Bayesian optimization with input warping

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BO with Warped Gaussian Processes

In this tutorial, we illustrate how to use learned input warping functions for robust Bayesian Optimization when the outcome may be non-stationary functions. When the lengthscales are non-stationarity in the raw input space, learning a warping function that maps raw inputs to a warped space where the lengthscales are stationary can be useful, because then standard stationary kernels can be used to effectively model the function.

In general, for a relatively simple setup (like this one), we recommend using Ax, since this will simplify your setup (including the amount of code you need to write) considerably. See Ax's Modular BoTorch tutorial tutorial. To use input warping with MODULAR_BOTORCH, we can pass the warp_tf, constructed as below, by adding input_transform=warp_tf argument to the Surrogate(...) call.

We consider use a Kumaraswamy CDF as the class of input warping function and learn the concentration parameters ($a>0$ and $b>0$). Kumaraswamy CDFs are quite flexible and map inputs in [0, 1] to outputs in [0, 1]. This work follows the Beta CDF input warping proposed by Snoek et al., but replaces the Beta distribution Kumaraswamy distribution, which has a differentiable and closed-form CDF.

$$K_\text{cdf}(x) = 1 - (1-x^a)^b$$

This enables maximum likelihood (or maximum a posteriori) estimation of the CDF hyperparameters using gradient methods to maximize the likelihood (or posterior probability) jointly with the GP hyperparameters. (Snoek et al. use a fully Bayesian treatment of the CDF parameters). Each input dimension is transformed using a separate warping function.

We use the Log Noisy Expected Improvement (qLogNEI) acquisition function to optimize a synthetic Hartmann6 test function. The standard problem is

$$f(x) = -\sum_{i=1}^4 \alpha_i \exp \left( -\sum_{j=1}^6 A_{ij} (x_j - P_{ij})^2 \right)$$

over $x \in [0,1]^6$ (parameter values can be found in botorch/test_functions/hartmann6.py). For this demonstration, We first warp each input dimension through a different inverse Kumaraswamy CDF.

Since BoTorch assumes a maximization problem, we will attempt to maximize $-f(x)$ to achieve $\max_{x} -f(x) = 3.32237$.

[1] J. Snoek, K. Swersky, R. S. Zemel, R. P. Adams. Input Warping for Bayesian Optimization of Non-Stationary Functions. Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):1674-1682, 2014.

# Install dependencies if we are running in colab
import sys
if 'google.colab' in sys.modules:
    %pip install botorch

import os
import torch


device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
dtype = torch.double
SMOKE_TEST = os.environ.get("SMOKE_TEST")

Problem setup

First, we define the sample parameters for the sigmoid functions that transform the respective inputs.

from torch.distributions import Kumaraswamy
import matplotlib.pyplot as plt

%matplotlib inline


fontdict = {"fontsize": 15}
torch.manual_seed(1234567890)
c1 = torch.rand(6, dtype=dtype, device=device) * 3 + 0.1
c0 = torch.rand(6, dtype=dtype, device=device) * 3 + 0.1
x = torch.linspace(0, 1, 101, dtype=dtype, device=device)
k = Kumaraswamy(concentration1=c1, concentration0=c0)
k_icdfs = k.icdf(x.unsqueeze(1).expand(101, 6))
fig, ax = plt.subplots(1, 1, figsize=(5, 5))

for i in range(6):
    ax.plot(x.cpu(), k_icdfs[:, i].cpu())
ax.set_xlabel("Raw Value", **fontdict)
ax.set_ylabel("Transformed Value", **fontdict)

Output:

Text(0, 0.5, 'Transformed Value')

from botorch.test_functions import Hartmann

neg_hartmann6 = Hartmann(negate=True)


def obj(X):
    X_warp = k.icdf(X)
    return neg_hartmann6(X_warp)

Output:

[KeOps] Warning : There were warnings or errors :
/bin/sh: brew: command not found
[KeOps] Warning : CUDA libraries not found or could not be loaded; Switching to CPU only.
[KeOps] Warning : There were warnings or errors :
/bin/sh: brew: command not found
[KeOps] Warning : OpenMP library not found, it must be downloaded through Homebrew for apple Silicon chips
[KeOps] Warning : OpenMP support is not available. Disabling OpenMP.

Initial design

The models are initialized with 14 points in $[0,1]^6$ drawn from a scrambled sobol sequence.

We observe the objectives with additive Gaussian noise with a standard deviation of 0.05.

from botorch.models import SingleTaskGP
from gpytorch.mlls.sum_marginal_log_likelihood import ExactMarginalLogLikelihood
from botorch.utils.sampling import draw_sobol_samples

NOISE_SE = 0.05
train_yvar = torch.tensor(NOISE_SE**2, device=device, dtype=dtype)

bounds = torch.tensor([[0.0] * 6, [1.0] * 6], device=device, dtype=dtype)


n = 14
# generate initial training data
train_x = draw_sobol_samples(
    bounds=bounds, n=n, q=1, seed=torch.randint(0, 10000, (1,)).item()
).squeeze(1)
exact_obj = obj(train_x).unsqueeze(-1)  # add output dimension

best_observed_value = exact_obj.max().item()
train_obj = exact_obj + NOISE_SE * torch.randn_like(exact_obj)

Input warping and model initialization

We initialize the Warp input transformation and pass it a SingleTaskGP to model the noiseless objective. The Warp object is a torch.nn.Module that contains the concentration parameters and applies the warping function in the Model's forward pass.

from botorch.models.transforms.input import Warp
from gpytorch.priors.torch_priors import LogNormalPrior


def initialize_model(train_x, train_obj, bounds):
    # initialize input_warping transformation
    warp_tf = Warp(
        d=train_x.shape[-1],
        indices=list(range(train_x.shape[-1])),
        # use a prior with median at 1.
        # when a=1 and b=1, the Kumaraswamy CDF is the identity function
        concentration1_prior=LogNormalPrior(0.0, 0.75**0.5),
        concentration0_prior=LogNormalPrior(0.0, 0.75**0.5),
        bounds=bounds,
    )
    # define the model for objective
    model = SingleTaskGP(
        train_X=train_x,
        train_Y=train_obj,
        train_Yvar=train_yvar.expand_as(train_obj),
        input_transform=warp_tf,
    ).to(train_x)
    mll = ExactMarginalLogLikelihood(model.likelihood, model)
    return mll, model

Define a helper function that performs the essential BO step

The helper function below takes an acquisition function as an argument, optimizes it, and returns the batch ${x_1, x_2, \ldots x_q}$ along with the observed function values. For this example, we'll use sequential $q=1$ optimization. A simple initialization heuristic is used to select the 20 restart initial locations from a set of 512 random points.

from botorch.optim import optimize_acqf


num_restarts = 20 if not SMOKE_TEST else 2
raw_samples = 512 if not SMOKE_TEST else 32


def optimize_acqf_and_get_observation(acq_func):
    """Optimizes the acquisition function, and returns a new candidate and a noisy observation."""
    # optimize
    candidates, _ = optimize_acqf(
        acq_function=acq_func,
        bounds=bounds,
        q=1,
        num_restarts=num_restarts,
        raw_samples=raw_samples,  # used for intialization heuristic
        options={"batch_limit": 5, "maxiter": 200},
    )
    # observe new values
    new_x = candidates.detach()
    exact_obj = obj(new_x).unsqueeze(-1)  # add output dimension
    train_obj = exact_obj + NOISE_SE * torch.randn_like(exact_obj)
    return new_x, train_obj

Perform Bayesian Optimization

The Bayesian optimization loop iterates the following steps:

1. given a surrogate model, choose a candidate point $x$
2. observe $f(x)$
3. update the surrogate model.

We do N_BATCH=50 rounds of optimization.

from botorch import fit_gpytorch_mll
from botorch.acquisition.logei import qLogNoisyExpectedImprovement
from botorch.exceptions import BadInitialCandidatesWarning

import warnings

warnings.filterwarnings("ignore", category=BadInitialCandidatesWarning)
warnings.filterwarnings("ignore", category=RuntimeWarning)

N_BATCH = 50 if not SMOKE_TEST else 5

torch.manual_seed(0)

best_observed = [best_observed_value]
mll, model = initialize_model(train_x, train_obj, bounds)

# run N_BATCH rounds of BayesOpt after the initial random batch
for iteration in range(1, N_BATCH + 1):

    # fit the models
    fit_gpytorch_mll(mll)
    ei = qLogNoisyExpectedImprovement(model=model, X_baseline=train_x)

    # optimize and get new observation
    new_x, new_obj = optimize_acqf_and_get_observation(ei)

    # update training points
    train_x = torch.cat([train_x, new_x])
    train_obj = torch.cat([train_obj, new_obj])

    # update progress
    best_value = obj(train_x).max().item()
    best_observed.append(best_value)

    mll, model = initialize_model(train_x, train_obj, bounds)

    print(".", end="")

Output:

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Plot the results

The plot below shows the log regret at each step of the optimization for each of the algorithms.

import numpy as np
from matplotlib import pyplot as plt

%matplotlib inline


GLOBAL_MAXIMUM = neg_hartmann6.optimal_value

iters = np.arange(N_BATCH + 1)
y_ei = np.log10(GLOBAL_MAXIMUM - np.asarray(best_observed))

fig, ax = plt.subplots(1, 1, figsize=(8, 6))

ax.plot(
    iters,
    y_ei,
    linewidth=1.5,
    alpha=0.6,
)

ax.set_xlabel("number of observations (beyond initial points)")
ax.set_ylabel("Log10 Regret")

Output:

Text(0, 0.5, 'Log10 Regret')