### A Critical Look at the Evaluation of GNNs under Heterophily: Are We Really Making Progress?

ICLR,

2023Node classification is a classical graph representation learning task on which Graph Neural Networks (GNNs) have recently achieved strong results. However, it is often believed that standard GNNs only work well for homophilous graphs, i.e., graphs where edges tend to connect nodes of the same class. Graphs without this property are called heterophilous, and it is typically assumed that specialized methods are required to achieve strong performance on such graphs. In this work, we challenge this assumption. First, we show that the standard datasets used for evaluating heterophily-specific models have serious drawbacks, making results obtained by using them unreliable. The most significant of these drawbacks is the presence of a large number of duplicate nodes in the datasets Squirrel and Chameleon, which leads to train-test data leakage. We show that removing duplicate nodes strongly affects GNN performance on these datasets. Then, we propose a set of heterophilous graphs of varying properties that we believe can serve as a better benchmark for evaluating the performance of GNNs under heterophily. We show that standard GNNs achieve strong results on these heterophilous graphs, almost always outperforming specialized models. Our datasets and the code for reproducing our experiments are available at https://github.com/yandex-research/heterophilous-graphs

### A View of Mini-batch SGD via Generating Functions: Conditions of Convergence, Phase Transitions, Benefit from Negative Momenta

ICLR,

2023Mini-batch SGD with momentum is a fundamental algorithm for learning large predictive models. In this paper we develop a new analytic framework to analyze noise-averaged properties of mini-batch SGD for linear models at constant learning rates, momenta and sizes of batches. Our key idea is to consider the dynamics of the second moments of model parameters for a special family of "Spectrally Expressible" approximations. This allows to obtain an explicit expression for the generating function of the sequence of loss values. By analyzing this generating function, we find, in particular, that 1) the SGD dynamics exhibits several convergent and divergent regimes depending on the spectral distributions of the problem; 2) the convergent regimes admit explicit stability conditions, and explicit loss asymptotics in the case of power-law spectral distributions; 3) the optimal convergence rate can be achieved at negative momenta. We verify our theoretical predictions by extensive experiments with MNIST and synthetic problems, and find a good quantitative agreement.

### Gradient Boosting Performs Gaussian Process Inference

ICLR,

2023This paper shows that gradient boosting based on symmetric decision trees can be equivalently reformulated as a kernel method that converges to the solution of a certain Kernel Ridge Regression problem. Thus, we obtain the convergence to a Gaussian Process' posterior mean, which, in turn, allows us to easily transform gradient boosting into a sampler from the posterior to provide better knowledge uncertainty estimates through Monte-Carlo estimation of the posterior variance. We show that the proposed sampler allows for better knowledge uncertainty estimates leading to improved out-of-domain detection.

### Understanding DDPM Latent Codes Through Optimal Transport

ICLR,

2023Diffusion models have recently outperformed alternative approaches to model the distribution of natural images. Such diffusion models allow for deterministic sampling via the probability flow ODE, giving rise to a latent space and an encoder map. While having important practical applications, such as the estimation of the likelihood, the theoretical properties of this map are not yet fully understood. In the present work, we partially address this question for the popular case of the VP-SDE (DDPM) approach. We show that, perhaps surprisingly, the DDPM encoder map coincides with the optimal transport map for common distributions; we support this claim by extensive numerical experiments using advanced tensor train solver for multidimensional Fokker-Planck equation. We provide additional theoretical evidence for the case of multivariate normal distributions.

### Decentralized Local Stochastic Extra-Gradient for Variational Inequalities

NeurIPS,

2022We consider distributed stochastic variational inequalities (VIs) on unbounded domains with the problem data that is heterogeneous (non-IID) and distributed across many devices. We make a very general assumption on the computational network that, in particular, covers the settings of fully decentralized calculations with time-varying networks and centralized topologies commonly used in Federated Learning. Moreover, multiple local updates on the workers can be made for reducing the communication frequency between the workers. We extend the stochastic extragradient method to this very general setting and theoretically analyze its convergence rate in the strongly-monotone, monotone, and non-monotone (when a Minty solution exists) settings. The provided rates explicitly exhibit the dependence on network characteristics (e.g., mixing time), iteration counter, data heterogeneity, variance, number of devices, and other standard parameters. As a special case, our method and analysis apply to distributed stochastic saddle-point problems (SPP), e.g., to the training of Deep Generative Adversarial Networks (GANs) for which decentralized training has been reported to be extremely challenging. In experiments for the decentralized training of GANs we demonstrate the effectiveness of our proposed approach.

# Publications

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