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Link > Use of Machine Learning Techniques for improved Monte Calro Integration

Use of Machine Learning Techniques for improved Monte Calro Integration
https://indico.cern.ch/event/568875/contributions/2397925/attachments/1459058/2253175/mcgbr-May12-2017.pdf

by Josh Bendavid (Caltech/LPC)
May 12, 2017
MB4BSM
SLAC

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  • Monte Calro integration
  • Monte Calro generation
  • camel function
  • General idea, sample from some generating probability density g($\bar{x}$) instead of uniformly:
  • VEGAS
    • Multidimensional functions are handled as a simple product of one-dimensional histograms
  • Foam
    • S. Jadach, physics/0203033
    • based on a single decision tree
    • hyper-rectangle
  • Boosted Decision Trees for Classification
  • GBRIntegration
    • Boosted Decision Treesのことらしい
  • 4D Camel Function Integration
  • Generative Deep Neural Networks
  • Generative Adversarial Networks
    • https://arxiv.org/abs/1406.2661
    • a known prior distribution $p(\bar{z})$ (e.g. an N-dimensional normal distribution)
    • generative network $\bar{G}$
      • $G(\bar{z}) = \bar{x}$
    • discriminator network D
    • training
      • D is trained to maximally discriminate
      • G is trained to minimize the discrimination power of D
  • $D_{KL}$
    • KL divergence with respect to the generating probability density function
      • Eq. (3)
    • can be approximated numerically from a finite data set
      • Eq. (4)
  • Implementation Details
    • Keras + TensorFlow
    • Generative model and regression for function approximation
      • 5 hidden layers of different sizes
    • Generative Model
      • modified tanh activation
        • 0.7 * tanh(x) + 0.3 * x
      • sigmoid activtion for Output layer
    • Regression Model
      • elu activation for intermediate layers
      • linear activation for output layer
  • results: 4D Camel Function Integration
    • Generative DNNの誤差はGBRIntegratorと同じオーダー
    • TODO: Generative DNN (staged)とは?
  • results: 9D Camel Function Integration
    • Generative DNN (staged)が最も良い結果を示している