Linalg Error during fit with Newton's Method

[bantin]

Certain parameter settings seem to give LinalgErrors when using Newton's method for the Continuous parameter update. This happens even with a simple gaussian LDS, for which Laplace-EM with Newton's method should be exact. This reproduces the error for me:

import ssm
import numpy as np
N, K, D, T = 10, 1, 30, 100
np.random.seed(seed=123)
true_slds = ssm.LDS(N, D, dynamics="gaussian", emissions="gaussian")
x, y = true_slds.sample(T)
fit_lds = ssm.LDS(N, D, dynamics="gaussian", emissions="gaussian")
fit_lds.fit(y, num_iters=20)

And the resulting error:

 File "/Users/bantin/Documents/Linderman-Shenoy/ssm/ssm/lds.py", line 618, in _fit_laplace_em_continuous_state_update
    tolerance=continuous_tolerance, maxiter=continuous_maxiter)
  File "/Users/bantin/Documents/Linderman-Shenoy/ssm/ssm/optimizers.py", line 162, in newtons_method_block_tridiag_hessian
    dx = -1.0 * solve_symm_block_tridiag(H_diag, H_lower_diag, g)
  File "/Users/bantin/Documents/Linderman-Shenoy/ssm/ssm/primitives.py", line 240, in solve_symm_block_tridiag
    x_flat = solveh_banded(J_banded, np.ravel(v), lower=True)
  File "/Users/Bantin/anaconda3/lib/python3.7/site-packages/autograd/tracer.py", line 48, in f_wrapped
    return f_raw(*args, **kwargs)
  File "/Users/Bantin/anaconda3/lib/python3.7/site-packages/scipy/linalg/basic.py", line 587, in solveh_banded
    raise LinAlgError("%d-th leading minor not positive definite" % info)
numpy.linalg.LinAlgError: 51-th leading minor not positive definite

This means that somehow the Hessian here is not PSD.

[bantin]

Per discussion with Scott (@slinderman) we've partially addressed this in 7e5deb2 by increasing the regularization for the dynamics covariance matrix update. For now, we'll switch the default continuous_optimizer to "lbfgs".

When debugging we also found the Newton's method is not always solving the resulting quadratic problem in a single step (as would be expected for a vanilla LDS where the expected log-joint is quadratic in x).