Contrastive Representation Learning (CRL) has achieved strong empirical success in multiple machine learning disciplines, yet its theoretical sample complexity remains poorly understood. Existing analyses usually assume that input tuples are identically and independently distributed, an assumption violated in most practical settings where contrastive tuples are constructed from a finite pool of labeled data, inducing dependencies among tuples. While one recent work analyzed this learning setting using U-Statistics to estimate the population risk, the techniques used therein require the risk of each class to concentrate uniformly, making excess risk bounds scale in the order of $ρ_{\min}^{-{1}/{2}}$ where $ρ_{\min}$ denotes the probability of the rarest class. Such a dependency can be overly pessimistic in the extreme multiclass settings where there are many tail classes which contribute minimally to the overall population risk. Our contributions are two-fold. Firstly, we improve upon the previous work and prove a bound with a sample complexity of the same order as the number of classes $R$, regardless of the distribution over classes. Furthermore, we formulate a different estimator that captures the concentration of the risk \textit{across classes}, enabling sharper bounds in extreme multi-class learning scenarios, especially where class distributions are long-tailed. Under mild assumptions on the class distributions, the resulting sample complexity is $\mathcal{O}(k)$ where $k$ is the number of samples per tuple.

UMAP has supplanted t-SNE as state-of-the-art for visualizing high-dimensional datasets in many disciplines, but the reason for its success is not well understood. In this work, we investigate UMAP's sampling based optimization scheme in detail. We derive UMAP's true loss function in closed form and find that it differs from the published one in a dataset size dependent way. As a consequence, we show that UMAP does not aim to reproduce its theoretically motivated high-dimensional UMAP similarities. Instead, it tries to reproduce similarities that only encode the knearest neighbor graph, thereby challenging the previous understanding of UMAP's effectiveness. Alternatively, we consider the implicit balancing of attraction and repulsion due to the negative sampling to be key to UMAP's success. We corroborate our theoretical findings on toy and single cell RNA sequencing data.

Network embedding plays a significant role in a variety of applications. To capture the topology of the network, most of the existing network embedding algorithms follow a sampling training procedure, which maximizes the similarity (e.g., embedding vectors' dot product) between positively sampled node pairs and minimizes the similarity between negatively sampled node pairs in the embedding space. Typically, close node pairs function as positive samples while distant node pairs are usually considered as negative samples. However, under different or even competing sampling strategies, some methods champion sampling distant node pairs as positive samples to encapsulate longer distance information in link prediction, whereas others advocate adding close nodes into the negative sample set to boost the performance of node recommendation. In this paper, we seek to understand the intrinsic relationships between these competing strategies. To this end, we identify two properties (discrimination and monotonicity) that given any node pair proximity distribution, node embeddings should embrace. Moreover, we quantify the empirical error of the trained similarity score w.r.t. the sampling strategy, which leads to an important finding that the discrimination property and the monotonicity property for all node pairs can not be satisfied simultaneously in real-world applications. Guided by such analysis, a simple yet novel model (SENSEI) is proposed, which seamlessly fulfills the discrimination property and the partial monotonicity within the top-K ranking list. Extensive experiments show that SENSEI outperforms the state-of-the-arts in plain network embedding.

We remind important related works to understand how our AdvInfoNCE stands and its role in rich literature. Our work is related to the literature on contrastive learning-based collaborative filtering (CL-based CF) methods, and theoretical understanding of contrastive loss in collaborative filtering. A.1 Contrastive Learning-based Collaborative Filtering The latest CL-based CF methods can roughly fall into two research lines. The second category, referred to as "loss-based" approaches, mainly focuses on the modification of contrastive loss. In loss-based CF models, interacted items serve as positive instances. The prevailing augmentation-based paradigm in CL-based CF methods is to employ user-item bipartite graph augmentations to generate contrasting views. These contrasting views are then treated as positive instances in the application of contrastive loss, such as InfoNCE loss, to further enhance collaborative filtering signals.

We adopt a residual network (ResNet) [23] based feature extractor, with ELU as the activation function. Following [15], we adopt group normalization and instance normalization for better stability of the networks. We adopt the "leave-one-out" training strategy for obtaining the results on each of the categories of MVTec-AD. All experiments are performed with the same settings and hyperparameters. We resize all images to 128 128, and do not perform any data augmentation.

Visual question answering (VQA) is designed to examine the visual-textual reasoning ability of an intelligent agent. However, recent observations show that many VQA models may only capture the biases between questions and answers in a dataset rather than showing real reasoning abilities. For example, given a question, some VQA models tend to output the answer that occurs frequently in the dataset and ignore the images. To reduce this tendency, existing methods focus on weakening the language bias. Meanwhile, only a few works also consider vision bias implicitly.

This section provides parameter estimation equations in the MM procedure Eq. (13) for the baseline intensity µand the decay parameter β, which were omitted in the main text due to space limitations. Below, we provide results for the exponential and power distributions. This section describes the details of the experiments. We have included the Sparse5and Dense10 data sets and the Python code to generate those as part of the final submission. B.1 Data generation Sparse5 The Sparse5 benchmark dataset is designed to have a simplest but nontrivial kind of causal structure, which is supposed to be easily reproduced by any Granger-causal learning algorithms.

In Figure 1 we demonstrate the common neighbor (CN) distribution among positive and negative test samples for ogbl-collab, ogbl-ppa, and ogbl-citation2. These results demonstrate that a vast majority of negative samples have no CNs. Since CNs is a typically good heuristic, this makes it easy to identify most negative samples. We further present the CN distribution of Cora, Citeseer, Pubmed, and ogbl-ddi in Figure 3. The CN distribution of Cora, Citeseer, and Pubmed are consistent with our previous observations on the OGB datasets in Figure 1.

Link prediction attempts to predict whether an unseen edge exists based on only a portion of edges of a graph. A flurry of methods have been introduced in recent years that attempt to make use of graph neural networks (GNNs) for this task. Furthermore, new and diverse datasets have also been created to better evaluate the effectiveness of these new models. However, multiple pitfalls currently exist that hinder our ability to properly evaluate these new methods. These pitfalls mainly include: (1) Lower than actual performance on multiple baselines, (2) A lack of a unified data split and evaluation metric on some datasets, and (3) An unrealistic evaluation setting that uses easy negative samples. To overcome these challenges, we first conduct a fair comparison across prominent methods and datasets, utilizing the same dataset and hyperparameter search settings. We then create a more practical evaluation setting based on a Heuristic Related Sampling Technique (HeaRT), which samples hard negative samples via multiple heuristics. The new evaluation setting helps promote new challenges and opportunities in link prediction by aligning the evaluation with real-world situations.