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Batch GP regression using BoTorch's functions #1259

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takafusui opened this issue Jun 13, 2022 · 14 comments
Closed

Batch GP regression using BoTorch's functions #1259

takafusui opened this issue Jun 13, 2022 · 14 comments
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bug Something isn't working upstream issue

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@takafusui
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takafusui commented Jun 13, 2022 •
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Hi,

I want to do a batch GP regression but have faced unexpected outcomes.
In short, what I want to do is the following:
First I construct the GP model based on the identical train_X and train_y and fit hyper parameters using fit_gpytorch_model.
Then I want to test the fitted GP model by changing test_X for batch_shape times.
To do so, instead of iterating this process batch_shape times, I want to parallelize this process in batch.
Specifically I expand train_X by .repeat() and have batch_shape x n x d and apply the same thing train_y to have batch_shape x n x 1, then instantiage GP model and test

    gp = SingleTaskGP(
        train_X, train_y,  # likelihood=likelihood,
        input_transform=InputStandardize(
            d=train_X_dim, batch_shape=torch.Size([N_batch_size])),
        outcome_transform=Standardize(
            m=train_y_dim, batch_shape=torch.Size([N_batch_size]))
    )
    mll = gpytorch.mlls.ExactMarginalLogLikelihood(gp.likelihood, gp)
    fit_gpytorch_model(mll)  # Using BoTorch's routine

    with torch.no_grad(), gpytorch.settings.fast_pred_var():
        # Total model output variance
        pred_test_X = gp.posterior(test_X_batch)
        pred_test_X_mean = pred_test_X.mean
        # Take the variance wrt. test_X for only one simulation batch
        var_E_pred_test_X = torch.var(pred_test_X_mean[0, :, :])

What I intend above is that I have the identical batch_shape number of GP models and test the identical GP model by test_X_batch (test_X_batch[0] != text_X_batch[1] and so on).

What I am confusing now is the following:
When I focus on only one batch, I can define the GP model

    gp_single = SingleTaskGP(
        train_X[0], train_y[0],
        input_transform=InputStandardize(d=train_X_dim),
        outcome_transform=Standardize(m=train_y_dim)
    )
    mll_single = gpytorch.mlls.ExactMarginalLogLikelihood(
        gp_single.likelihood, gp_single)
    fit_gpytorch_model(mll_single)  # Using BoTorch's routine
    with torch.no_grad(), gpytorch.settings.fast_pred_var():
        # Total model output variance
        pred_test_X_single = gp_single.posterior(test_X_batch[0])
        pred_test_X_mean_single = pred_test_X_single.mean
        # Take the variance wrt. test_X for only one simulation batch
        var_E_pred_test_X_single = torch.var(pred_test_X_mean_single[:, :])

and the resulted predictive mean or variance should be the same but they are not.

What am I missing here? I am confusing...

Thank you very much in advance.
@Balandat
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Can you provide the data under which you observe this behavior? How different are the predicted outcomes?

Minimal differences are always expected due to the fact that the optimization of the parameters may produce slightly different results in different fitting steps. If the differences are large, it may be that the data is such that the MLL is multimodal and the MAP optimization finds different hyperparameters for the batch vs. non batch version.

Note also that fitting a batched model under the hood actually runs a number if individual fits and then reassembles the batched model (https://github.com/pytorch/botorch/blob/main/botorch/fit.py#L86-L95) since fitting the batched model results in a larger optimization problem to be solved (with the optimizer not being aware that the different batches are actually independent optimization problems).

@takafusui
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Thank you very much for your reply. Below I attached a code snippet.
To me, the difference between the two models is considerably large...

Sorry at first that I simply copy and paste thus the code is super nasty, especially when reading train_X, train_y, and test_X, but I hope you understand what I want to do.

Thank you very much.

import torch
import gpytorch
from botorch.models import SingleTaskGP
from botorch.fit import fit_gpytorch_model
from botorch.models.transforms.outcome import Standardize
from botorch.models.transforms.input import InputStandardize

torch.set_default_dtype(torch.float64)
torch.manual_seed(123)

# --------------------------------------------------------------------------- #
# Raw training data
# --------------------------------------------------------------------------- #
train_X = torch.tensor([
    [2.8600e-02, 1.6835e+01, 1.8205e+00, 1.0802e+01, 7.4367e-01, 5.7300e-02,
     6.2900e-03],
    [3.8920e-02, 2.5385e+01, 1.7935e+00, 3.8625e+00, 5.5937e-01, 2.3000e-03,
     9.7500e-03],
    [1.6120e-02, 3.4115e+01, 1.5485e+00, 3.0255e+00, 9.0138e-01, 6.2700e-02,
     5.1700e-03],
    [4.1320e-02, 1.8545e+01, 1.6975e+00, 5.3475e+00, 6.0877e-01, 8.8500e-02,
     1.2900e-03],
    [2.7400e-02, 2.9885e+01, 1.7985e+00, 1.6485e+00, 9.0517e-01, 2.4500e-02,
     1.6500e-03],
    [3.9720e-02, 2.6195e+01, 1.9685e+00, 3.4575e+00, 9.6598e-01, 8.9000e-03,
     2.5900e-03],
    [3.1320e-02, 2.1875e+01, 1.9915e+00, 7.9395e+00, 5.7363e-01, 2.1300e-02,
     6.9900e-03],
    [1.7320e-02, 3.3845e+01, 1.5105e+00, 1.1476e+01, 8.6813e-01, 4.7900e-02,
     4.1100e-03],
    [1.1640e-02, 2.8805e+01, 1.7735e+00, 9.2085e+00, 7.7882e-01, 4.2700e-02,
     7.3700e-03],
    [1.5960e-02, 5.8550e+00, 1.7575e+00, 1.3043e+01, 8.9758e-01, 8.1100e-02,
     8.1300e-03],
    [1.4360e-02, 1.3235e+01, 1.9415e+00, 2.1615e+00, 6.2872e-01, 8.6100e-02,
     1.5500e-03],
    [2.2280e-02, 6.1250e+00, 1.9335e+00, 7.1295e+00, 8.7003e-01, 3.5300e-02,
     7.4700e-03],
    [1.3000e-02, 4.0235e+01, 1.6335e+00, 1.3475e+01, 5.8313e-01, 7.5900e-02,
     7.7500e-03],
    [1.4200e-02, 1.8995e+01, 1.7045e+00, 1.3771e+01, 6.5058e-01, 2.5100e-02,
     1.7500e-03],
    [1.4680e-02, 3.6635e+01, 1.6955e+00, 5.2395e+00, 7.4557e-01, 1.1300e-02,
     1.2100e-03],
    [4.2680e-02, 3.3305e+01, 1.7695e+00, 1.2934e+01, 7.2182e-01, 5.8100e-02,
     2.8700e-03],
    [4.5720e-02, 2.2325e+01, 1.8485e+00, 1.4284e+01, 8.1967e-01, 9.0500e-02,
     8.6500e-03],
    [4.7000e-02, 1.7645e+01, 1.6905e+00, 6.1845e+00, 5.3563e-01, 8.7700e-02,
     5.4700e-03],
    [1.0040e-02, 2.3675e+01, 1.9355e+00, 7.7775e+00, 6.1067e-01, 2.2700e-02,
     6.4500e-03],
    [4.4520e-02, 3.8975e+01, 1.6505e+00, 1.9185e+00, 8.2062e-01, 7.3300e-02,
     3.7300e-03],
    [1.2680e-02, 3.2135e+01, 1.5555e+00, 6.2655e+00, 7.3987e-01, 7.4900e-02,
     7.1700e-03],
    [4.8520e-02, 4.6175e+01, 1.7865e+00, 5.7525e+00, 6.5153e-01, 5.9500e-02,
     9.2500e-03],
    [4.6040e-02, 7.2050e+00, 1.6425e+00, 1.2989e+01, 6.0023e-01, 8.1700e-02,
     3.6100e-03],
    [4.4440e-02, 7.7450e+00, 1.7055e+00, 1.2098e+01, 7.9688e-01, 7.0000e-04,
     4.5500e-03],
    [1.7880e-02, 2.7635e+01, 1.6135e+00, 1.3204e+01, 7.7977e-01, 6.7100e-02,
     5.5000e-04],
    [1.3480e-02, 1.3595e+01, 1.6095e+00, 1.0046e+01, 9.3558e-01, 3.9000e-03,
     3.5900e-03],
    [1.0120e-02, 1.1705e+01, 1.8235e+00, 7.6155e+00, 8.1017e-01, 4.6700e-02,
     4.0100e-03],
    [2.9880e-02, 4.5815e+01, 1.6295e+00, 6.7785e+00, 5.6127e-01, 2.9000e-03,
     8.9900e-03],
    [2.4600e-02, 2.8445e+01, 1.5515e+00, 1.2502e+01, 9.1087e-01, 4.0100e-02,
     5.7000e-04],
    [2.0120e-02, 1.0895e+01, 1.5945e+00, 7.8045e+00, 6.1638e-01, 3.6100e-02,
     7.6300e-03],
    [3.4760e-02, 2.5835e+01, 1.7585e+00, 1.3421e+01, 9.2702e-01, 8.3900e-02,
     9.9900e-03],
    [2.3800e-02, 1.6565e+01, 1.8465e+00, 3.0525e+00, 6.2113e-01, 2.1100e-02,
     6.5300e-03],
    [2.9400e-02, 7.8350e+00, 1.5445e+00, 3.1605e+00, 7.9497e-01, 5.0000e-04,
     4.7700e-03],
    [1.4040e-02, 2.7545e+01, 1.8765e+00, 4.1325e+00, 8.4437e-01, 5.6700e-02,
     3.2100e-03],
    [3.3880e-02, 4.5005e+01, 1.8975e+00, 6.7245e+00, 7.0568e-01, 2.5900e-02,
     7.4300e-03],
    [3.3080e-02, 4.2305e+01, 1.6755e+00, 3.2415e+00, 7.8547e-01, 7.0700e-02,
     4.9000e-04],
    [4.6840e-02, 2.3945e+01, 1.7835e+00, 9.6675e+00, 7.3798e-01, 3.7300e-02,
     4.4900e-03],
    [4.0280e-02, 1.8365e+01, 1.7795e+00, 1.1989e+01, 9.5173e-01, 8.9100e-02,
     2.8100e-03],
    [4.9400e-02, 4.2485e+01, 1.6705e+00, 7.0485e+00, 8.5292e-01, 2.9900e-02,
     8.6900e-03],
    [2.6680e-02, 4.3565e+01, 1.5755e+00, 6.8595e+00, 6.9427e-01, 5.9900e-02,
     4.3100e-03],
    [1.7560e-02, 2.4935e+01, 1.7725e+00, 6.0225e+00, 5.5842e-01, 4.1000e-03,
     1.8100e-03],
    [2.5720e-02, 3.5555e+01, 1.5295e+00, 1.6215e+00, 8.2917e-01, 9.7900e-02,
     6.0900e-03],
    [2.2200e-02, 1.9625e+01, 1.5705e+00, 1.1341e+01, 5.2043e-01, 4.5300e-02,
     6.8100e-03],
    [1.1400e-02, 2.7005e+01, 1.7155e+00, 5.7255e+00, 6.5248e-01, 6.6700e-02,
     3.0500e-03],
    [1.4600e-02, 2.5475e+01, 1.5875e+00, 4.9965e+00, 7.0283e-01, 3.1300e-02,
     1.5900e-03],
    [4.6520e-02, 4.5365e+01, 1.7465e+00, 5.1045e+00, 8.1207e-01, 5.2500e-02,
     3.7900e-03],
    [1.8760e-02, 1.8185e+01, 1.7875e+00, 1.3960e+01, 6.0308e-01, 8.6300e-02,
     2.7500e-03],
    [2.8760e-02, 2.4125e+01, 1.7915e+00, 1.4851e+01, 6.0972e-01, 6.0300e-02,
     3.8100e-03],
    [4.7560e-02, 3.0695e+01, 1.7255e+00, 1.2421e+01, 5.1092e-01, 9.4700e-02,
     9.7700e-03],
    [4.4600e-02, 1.1345e+01, 1.8085e+00, 1.3582e+01, 9.7452e-01, 1.1900e-02,
     5.1000e-04],
    [1.9000e-02, 8.4650e+00, 1.7245e+00, 1.0505e+01, 6.8288e-01, 7.9100e-02,
     2.0700e-03],
    [2.3880e-02, 2.8715e+01, 1.6665e+00, 9.5595e+00, 9.2988e-01, 7.0500e-02,
     2.8300e-03],
    [1.5720e-02, 2.5925e+01, 1.6615e+00, 3.3765e+00, 7.6362e-01, 8.9300e-02,
     3.1500e-03],
    [3.6040e-02, 8.5550e+00, 1.8885e+00, 7.2645e+00, 5.9262e-01, 9.9100e-02,
     5.3700e-03],
    [4.3880e-02, 4.6535e+01, 1.6985e+00, 4.3485e+00, 6.7338e-01, 7.8900e-02,
     5.3500e-03],
    [2.7000e-02, 1.5575e+01, 1.7195e+00, 9.7215e+00, 9.1848e-01, 1.9700e-02,
     6.8300e-03],
    [4.5960e-02, 2.7905e+01, 1.6595e+00, 1.2637e+01, 6.0783e-01, 5.1300e-02,
     3.9300e-03],
    [3.6760e-02, 2.0975e+01, 1.6845e+00, 9.8025e+00, 9.2512e-01, 5.6900e-02,
     7.9100e-03],
    [3.3800e-02, 9.5450e+00, 1.7605e+00, 9.5325e+00, 7.3133e-01, 6.4300e-02,
     4.2900e-03],
    [4.5560e-02, 4.8875e+01, 1.8445e+00, 2.4315e+00, 6.8478e-01, 2.6300e-02,
     7.0000e-05],
    [3.4360e-02, 4.3025e+01, 1.5205e+00, 1.3286e+01, 5.7647e-01, 7.6100e-02,
     3.0900e-03],
    [2.5960e-02, 2.2865e+01, 1.5075e+00, 1.3825e+01, 8.6242e-01, 2.9300e-02,
     6.7300e-03],
    [1.6840e-02, 1.4315e+01, 1.8915e+00, 1.3664e+01, 7.7407e-01, 3.3900e-02,
     7.2100e-03],
    [3.1240e-02, 9.3650e+00, 1.8275e+00, 6.0765e+00, 9.6693e-01, 5.2900e-02,
     4.1300e-03],
    [1.2200e-02, 2.0345e+01, 1.9895e+00, 1.2665e+01, 7.6553e-01, 3.4300e-02,
     2.4300e-03],
    [4.5080e-02, 3.1595e+01, 1.9435e+00, 3.7545e+00, 7.4652e-01, 2.4700e-02,
     6.1500e-03],
    [1.8840e-02, 2.4575e+01, 1.8285e+00, 3.6465e+00, 6.0593e-01, 3.9100e-02,
     3.4500e-03],
    [2.3320e-02, 4.9055e+01, 1.8185e+00, 2.8095e+00, 7.5317e-01, 2.4300e-02,
     5.7300e-03],
    [3.5000e-02, 3.7445e+01, 1.8875e+00, 2.1345e+00, 6.0213e-01, 1.8300e-02,
     9.8900e-03],
    [1.3720e-02, 5.4050e+00, 1.9215e+00, 8.9385e+00, 7.2277e-01, 3.1900e-02,
     5.8700e-03],
    [2.6200e-02, 3.3125e+01, 1.7395e+00, 2.1075e+00, 7.5033e-01, 8.9500e-02,
     7.4100e-03],
    [2.1560e-02, 1.4945e+01, 1.5015e+00, 1.4041e+01, 7.8262e-01, 7.0100e-02,
     9.3100e-03],
    [3.6840e-02, 2.8625e+01, 1.9425e+00, 6.1575e+00, 9.5932e-01, 8.4500e-02,
     9.1000e-04],
    [3.4200e-02, 1.5845e+01, 1.6165e+00, 1.4743e+01, 6.4297e-01, 3.9700e-02,
     2.7700e-03],
    [3.3000e-02, 3.4835e+01, 1.8705e+00, 1.9455e+00, 7.6838e-01, 3.2100e-02,
     7.1100e-03],
    [4.7640e-02, 3.7355e+01, 1.8645e+00, 1.5675e+00, 7.0947e-01, 4.4500e-02,
     7.7700e-03],
    [4.8920e-02, 4.6085e+01, 1.7545e+00, 1.3367e+01, 5.1853e-01, 5.9000e-03,
     2.1900e-03],
    [3.1480e-02, 4.0865e+01, 1.9495e+00, 6.6435e+00, 7.6267e-01, 7.9900e-02,
     2.4700e-03],
    [1.7640e-02, 1.8095e+01, 1.7905e+00, 8.1015e+00, 7.8927e-01, 7.1700e-02,
     2.6300e-03],
    [2.2760e-02, 2.9975e+01, 1.6065e+00, 9.4785e+00, 8.8048e-01, 5.9700e-02,
     8.0700e-03],
    [2.2520e-02, 4.5995e+01, 1.5325e+00, 9.0735e+00, 8.9948e-01, 6.2900e-02,
     8.3000e-04],
    [3.5720e-02, 1.5755e+01, 1.5475e+00, 3.8355e+00, 7.2087e-01, 8.1900e-02,
     7.3100e-03],
    [2.4040e-02, 4.2665e+01, 1.7715e+00, 1.1585e+01, 6.6863e-01, 5.8900e-02,
     9.7900e-03],
    [3.5160e-02, 3.0245e+01, 1.5315e+00, 1.1396e+01, 8.6908e-01, 4.0300e-02,
     1.1300e-03],
    [1.0200e-02, 2.9615e+01, 1.9055e+00, 1.2854e+01, 6.1257e-01, 3.2700e-02,
     9.1100e-03],
    [4.5320e-02, 3.9155e+01, 1.6675e+00, 1.0666e+01, 8.8617e-01, 9.5900e-02,
     4.5000e-04],
    [2.5480e-02, 6.6650e+00, 1.9295e+00, 2.5935e+00, 6.6482e-01, 8.8700e-02,
     7.5100e-03],
    [4.8600e-02, 1.9445e+01, 1.5695e+00, 1.2826e+01, 8.6147e-01, 1.5700e-02,
     1.1700e-03],
    [3.7960e-02, 2.9075e+01, 1.8725e+00, 8.7765e+00, 5.2992e-01, 8.5000e-03,
     6.8500e-03],
    [4.3960e-02, 4.3115e+01, 1.6835e+00, 5.6985e+00, 9.2417e-01, 4.8500e-02,
     6.2100e-03],
    [1.5080e-02, 4.4285e+01, 1.8575e+00, 1.9995e+00, 7.0852e-01, 2.4100e-02,
     8.8100e-03],
    [3.0840e-02, 1.6475e+01, 1.5965e+00, 1.2259e+01, 5.3373e-01, 2.7100e-02,
     9.7300e-03],
    [3.6920e-02, 3.3755e+01, 1.8715e+00, 1.1530e+01, 6.6292e-01, 5.0100e-02,
     6.0300e-03],
    [4.1000e-02, 4.5185e+01, 1.9745e+00, 5.8065e+00, 6.7717e-01, 2.7300e-02,
     7.3000e-04],
    [2.2920e-02, 1.2515e+01, 1.7815e+00, 7.6965e+00, 9.0897e-01, 6.7000e-03,
     9.3700e-03],
    [1.3640e-02, 8.1950e+00, 1.6735e+00, 1.8105e+00, 5.1663e-01, 4.1300e-02,
     4.7500e-03],
    [3.2200e-02, 3.2405e+01, 1.9945e+00, 1.7025e+00, 8.8807e-01, 3.5900e-02,
     6.3000e-04],
    [4.7880e-02, 2.1605e+01, 1.6365e+00, 4.3755e+00, 6.6387e-01, 4.9900e-02,
     9.1500e-03],
    [1.1720e-02, 4.2845e+01, 1.8175e+00, 8.9655e+00, 6.6578e-01, 4.0700e-02,
     3.8700e-03],
    [3.3240e-02, 4.6985e+01, 1.9255e+00, 1.0855e+01, 6.0403e-01, 5.3500e-02,
     3.4700e-03],
    [4.2840e-02, 2.8175e+01, 1.6345e+00, 4.8885e+00, 9.6122e-01, 9.1900e-02,
     4.9700e-03],
    [1.1000e-02, 8.2850e+00, 1.7085e+00, 1.3097e+01, 6.5532e-01, 7.9700e-02,
     2.2900e-03],
    [4.2360e-02, 9.7250e+00, 1.7315e+00, 9.9105e+00, 6.1733e-01, 5.5000e-03,
     2.7000e-04],
    [2.4120e-02, 2.2505e+01, 1.8685e+00, 6.9135e+00, 8.5957e-01, 9.9300e-02,
     7.5300e-03],
    [2.8920e-02, 2.3315e+01, 1.6575e+00, 1.5135e+00, 8.6528e-01, 7.7500e-02,
     9.9000e-04],
    [4.7480e-02, 3.8435e+01, 1.5925e+00, 6.4275e+00, 5.1187e-01, 3.7000e-03,
     3.9700e-03],
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    [3.2600e-02, 3.9515e+01, 1.7675e+00, 7.5345e+00, 8.3583e-01, 4.7700e-02,
     7.4900e-03],
    [1.8440e-02, 3.9335e+01, 1.7825e+00, 8.3175e+00, 6.2302e-01, 2.3300e-02,
     3.3500e-03],
    [1.7400e-02, 3.5015e+01, 1.6145e+00, 1.0100e+01, 7.0093e-01, 3.0500e-02,
     3.5300e-03],
    [2.5560e-02, 1.1525e+01, 1.6215e+00, 1.1909e+01, 6.3348e-01, 5.4700e-02,
     1.5700e-03],
    [2.2440e-02, 1.4045e+01, 1.8595e+00, 2.8365e+00, 7.5887e-01, 3.0900e-02,
     2.9100e-03]])

train_y = torch.tensor([
    [ 2.1743e+00],
    [ 4.6957e+00],
    [-1.1596e-01],
    [ 1.7808e+00],
    [ 2.9374e+00],
    [ 4.5322e+00],
    [ 3.5781e+00],
    [ 1.1810e+00],
    [ 7.7894e-01],
    [ 1.0616e+00],
    [-7.1864e-01],
    [ 2.6069e+00],
    [-8.9134e-02],
    [ 2.0863e+00],
    [ 2.3849e+00],
    [ 3.2236e+00],
    [ 2.2584e+00],
    [ 2.4323e+00],
    [ 1.5081e+00],
    [ 1.9815e+00],
    [-4.5730e-01],
    [ 3.3043e+00],
    [ 4.0333e+00],
    [ 5.3843e+00],
    [ 7.1433e-01],
    [ 2.3193e+00],
    [ 1.1071e+00],
    [ 3.8495e+00],
    [ 2.3801e+00],
    [ 2.5313e+00],
    [ 1.1480e+00],
    [ 2.9748e+00],
    [ 4.0386e+00],
    [ 2.9669e-01],
    [ 3.2690e+00],
    [ 1.3723e+00],
    [ 4.5411e+00],
    [ 1.9854e+00],
    [ 4.5231e+00],
    [ 1.4644e+00],
    [ 2.7771e+00],
    [-1.2194e+00],
    [ 2.0792e+00],
    [-1.1882e-01],
    [ 1.7628e+00],
    [ 3.4556e+00],
    [ 2.9343e-01],
    [ 1.9780e+00],
    [ 2.3418e+00],
    [ 5.3219e+00],
    [ 1.2843e+00],
    [ 9.1304e-01],
    [-1.0475e+00],
    [ 1.9564e+00],
    [ 2.2030e+00],
    [ 3.0827e+00],
    [ 3.8750e+00],
    [ 2.5148e+00],
    [ 2.9980e+00],
    [ 4.4516e+00],
    [ 1.8720e+00],
    [ 2.8161e+00],
    [ 1.9146e+00],
    [ 2.8882e+00],
    [ 1.4244e+00],
    [ 4.5422e+00],
    [ 1.6399e+00],
    [ 2.2735e+00],
    [ 3.7304e+00],
    [ 2.0863e+00],
    [-4.8089e-01],
    [ 1.1216e+00],
    [ 1.4910e+00],
    [ 3.6392e+00],
    [ 2.8222e+00],
    [ 3.5203e+00],
    [ 5.7625e+00],
    [ 1.3012e+00],
    [ 5.5025e-01],
    [ 9.8848e-01],
    [ 1.1194e+00],
    [ 1.4845e+00],
    [ 1.2226e+00],
    [ 3.3138e+00],
    [ 1.0426e+00],
    [ 2.2471e+00],
    [ 1.3411e+00],
    [ 5.7177e+00],
    [ 4.5401e+00],
    [ 3.1662e+00],
    [ 1.4584e+00],
    [ 3.6067e+00],
    [ 2.8628e+00],
    [ 4.0613e+00],
    [ 2.6778e+00],
    [ 1.7796e+00],
    [ 2.7138e+00],
    [ 3.9396e+00],
    [ 8.7683e-01],
    [ 2.4370e+00],
    [ 1.3658e+00],
    [ 6.0429e-01],
    [ 5.2041e+00],
    [-3.3878e-01],
    [ 2.7597e-01],
    [ 5.7402e+00],
    [ 2.3492e+00],
    [-6.2418e-01],
    [ 1.7953e-01],
    [ 1.2476e+00],
    [ 3.8466e+00],
    [ 3.1815e+00],
    [ 1.6956e+00],
    [ 1.1026e+00],
    [ 3.1850e+00],
    [ 1.8288e+00],
    [ 2.0765e+00],
    [ 7.2247e-01],
    [ 3.3432e+00],
    [ 2.7565e+00],
    [ 8.4799e-01],
    [ 7.0737e-01],
    [ 2.2225e+00],
    [ 2.1320e+00],
    [ 2.9315e+00],
    [ 5.2570e+00],
    [ 3.5410e+00],
    [ 6.3282e-01],
    [ 5.0961e+00],
    [ 5.1952e+00],
    [ 9.2022e-01],
    [ 4.7569e+00],
    [ 2.9478e+00],
    [ 3.3915e+00],
    [ 1.4620e+00],
    [ 2.0248e+00],
    [ 3.7657e+00],
    [ 9.2594e-01],
    [ 6.2498e-01],
    [ 4.4038e+00],
    [ 9.6859e-01],
    [ 2.8639e+00],
    [ 5.4233e+00],
    [-1.3157e-01],
    [ 1.6851e+00],
    [ 1.8201e+00],
    [ 2.2499e+00],
    [ 1.8626e+00],
    [ 4.6094e+00],
    [ 5.2024e+00],
    [ 9.0005e-01],
    [ 3.8737e+00],
    [ 2.1583e+00],
    [ 2.6656e+00],
    [ 5.3761e+00],
    [ 3.6815e+00],
    [ 1.0891e+00],
    [ 1.5778e-01],
    [ 2.7724e+00],
    [ 2.2021e+00],
    [ 4.6330e+00],
    [ 5.4431e+00],
    [ 1.7554e+00],
    [ 3.7453e+00],
    [ 1.9830e+00],
    [ 1.4433e+00],
    [ 9.8675e-01],
    [ 5.6188e-01],
    [ 2.0979e+00],
    [ 4.5619e+00],
    [ 1.9540e+00],
    [-1.1324e+00],
    [ 3.6509e-01],
    [ 4.2373e-01],
    [ 3.7806e+00],
    [ 1.7086e+00],
    [ 7.6614e-01],
    [ 1.1420e+00],
    [ 1.6897e+00],
    [ 3.5732e+00],
    [ 3.8650e+00],
    [ 5.6612e-01],
    [ 3.1652e+00],
    [ 2.5904e+00],
    [-4.4512e-01],
    [ 2.4177e+00],
    [ 1.4581e+00],
    [ 2.4941e+00],
    [ 1.1331e+00],
    [ 1.4996e+00],
    [ 1.4506e+00],
    [ 1.2101e+00],
    [ 4.3661e+00],
    [ 3.4513e+00],
    [ 2.8402e+00],
    [ 2.5823e+00],
    [ 4.6850e+00],
    [ 2.4991e+00],
    [ 2.3529e+00],
    [ 1.9080e+00],
    [ 2.5847e+00],
    [ 3.8108e+00],
    [-6.1697e-01],
    [ 1.6368e+00],
    [ 2.2007e+00],
    [ 3.1122e+00],
    [ 2.5771e+00],
    [ 4.9207e+00],
    [ 1.9262e+00],
    [ 3.6927e+00],
    [ 2.2270e+00],
    [ 1.6163e+00],
    [ 3.3793e+00],
    [ 4.7123e+00],
    [ 3.6472e+00],
    [ 2.6794e+00],
    [ 1.8863e+00],
    [ 1.4135e+00],
    [ 2.1401e+00],
    [ 4.4333e+00],
    [ 3.4774e+00],
    [ 1.6868e+00],
    [ 2.2722e-01],
    [ 4.5379e+00],
    [ 2.6463e+00],
    [ 1.7861e+00],
    [ 1.5696e+00],
    [ 1.0179e+00],
    [ 1.7176e+00],
    [ 2.0878e+00],
    [ 2.1977e+00],
    [ 1.3079e+00],
    [ 3.0531e+00],
    [ 3.0415e+00],
    [ 3.0105e+00],
    [ 1.9865e+00],
    [ 2.5631e+00],
    [ 4.6996e-01],
    [ 2.7671e+00],
    [ 3.5494e+00],
    [-1.6034e+00],
    [ 4.5454e+00],
    [ 1.4066e+00],
    [ 4.3003e+00],
    [ 1.6961e+00],
    [ 1.7685e+00],
    [-2.6783e-01],
    [-1.1119e-01],
    [ 1.8734e+00],
    [ 4.8617e+00],
    [ 3.6154e+00],
    [-1.5843e-01],
    [ 1.0183e+00],
    [ 5.3973e+00],
    [ 3.3901e+00],
    [ 2.1614e+00],
    [ 4.1328e+00],
    [ 4.9674e-01],
    [ 2.2679e+00],
    [ 2.7528e+00],
    [ 2.1096e+00],
    [ 4.1187e+00],
    [ 1.1763e+00],
    [ 8.4389e-01],
    [ 8.1070e-01],
    [ 1.8023e+00],
    [-2.8910e-03],
    [ 2.3066e+00],
    [ 4.6830e+00],
    [ 5.0495e+00],
    [ 7.8840e-01],
    [ 2.1122e+00],
    [ 2.4062e+00],
    [ 9.1734e-01],
    [ 1.5164e-01],
    [ 4.1833e+00],
    [ 1.9462e+00],
    [ 1.0708e+00],
    [ 8.6457e-01],
    [ 9.5329e-01],
    [ 2.9187e+00],
    [ 1.7403e+00],
    [-2.7705e-01],
    [ 2.5634e+00],
    [ 2.7781e+00],
    [ 1.0038e+00],
    [ 2.3161e+00],
    [ 9.4054e-01],
    [ 5.3699e+00],
    [-1.7289e-01],
    [ 1.7415e+00],
    [ 1.8357e+00],
    [ 1.4147e+00],
    [ 2.6790e+00],
    [ 6.5715e-01],
    [ 5.4224e+00],
    [ 4.0507e+00],
    [ 2.4618e+00],
    [ 3.7224e+00],
    [-3.1234e-01],
    [ 5.2510e-01],
    [ 1.0904e+00],
    [ 2.9703e+00],
    [ 3.1230e+00],
    [ 4.1744e+00],
    [ 3.1950e+00],
    [ 1.0616e+00],
    [ 4.1007e+00],
    [ 1.9450e-01],
    [ 2.0939e+00],
    [ 9.7366e-01],
    [ 4.9941e+00],
    [ 1.3886e+00],
    [ 1.2702e+00],
    [ 1.1623e+00],
    [ 1.0914e+00],
    [ 4.7595e+00],
    [ 5.5165e+00],
    [ 1.8937e+00],
    [ 3.1651e-01],
    [ 2.9186e+00],
    [ 3.6607e-01],
    [ 2.5902e+00],
    [ 4.1939e+00],
    [ 3.2037e+00],
    [ 4.8482e+00],
    [ 2.9179e+00],
    [ 2.6266e+00],
    [ 3.6330e+00],
    [ 2.1107e+00],
    [ 2.0435e+00],
    [ 4.0294e+00],
    [ 4.1403e+00],
    [ 3.9560e+00],
    [ 3.6491e+00],
    [ 6.3370e-02],
    [ 2.6406e+00],
    [ 5.4708e+00],
    [ 2.5947e+00],
    [ 1.8416e+00],
    [ 2.0929e-01],
    [ 3.5463e+00],
    [ 1.3630e+00],
    [ 2.9029e+00],
    [ 4.5957e+00],
    [ 2.2284e+00],
    [ 3.8843e+00],
    [ 9.7586e-01],
    [ 1.5264e+00],
    [ 1.3026e+00],
    [ 4.8512e+00],
    [ 2.3869e+00],
    [ 3.2765e+00],
    [ 1.3181e+00],
    [ 4.4726e+00],
    [ 2.9717e+00],
    [ 7.7538e-01],
    [ 9.5782e-02],
    [ 5.9856e-01],
    [ 1.8793e+00],
    [ 1.7033e+00],
    [-7.3544e-02],
    [ 2.6673e+00],
    [ 5.2618e+00],
    [ 3.2704e+00],
    [ 7.3382e-01],
    [ 3.1594e+00],
    [ 2.8281e+00],
    [ 1.5330e+00],
    [ 2.9248e+00],
    [ 1.3769e+00],
    [ 3.3507e+00],
    [ 1.2081e+00],
    [ 4.9458e+00],
    [ 2.9866e+00],
    [ 2.2538e+00],
    [ 4.1918e+00],
    [ 9.6699e-01],
    [ 2.6238e+00],
    [ 4.0743e-01],
    [ 2.4295e+00],
    [ 3.8483e+00],
    [ 1.4629e+00],
    [ 3.5258e+00],
    [ 1.5203e+00],
    [ 2.9767e+00],
    [-3.1828e-02],
    [ 8.7529e-01],
    [ 5.9369e-01],
    [-1.3175e+00],
    [-8.4835e-02],
    [ 3.9937e+00],
    [ 2.5711e+00],
    [ 9.8123e-01],
    [ 1.1162e+00],
    [ 4.3127e+00],
    [ 9.8791e-01],
    [ 2.6222e+00],
    [ 6.3274e-01],
    [ 4.0730e+00],
    [ 1.8950e+00],
    [ 2.3615e+00],
    [ 1.8567e+00],
    [ 1.4608e+00],
    [ 4.2874e-01],
    [ 5.3665e+00],
    [ 5.3335e+00],
    [ 1.8681e+00],
    [ 4.6781e+00],
    [ 4.4942e-01],
    [ 8.0698e-01],
    [ 2.1236e+00],
    [ 1.1974e+00],
    [ 2.1510e+00],
    [ 1.5997e+00],
    [ 4.6629e+00],
    [ 1.5502e+00],
    [ 4.4369e+00],
    [ 3.4827e+00],
    [ 1.8196e+00],
    [-4.9467e-01],
    [ 4.7623e+00],
    [ 4.2460e+00],
    [ 4.3793e+00],
    [ 9.7443e-01],
    [ 3.8735e+00],
    [ 2.1834e+00],
    [ 5.5092e+00],
    [ 2.9697e+00],
    [-1.0918e+00],
    [ 2.2256e+00],
    [-8.5351e-01],
    [-1.5524e-01],
    [ 2.9838e+00],
    [ 3.5454e+00],
    [ 4.4012e+00],
    [ 2.9224e+00],
    [ 9.2362e-01],
    [ 1.3193e+00],
    [ 3.7343e+00],
    [ 4.0905e-01],
    [ 2.7693e+00],
    [ 1.5760e+00],
    [ 1.5499e+00],
    [ 2.1830e+00],
    [-7.6546e-01],
    [ 1.4411e+00],
    [ 3.5230e+00],
    [ 3.5090e+00],
    [ 4.4504e-01],
    [ 4.2744e+00],
    [ 1.9580e+00],
    [ 1.7152e+00],
    [ 5.0790e+00],
    [ 4.8307e+00],
    [ 2.6501e+00],
    [ 3.0501e+00],
    [ 3.0828e+00],
    [ 2.8142e+00],
    [ 9.9297e-01],
    [ 1.0842e-01],
    [ 5.0652e+00],
    [ 5.3247e+00],
    [ 2.4860e+00],
    [ 1.9157e+00],
    [ 1.9875e+00],
    [ 4.0103e+00],
    [ 2.6489e+00],
    [ 3.6781e+00],
    [ 2.4142e+00],
    [ 3.3584e+00],
    [ 3.1605e+00],
    [ 3.6403e-01],
    [ 5.0327e+00],
    [ 1.7275e+00],
    [ 2.1059e+00],
    [ 1.3904e+00],
    [ 1.6000e+00],
    [ 5.7289e-01],
    [ 3.3504e+00],
    [ 3.7420e+00],
    [ 2.8768e+00],
    [-3.6229e-01],
    [ 5.4868e-01],
    [ 8.2206e-01],
    [ 2.0763e+00],
    [ 2.2001e+00],
    [ 4.2265e+00],
    [ 5.2503e+00],
    [ 2.5936e+00],
    [-1.1987e+00],
    [ 2.7531e+00],
    [ 3.3446e+00],
    [ 1.9106e+00],
    [ 3.0574e+00],
    [ 2.2389e+00],
    [ 2.2253e+00],
    [ 1.9534e+00],
    [ 2.4863e+00],
    [ 2.6057e+00]
])

# --------------------------------------------------------------------------- #
# Test points drawn from uniform distribution
# --------------------------------------------------------------------------- #
test_X = torch.tensor([
    [1.8321e-02, 4.7062e+01, 1.7140e+00, 7.1926e+00, 5.8119e-01, 9.0300e-02,
     5.8830e-03],
    [4.8415e-02, 3.9068e+01, 1.9456e+00, 4.2814e+00, 7.8691e-01, 6.2157e-02,
     2.3631e-03],
    [3.9575e-02, 3.8121e+01, 1.8828e+00, 4.5891e+00, 5.3743e-01, 3.1606e-02,
     6.9375e-03],
    [3.5756e-02, 2.3752e+01, 1.5028e+00, 3.9136e+00, 5.5181e-01, 2.1607e-03,
     4.1010e-03],
    [2.6394e-02, 4.9054e+01, 1.6090e+00, 2.7706e+00, 5.1480e-01, 4.8289e-02,
     8.8922e-03],
    [3.1953e-02, 5.6796e+00, 1.7372e+00, 3.5388e+00, 7.3318e-01, 4.0067e-02,
     2.1139e-03],
    [1.2796e-02, 4.6932e+01, 1.7391e+00, 1.3389e+01, 5.0022e-01, 3.8733e-02,
     2.2983e-03],
    [2.7452e-02, 4.0514e+01, 1.6678e+00, 1.1670e+01, 5.0164e-01, 3.6830e-02,
     9.5681e-03],
    [2.0536e-02, 2.5692e+01, 1.9338e+00, 1.0863e+01, 9.4334e-01, 8.1506e-02,
     8.1311e-03],
    [2.3917e-02, 3.4758e+01, 1.9517e+00, 1.3517e+01, 7.5617e-01, 8.4040e-02,
     8.5139e-03],
    [1.6532e-02, 1.0795e+01, 1.9025e+00, 7.7135e+00, 9.5760e-01, 2.9464e-02,
     2.2375e-05],
    [1.0950e-02, 2.4599e+01, 1.7787e+00, 1.4037e+01, 7.1841e-01, 3.0620e-02,
     8.0031e-03],
    [4.6241e-02, 4.8122e+01, 1.9169e+00, 4.3702e+00, 8.8714e-01, 5.3552e-03,
     8.6029e-03],
    [4.1012e-02, 3.6737e+01, 1.8144e+00, 6.2366e+00, 9.4705e-01, 9.0988e-02,
     8.3032e-03],
    [1.2032e-02, 3.5333e+01, 1.8935e+00, 5.4062e+00, 9.2938e-01, 8.2844e-02,
     4.1949e-03],
    [3.9128e-02, 2.6126e+01, 1.8645e+00, 9.9896e+00, 5.3418e-01, 3.7173e-02,
     1.7980e-03],
    [4.9210e-02, 3.5911e+01, 1.9261e+00, 1.4822e+01, 6.1260e-01, 7.5400e-02,
     3.4626e-03],
    [4.7416e-02, 4.1533e+01, 1.8375e+00, 1.3124e+01, 7.7375e-01, 6.5459e-02,
     8.2843e-03],
    [1.2567e-02, 1.5018e+01, 1.9560e+00, 1.0933e+01, 6.5984e-01, 1.7812e-03,
     8.3674e-03],
    [2.6310e-02, 2.5926e+01, 1.9045e+00, 1.2906e+01, 5.4491e-01, 9.5597e-02,
     3.4892e-03],
    [2.1398e-02, 2.6054e+01, 1.6773e+00, 7.8091e+00, 9.1517e-01, 8.9972e-02,
     7.8526e-03],
    [1.3060e-02, 4.7905e+01, 1.6218e+00, 5.4807e+00, 8.9210e-01, 7.3208e-02,
     7.5685e-03],
    [2.5500e-02, 2.2278e+01, 1.5637e+00, 1.9580e+00, 7.6607e-01, 2.8287e-02,
     3.9312e-03],
    [1.7688e-02, 2.5906e+01, 1.8513e+00, 3.3983e+00, 7.4134e-01, 8.4010e-02,
     1.4166e-03],
    [2.1300e-02, 2.6385e+01, 1.7120e+00, 1.2834e+01, 6.9246e-01, 8.4185e-02,
     1.2138e-03],
    [3.2359e-02, 1.4497e+01, 1.8747e+00, 1.0565e+01, 6.0004e-01, 1.3177e-02,
     6.5252e-03],
    [4.9294e-02, 1.0597e+01, 1.6047e+00, 1.1463e+01, 8.6255e-01, 2.6204e-02,
     7.1979e-03],
    [3.8403e-02, 4.7241e+01, 1.8809e+00, 9.7144e+00, 5.4558e-01, 9.8699e-02,
     7.3356e-03],
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    [1.9601e-02, 3.0567e+01, 1.5801e+00, 9.6260e+00, 5.6296e-01, 4.2657e-02,
     2.9236e-03],
    [3.1617e-02, 3.9291e+01, 1.5970e+00, 1.1365e+01, 9.3126e-01, 5.0273e-02,
     8.9956e-03],
    [3.9967e-02, 1.5312e+01, 1.6296e+00, 6.8985e+00, 8.2244e-01, 6.9177e-02,
     4.2328e-03],
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     6.8548e-03],
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     6.9804e-03],
    [3.3894e-02, 1.6633e+01, 1.6039e+00, 2.0147e+00, 7.5756e-01, 3.7599e-02,
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    [2.1021e-02, 3.2447e+01, 1.7043e+00, 2.5194e+00, 7.6357e-01, 6.8875e-03,
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     9.0819e-03],
    [2.9722e-02, 4.0349e+01, 1.6723e+00, 1.4239e+01, 6.0001e-01, 4.2330e-02,
     8.2809e-03],
    [2.6385e-02, 5.2592e+00, 1.5566e+00, 1.2875e+01, 6.5092e-01, 9.6114e-02,
     6.7690e-03],
    [1.5758e-02, 3.1616e+01, 1.5416e+00, 5.4714e+00, 6.1695e-01, 7.3729e-02,
     2.4934e-03],
    [4.7913e-02, 2.5459e+01, 1.8857e+00, 1.4194e+01, 8.6780e-01, 9.1611e-02,
     7.5849e-03],
    [2.8578e-02, 2.6035e+01, 1.5648e+00, 4.0283e+00, 7.5531e-01, 5.4560e-02,
     4.5542e-03],
    [4.0676e-02, 4.7872e+01, 1.5023e+00, 9.0545e+00, 5.3510e-01, 9.4142e-02,
     3.7660e-03],
    [2.1507e-02, 3.5742e+01, 1.6711e+00, 6.6870e+00, 6.6944e-01, 2.6188e-02,
     7.8780e-03],
    [1.8989e-02, 4.7017e+01, 1.9703e+00, 6.0397e+00, 6.5207e-01, 6.9744e-02,
     5.9859e-03],
    [2.4772e-02, 4.4031e+01, 1.9335e+00, 1.2951e+01, 5.1090e-01, 1.7484e-02,
     3.8018e-03],
    [2.2766e-02, 4.7913e+01, 1.7921e+00, 6.0751e+00, 9.3432e-01, 8.6951e-02,
     8.3022e-04],
    [1.5177e-02, 4.5629e+01, 1.7351e+00, 4.6328e+00, 9.6763e-01, 5.5239e-02,
     8.2445e-03],
    [1.4384e-02, 3.2312e+01, 1.6139e+00, 1.1928e+01, 6.9921e-01, 2.0362e-02,
     6.6831e-03],
    [1.8165e-02, 3.9065e+01, 1.5667e+00, 6.7476e+00, 5.8482e-01, 4.2365e-02,
     5.9453e-03],
    [2.4402e-02, 1.5755e+01, 1.7856e+00, 7.0201e+00, 7.9229e-01, 5.4919e-02,
     8.0324e-03],
    [2.6143e-02, 4.7044e+01, 1.9153e+00, 6.2572e+00, 7.1250e-01, 6.5391e-02,
     1.3917e-03],
    [4.5390e-02, 4.2065e+01, 1.8331e+00, 3.3236e+00, 5.8644e-01, 8.6719e-02,
     5.5766e-03],
    [1.9162e-02, 2.8164e+01, 1.9470e+00, 4.7394e+00, 8.9120e-01, 8.7844e-02,
     7.3705e-03],
    [2.4820e-02, 3.5216e+01, 1.9319e+00, 5.6319e+00, 7.1064e-01, 6.9953e-02,
     6.2459e-03],
    [2.4998e-02, 4.4993e+01, 1.9388e+00, 9.0490e+00, 6.4311e-01, 9.7268e-02,
     8.1623e-03],
    [1.5744e-02, 6.2258e+00, 1.7003e+00, 3.1043e+00, 5.8521e-01, 5.3468e-02,
     4.8789e-03],
    [4.9296e-02, 1.5692e+01, 1.5830e+00, 1.3553e+01, 5.4428e-01, 8.5186e-02,
     6.3130e-03],
    [3.4269e-02, 8.8232e+00, 1.5876e+00, 2.6158e+00, 8.4173e-01, 3.9901e-02,
     5.7050e-03],
    [2.7714e-02, 3.1355e+01, 1.8334e+00, 4.8824e+00, 6.6533e-01, 7.4490e-02,
     1.5618e-03],
    [4.6574e-02, 2.7038e+01, 1.7181e+00, 1.0638e+01, 7.5525e-01, 6.8522e-03,
     2.7013e-03],
    [2.3907e-02, 2.5771e+01, 1.9863e+00, 1.0740e+01, 9.5443e-01, 4.1909e-02,
     7.0699e-03],
    [3.6940e-02, 1.4438e+01, 1.8485e+00, 1.2743e+01, 6.0804e-01, 9.3177e-02,
     3.4116e-03],
    [3.0518e-02, 1.9100e+01, 1.6450e+00, 7.2850e+00, 8.4250e-01, 2.2250e-02,
     8.1941e-03],
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     7.2617e-03],
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     2.4469e-03],
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     9.3045e-04],
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     5.8759e-03],
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     1.5029e-03],
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     6.1124e-03],
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     7.8475e-03],
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     9.2423e-03],
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     1.0658e-03],
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     6.6781e-03],
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     4.1006e-03],
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     9.0567e-03],
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     1.7118e-03],
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     4.1141e-05],
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     3.2026e-03],
    [3.1880e-02, 1.1306e+01, 1.9377e+00, 1.0528e+01, 7.7939e-01, 4.7488e-02,
     1.2086e-04],
    [4.2516e-02, 3.1648e+01, 1.8655e+00, 1.4394e+01, 9.3722e-01, 2.9147e-02,
     5.5092e-03]
])

# --------------------------------------------------------------------------- #
# Expand one dimension to be batched
# --------------------------------------------------------------------------- #
N_batch_size = 50
train_X_batch = train_X.repeat(N_batch_size, 1, 1)
train_y_batch = train_y.repeat(N_batch_size, 1, 1)
test_X_batch = test_X.repeat(N_batch_size, 1, 1)

# --------------------------------------------------------------------------- #
# Instantiate GP models and fit
# --------------------------------------------------------------------------- #
train_X_dim = train_X_batch.shape[-1]
train_y_dim = train_y_batch.shape[-1]
# Get the batch size
N_batch_size = train_X_batch.shape[0]

# Train the batched GP model
gp = SingleTaskGP(
    train_X_batch, train_y_batch,
    input_transform=InputStandardize(
        d=train_X_dim, batch_shape=torch.Size([N_batch_size])),
    outcome_transform=Standardize(
        m=train_y_dim, batch_shape=torch.Size([N_batch_size]))
)
mll = gpytorch.mlls.ExactMarginalLogLikelihood(gp.likelihood, gp)
fit_gpytorch_model(mll)  # Using BoTorch's routine

with torch.no_grad(), gpytorch.settings.fast_pred_var():
    # Total model output variance
    pred_test_X = gp.posterior(test_X_batch)
    pred_test_X_mean = pred_test_X.mean
    # Take the variance wrt. test_X for only one simulation batch
    var_E_pred_test_X = torch.var(pred_test_X_mean[0, :, :])

print(var_E_pred_test_X)

# Train the non-batched GP model
gp_single = SingleTaskGP(
    train_X_batch[0], train_y_batch[0],
    input_transform=InputStandardize(d=train_X_dim),
    outcome_transform=Standardize(m=train_y_dim)
)
mll_single = gpytorch.mlls.ExactMarginalLogLikelihood(
    gp_single.likelihood, gp_single)
fit_gpytorch_model(mll_single)  # Using BoTorch's routine

with torch.no_grad(), gpytorch.settings.fast_pred_var():
    # Total model output variance
    pred_test_X_single = gp_single.posterior(test_X_batch[0])
    pred_test_X_mean_single = pred_test_X_single.mean
    # Take the variance wrt. test_X for only one simulation batch
    var_E_pred_test_X_single = torch.var(pred_test_X_mean_single[:, :])

print(var_E_pred_test_X_single)

@saitcakmak
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The hyper parameters of the two GP models seem to be quite different. Looking into what's causing this

@takafusui
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Thank you. For me, the mean and the variance from the non-batched GP model are much more reliable and consistent with what I expect when further post-processing.

@saitcakmak
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Note also that fitting a batched model under the hood actually runs a number if individual fits and then reassembles the batched model (https://github.com/pytorch/botorch/blob/main/botorch/fit.py#L86-L95) since fitting the batched model results in a larger optimization problem to be solved (with the optimizer not being aware that the different batches are actually independent optimization problems).

@Balandat , this is not happening in this case since the MLL is not a SumMLL. We're trying to fit it in one large batch, which I suspect is the underlying reason for the poor fit

@Balandat
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I see. We should consider automating the splitting in this case.

@takafusui the issue here is that the larger much higher-dim problem for the batched mod is a lot harder and so you get worse fits. for now I recommend fitting individual models and then assembling them into a batched model manually if you want to batch-evaluate.

@saitcakmak
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TL;DR: This is a larger bug with how prior terms are added in ExactMarginalLogLikelihood.

Looks like this is a larger bug with the MLL. I collect the state dicts produced by fitting the single model and the batch model and load them (after appropriate reshaping) to both single and batch models. Single model produces smaller MLL with the single state dict and batch model produces smaller MLL with the batch state dict. Looking closer, the log probability of observations on the prior is the same across the two models. Where things differ is the priors added in https://github.com/cornellius-gp/gpytorch/blob/master/gpytorch/mlls/exact_marginal_log_likelihood.py#L42-L43.

The problem here is that we add the sum of all prior terms to all results. The res is a 50-dim tensor and the evaluation of the priors produces a 50, 50 x 1 or 50 x 1 x 7 (d) terms. So, what ends up happening is the priors are accounted for N_batch times in the MLL. As a result MAP inference ends up being heavily skewed towards the prior whenever we batch train.

@Balandat
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Yeah the inconsistency with gpytorch prior shapes has been a long standing issue, unfortunately: cornellius-gp/gpytorch#1317, cornellius-gp/gpytorch#1318. We really ought to fix this...

@Balandat Balandat added bug Something isn't working upstream issue labels Jun 14, 2022
@saitcakmak saitcakmak mentioned this issue Jun 14, 2022
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@saitcakmak
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Looks like cornellius-gp/gpytorch#1318 is the same issue, though cornellius-gp/gpytorch#1317 is different. I linked a PR that worked in my testing NB.

@takafusui
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@takafusui the issue here is that the larger much higher-dim problem for the batched mod is a lot harder and so you get worse fits. for now I recommend fitting individual models and then assembling them into a batched model manually if you want to batch-evaluate.

Thank you very much for your close investigations. I would like to clarify: do you recommend using botorch.models.model.ModelList or something else?

@saitcakmak
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I would like to clarify: do you recommend using botorch.models.model.ModelList or something else?

Since you're only using GP models, you can use ModelListGP instead (ModelList would also work but the GP version is preferred). If you pair that with a SumMarginalLogLikelihood, then you can train it very easily with fit_gpytorch_model as if it is a single model. One thing worth noting is that ModelListGP will act like a single GP model with n_batch outputs (for posterior calls etc). To access the individual models, you can use model_list.models[<index of individual model>].

@saitcakmak
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Example usage from https://botorch.org/tutorials/multi_objective_bo:

def initialize_model(train_x, train_obj):
    # define models for objective and constraint
    train_x = normalize(train_x, problem.bounds)
    models = []
    for i in range(train_obj.shape[-1]):
        train_y = train_obj[..., i:i+1]
        train_yvar = torch.full_like(train_y, NOISE_SE[i] ** 2)
        models.append(
            FixedNoiseGP(train_x, train_y, train_yvar, outcome_transform=Standardize(m=1))
        )
    model = ModelListGP(*models)
    mll = SumMarginalLogLikelihood(model.likelihood, model)
    return mll, model

@takafusui
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Thank you very much for the series of comments/investigations/suggestions. I really appreciate them.

@saitcakmak
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Closing this as the bug has been fixed since GPyTorch 1.8.1

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