Parallel filtering and smoothing in an LG-SSM#
This notebook shows how can reduce the cost of inference from O(T) to O(log T) time, if we have a GPU device.
This code borrows heavily from this example notebook from Adrien Correnflos. Some small changes have been made so it works with dynamax.
If you have a GPU, you should be able to get a speedup curve like this:
Setup#
Show code cell content
%%capture
try:
import dynamax
except ModuleNotFoundError:
print('installing dynamax')
%pip install -q dynamax[notebooks]
import dynamax
import jax
from jax import numpy as jnp
from jax import random as jr
from jax import block_until_ready
from matplotlib import pyplot as plt
import time
from dynamax.linear_gaussian_ssm import lgssm_smoother, parallel_lgssm_smoother
from dynamax.linear_gaussian_ssm import LinearGaussianSSM
print(jax.devices())
print(jax.devices()[0].platform)
if jax.devices()[0].platform == 'cpu':
cpu_mode = True
else:
cpu_mode = False
[CpuDevice(id=0)]
cpu
Model#
The model is a simple tracking model (see Example 3.6 in Bayesian Filtering and Smoothing (S. Särkkä, 2013).
dt = 0.1
F = jnp.eye(4) + dt * jnp.eye(4, k=2)
Q = 1. * jnp.kron(jnp.array([[dt**3/3, dt**2/2],
[dt**2/2, dt]]),
jnp.eye(2))
H = jnp.eye(2, 4)
R = 0.5 ** 2 * jnp.eye(2)
μ0 = jnp.array([0.,0.,1.,-1.])
Σ0 = jnp.eye(4)
latent_dim = 4
observation_dim = 2
input_dim = 0
lgssm = LinearGaussianSSM(latent_dim, observation_dim)
params, _ = lgssm.initialize(
initial_mean=μ0,
initial_covariance= Σ0,
dynamics_weights=F,
dynamics_covariance=Q,
emission_weights=H,
emission_covariance=R)
Test parallel inference on a single sequence#
num_timesteps = 100
key = jr.PRNGKey(0)
key, subkey = jr.split(key)
z,emissions = lgssm.sample(params, subkey, num_timesteps)
ssm_posterior = lgssm_smoother(params, emissions)
parallel_posterior = parallel_lgssm_smoother(params, emissions)
print(parallel_posterior.filtered_means.shape)
print(parallel_posterior.filtered_covariances.shape)
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
Cell In[6], line 5
2 key = jr.PRNGKey(0)
4 key, subkey = jr.split(key)
----> 5 z,emissions = lgssm.sample(params, subkey, num_timesteps)
6 ssm_posterior = lgssm_smoother(params, emissions)
7 parallel_posterior = parallel_lgssm_smoother(params, emissions)
File ~/work/dynamax/dynamax/dynamax/linear_gaussian_ssm/models.py:209, in LinearGaussianSSM.sample(self, params, key, num_timesteps, inputs)
202 def sample(
203 self,
204 params: ParamsLGSSM,
(...)
207 inputs: Optional[Float[Array, "ntime input_dim"]] = None
208 ) -> PosteriorGSSMFiltered:
--> 209 return lgssm_joint_sample(params, key, num_timesteps, inputs)
File ~/work/dynamax/dynamax/dynamax/linear_gaussian_ssm/inference.py:421, in lgssm_joint_sample(params, key, num_timesteps, inputs)
418 # Sample the initial state
419 key1, key2 = jr.split(key)
--> 421 initial_state, initial_emission = _sample_initial(key1, params, inputs)
423 # Sample the remaining emissions and states
424 next_keys = jr.split(key2, num_timesteps - 1)
File ~/work/dynamax/dynamax/dynamax/linear_gaussian_ssm/inference.py:397, in lgssm_joint_sample.<locals>._sample_initial(key, params, inputs)
394 def _sample_initial(key, params, inputs):
395 key1, key2 = jr.split(key)
--> 397 initial_state = MVN(params.initial.mean, params.initial.cov).sample(seed=key1)
399 H0, D0, d0, R0 = _get_params(params, num_timesteps, 0)[4:]
400 u0 = tree_map(lambda x: x[0], inputs)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/decorator.py:232, in decorate.<locals>.fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/distribution.py:342, in _DistributionMeta.__new__.<locals>.wrapped_init(***failed resolving arguments***)
339 # Note: if we ever want to have things set in `self` before `__init__` is
340 # called, here is the place to do it.
341 self_._parameters = None
--> 342 default_init(self_, *args, **kwargs)
343 # Note: if we ever want to override things set in `self` by subclass
344 # `__init__`, here is the place to do it.
345 if self_._parameters is None:
346 # We prefer subclasses will set `parameters = dict(locals())` because
347 # this has nearly zero overhead. However, failing to do this, we will
348 # resolve the input arguments dynamically and only when needed.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/mvn_full_covariance.py:191, in MultivariateNormalFullCovariance.__init__(self, loc, covariance_matrix, validate_args, allow_nan_stats, name)
185 # No need to validate that covariance_matrix is non-singular.
186 # LinearOperatorLowerTriangular has an assert_non_singular method that
187 # is called by the Bijector.
188 # However, cholesky() ignores the upper triangular part, so we do need
189 # to separately assert symmetric.
190 scale_tril = tf.linalg.cholesky(covariance_matrix)
--> 191 super(MultivariateNormalFullCovariance, self).__init__(
192 loc=loc,
193 scale_tril=scale_tril,
194 validate_args=validate_args,
195 allow_nan_stats=allow_nan_stats,
196 name=name)
197 self._parameters = parameters
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/decorator.py:232, in decorate.<locals>.fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/distribution.py:342, in _DistributionMeta.__new__.<locals>.wrapped_init(***failed resolving arguments***)
339 # Note: if we ever want to have things set in `self` before `__init__` is
340 # called, here is the place to do it.
341 self_._parameters = None
--> 342 default_init(self_, *args, **kwargs)
343 # Note: if we ever want to override things set in `self` by subclass
344 # `__init__`, here is the place to do it.
345 if self_._parameters is None:
346 # We prefer subclasses will set `parameters = dict(locals())` because
347 # this has nearly zero overhead. However, failing to do this, we will
348 # resolve the input arguments dynamically and only when needed.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/mvn_tril.py:228, in MultivariateNormalTriL.__init__(self, loc, scale_tril, validate_args, allow_nan_stats, experimental_use_kahan_sum, name)
221 linop_cls = (KahanLogDetLinOpTriL if experimental_use_kahan_sum else
222 tf.linalg.LinearOperatorLowerTriangular)
223 scale = linop_cls(
224 scale_tril,
225 is_non_singular=True,
226 is_self_adjoint=False,
227 is_positive_definite=False)
--> 228 super(MultivariateNormalTriL, self).__init__(
229 loc=loc,
230 scale=scale,
231 validate_args=validate_args,
232 allow_nan_stats=allow_nan_stats,
233 experimental_use_kahan_sum=experimental_use_kahan_sum,
234 name=name)
235 self._parameters = parameters
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/decorator.py:232, in decorate.<locals>.fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/distribution.py:342, in _DistributionMeta.__new__.<locals>.wrapped_init(***failed resolving arguments***)
339 # Note: if we ever want to have things set in `self` before `__init__` is
340 # called, here is the place to do it.
341 self_._parameters = None
--> 342 default_init(self_, *args, **kwargs)
343 # Note: if we ever want to override things set in `self` by subclass
344 # `__init__`, here is the place to do it.
345 if self_._parameters is None:
346 # We prefer subclasses will set `parameters = dict(locals())` because
347 # this has nearly zero overhead. However, failing to do this, we will
348 # resolve the input arguments dynamically and only when needed.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/mvn_linear_operator.py:205, in MultivariateNormalLinearOperator.__init__(self, loc, scale, validate_args, allow_nan_stats, experimental_use_kahan_sum, name)
202 if loc is not None:
203 bijector = shift_bijector.Shift(
204 shift=loc, validate_args=validate_args)(bijector)
--> 205 super(MultivariateNormalLinearOperator, self).__init__(
206 # TODO(b/137665504): Use batch-adding meta-distribution to set the batch
207 # shape instead of tf.zeros.
208 # We use `Sample` instead of `Independent` because `Independent`
209 # requires concatenating `batch_shape` and `event_shape`, which loses
210 # static `batch_shape` information when `event_shape` is not statically
211 # known.
212 distribution=sample.Sample(
213 normal.Normal(
214 loc=tf.zeros(batch_shape, dtype=dtype),
215 scale=tf.ones([], dtype=dtype)),
216 event_shape,
217 experimental_use_kahan_sum=experimental_use_kahan_sum),
218 bijector=bijector,
219 validate_args=validate_args,
220 name=name)
221 self._parameters = parameters
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/decorator.py:232, in decorate.<locals>.fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/distribution.py:342, in _DistributionMeta.__new__.<locals>.wrapped_init(***failed resolving arguments***)
339 # Note: if we ever want to have things set in `self` before `__init__` is
340 # called, here is the place to do it.
341 self_._parameters = None
--> 342 default_init(self_, *args, **kwargs)
343 # Note: if we ever want to override things set in `self` by subclass
344 # `__init__`, here is the place to do it.
345 if self_._parameters is None:
346 # We prefer subclasses will set `parameters = dict(locals())` because
347 # this has nearly zero overhead. However, failing to do this, we will
348 # resolve the input arguments dynamically and only when needed.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/distributions/transformed_distribution.py:244, in _TransformedDistribution.__init__(self, distribution, bijector, kwargs_split_fn, validate_args, parameters, name)
238 self._zero = tf.constant(0, dtype=tf.int32, name='zero')
240 # We don't just want to check isinstance(JointDistribution) because
241 # TransformedDistributions with multipart bijectors are effectively
242 # joint but don't inherit from JD. The 'duck-type' test is that
243 # JDs have a structured dtype.
--> 244 dtype = self.bijector.forward_dtype(self.distribution.dtype)
245 self._is_joint = tf.nest.is_nested(dtype)
247 super(_TransformedDistribution, self).__init__(
248 dtype=dtype,
249 reparameterization_type=self._distribution.reparameterization_type,
(...)
252 parameters=parameters,
253 name=name)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/bijectors/bijector.py:1705, in Bijector.forward_dtype(self, dtype, name, **kwargs)
1701 input_dtype = nest_util.broadcast_structure(
1702 self.forward_min_event_ndims, self.dtype)
1703 else:
1704 # Make sure inputs are compatible with statically-known dtype.
-> 1705 input_dtype = nest.map_structure_up_to(
1706 self.forward_min_event_ndims,
1707 lambda x: dtype_util.convert_to_dtype(x, dtype=self.dtype),
1708 nest_util.coerce_structure(self.forward_min_event_ndims, dtype),
1709 check_types=False)
1711 output_dtype = self._forward_dtype(input_dtype, **kwargs)
1712 try:
1713 # kwargs may alter dtypes themselves, but we currently require
1714 # structure to be statically known.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/python/internal/backend/jax/nest.py:324, in map_structure_up_to(shallow_structure, func, *structures, **kwargs)
323 def map_structure_up_to(shallow_structure, func, *structures, **kwargs):
--> 324 return map_structure_with_tuple_paths_up_to(
325 shallow_structure,
326 lambda _, *args: func(*args), # Discards path.
327 *structures,
328 **kwargs)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/python/internal/backend/jax/nest.py:353, in map_structure_with_tuple_paths_up_to(shallow_structure, func, expand_composites, *structures, **kwargs)
350 for input_tree in structures:
351 assert_shallow_structure(
352 shallow_structure, input_tree, check_types=check_types)
--> 353 return dm_tree.map_structure_with_path_up_to(
354 shallow_structure, func, *structures, **kwargs)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tree/__init__.py:778, in map_structure_with_path_up_to(***failed resolving arguments***)
776 results = []
777 for path_and_values in _multiyield_flat_up_to(shallow_structure, *structures):
--> 778 results.append(func(*path_and_values))
779 return unflatten_as(shallow_structure, results)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/python/internal/backend/jax/nest.py:326, in map_structure_up_to.<locals>.<lambda>(_, *args)
323 def map_structure_up_to(shallow_structure, func, *structures, **kwargs):
324 return map_structure_with_tuple_paths_up_to(
325 shallow_structure,
--> 326 lambda _, *args: func(*args), # Discards path.
327 *structures,
328 **kwargs)
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/bijectors/bijector.py:1707, in Bijector.forward_dtype.<locals>.<lambda>(x)
1701 input_dtype = nest_util.broadcast_structure(
1702 self.forward_min_event_ndims, self.dtype)
1703 else:
1704 # Make sure inputs are compatible with statically-known dtype.
1705 input_dtype = nest.map_structure_up_to(
1706 self.forward_min_event_ndims,
-> 1707 lambda x: dtype_util.convert_to_dtype(x, dtype=self.dtype),
1708 nest_util.coerce_structure(self.forward_min_event_ndims, dtype),
1709 check_types=False)
1711 output_dtype = self._forward_dtype(input_dtype, **kwargs)
1712 try:
1713 # kwargs may alter dtypes themselves, but we currently require
1714 # structure to be statically known.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/tensorflow_probability/substrates/jax/internal/dtype_util.py:247, in convert_to_dtype(tensor_or_dtype, dtype, dtype_hint)
245 elif isinstance(tensor_or_dtype, np.ndarray):
246 dt = base_dtype(dtype or dtype_hint or tensor_or_dtype.dtype)
--> 247 elif np.issctype(tensor_or_dtype):
248 dt = base_dtype(dtype or dtype_hint or tensor_or_dtype)
249 else:
250 # If this is a Python object, call `convert_to_tensor` and grab the dtype.
251 # Note that this will add ops in graph-mode; we may want to consider
252 # other ways to handle this case.
File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/numpy/__init__.py:397, in __getattr__(attr)
394 raise AttributeError(__former_attrs__[attr])
396 if attr in __expired_attributes__:
--> 397 raise AttributeError(
398 f"`np.{attr}` was removed in the NumPy 2.0 release. "
399 f"{__expired_attributes__[attr]}"
400 )
402 if attr == "chararray":
403 warnings.warn(
404 "`np.chararray` is deprecated and will be removed from "
405 "the main namespace in the future. Use an array with a string "
406 "or bytes dtype instead.", DeprecationWarning, stacklevel=2)
AttributeError: `np.issctype` was removed in the NumPy 2.0 release. Use `issubclass(rep, np.generic)` instead.
assert jnp.allclose(parallel_posterior.filtered_means, ssm_posterior.filtered_means, atol=1e-3)
assert jnp.allclose(parallel_posterior.filtered_covariances, ssm_posterior.filtered_covariances, atol=1e-3)
assert jnp.allclose(parallel_posterior.smoothed_means, ssm_posterior.smoothed_means, atol=1e-3)
assert jnp.allclose(parallel_posterior.smoothed_covariances, ssm_posterior.smoothed_covariances, atol=1e-3)
plt.figure(figsize=(6,6))
plt.plot(*emissions.T,'.', label="observations")
plt.plot(*ssm_posterior.filtered_means[:,:2].T, color="C2", label="serial filtering")
plt.plot(*parallel_posterior.filtered_means[:,:2].T, "--", color="C3",label="parallel filtering");
plt.legend();
plt.figure(figsize=(6,6))
plt.plot(*emissions.T,'.', label="observations")
plt.plot(*ssm_posterior.smoothed_means[:,:2].T, color="C2", label="serial smoothing")
plt.plot(*parallel_posterior.smoothed_means[:,:2].T, "--", color="C3",label="parallel smoothing")
plt.legend();
Timing comparison#
key = jr.PRNGKey(0)
if cpu_mode:
Ts = [100, 200, 500]
num_repeats = 1
else:
Ts = [100, 1_000, 10_000, 100_000]
num_repeats = 3
serial_smoothing_durations = []
parallel_smoothing_durations = []
compiled = False
for T in Ts:
key, subkey = jr.split(key)
z,emissions = lgssm.sample(params, subkey, T)
if not compiled:
ssm_posterior = block_until_ready(lgssm_smoother(params, emissions))
parallel_posterior = block_until_ready(parallel_lgssm_smoother(params, emissions))
start = time.time()
for _ in range(num_repeats):
ssm_posterior = block_until_ready(lgssm_smoother(params, emissions))
end = time.time()
mean_time = (end-start)/num_repeats
serial_smoothing_durations.append(mean_time)
print(f"Num timesteps={T}, \t time serial = {mean_time}")
start = time.time()
for _ in range(num_repeats):
parallel_posterior = block_until_ready(parallel_lgssm_smoother(params, emissions))
end = time.time()
mean_time = (end-start)/num_repeats
parallel_smoothing_durations.append(mean_time)
print(f"Num timesteps={T}, \t time parallel = {mean_time}")
Num timesteps=100, time serial = 0.5520718097686768
Num timesteps=100, time parallel = 0.27495622634887695
Num timesteps=200, time serial = 0.5372998714447021
Num timesteps=200, time parallel = 0.29311299324035645
Num timesteps=500, time serial = 0.5538558959960938
Num timesteps=500, time parallel = 0.34401488304138184
plt.figure(figsize=(5, 5))
plt.loglog(Ts, serial_smoothing_durations, '-o', label='serial')
plt.loglog(Ts, parallel_smoothing_durations, '-o', label='parallel')
plt.xticks(Ts)
plt.xlabel("seq. length")
plt.ylabel("time per forward pass (s)")
plt.grid(True)
plt.legend()
plt.tight_layout()
