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Improving sampling accuracy of stochastic gradient MCMC methods via non-uniform subsampling of gradients

  • *Corresponding author

    *Corresponding author

© 2021 The Author(s). Published by AIMS, LLC. This is an Open Access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Abstract Full Text(HTML) Figure(6) / Table(3) Related Papers Cited by
  • Many Markov Chain Monte Carlo (MCMC) methods leverage gradient information of the potential function of target distribution to explore sample space efficiently. However, computing gradients can often be computationally expensive for large scale applications, such as those in contemporary machine learning. Stochastic Gradient (SG-)MCMC methods approximate gradients by stochastic ones, commonly via uniformly subsampled data points, and achieve improved computational efficiency, however at the price of introducing sampling error. We propose a non-uniform subsampling scheme to improve the sampling accuracy. The proposed exponentially weighted stochastic gradient (EWSG) is designed so that a non-uniform-SG-MCMC method mimics the statistical behavior of a batch-gradient-MCMC method, and hence the inaccuracy due to SG approximation is reduced. EWSG differs from classical variance reduction (VR) techniques as it focuses on the entire distribution instead of just the variance; nevertheless, its reduced local variance is also proved. EWSG can also be viewed as an extension of the importance sampling idea, successful for stochastic-gradient-based optimizations, to sampling tasks. In our practical implementation of EWSG, the non-uniform subsampling is performed efficiently via a Metropolis-Hastings chain on the data index, which is coupled to the MCMC algorithm. Numerical experiments are provided, not only to demonstrate EWSG's effectiveness, but also to guide hyperparameter choices, and validate our non-asymptotic global error bound despite of approximations in the implementation. Notably, while statistical accuracy is improved, convergence speed can be comparable to the uniform version, which renders EWSG a practical alternative to VR (but EWSG and VR can be combined too).

    Mathematics Subject Classification: Primary: 65C05, 65C40, 60J20; Secondary: 62D05, 37A50, 37A60, 90C15.

    Citation:

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  • Figure 1.  Performance quantification (Gaussian target)

    Figure 2.  BLR learning curve

    Figure 3.  BNN learning curve. Shade: one standard deviation.

    Figure 4.  KL divergence

    Figure 5.  Posterior prediction of mean (left) and standard deviation (right) of log likelihood on test data set generated by SGHMC, EWSG and EWSG-VR on two Bayesian logistic regression tasks. Statistics are computed based on 1000 independent simulations. Minibatch size $ b = 1 $ for all methods except FG. $ M = 1 $ for EWSG and EWSG-VR

    Figure 6.  (a) Histogram of data used in each iteration for FlyMC algorithm. (b) Autocorrelation plot of FlyMC, EWSG and MH. (c) Samples of EWSG. (d) Samples of FlyMC

    Table 1.  Accuracy, log likelihood and wall time of various algorithms on test data after one data pass (mean $ \pm $ std)

    Method SGLD pSGLD SGHMC EWSG FlyMC
    Accuracy(%) 75.283 $ \pm $ 0.016 75.126 $ \pm $ 0.020 75.268 $ \pm $ 0.017 75.306 $ \pm $ 0.016 75.199 $ \pm $ 0.080
    Log Likelihood -0.525 $ \pm $ 0.000 -0.526 $ \pm $ 0.000 -0.525 $ \pm $ 0.000 -0.523 $ \pm $ 0.000 -0.523 $ \pm $ 0.000
    Wall Time (s) 3.085 $ \pm $ 0.283 4.312 $ \pm $ 0.359 3.145 $ \pm $ 0.307 3.755 $ \pm $ 0.387 291.295 $ \pm $ 56.368
     | Show Table
    DownLoad: CSV

    Table 2.  Test error (mean $ \pm $ standard deviation) after 200 epoches

    Method Test Error(%), MLP Test Error(%), CNN
    SGLD 1.976 $ \pm $ 0.055 0.848 $ \pm $ 0.060
    pSGLD 1.821 $ \pm $ 0.061 0.860 $ \pm $ 0.052
    SGHMC 1.833 $ \pm $ 0.073 0.778 $ \pm $ 0.040
    CP-SGHMC 1.835 $ \pm $ 0.047 0.772 $ \pm $ 0.055
    EWSG 1.793 $ \pm $ 0.100 0.753 $ \pm $ 0.035
     | Show Table
    DownLoad: CSV

    Table 3.  Test errors of EWSG (top of each cell) and SGHMC (bottom of each cell) after 200 epoches. $ b $ is minibatch size for EWSG, and minibatch size of SGHMC is set as $ b\times(M+1) $ to ensure the same number of data used per parameter update for both algorithms. Step size is set $ h = \frac{10}{b(M+1)} $ as suggested in [12], different from that used to produce Table 2. Results with smaller test error is highlighted in boldface

    $ b $ $ M+1=2 $ $ M+1=5 $ $ M+1=10 $
    $ 100 $ 1.86% 1.83% 1.80%
    1.94% 1.92% 1.97%
    $ 200 $ 1.90% 1.87% 1.80%
    1.87% 1.97% 2.07%
    $ 500 $ 1.79% 2.01% 2.36%
    1.97% 2.17% 2.37%
     | Show Table
    DownLoad: CSV
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