Contour-Stochastic-Gradient-Langevin-Dynamics

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Contour-Stochastic-Gradient-Langevin-Dynamics

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An elegant adaptive importance sampling algorithms for simulations of multi-modal distributions (NeurIPS'20)

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# Contour Stochastic Gradient Langevin Dynamics

Simulations of multi-modal distributions can be very **costly** and often lead to **unreliable** predictions. To accelerate the computations, we propose to sample from a **flattened** distribution to **accelerate the computations** and estimate the **importance weights** between the original distribution and the flattened distribution to **ensure the correctness** of the distribution.

We refer interested readers to [blog](https://waynedw.github.io/posts/CSGLD/) here. For Chinese readers, you may also find this blog interesting [知乎](https://zhuanlan.zhihu.com/p/267633636).

| Methods   |      Speed      | Special features  | Cost |
|----------|:-------------:|:-------------:|:-------------:|
| SGLD (ICML'11) |  Extremely slow | None | None |
| Cycic SGLD (ICLR'20) |    Medium   | Cyclic learning rates  | More cycles |
| Replica exchange SGLD (ICML'20) | Fast | Swaps/Jumps | Parallel chains |
| Contour SGLD (NeurIPS'20) | Fast | Bouncy moves | Latent vector |

The following is a demo to show how the latent vector is gradually estimated
<p float="left">
  <img src="figures/CSGLD_with_PDF.gif" width="200" title="SGLD"/>
  <img src="figures/CSGLD_PDF.gif" width="200" alt="Made with Angular" title="Angular" /> 
</p>

Although this version of CSGLD has a global statbility condition, it doesn't handle **high-loss** problems appropriately. For a more scalable version (called ICSGLD), please check the paper [\[Link\]](https://openreview.net/pdf?id=IK9ap6nxXr2) and the code [\[Link\]](https://github.com/WayneDW/Interacting-Contour-Stochastic-Gradient-Langevin-Dynamics) here. Since importance sampling suffers a lot from the **large variance** issue, the conditional independence of ICSGLD will provide the algorithm a **perfect variance reduction** for estimating the density of states.

```
@inproceedings{CSGLD,
  title={A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions},
  author={Wei Deng and Guang Lin and Faming Liang},
  booktitle={Advances in Neural Information Processing Systems},
  year={2020}
}
```

## References:

1. Max Welling, Yee Whye Teh. [Bayesian Learning via Stochastic Gradient Langevin Dynamics](https://pdfs.semanticscholar.org/aeed/631d6a84100b5e9a021ec1914095c66de415.pdf). ICML'11

2. R. Zhang, C. Li, J. Zhang, C. Chen, A. Wilson. [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://arxiv.org/pdf/1902.03932.pdf). ICLR'20

3. W. Deng, Q. Feng, L. Gao, F. Liang, G. Lin. [Non-convex Learning via Replica Exchange Stochastic Gradient MCMC](https://arxiv.org/pdf/2008.05367.pdf). ICML'20.

4. W. Deng, G. Lin, F. Liang. [A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions](https://arxiv.org/pdf/2010.09800.pdf). NeurIPS'20.

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