Summary of Convergence Of Unadjusted Langevin in High Dimensions: Delocalization Of Bias, by Yifan Chen et al.
Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias
by Yifan Chen, Xiaoou Cheng, Jonathan Niles-Weed, Jonathan Weare
First submitted to arxiv on: 20 Aug 2024
Categories
- Main: Machine Learning (stat.ML)
- Secondary: Machine Learning (cs.LG); Probability (math.PR); Computation (stat.CO)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The paper presents an analysis of the unadjusted Langevin algorithm for sampling probability distributions in high-dimensional settings. The algorithm is commonly used but existing analyses suggest that it requires a large number of iterations to converge, scaling with dimension d or √d. However, the authors show that when focusing on a small number of variables (K-marginals), the algorithm can converge quickly, often requiring only a number of iterations proportional to K, up to logarithmic terms in d. This effect is referred to as delocalization of bias and is shown to hold for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. The authors use a novel W2,ℓ∞ metric to measure convergence and address the lack of a one-step contraction property in this metric. They also explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary The paper looks at how an algorithm for sampling probability distributions works in high-dimensional settings. The algorithm is important but needs many iterations to give good results, especially as the number of variables gets very large. But what if we only care about a small number of these variables? The authors show that the algorithm can be much faster then, requiring fewer iterations. This is called delocalization of bias and it happens for certain types of distributions, like Gaussian ones. The authors use a new way to measure how well the algorithm works and figure out why this delocalization effect happens. |
Keywords
» Artificial intelligence » Probability
