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Add jitter to Cholesky factorization in Gaussian ops #3151

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merged 11 commits into from
Oct 30, 2022
Merged

Add jitter to Cholesky factorization in Gaussian ops #3151

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be0e68d
Attempt to add jitter to Gaussian ops
fritzo Oct 27, 2022
102abf5
Isolate changes to pyro.ops.tensor_utils
fritzo Oct 28, 2022
ae184e8
Add a warning
fritzo Oct 28, 2022
fd40b54
Simplify
fritzo Oct 28, 2022
d78b83c
Simplify
fritzo Oct 28, 2022
d542b83
Simplify to fixed jitter based on finfo.eps
fritzo Oct 28, 2022
2f5e184
Simplify
fritzo Oct 28, 2022
fdb1be1
Rename cholesky -> safe_cholesky
fritzo Oct 28, 2022
2da3c75
Allow disabling jitter
fritzo Oct 28, 2022
152b913
Address review comment
fritzo Oct 30, 2022
3e273e7
Switch to column-wise max, increase jitter
fritzo Oct 30, 2022
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10 changes: 5 additions & 5 deletions pyro/distributions/hmm.py
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Original file line number Diff line number Diff line change
Expand Up @@ -20,7 +20,7 @@
)
from pyro.ops.indexing import Vindex
from pyro.ops.special import safe_log
from pyro.ops.tensor_utils import cholesky, cholesky_solve
from pyro.ops.tensor_utils import cholesky_solve, safe_cholesky

from . import constraints
from .torch import Categorical, Gamma, Independent, MultivariateNormal
Expand Down Expand Up @@ -628,9 +628,9 @@ def filter(self, value):

# Convert to a distribution
precision = logp.precision
loc = cholesky_solve(logp.info_vec.unsqueeze(-1), cholesky(precision)).squeeze(
-1
)
loc = cholesky_solve(
logp.info_vec.unsqueeze(-1), safe_cholesky(precision)
).squeeze(-1)
return MultivariateNormal(
loc, precision_matrix=precision, validate_args=self._validate_args
)
Expand Down Expand Up @@ -928,7 +928,7 @@ def filter(self, value):
gamma_dist.concentration, gamma_dist.rate, validate_args=self._validate_args
)
# Conditional of last state on unit scale
scale_tril = cholesky(logp.precision)
scale_tril = safe_cholesky(logp.precision)
loc = cholesky_solve(logp.info_vec.unsqueeze(-1), scale_tril).squeeze(-1)
mvn = MultivariateNormal(
loc, scale_tril=scale_tril, validate_args=self._validate_args
Expand Down
4 changes: 2 additions & 2 deletions pyro/distributions/transforms/cholesky.py
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Original file line number Diff line number Diff line change
Expand Up @@ -89,7 +89,7 @@ def log_abs_det_jacobian(self, x, y):

class CholeskyTransform(Transform):
r"""
Transform via the mapping :math:`y = cholesky(x)`, where `x` is a
Transform via the mapping :math:`y = safe_cholesky(x)`, where `x` is a
positive definite matrix.
"""
bijective = True
Expand All @@ -116,7 +116,7 @@ def log_abs_det_jacobian(self, x, y):

class CorrMatrixCholeskyTransform(CholeskyTransform):
r"""
Transform via the mapping :math:`y = cholesky(x)`, where `x` is a
Transform via the mapping :math:`y = safe_cholesky(x)`, where `x` is a
correlation matrix.
"""
bijective = True
Expand Down
10 changes: 5 additions & 5 deletions pyro/ops/gaussian.py
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Original file line number Diff line number Diff line change
Expand Up @@ -9,7 +9,7 @@
from torch.nn.functional import pad

from pyro.distributions.util import broadcast_shape
from pyro.ops.tensor_utils import cholesky, matmul, matvecmul, triangular_solve
from pyro.ops.tensor_utils import matmul, matvecmul, safe_cholesky, triangular_solve


class Gaussian:
Expand Down Expand Up @@ -154,7 +154,7 @@ def rsample(
"""
Reparameterized sampler.
"""
P_chol = cholesky(self.precision)
P_chol = safe_cholesky(self.precision)
loc = self.info_vec.unsqueeze(-1).cholesky_solve(P_chol).squeeze(-1)
shape = sample_shape + self.batch_shape + (self.dim(), 1)
if noise is None:
Expand Down Expand Up @@ -254,7 +254,7 @@ def marginalize(self, left=0, right=0) -> "Gaussian":
P_aa = self.precision[..., a, a]
P_ba = self.precision[..., b, a]
P_bb = self.precision[..., b, b]
P_b = cholesky(P_bb)
P_b = safe_cholesky(P_bb)
P_a = triangular_solve(P_ba, P_b, upper=False)
P_at = P_a.transpose(-1, -2)
precision = P_aa - matmul(P_at, P_a)
Expand All @@ -277,7 +277,7 @@ def event_logsumexp(self) -> torch.Tensor:
Integrates out all latent state (i.e. operating on event dimensions).
"""
n = self.dim()
chol_P = cholesky(self.precision)
chol_P = safe_cholesky(self.precision)
chol_P_u = triangular_solve(
self.info_vec.unsqueeze(-1), chol_P, upper=False
).squeeze(-1)
Expand Down Expand Up @@ -550,7 +550,7 @@ def gaussian_tensordot(x: Gaussian, y: Gaussian, dims: int = 0) -> Gaussian:
b = xb + yb

# Pbb + Qbb needs to be positive definite, so that we can malginalize out `b` (to have a finite integral)
L = cholesky(Pbb + Qbb)
L = safe_cholesky(Pbb + Qbb)
LinvB = triangular_solve(B, L, upper=False)
LinvBt = LinvB.transpose(-2, -1)
Linvb = triangular_solve(b.unsqueeze(-1), L, upper=False)
Expand Down
13 changes: 12 additions & 1 deletion pyro/ops/tensor_utils.py
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Original file line number Diff line number Diff line change
Expand Up @@ -7,6 +7,7 @@
from torch.fft import irfft, rfft

_ROOT_TWO_INVERSE = 1.0 / math.sqrt(2.0)
CHOLESKY_JITTER = 1.0 # in units of finfo.eps


def as_complex(x):
Expand Down Expand Up @@ -393,9 +394,19 @@ def inverse_haar_transform(x):
return x


def cholesky(x):
def safe_cholesky(x):
if x.size(-1) == 1:
if CHOLESKY_JITTER:
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@fehiepsi fehiepsi Oct 28, 2022 •
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Do you intend to clamp by (CHOLESKY_JITTER * finfo(x.dtype).eps) ** 2 here?

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My intention was to scale by about finfo(x.dtype).eps * x.max() so that the jitter was just barely detectable by the largest matrix entry before Cholesky factorizing. That way if we set RELATIVE_CHOLESKY_JITTER = 1/2 then jitter will only affect matrix entries less than half the size of the max. And it kindof makes sense to me that each additional bit of precision would mean we would need to add half as much jitter, thus jitter would be proportional to finfo(x.dtype).eps. Mostly the proportional think helps us keep a constant RELATIVE_CHOLESKY_JITTER across float32 and float64.

What's your intuition behind the square here, is that to keep constant error post-Cholesky-factorization?

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I missed the x_max term in the above comment. Using square seems to be more consistent w.r.t. to cases x.size(-1) > 1 - but I like your clamp by tiny better.

Re x_max: using global max makes sense, but I feel that it might be better to use max of rows instead, e.g. considering the diagonal matrix [0.0001, 10000], the global jitter is large w.r.t. the first diagonal term.

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Nice idea! I've switched to using a row-wise max. This required increasing CHOLESKY_RELATIVE_JITTER from 1.0 to 4.0, but this way still seems better 👍

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is this really preferred? this changes the eigenvalues and eigenvectors as opposed to jitter that is proportional to the identity (which only changes eigenvalues)

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@fritzo fritzo Oct 30, 2022 •
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@martinjankowiak that's a good point, I didn't know about eigenvector preservation. I'd be ok with either version.

One thing I like about @fehiepsi's solution is that users can on their side rotate the system before performing Gaussian ops, e.g. I'm approximately diagonalizing via QR

evals, evecs = torch.linalg.eig(transition_matrix)
Q, R = torch.linalg.qr(evecs.real)
transition_matrix = Q.T @ transition_matrix @ Q

which shrinks my diagonal perturbations

- [5.45, 1.81, 0.86, 0.76] * eps * CHOLESKY_RELATIVE_JITTER
+ [4.91, 1.67, 0.21, 0.17] * eps * CHOLESKY_RELATIVE_JITTER

x = x.clamp(min=torch.finfo(x.dtype).tiny)
return x.sqrt()

if CHOLESKY_JITTER:
# Add adaptive jitter.
x = x.clone()
x_max = x.data.reshape(*x.shape[:-2], -1).abs().max(-1, True).values
jitter = CHOLESKY_JITTER * torch.finfo(x.dtype).eps * x_max
x.data.diagonal(dim1=-1, dim2=-2).add_(jitter)

return torch.linalg.cholesky(x)


Expand Down
55 changes: 54 additions & 1 deletion tests/ops/test_gaussian.py
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Original file line number Diff line number Diff line change
Expand Up @@ -15,6 +15,7 @@
AffineNormal,
Gaussian,
gaussian_tensordot,
matrix_and_gaussian_to_gaussian,
matrix_and_mvn_to_gaussian,
mvn_to_gaussian,
sequential_gaussian_filter_sample,
Expand Down Expand Up @@ -378,7 +379,7 @@ def test_gaussian_tensordot(
nc = y_dim - dot_dims
try:
torch.linalg.cholesky(x.precision[..., na:, na:] + y.precision[..., :nb, :nb])
except RuntimeError:
except Exception:
pytest.skip("Cannot marginalize the common variables of two Gaussians.")

z = gaussian_tensordot(x, y, dot_dims)
Expand Down Expand Up @@ -557,3 +558,55 @@ def test_sequential_gaussian_filter_sample_antithetic(
)
expected = torch.stack([sample, mean, 2 * mean - sample])
assert torch.allclose(sample3, expected)


@pytest.mark.filterwarnings("ignore:Singular matrix in cholesky")
@pytest.mark.parametrize("num_steps", [10, 100, 1000, 10000, 100000, 1000000])
def test_sequential_gaussian_filter_sample_stability(num_steps):
# This tests long-chain filtering at low precision.
zero = torch.zeros((), dtype=torch.float)
eye = torch.eye(4, dtype=torch.float)
noise = torch.randn(num_steps, 4, dtype=torch.float, requires_grad=True)
trans_matrix = torch.tensor(
[
[
0.8571434617042542,
-0.23285813629627228,
0.05360094830393791,
-0.017088839784264565,
],
[
0.7609677314758301,
0.6596274971961975,
-0.022656921297311783,
0.05166701227426529,
],
[
3.0979342460632324,
5.446939945220947,
-0.3425334692001343,
0.01096670888364315,
],
[
-1.8180007934570312,
-0.4965082108974457,
-0.006048532668501139,
-0.08525419235229492,
],
],
dtype=torch.float,
requires_grad=True,
)

init = Gaussian(zero, zero.expand(4), eye)
trans = matrix_and_gaussian_to_gaussian(
trans_matrix, Gaussian(zero, zero.expand(4), eye)
).expand((num_steps - 1,))

# Check numerically stabilized value.
x = sequential_gaussian_filter_sample(init, trans, (), noise)
assert torch.isfinite(x).all()

# Check gradients.
grads = torch.autograd.grad(x.sum(), [trans_matrix, noise])
assert all(torch.isfinite(g).all() for g in grads)