Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity
Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity
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DOI：
10.3150/21-bej1343
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发表时间：
2019-07
期刊：
影响因子：
1.5
通讯作者：
Wenlong Mou;Nicolas Flammarion;M. Wainwright;P. Bartlett
中科院分区：
文献类型：
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作者：
Wenlong Mou;Nicolas Flammarion;M. Wainwright;P. Bartlett
我们提出了对Langevin扩散的Euler-Maruyama离散化的改进分析。我们的分析不需要全球合同，并且会产生对时间范围的多项式依赖。与现有方法相比，我们做出了额外的平滑度假设，并将现有利率从$ O（\ eta）$（\ eta^2）$提高到kl Divergence。该结果与数值SDE的正确顺序相匹配，而没有指数时间依赖性。当应用于采样和学习算法时，此结果同时根据Dalayan的方法改进了所有这些方法。
We present an improved analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an additional smoothness assumption, and improve the existing rate from $O(\eta)$ to $O(\eta^2)$ in terms of the KL divergence. This result matches the correct order for numerical SDEs, without suffering from exponential time dependence. When applied to algorithms for sampling and learning, this result simultaneously improves all those methods based on Dalayan's approach.