Assorted PairwiseGP stability improvements

Summary:
Main changes include

- Prior update: updated prior for better model fit and better numerical stability
- Utility heuristic initialization: previously we initialize the latent utility (i.e., the latent function value) randomly, which may lead to extreme likelihood values and unnecessarily longer optimization time. We now use comparison-winning-count-based heuristics to initialize the utility weights.
- Ensuring covariance is PSD: despite the numerical instability of working on logit/probit scale, at the minimum, the covariance between training datapoints should be PSD by definition (e.g., when using a scaled RBF kernel). If this assumption is not hold, the accumulation of error is going to lead to many other undesirable consequences downstream. To resolve this, check and add jitter to guarantee the PSD-ness of covariance matrices.

Differential Revision: D44137937

fbshipit-source-id: 337284b4ef0d58878507219344e6a7e557b9f5b1

botorch/models/pairwise_gp.py

@@ -49,6 +49,7 @@
 from gpytorch.priors.torch_priors import GammaPrior
 from linear_operator.operators import LinearOperator, RootLinearOperator
 from linear_operator.utils.cholesky import psd_safe_cholesky
+from linear_operator.utils.errors import NotPSDError
 from scipy import optimize
 from torch import float32, float64, Tensor
 from torch.nn.modules.module import _IncompatibleKeys
@@ -84,6 +85,34 @@ def _scaled_psd_safe_cholesky(
     return chol
 
 
+def _ensure_psd_with_jitter(
+    M: Tensor,
+    scale: Union[float, Tensor] = 1.0,
+    jitter: float = 1e-8,
+    max_tries: int = 3,
+) -> Tensor:
+    scaled_M = M / scale
+    new_jitter = 0
+    for i in range(max_tries):
+        scaled_M = scaled_M + new_jitter * torch.diag_embed(
+            torch.ones(
+                scaled_M.shape[:-1], device=scaled_M.device, dtype=scaled_M.dtype
+            )
+        )
+        _, info = torch.linalg.cholesky_ex(scaled_M)
+        is_psd = (info == 0).all()
+        if is_psd:
+            break
+        else:
+            new_jitter = jitter * (10**i) - new_jitter
+    if not is_psd:
+        raise NotPSDError(
+            "Matrix not positive definite after repeatedly adding jitter "
+            f"up to {jitter * (10**i):.1e}."
+        )
+    return scaled_M * scale
+
+
 # Why we subclass GP even though it provides no functionality:
 # if this subclassing is removed, we get the following GPyTorch error:
 # "RuntimeError: All MarginalLogLikelihood objects must be given a GP object as
@@ -214,7 +243,7 @@ def __init__(
         # at 0 or 1
         if covar_module is None:
             os_lb, os_ub = 1e-2, 1e2
-            ls_prior = GammaPrior(1.2, 0.5)
+            ls_prior = GammaPrior(concentration=3.0, rate=3.5)
             ls_prior_mode = (ls_prior.concentration - 1) / ls_prior.rate
             covar_module = ScaleKernel(
                 RBFKernel(
@@ -224,6 +253,7 @@ def __init__(
                     lengthscale_constraint=GreaterThan(
                         lower_bound=1e-4, transform=None, initial_value=ls_prior_mode
                     ),
+                    dtype=torch.float64,
                 ),
                 outputscale_prior=SmoothedBoxPrior(a=os_lb, b=os_ub),
                 # make sure we won't get extreme values for the output scale
@@ -232,6 +262,7 @@ def __init__(
                     upper_bound=os_ub * 2.0,
                     initial_value=1.0,
                 ),
+                dtype=torch.float64,
             )
         if not isinstance(covar_module, ScaleKernel):
             raise UnsupportedError("PairwiseGP must be used with a ScaleKernel.")
@@ -290,8 +321,16 @@ def _has_no_data(self):
 
     def _calc_covar(self, X1: Tensor, X2: Tensor) -> Union[Tensor, LinearOperator]:
         r"""Calculate the covariance matrix given two sets of datapoints"""
-        covar = self.covar_module(X1, X2)
-        return covar.to_dense()
+        covar = self.covar_module(X1, X2).to_dense()
+        # making sure covar is PSD when it's a covariance matrix
+        if X1 is X2:
+            scale = self.covar_module.outputscale.unsqueeze(-1).unsqueeze(-1).detach()
+            covar = _ensure_psd_with_jitter(
+                M=covar,
+                scale=scale,
+                jitter=self._jitter,
+            )
+        return covar
 
     def _update_covar(self, datapoints: Tensor) -> None:
         r"""Update values derived from the data and hyperparameters
@@ -302,9 +341,10 @@ def _update_covar(self, datapoints: Tensor) -> None:
             datapoints: (Transformed) datapoints for finding f_max
         """
         self.covar = self._calc_covar(datapoints, datapoints)
+        scale = self.covar_module.outputscale.unsqueeze(-1).unsqueeze(-1).detach()
         self.covar_chol = _scaled_psd_safe_cholesky(
             M=self.covar,
-            scale=self.covar_module.outputscale.unsqueeze(-1).unsqueeze(-1),
+            scale=scale,
             jitter=self._jitter,
         )
         self.covar_inv = torch.cholesky_inverse(self.covar_chol)
@@ -465,8 +505,16 @@ def _update(self, datapoints: Tensor, **kwargs) -> None:
                     .cpu()
                     .numpy()
                 )
-                # initialize x0 using std normal but clip by 3 std to keep it bounded
-                x0 = np.random.standard_normal(init_x0_size).clip(min=-3, max=3)
+                # Heuristic intialization using winning count with perturbation
+                # to avoid extreme or unprobable likelihood values
+                win_count = self.D.sum(dim=-2).detach().numpy()
+                wc_mean, wc_std = (
+                    win_count.mean(axis=-1)[..., np.newaxis],
+                    win_count.std(axis=-1)[..., np.newaxis],
+                )
+                x0 = (win_count - wc_mean) / wc_std
+                # adding random perturbation to in case get stuck at strange init values
+                x0 = x0 + 0.05 * np.random.standard_normal(init_x0_size)
                 # scale x0 to be on roughly the right scale
                 x0 = x0 * sqrt_scale
             else:
@@ -933,14 +981,15 @@ def forward(self, datapoints: Tensor) -> MultivariateNormal:
 
             output_mean, output_covar = pred_mean, pred_covar
 
+        scale = self.covar_module.outputscale.unsqueeze(-1).unsqueeze(-1).detach()
         post = MultivariateNormal(
             mean=output_mean,
             # output_covar is sometimes non-PSD
             # perform a cholesky decomposition to check and amend
             covariance_matrix=RootLinearOperator(
                 _scaled_psd_safe_cholesky(
                     output_covar,
-                    scale=self.covar_module.outputscale.unsqueeze(-1).unsqueeze(-1),
+                    scale=scale,
                     jitter=self._jitter,
                 )
             ),