity with hardware-friendly regular operations. One contribution of this work is the quantitative analysis of the error-complexity tradeoff of various sparsification techniques for building efficient transformers. Our findings highlight the superiority of using locality-sensitive hashing (LSH), especially OR & AND-construction LSH, in kernel approximation for large-scale point cloud data with local inductive bias. Based on this finding, we propose LSH-based Efficient Point Transformer (**HEPT**), which combines E$^2$LSH with OR & AND constructions and is built upon regular computations. HEPT demonstrates remarkable performance on two critical yet time-consuming HEP tasks, significantly outperforming existing GNNs and transformers in accuracy and computational speed, marking a significant advancement in geometric deep learning and large-scale scientific data processing. Our code is available at https://github.com/Graph-COM/HEPT.

Optimization

Motahareh Sohrabi, Juan Ramirez, Tianyue Zhang, Simon Lacoste-Julien, Jose Gallego-Posada

Constrained optimization offers a powerful framework to prescribe desired behaviors in neural network models. Typically, constrained problems are solved via their min-max Lagrangian formulations, which exhibit unstable oscillatory dynamics when optimized using gradient descent-ascent. The adoption of constrained optimization techniques in the machine learning community is currently limited by the lack of reliable, general-purpose update schemes for the Lagrange multipliers. This paper proposes the νPI algorithm and contributes an optimization perspective on Lagrange multiplier updates based on PI controllers, extending the work of Stooke, Achiam and Abbeel (2020). We provide theoretical and empirical insights explaining the inability of momentum methods to address the shortcomings of gradient descent-ascent, and contrast this with the empirical success of our proposed νPI controller. Moreover, we prove that νPI generalizes popular momentum methods for single-objective minimization. Our experiments demonstrate that νPI reliably stabilizes the multiplier dynamics and its hyperparameters enjoy robust and predictable behavior.

Applications · Physics

Zhuo Chen, Jacob McCarran, Esteban Vizcaino, Marin Soljačić, Di Luo

Partial differential equations (PDEs) are instrumental for modeling dynamical systems in science and engineering. The advent of neural networks has initiated a significant shift in tackling these complexities though challenges in accuracy persist, especially for initial value problems. In this paper, we introduce the *Time-Evolving Natural Gradient (TENG)*, generalizing time-dependent variational principles and optimization-based time integration, leveraging natural gradient optimization to obtain high accuracy in neural-network-based PDE solutions. Our comprehensive development includes algorithms like TENG-Euler and its high-order variants, such as TENG-Heun, tailored for enhanced precision and efficiency. TENG's effectiveness is further validated through its performance, surpassing current leading methods and achieving *machine precision* in step-by-step optimizations across a spectrum of PDEs, including the heat equation, Allen-Cahn equation, and Burgers' equation.

Yash Akhauri, Mohamed Abdelfattah

Predictor-based methods have substantially enhanced Neural Architecture Search (NAS) optimization. The efficacy of these predictors is largely influenced by the method of encoding neural network architectures. While traditional encodings used an adjacency matrix describing the graph structure of a neural network, novel encodings embrace a variety of approaches from unsupervised pretraining of latent representations to vectors of zero-cost proxies. In this paper, we categorize and investigate neural encodings from three main types: structural, learned, and score-based. Furthermore, we extend these encodings and introduce *unified encodings*, that extend NAS predictors to multiple search spaces. Our analysis draws from experiments conducted on over 1.5 million neural network architectures on NAS spaces such as NASBench-101 (NB101), NB201, NB301, Network Design Spaces (NDS), and TransNASBench-101. Building on our study, we present our predictor **FLAN**: **Fl**ow **A**ttention for **N**AS. FLAN integrates critical insights on predictor design, transfer learning, and *unified encodings* to enable more than an order of magnitude cost reduction for training NAS accuracy predictors. Our implementation and encodings for all neural networks are open-sourced at https://github.com/abdelfattah-lab/flan_nas.

Deep Learning · Robustness

Fengchun Qiao, Xi Peng

Ensemble of deep neural networks has achieved great success in hedging against single-model failure under distribution shift. However, existing techniques suffer from producing redundant models, limiting predictive diversity and yielding compromised generalization performance. Existing ensemble pruning methods can only guarantee predictive diversity for in-distribution data, which may not transfer well to out-of-distribution (OoD) data. To address this gap, we propose a principled optimization framework for ensemble pruning under distribution shifts. Since the annotations of test data are not available, we explore relationships between prediction distributions of the models, encapsulated in a topology graph. By incorporating this topology into a combinatorial optimization framework, complementary models with high predictive diversity are selected with theoretical guarantees. Our approach is model-agnostic and can be applied on top of a broad spectrum of off-the-shelf ensembling methods for improved generalization performance. Experiments on common benchmarks demonstrate the superiority of our approach in both multi- and single-source OoD generalization. The source codes are publicly available at: https://github.com/joffery/TEP.

Optimization · Optimization and Learning under Uncertainty

Yunsheng Tian, Ane Zuniga, Xinwei Zhang, Johannes P. Dürholt, Payel Das, Jie Chen, Wojciech Matusik, Mina Konakovic Lukovic

Bayesian optimization has been successfully applied to optimize black-box functions where the number of evaluations is severely limited. However, in many real-world applications, it is hard or impossible to know in advance which designs are feasible due to some physical or system limitations. These issues lead to an even more challenging problem of optimizing an unknown function with unknown constraints. In this paper, we observe that in such scenarios optimal solution typically lies on the boundary between feasible and infeasible regions of the design space, making it considerably more difficult than that with interior optima. Inspired by this observation, we propose BE-CBO, a new Bayesian optimization method that efficiently explores the boundary between feasible and infeasible designs. To identify the boundary, we learn the constraints with an ensemble of neural networks that outperform the standard Gaussian Processes for capturing complex boundaries. Our method demonstrates superior performance against state-of-the-art methods through comprehensive experiments on synthetic and real-world benchmarks. Code available at: https://github.com/yunshengtian/BE-CBO

Deep Learning

Jintong Gao, He Zhao, Dandan Guo, Hongyuan Zha

A well-trained deep neural network on balanced datasets usually exhibits the Neural Collapse (NC) phenomenon, which is an informative indicator of the model achieving good performance. However, NC is usually hard to be achieved for a model trained on long-tailed datasets, leading to the deteriorated performance of test data. This work aims to induce the NC phenomenon in imbalanced learning from the perspective of distribution matching. By enforcing the distribution of last-layer representations to align the ideal distribution of the ETF structure, we develop a Distribution Alignment Optimization (DisA) loss, acting as a plug-and-play method can be combined with most of the existing long-tailed methods, we further instantiate it to the cases of fixing classifier and learning classifier. The extensive experiments show the effectiveness of DisA, providing a promising solution to the imbalanced issue. Our code is available at DisA.

Deep Learning · Algorithms

Zhengyang Hu, Song Kang, Qunsong Zeng, Kaibin Huang, Yanchao Yang

Estimating mutual correlations between random variables or data streams is essential for intelligent behavior and decision-making. As a fundamental quantity for measuring statistical relationships, mutual information has been extensively studied and utilized for its generality and equitability. However, existing methods often lack the efficiency needed for real-time applications, such as test-time optimization of a neural network, or the differentiability required for end-to-end learning, like histograms. We introduce a neural network called InfoNet, which directly outputs mutual information estimations of data streams by leveraging the attention mechanism and the computational efficiency of deep learning infrastructures. By maximizing a dual formulation of mutual information through large-scale simulated training, our approach circumvents time-consuming test-time optimization and offers generalization ability. We evaluate the effectiveness and generalization of our proposed mutual information estimation scheme on various families of distributions and applications. Our results demonstrate that InfoNet and its training process provide a graceful efficiency-accuracy trade-off and order-preserving properties. We will make the code and models available as a comprehensive toolbox to facilitate studies in different fields requiring real-time mutual information estimation.

Social Aspects · Trustworthy Machine Learning

Dennis Frauen, Valentyn Melnychuk, Stefan Feuerriegel

Algorithmic decision-making in practice must be fair for legal, ethical, and societal reasons. To achieve this, prior research has contributed various approaches that ensure fairness in machine learning predictions, while comparatively little effort has focused on fairness in decision-making, specifically off-policy learning. In this paper, we propose a novel framework for fair off-policy learning: we learn decision rules from observational data under different notions of fairness, where we explicitly assume that observational data were collected under a different -- potentially discriminatory -- behavioral policy. Importantly, our framework applies to different fairness notions for off-policy learning, where fairness is formalized based on actions or policy values. As our main contribution, we propose a neural network-based framework to learn optimal policies under different fairness notions. We further provide theoretical guarantees in the form of generalization bounds for the finite-sample version of our framework. We demonstrate the effectiveness of our framework through extensive numerical experiments using both simulated and real-world data. Altogether, our work enables algorithmic decision-making in a wide array of practical applications where fairness must be ensured.

Deep Learning · Graph Neural Networks

Yuchen Zhang, Tianle Zhang, Kai Wang, Ziyao Guo, Yuxuan Liang, Xavier Bresson, Wei Jin, Yang You

Graph condensation aims to reduce the size of a large-scale graph dataset by synthesizing a compact counterpart without sacrificing the performance of Graph Neural Networks (GNNs) trained on it, which has shed light on reducing the computational cost for training GNNs. Nevertheless, existing methods often fall short of accurately replicating the original graph for certain datasets, thereby failing to achieve the objective of lossless condensation. To understand this phenomenon, we investigate the potential reasons and reveal that the previous state-of-the-art trajectory matching method provides biased and restricted supervision signals from the original graph when optimizing the condensed one. This significantly limits both the scale and efficacy of the condensed graph. In this paper, we make the first attempt toward *lossless graph condensation* by bridging the previously neglected supervision signals. Specifically, we employ a curriculum learning strategy to train expert trajectories with more diverse supervision signals from the original graph, and then effectively transfer the information into the condensed graph with expanding window matching. Moreover, we design a loss function to further extract knowledge from the expert trajectories. Theoretical analysis justifies the design of our method and extensive experiments verify its superiority across different datasets. Code is released at https://github.com/NUS-HPC-AI-Lab/GEOM.

Optimization · Optimization and Learning under Uncertainty

Yucen Li, Tim G. J. Rudner, Andrew Gordon Wilson

Bayesian optimization is a highly efficient approach to optimizing objective functions which are expensive to query. These objectives are typically represented by Gaussian process (GP) surrogate models which are easy to optimize and support exact inference. While standard GP surrogates have been well-established in Bayesian optimization, Bayesian neural networks (BNNs) have recently become practical function approximators, with many benefits over standard GPs such as the ability to naturally handle non-stationarity and learn representations for high-dimensional data. In this paper, we study BNNs as alternatives to standard GP surrogates for optimization. We consider a variety of approximate inference procedures for finite-width BNNs, including high-quality Hamiltonian Monte Carlo, low-cost stochastic MCMC, and heuristics such as deep ensembles. We also consider infinite-width BNNs, linearized Laplace approximations, and partially stochastic models such as deep kernel learning. We evaluate this collection of surrogate models on diverse problems with varying dimensionality, number of objectives, non-stationarity, and discrete and continuous inputs. We find: (i) the ranking of methods is highly problem dependent, suggesting the need for tailored inductive biases; (ii) HMC is the most successful approximate inference procedure for fully stochastic BNNs; (iii) full stochasticity may be unnecessary as deep kernel learning is relatively competitive; (iv) deep ensembles perform relatively poorly; (v) infinite-width BNNs are particularly promising, especially in high dimensions.

Deep Learning · Other Representation Learning

Yuan Gao, WEIZHONG ZHANG, Wenhan Luo, Lin Ma, Jin-Gang Yu, Gui-Song Xia, Jiayi Ma

We aim at exploiting additional auxiliary labels from an independent (auxiliary) task to boost the primary task performance which we focus on, while preserving a single task inference cost of the primary task. While most existing auxiliary learning methods are optimization-based relying on loss weights/gradients manipulation, our method is architecture-based with a flexible asymmetric structure for the primary and auxiliary tasks, which produces different networks for training and inference. Specifically, starting from two single task networks/branches (each representing a task), we propose a novel method with evolving networks where only primary-to-auxiliary links exist as the cross-task connections after convergence. These connections can be removed during the primary task inference, resulting in a single-task inference cost. We achieve this by formulating a Neural Architecture Search (NAS) problem, where we initialize bi-directional connections in the search space and guide the NAS optimization converging to an architecture with only the single-side primary-to-auxiliary connections. Moreover, our method can be incorporated with optimization-based auxiliary learning approaches. Extensive experiments with six tasks on NYU v2, CityScapes, and Taskonomy datasets using VGG, ResNet, and ViT backbones validate the promising performance. The codes are available at https://github.com/ethanygao/Aux-NAS.

Deep Learning

Renbo Tu, Colin White, Jean Kossaifi, Boris Bonev, Gennady Pekhimenko, Kamyar Azizzadenesheli, anima anandkumar

Neural operators, such as Fourier Neural Operators (FNO), form a principled approach for learning solution operators for partial differential equations (PDE) and other mappings between function spaces. However, many real-world problems require high-resolution training data, and the training time and limited GPU memory pose big barriers. One solution is to train neural operators in mixed precision to reduce the memory requirement and increase training speed. However, existing mixed-precision training techniques are designed for standard neural networks, and we find that their direct application to FNO leads to numerical overflow and poor memory efficiency. Further, at first glance, it may appear that mixed precision in FNO will lead to drastic accuracy degradation since reducing the precision of the Fourier transform yields poor results in classical numerical solvers. We show that this is not the case; in fact, we prove that reducing the precision in FNO still guarantees a good approximation bound, when done in a targeted manner. Specifically, we build on the intuition that neural operator learning inherently induces an approximation error, arising from discretizing the infinite-dimensional ground-truth input function, implying that training in full precision is not needed. We formalize this intuition by rigorously characterizing the approximation and precision errors of FNO and bounding these errors for general input functions. We prove that the precision error is asymptotically comparable to the approximation error. Based on this, we design a simple method to optimize the memory-intensive half-precision tensor contractions by greedily finding the optimal contraction order. Through extensive experiments on different state-of-the-art neural operators, datasets, and GPUs, we demonstrate that our approach reduces GPU memory usage by up to 50% and improves throughput by 58% with little or no reduction in accuracy.

Deep Learning · Graph Neural Networks

Mert Kosan, Samidha Verma, Burouj Armgaan, Khushbu Pahwa, Ambuj K Singh, Sourav Medya, Sayan Ranu

Numerous explainability methods have been proposed to shed light on the inner workings of GNNs. Despite the inclusion of empirical evaluations in all the proposed algorithms, the interrogative aspects of these evaluations lack diversity. As a result, various facets of explainability pertaining to GNNs, such as a comparative analysis of counterfactual reasoners, their stability to variational factors such as different GNN architectures, noise, stochasticity in non-convex loss surfaces, feasibility amidst domain constraints, and so forth, have yet to be formally investigated. Motivated by this need, we present a benchmarking study on perturbation-based explainability methods for GNNs, aiming to systematically evaluate and compare a wide range of explainability techniques. Among the key findings of our study, we identify the Pareto-optimal methods that exhibit superior efficacy and stability in the presence of noise. Nonetheless, our study reveals thatall algorithms are affected by stability issues when faced with noisy data. Furthermore, we have established that the current generation of counterfactual explainers often fails to provide feasible recourses due to violations of topological constraints encoded by domain-specific considerations. Overall, this benchmarking study empowers stakeholders in the field of GNNs with a comprehensive understanding of the state-of-the-art explainability methods, potential research problems for further enhancement, and the implications of their application in real-world scenarios.

Zahra Kadkhodaie, Florentin Guth, Eero Simoncelli, Stéphane Mallat

Deep neural networks (DNNs) trained for image denoising are able to generate high-quality samples with score-based reverse diffusion algorithms. These impressive capabilities seem to imply an escape from the curse of dimensionality, but recent reports of memorization of the training set raise the question of whether these networks are learning the "true" continuous density of the data. Here, we show that two DNNs trained on non-overlapping subsets of a dataset learn nearly the same score function, and thus the same density, when the number of training images is large enough. In this regime of strong generalization, diffusion-generated images are distinct from the training set, and are of high visual quality, suggesting that the inductive biases of the DNNs are well-aligned with the data density. We analyze the learned denoising functions and show that the inductive biases give rise to a shrinkage operation in a basis adapted to the underlying image. Examination of these bases reveals oscillating harmonic structures along contours and in homogeneous regions. We demonstrate that trained denoisers are inductively biased towards these geometry-adaptive harmonic bases since they arise not only when the network is trained on photographic images, but also when it is trained on image classes supported on low-dimensional manifolds for which the harmonic basis is suboptimal. Finally, we show that when trained on regular image classes for which the optimal basis is known to be geometry-adaptive and harmonic, the denoising performance of the networks is near-optimal.

Zaishuo Xia, Han Yang, Binghui Wang, Jinyuan Jia

Graph classification, which aims to predict a label for a graph, has many real-world applications such as malware detection, fraud detection, and healthcare. However, many studies show an attacker could carefully perturb the structure and/or node features in a graph such that a graph classifier misclassifies the perturbed graph. Such vulnerability impedes the deployment of graph classification in security/safety-critical applications. Existing empirical defenses lack formal robustness guarantees and could be broken by adaptive or unknown attacks. Existing provable defenses have the following limitations: 1) they achieve sub-optimal robustness guarantees for graph structure perturbation, 2) they cannot provide robustness guarantees for arbitrarily node feature perturbations, 3) their robustness guarantees are probabilistic, meaning they could be incorrect with a non-zero probability, and 4) they incur large computation costs. We aim to address those limitations in this work. We propose GNNCert, a certified defense against both graph structure and node feature perturbations for graph classification. Our GNNCert provably predicts the same label for a graph when the number of perturbed edges and the number of nodes with perturbed features are bounded. Our results on 8 benchmark datasets show that GNNCert outperforms three state-of-the-art methods.