Shunan Sheng

Shunan Sheng, Bohan Wu, Bennett Zhu et al.

Structured variational inference constitutes a core methodology in modern statistical applications. Unlike mean-field variational inference, the approximate posterior is assumed to have interdependent structure. We consider the natural setting of star-structured variational inference, where a root variable impacts all the other ones. We prove the first results for existence, uniqueness, and self-consistency of the variational approximation. In turn, we derive quantitative approximation error bounds for the variational approximation to the posterior, extending prior work from the mean-field setting to the star-structured setting. We also develop a gradient-based algorithm with provable guarantees for computing the variational approximation using ideas from optimal transport theory. We explore the implications of our results for Gaussian measures and hierarchical Bayesian models, including generalized linear models with location family priors and spike-and-slab priors with one-dimensional debiasing. As a by-product of our analysis, we develop new stability results for star-separable transport maps which might be of independent interest.

Shunan Sheng, Bohan Wu, Alberto González-Sanz

Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. It has been empirically observed that MFVI optimizers often suffer from mode collapse. Specifically, when the target measure $π$ is a mixture $π= w P_0 + (1 - w) P_1$, the MFVI optimizer tends to place most of its mass near a single component of the mixture. This work provides the first theoretical explanation of mode collapse in MFVI. We introduce the notion to capture the separatedness of the two mixture components -- called $\varepsilon$-separateness -- and derive explicit bounds on the fraction of mass that any MFVI optimizer assigns to each component when $P_0$ and $P_1$ are $\varepsilon$-separated for sufficiently small $\varepsilon$. Our results suggest that the occurrence of mode collapse crucially depends on the relative position of the components. To address this issue, we propose the rotational variational inference (RoVI), which augments MFVI with a rotation matrix. The numerical studies support our theoretical findings and demonstrate the benefits of RoVI.

Jiawei Ren, Cunjun Yu, Shunan Sheng et al.

Deep classifiers have achieved great success in visual recognition. However, real-world data is long-tailed by nature, leading to the mismatch between training and testing distributions. In this paper, we show that the Softmax function, though used in most classification tasks, gives a biased gradient estimation under the long-tailed setup. This paper presents Balanced Softmax, an elegant unbiased extension of Softmax, to accommodate the label distribution shift between training and testing. Theoretically, we derive the generalization bound for multiclass Softmax regression and show our loss minimizes the bound. In addition, we introduce Balanced Meta-Softmax, applying a complementary Meta Sampler to estimate the optimal class sample rate and further improve long-tailed learning. In our experiments, we demonstrate that Balanced Meta-Softmax outperforms state-of-the-art long-tailed classification solutions on both visual recognition and instance segmentation tasks.