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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

### キーワード

• Monte Calro integration
• Monte Calro generation
• camel function
• (補足) 4Dや9Dの関数の例として使用
• 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
• 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
• 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