Optimizing Unnormalized Statistical Models Through Compositional Optimization

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.