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Noise Contrastive Estimation (NCE) is a widely used method for training generative models, typically used as an alternative to Maximum Likelihood Estimation (MLE) when exact computations of probability are hard. NCE trains generative models by discriminating between data and appropriately chosen noise distributions. Although NCE is statistically consistent, it suffers from slow convergence and high variance when there is small overlap between the noise and data distributions. Both these problems are related to the flatness of the NCE loss landscape. We propose an innovative approach to circumvent slow convergence rates by quick inference of the optimal normalizing constant at every gradient step. This allows the rest of the parameters to have more freedom during NCE optimization. We analyze the use of both binary search and the Bennett Acceptance Ratio (BAR) for quick computation of the normalizing constant and show improved performance for both methods on convex and non-convex settings. 

Noise Contrastive Estimation (NCE) is a popular approach for learning probability density functions parameterized up to a constant of proportionality. The main idea is to design a classification problem for distinguishing training data from samples from an easy-to-sample noise distribution q, in a manner that avoids having to calculate a partition function. It is well-known that the choice of q can severely impact the computational and statistical efficiency of NCE. In practice, a common choice for q is a Gaussian which matches the mean and covariance of the data. In this paper, we show that such a choice can result in an exponentially bad (in the ambient dimension) conditioning of the Hessian of the loss, even for very simple data distributions. As a consequence, both the statistical and algorithmic complexity for such a choice of q will be problematic in practice, suggesting that more complex and tailored noise distributions are essential to the success of NCE.