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Fit Like You Sample: Sample-Efficient Generalized Score Matching from Fast Mixing Markov Chains
Jun 20, 2023
Score matching is an approach to learning probability distributions parametrized up to a constant of proportionality (e.g. Energy-Based Models). The idea is to fit the score of the distribution, rather than the likelihood, thus avoiding the need to evaluate the constant of proportionality. While there's a clear algorithmic benefit, the statistical "cost'' can be steep: recent work by Koehler et al. 2022 showed that for distributions that have poor isoperimetric properties (a large Poincar\'e or log-Sobolev constant), score matching is substantially statistically less efficient than maximum likelihood. However, many natural realistic distributions, e.g. multimodal distributions as simple as a mixture of two Gaussians in one dimension -- have a poor Poincar\'e constant. In this paper, we show a close connection between the mixing time of an arbitrary Markov process with generator $\mathcal{L}$ and an appropriately chosen generalized score matching loss that tries to fit $\frac{\mathcal{O} p}{p}$. If $\mathcal{L}$ corresponds to a Markov process corresponding to a continuous version of simulated tempering, we show the corresponding generalized score matching loss is a Gaussian-convolution annealed score matching loss, akin to the one proposed in Song and Ermon 2019. Moreover, we show that if the distribution being learned is a finite mixture of Gaussians in $d$ dimensions with a shared covariance, the sample complexity of annealed score matching is polynomial in the ambient dimension, the diameter the means, and the smallest and largest eigenvalues of the covariance -- obviating the Poincar\'e constant-based lower bounds of the basic score matching loss shown in Koehler et al. 2022. This is the first result characterizing the benefits of annealing for score matching -- a crucial component in more sophisticated score-based approaches like Song and Ermon 2019.
Characterizing Out-of-Distribution Error via Optimal Transport
May 25, 2023
Out-of-distribution (OOD) data poses serious challenges in deployed machine learning models, so methods of predicting a model's performance on OOD data without labels are important for machine learning safety. While a number of methods have been proposed by prior work, they often underestimate the actual error, sometimes by a large margin, which greatly impacts their applicability to real tasks. In this work, we identify pseudo-label shift, or the difference between the predicted and true OOD label distributions, as a key indicator to this underestimation. Based on this observation, we introduce a novel method for estimating model performance by leveraging optimal transport theory, Confidence Optimal Transport (COT), and show that it provably provides more robust error estimates in the presence of pseudo-label shift. Additionally, we introduce an empirically-motivated variant of COT, Confidence Optimal Transport with Thresholding (COTT), which applies thresholding to the individual transport costs and further improves the accuracy of COT's error estimates. We evaluate COT and COTT on a variety of standard benchmarks that induce various types of distribution shift -- synthetic, novel subpopulation, and natural -- and show that our approaches significantly outperform existing state-of-the-art methods with an up to 3x lower prediction error.
Powering Comparative Classification with Sentiment Analysis via Domain Adaptive Knowledge Transfer
Sep 07, 2021
We study Comparative Preference Classification (CPC) which aims at predicting whether a preference comparison exists between two entities in a given sentence and, if so, which entity is preferred over the other. High-quality CPC models can significantly benefit applications such as comparative question answering and review-based recommendations. Among the existing approaches, non-deep learning methods suffer from inferior performances. The state-of-the-art graph neural network-based ED-GAT (Ma et al., 2020) only considers syntactic information while ignoring the critical semantic relations and the sentiments to the compared entities. We proposed sentiment Analysis Enhanced COmparative Network (SAECON) which improves CPC ac-curacy with a sentiment analyzer that learns sentiments to individual entities via domain adaptive knowledge transfer. Experiments on the CompSent-19 (Panchenko et al., 2019) dataset present a significant improvement on the F1 scores over the best existing CPC approaches.