Learning unnormalized statistical models (e.g., energy-based models) is computationally challenging due to the complexity of handling the partition function. To eschew this complexity, noise-contrastive estimation (NCE) has been proposed by formulating the objective as the logistic loss between the real data and the artificial noise. However, previous research indicates that NCE may perform poorly in many tasks due to its flat loss landscape and slow convergence. In this paper, we study a direct approach for optimizing the negative log-likelihood of unnormalized models through the lens of compositional optimization. To tackle the partition function, a noise distribution is introduced such that the log partition function can be expressed as a compositional function whose inner function can be estimated using stochastic samples. Consequently, the objective can be optimized via stochastic compositional optimization algorithms. Despite being a simple method, we demonstrate it is more favorable than NCE by (1) establishing a fast convergence rate and quantifying its dependence on the noise distribution through the variance of stochastic estimators; (2) developing better results in Gaussian mean estimation by showing our method has a much favorable loss landscape and enjoys faster convergence; (3) demonstrating better performance on various applications, including density estimation, out-of-distribution detection, and real image generation.

Contrastive Learning (CL) plays a crucial role in molecular representation learning, enabling unsupervised learning from large scale unlabeled molecule datasets. It has inspired various applications in molecular property prediction and drug design. However, existing molecular representation learning methods often introduce potential false positive and false negative pairs through conventional graph augmentations like node masking and subgraph removal. The issue can lead to suboptimal performance when applying standard contrastive learning techniques to molecular datasets. To address the issue of false positive and negative pairs in molecular representation learning, we propose a novel probability-based contrastive learning (CL) framework. Unlike conventional methods, our approach introduces a learnable weight distribution via Bayesian modeling to automatically identify and mitigate false positive and negative pairs. This method is particularly effective because it dynamically adjusts to the data, improving the accuracy of the learned representations. Our model is learned by a stochastic expectation-maximization process, which optimizes the model by iteratively refining the probability estimates of sample weights and updating the model parameters. Experimental results indicate that our method outperforms existing approaches in 13 out of 15 molecular property prediction benchmarks in MoleculeNet dataset and 8 out of 12 benchmarks in the QM9 benchmark, achieving new state-of-the-art results on average.

Learning unnormalized statistical models (e.g., energy-based models) is computationally challenging due to the complexity of handling the partition function. To eschew this complexity, noise-contrastive estimation (NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. However, as found in previous works, NCE may perform poorly in many tasks due to its flat loss landscape and slow convergence. In this paper, we study a direct approach for optimizing the negative log-likelihood of unnormalized models from the perspective of compositional optimization. To tackle the partition function, a noise distribution is introduced such that the log partition function can be written as a compositional function whose inner function can be estimated with stochastic samples. Hence, the objective can be optimized by stochastic compositional optimization algorithms. Despite being a simple method, we demonstrate that it is more favorable than NCE by (1) establishing a fast convergence rate and quantifying its dependence on the noise distribution through the variance of stochastic estimators; (2) developing better results for one-dimensional Gaussian mean estimation by showing our objective has a much favorable loss landscape and hence our method enjoys faster convergence; (3) demonstrating better performance on multiple applications, including density estimation, out-of-distribution detection, and real image generation.