README.md

MLMC-MIMC-SGD

WARNING work in progress

Code for the numerical experiments in the paper Multi-Index Antithetic Stochastic Gradient Algorithm

This repository contains Stochastic MCMC methods for Bayesian regression with variance reduction (MLMC with antithetic samples, MIMC with antithetic samples, control variate using MAP estimate) using Pytorch:

• MASGA (Multi-index Antithetic Stochastic Gradient Algorithm). Multi-index and antithetic samples at the level of the time discretisation and subsampling on the Stochastic Langevin SDE. For mathematical details, see

    @misc{majka2020multiindex,
        title={Multi-index Antithetic Stochastic Gradient Algorithm},
        author={Mateusz B. Majka and Marc Sabate-Vidales and Łukasz Szpruch},
        year={2020},
        eprint={2006.06102},
        archivePrefix={arXiv},
        primaryClass={stat.ML}
    }
  

• Stochastic Langevin Dynamics using Control Variate to reduce the variance a MAP estimator. This algorithm is taken from the paper Control Variates for Stochastic Gradient MCMC and is used for benchmarking purposes.

Acknowledgements

MLMC and MIMC abstract classes initially based on https://bitbucket.org/pefarrell/pymlmc/src/master/.

Files

Running the code

• Bayesian logistic regression with Gaussian prior on covertype dataset (the code is already prepared to download it). The code samples from the posterior using antithetic MLMC on the discretised Stochastic Langeving SDE and approximates E(F(X)), with F(X) = |X|^2. The code returns various plots specifying computational costs necessary to achieve different Mean Squared Errors.

python logistic_masga_mlmc_subsampling.py --prior Gaussian --T 5 --n_steps 100 --device 1 --s0 32 --type_data covtype --Lmin 0 --N 10000

• Bayesian logistic regression with Mixture of two Gaussians as the prior on a synthetic dataset:

python logistic_masga_mlmc_subsampling.py --prior MixtureGaussians --T 5 --n_steps 100 --device 1 --s0 2 --dim 2 --data_size 512 --type_data synthetic --Lmin 0 --N 10000