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Add jitter to Cholesky factorization in Gaussian ops #3151

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Merged
fehiepsi 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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8 changes: 8 additions & 0 deletions pyro/ops/tensor_utils.py
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Original file line number Diff line number Diff line change
Expand Up @@ -395,7 +395,15 @@ def inverse_haar_transform(x):

def cholesky(x):
if x.size(-1) == 1:
x = x.clamp(min=torch.finfo(x.dtype).tiny)
return x.sqrt()

# Add adaptive jitter.
x = x.clone()

@martinjankowiak martinjankowiak Oct 28, 2022

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are you sure you need/want clones? fwiw i do this in millipede, which is similar to what's done in gpytorch

@fritzo fritzo Oct 28, 2022 •
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Yup this .clone() is needed because we're mutating the matrix. Nice, I'll rename this to safe_cholesky() as in millipede.

Thanks, I did try various gpytorch-style lazy tactics that avoid adding jitter until a failure has occurred. I found that since the Cholesky error happens only late in the filtering process, by that point the filter state had already been corrupted by nearly-singular matrices that just barely didn't trigger an error. The best solution I've found so far is to add a tiny amount of noise to all matrices so that error doesn't build up during filtering. Other solutions include using svd or pinv or ldl_factor, but they were more expensive.

Note the core piece of linear algebra is in Gaussian.marginalize() which is repeatedly called in the filter pass of sequential_gaussian_filter_sample(). It's just the blockwise symmetric matrix inverse formula:

# in Gaussian.marginalize():
P_aa = self.precision[..., a, a]
P_ba = self.precision[..., b, a]
P_bb = self.precision[..., b, b]
P_b = safe_cholesky(P_bb)                       # Note if we add a little jitter here...
P_a = triangular_solve(P_ba, P_b, upper=False)  # ...then this is smaller...
P_at = P_a.transpose(-1, -2)
precision = P_aa - matmul(P_at, P_a)            # ...so this is even better conditioned.

This code has the nice property that if we add a little bit of jitter before Cholesky factorizing, the next precision matrix becomes only better-conditioned. Empirically this allowed me to get away with much smaller jitter than was needed if I waited for an error to occur.

BTW it looks like you could speed up millipede by switching from try: c = cholesky() to the faster c, info = cholesky_ex(); if not info.any(): return c, which is used in gpytorch. The only reason I'm not using cholesky_ex() here is that I found the decision-based version was too unstable.

x_max = x.data.reshape(*x.shape[:-2], -1).abs().max(-1, True).values
jitter = x_max * torch.finfo(x.dtype).eps
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)