Sampling from an unnormalized target distribution is an essential problem with many applications in probabilistic inference. Stein Variational Gradient Descent (SVGD) has been shown to be a powerful method that iteratively updates a set of particles to approximate the distribution of interest. Furthermore, when analysing its asymptotic properties, SVGD reduces exactly to a single-objective optimization problem and can be viewed as a probabilistic version of this single-objective optimization problem. A natural question then arises: "Can we derive a probabilistic version of the multi-objective optimization?". To answer this question, we propose Stochastic Multiple Target Sampling Gradient Descent (MT-SGD), enabling us to sample from multiple unnormalized target distributions. Specifically, our MT-SGD conducts a flow of intermediate distributions gradually orienting to multiple target distributions, which allows the sampled particles to move to the joint high-likelihood region of the target distributions. Interestingly, the asymptotic analysis shows that our approach reduces exactly to the multiple-gradient descent algorithm for multi-objective optimization, as expected. Finally, we conduct comprehensive experiments to demonstrate the merit of our approach to multi-task learning.

翻译：从未归一化的目标分布中采样是概率推理中具有诸多应用的基础问题。Stein变分梯度下降（SVGD）已被证明是一种强大的方法，它通过迭代更新一组粒子来逼近感兴趣的分布。此外，在分析其渐近性质时，SVGD精确地简化为单目标优化问题，并可被视为该单目标优化问题的概率版本。由此自然产生一个问题：“我们能否推导出多目标优化的概率版本？”为回答此问题，我们提出了随机多目标采样梯度下降（MT-SGD），使其能够从多个未归一化的目标分布中采样。具体而言，我们的MT-SGD引导一个中间分布流逐步趋向多个目标分布，从而使采样粒子移动至目标分布的联合高似然区域。有趣的是，渐近分析表明，正如预期，我们的方法精确地简化为多目标优化的多梯度下降算法。最后，我们进行了全面的实验，以证明我们的方法在多任务学习中的优势。