Tuan Anh Le

automatic statistician (wip)

27 February 2017

notes on (Duvenaud, Lloyd, Grosse, Tenenbaum, & Ghahramani, n.d.) and (Lloyd, Duvenaud, Grosse, Tenenbaum, & Ghahramani, 2014). these papers are part of the automatic statistician project.

summary

understanding: 5/10
code: https://github.com/jamesrobertlloyd/gp-structure-search, https://github.com/jamesrobertlloyd/gpss-research

the problem: find good kernels for GPs.

kernels are closed under addition and multiplication. we form a generative grammar for kernel construction: \begin{align} \mathcal S &\to \mathcal S + \mathcal B \\ \mathcal S &\to \mathcal S \times \mathcal B \\ \mathcal B &\to \mathcal B’ \end{align} where is a subexpression (e.g. ) and is a base kernel; one of (exponentiated square difference), (periodic), (linear), (rational quadratic).

how do we search over expressions? we use greedy search: at each stage, we choose the highest scoring kernel and expand it by applying all possible operators.

how do we score kernels? using the bayesian information criterion: \begin{align} \mathrm{BIC}(M) = -2 \log p(D \given M) + |M| \log n. \end{align} here, is kernel, is data, is number of data points is number of kernel parameters, is the evidence.

? this is some sort of approximation to if we had a prior on .


references

  1. Duvenaud, D., Lloyd, J. R., Grosse, R., Tenenbaum, J. B., & Ghahramani, Z. Structure Discovery in Nonparametric Regression through Compositional Kernel Search. In 30th International Conference on Machine Learning (June 2013). International Machine Learning Society.
    @inproceedings{duvenaud2013structure,
      title = {Structure Discovery in Nonparametric Regression through Compositional Kernel Search},
      author = {Duvenaud, David and Lloyd, James Robert and Grosse, Roger and Tenenbaum, Joshua B and Ghahramani, Zoubin},
      booktitle = {30th International Conference on Machine Learning (June 2013)},
      organization = {International Machine Learning Society}
    }
    
  2. Lloyd, J. R., Duvenaud, D., Grosse, R., Tenenbaum, J., & Ghahramani, Z. (2014). Automatic Construction and Natural-Language Description of Nonparametric Regression Models. In Twenty-Eighth AAAI Conference on Artificial Intelligence.
    @inproceedings{lloyd2014automatic,
      title = {Automatic Construction and Natural-Language Description of Nonparametric Regression Models},
      author = {Lloyd, James Robert and Duvenaud, David and Grosse, Roger and Tenenbaum, Joshua and Ghahramani, Zoubin},
      booktitle = {Twenty-Eighth AAAI Conference on Artificial Intelligence},
      year = {2014}
    }
    

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