Example: Hamiltonian Monte Carlo with Energy Conserving Subsampling

This example illustrates the use of data subsampling in HMC using Energy Conserving Subsampling. Data subsampling is applicable when the likelihood factorizes as a product of N terms.


  1. Hamiltonian Monte Carlo with energy conserving subsampling, Dang, K. D., Quiroz, M., Kohn, R., Minh-Ngoc, T., & Villani, M. (2019)

import argparse
import time

import matplotlib.pyplot as plt
import numpy as np

from jax import random
import jax.numpy as jnp

import numpyro
import numpyro.distributions as dist
from numpyro.examples.datasets import HIGGS, load_dataset
from numpyro.infer import HMC, HMCECS, MCMC, NUTS, SVI, Trace_ELBO, autoguide

def model(data, obs, subsample_size):
    n, m = data.shape
    theta = numpyro.sample("theta", dist.Normal(jnp.zeros(m), 0.5 * jnp.ones(m)))
    with numpyro.plate("N", n, subsample_size=subsample_size):
        batch_feats = numpyro.subsample(data, event_dim=1)
        batch_obs = numpyro.subsample(obs, event_dim=0)
            "obs", dist.Bernoulli(logits=theta @ batch_feats.T), obs=batch_obs

def run_hmcecs(hmcecs_key, args, data, obs, inner_kernel):
    svi_key, mcmc_key = random.split(hmcecs_key)

    # find reference parameters for second order taylor expansion to estimate likelihood (taylor_proxy)
    optimizer = numpyro.optim.Adam(step_size=1e-3)
    guide = autoguide.AutoDelta(model)
    svi = SVI(model, guide, optimizer, loss=Trace_ELBO())
    svi_result = svi.run(svi_key, args.num_svi_steps, data, obs, args.subsample_size)
    params, losses = svi_result.params, svi_result.losses
    ref_params = {"theta": params["theta_auto_loc"]}

    # taylor proxy estimates log likelihood (ll) by
    # taylor_expansion(ll, theta_curr) +
    #     sum_{i in subsample} ll_i(theta_curr) - taylor_expansion(ll_i, theta_curr) around ref_params
    proxy = HMCECS.taylor_proxy(ref_params)

    kernel = HMCECS(inner_kernel, num_blocks=args.num_blocks, proxy=proxy)
    mcmc = MCMC(kernel, num_warmup=args.num_warmup, num_samples=args.num_samples)

    mcmc.run(mcmc_key, data, obs, args.subsample_size)
    return losses, mcmc.get_samples()

def run_hmc(mcmc_key, args, data, obs, kernel):
    mcmc = MCMC(kernel, num_warmup=args.num_warmup, num_samples=args.num_samples)
    mcmc.run(mcmc_key, data, obs, None)
    return mcmc.get_samples()

def main(args):
    assert (
        11_000_000 >= args.num_datapoints
    ), "11,000,000 data points in the Higgs dataset"
    # full dataset takes hours for plain hmc!
    if args.dataset == "higgs":
        _, fetch = load_dataset(
            HIGGS, shuffle=False, num_datapoints=args.num_datapoints
        data, obs = fetch()
        data, obs = (np.random.normal(size=(10, 28)), np.ones(10))

    hmcecs_key, hmc_key = random.split(random.PRNGKey(args.rng_seed))

    # choose inner_kernel
    if args.inner_kernel == "hmc":
        inner_kernel = HMC(model)
        inner_kernel = NUTS(model)

    start = time.time()
    losses, hmcecs_samples = run_hmcecs(hmcecs_key, args, data, obs, inner_kernel)
    hmcecs_runtime = time.time() - start

    start = time.time()
    hmc_samples = run_hmc(hmc_key, args, data, obs, inner_kernel)
    hmc_runtime = time.time() - start

    summary_plot(losses, hmc_samples, hmcecs_samples, hmc_runtime, hmcecs_runtime)

def summary_plot(losses, hmc_samples, hmcecs_samples, hmc_runtime, hmcecs_runtime):
    fig, ax = plt.subplots(2, 2)
    ax[0, 0].plot(losses, "r")
    ax[0, 0].set_title("SVI losses")
    ax[0, 0].set_ylabel("ELBO")

    if hmc_runtime > hmcecs_runtime:
        ax[0, 1].bar([0], hmc_runtime, label="hmc", color="b")
        ax[0, 1].bar([0], hmcecs_runtime, label="hmcecs", color="r")
        ax[0, 1].bar([0], hmcecs_runtime, label="hmcecs", color="r")
        ax[0, 1].bar([0], hmc_runtime, label="hmc", color="b")
    ax[0, 1].set_title("Runtime")
    ax[0, 1].set_ylabel("Seconds")
    ax[0, 1].legend()
    ax[0, 1].set_xticks([])

    ax[1, 0].plot(jnp.sort(hmc_samples["theta"].mean(0)), "or")
    ax[1, 0].plot(jnp.sort(hmcecs_samples["theta"].mean(0)), "b")
    ax[1, 0].set_title(r"$\mathrm{\mathbb{E}}[\theta]$")

    ax[1, 1].plot(jnp.sort(hmc_samples["theta"].var(0)), "or")
    ax[1, 1].plot(jnp.sort(hmcecs_samples["theta"].var(0)), "b")
    ax[1, 1].set_title(r"Var$[\theta]$")

    for a in ax[1, :]:

    fig.savefig("hmcecs_plot.pdf", bbox_inches="tight")

if __name__ == "__main__":
    parser = argparse.ArgumentParser(
        "Hamiltonian Monte Carlo with Energy Conserving Subsampling"
    parser.add_argument("--subsample_size", type=int, default=1300)
    parser.add_argument("--num_svi_steps", type=int, default=5000)
    parser.add_argument("--num_blocks", type=int, default=100)
    parser.add_argument("--num_warmup", type=int, default=500)
    parser.add_argument("--num_samples", type=int, default=500)
    parser.add_argument("--num_datapoints", type=int, default=1_500_000)
        "--dataset", type=str, choices=["higgs", "mock"], default="higgs"
        "--inner_kernel", type=str, choices=["nuts", "hmc"], default="nuts"
    parser.add_argument("--device", default="cpu", type=str, choices=["cpu", "gpu"])
        "--rng_seed", default=37, type=int, help="random number generator seed"

    args = parser.parse_args()



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