离散松弛连续变量的易处理变分推断
Discretely Relaxing Continuous Variables for tractable Variational Inference
摘要：
我们探索了一个新的研究方向，在贝叶斯变分推断与离散潜变量先验，我们利用克罗内克矩阵代数的证据下界（ELBO）的有效和准确的计算。所提出的“DIRECT”方法与其前辈相比具有以下几个优点：（i）它可以精确地计算ELBO梯度（即无偏、零方差梯度估计），消除了对高方差随机梯度估计器的需要，并使得能够使用拟牛顿优化方法;（ii）其训练复杂性与训练点的数量无关，允许在大数据集上进行推断;以及（iii）其后验样本由稀疏和低精度量化整数组成，这允许在硬件有限的设备上进行快速推断。此外，我们的DIRECT模型可以精确地计算参数化预测后验的统计矩，而不依赖于蒙特卡罗采样。DIRECT方法并不适用于所有的可能性，但是，我们确定了一个流行的模型结构，这是实用的，并证明了准确的推理，使用隐变量离散化为极低精度的4位量化整数。虽然数值研究中考虑的ELBO计算需要超过$10^{2352}$的对数似然评估，但我们在几秒钟内就在超过200万个点的数据集上进行了训练。
英文摘要：
We explore a new research direction in Bayesian variational inference with discrete latent variable priors where we exploit Kronecker matrix algebra for efficient and exact computations of the evidence lower bound (ELBO). The proposed "DIRECT" approach has several advantages over its predecessors; (i) it can exactly compute ELBO gradients (i.e. unbiased, zero-variance gradient estimates), eliminating the need for high-variance stochastic gradient estimators and enabling the use of quasi-Newton optimization methods; (ii) its training complexity is independent of the number of training points, permitting inference on large datasets; and (iii) its posterior samples consist of sparse and low-precision quantized integers which permit fast inference on hardware limited devices. In addition, our DIRECT models can exactly compute statistical moments of the parameterized predictive posterior without relying on Monte Carlo sampling. The DIRECT approach is not practical for all likelihoods, however, we identify a popular model structure which is practical, and demonstrate accurate inference using latent variables discretized as extremely low-precision 4-bit quantized integers. While the ELBO computations considered in the numerical studies require over $10^{2352}$ log-likelihood evaluations, we train on datasets with over two-million points in just seconds.
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