Online learning for an MLP using extended Kalman filtering#
We perform sequential (recursive) Bayesian inference for the parameters of a multi layer perceptron (MLP) using the extended Kalman filter. To do this, we treat the parameters of the model as the unknown hidden states. We assume that these are approximately constant over time (we add a small amount of Gaussian drift, for numerical stability.) The graphical model is shown below.
The model has the following form
\[\begin{align*}
\theta_t &= \theta_{t-1} + q_t, \; q_t \sim N(0, 0.01 I) \\
y_t &= h(\theta_t, x_t) + r_t, \; r_t \sim N(0, \sigma^2)
\end{align*}\]
This is a NLG-SSM, where \(h\) is the nonlinear observation model. For details, see sec 17.5.2 of Probabilistic Machine Learning: Advanced Topics.
For a video of the training in action see, https://github.com/probml/probml-data/blob/main/data/ekf_mlp_demo.mp4
Setup#
%%capture
try:
import dynamax
except ModuleNotFoundError:
print('installing dynamax')
%pip install -q dynamax[notebooks]
import dynamax
from jax import vmap
import jax.numpy as jnp
import flax.linen as nn
import jax.random as jr
from jax.flatten_util import ravel_pytree
from matplotlib import pyplot as plt
from typing import Sequence
from functools import partial
from dynamax.nonlinear_gaussian_ssm import ParamsNLGSSM, extended_kalman_filter
Data#
def sample_observations(f, x_min, x_max, x_var=0.1, y_var=3.0, num_obs=200, key=0):
"""Generate random training set for MLP given true function and
distribution parameters.
Args:
f (Callable): True function.
x_min (float): Min x-coordinate to sample from.
x_max (float): Max x-coordinate to sample from.
x_var (float, optional): Sampling variance in x-coordinate. Defaults to 0.1.
y_var (float, optional): Sampling variance in y-coordinate. Defaults to 3.0.
num_obs (int, optional): Number of training data to generate. Defaults to 200.
key (int, optional): Random key. Defaults to 0.
Returns:
x (num_obs,): x-coordinates of generated data
y (num_obs,): y-coordinates of generated data
"""
if isinstance(key, int):
key = jr.PRNGKey(key)
keys = jr.split(key, 3)
# Generate noisy x coordinates
x_noise = jr.normal(keys[0], (num_obs,)) * x_var
x = jnp.linspace(x_min, x_max, num_obs) + x_noise
# Generate noisy y coordinates
y_noise = jr.normal(keys[1], (num_obs,)) * y_var
y = f(x) + y_noise
# Random shuffle (x, y) coordinates
shuffled_idx = jr.permutation(keys[2], jnp.arange(num_obs))
x, y = x[shuffled_idx], y[shuffled_idx]
return x, y
# Generate training set.
# Note that we view the x-coordinates of training data as control inputs
# and the y-coordinates of training data as emissions.
f = lambda x: x - 10 * jnp.cos(x) * jnp.sin(x) + x**3
y_var = 3.0
inputs, emissions = sample_observations(f, x_min=-3, x_max=3, y_var=y_var)
Neural network#
class MLP(nn.Module):
features: Sequence[int]
@nn.compact
def __call__(self, x):
for feat in self.features[:-1]:
x = nn.sigmoid(nn.Dense(feat)(x))
x = nn.Dense(self.features[-1])(x)
return x
def get_mlp_flattened_params(model_dims, key=0):
"""Generate MLP model, initialize it using dummy input, and
return the model, its flattened initial parameters, function
to unflatten parameters, and apply function for the model.
Args:
model_dims (List): List of [input_dim, hidden_dim, ..., output_dim]
key (PRNGKey): Random key. Defaults to 0.
Returns:
model: MLP model with given feature dimensions.
flat_params: Flattened parameters initialized using dummy input.
unflatten_fn: Function to unflatten parameters.
apply_fn: fn(flat_params, x) that returns the result of applying the model.
"""
if isinstance(key, int):
key = jr.PRNGKey(key)
# Define MLP model
input_dim, features = model_dims[0], model_dims[1:]
model = MLP(features)
dummy_input = jnp.ones((input_dim,))
# Initialize parameters using dummy input
params = model.init(key, dummy_input)
flat_params, unflatten_fn = ravel_pytree(params)
# Define apply function
def apply(flat_params, x, model, unflatten_fn):
return model.apply(unflatten_fn(flat_params), jnp.atleast_1d(x))
apply_fn = partial(apply, model=model, unflatten_fn=unflatten_fn)
return model, flat_params, unflatten_fn, apply_fn
input_dim, hidden_dim, output_dim = 1, 6, 1
model_dims = [input_dim, hidden_dim, output_dim]
_, flat_params, _, apply_fn = get_mlp_flattened_params(model_dims)
Online inference#
# Note that the dynamics function is the identity function
# and the emission function is the model apply function
state_dim, emission_dim = flat_params.size, output_dim
ekf_params = ParamsNLGSSM(
initial_mean=flat_params,
initial_covariance=jnp.eye(state_dim) * 100,
dynamics_function=lambda x, u: x,
dynamics_covariance=jnp.eye(state_dim) * 1e-4,
emission_function=apply_fn,
emission_covariance=jnp.eye(emission_dim) * y_var**2,
)
# Run EKF on training set to train MLP
ekf_post = extended_kalman_filter(ekf_params, emissions, inputs=inputs)
w_means, w_covs = ekf_post.filtered_means, ekf_post.filtered_covariances
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
Cell In[9], line 14
4 ekf_params = ParamsNLGSSM(
5 initial_mean=flat_params,
6 initial_covariance=jnp.eye(state_dim) * 100,
(...)
10 emission_covariance=jnp.eye(emission_dim) * y_var**2,
11 )
13 # Run EKF on training set to train MLP
---> 14 ekf_post = extended_kalman_filter(ekf_params, emissions, inputs=inputs)
15 w_means, w_covs = ekf_post.filtered_means, ekf_post.filtered_covariances
File ~/work/dynamax/dynamax/dynamax/nonlinear_gaussian_ssm/inference_ekf.py:153, in extended_kalman_filter(params, emissions, num_iter, inputs, output_fields)
151 # Run the extended Kalman filter
152 carry = (0.0, params.initial_mean, params.initial_covariance)
--> 153 (ll, *_), outputs = lax.scan(_step, carry, jnp.arange(num_timesteps))
154 outputs = {"marginal_loglik": ll, **outputs}
155 posterior_filtered = PosteriorGSSMFiltered(
156 **outputs,
157 )
[... skipping hidden 9 frame]
File ~/work/dynamax/dynamax/dynamax/nonlinear_gaussian_ssm/inference_ekf.py:130, in extended_kalman_filter.<locals>._step(carry, t)
128 # Update the log likelihood
129 H_x = H(pred_mean, u)
--> 130 ll += MVN(h(pred_mean, u), H_x @ pred_cov @ H_x.T + R).log_prob(jnp.atleast_1d(y))
132 # Condition on this emission
133 filtered_mean, filtered_cov = _condition_on(pred_mean, pred_cov, h, H, R, u, y, num_iter)
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.
Plot results#
def plot_mlp_prediction(f, obs, x_grid, w_mean, w_cov, ax, num_samples=100, x_lim=(-3, 3), y_lim=(-30, 30), key=0):
if isinstance(key, int):
key = jr.PRNGKey(key)
# Plot observations (training set)
ax.plot(obs[0], obs[1], "ok", fillstyle="none", ms=4, alpha=0.5, label="Data")
# Indicate uncertainty through sampling
w_samples = jr.multivariate_normal(key, w_mean, w_cov, (num_samples,))
y_samples = vmap(vmap(f, in_axes=(None, 0)), in_axes=(0, None))(w_samples, x_grid)
for y_sample in y_samples:
ax.plot(x_grid, y_sample, color="gray", alpha=0.07)
# Plot prediction on grid using filtered mean of MLP params
# y_mean = vmap(f, in_axes=(None, 0))(w_mean, x_grid)
y_mean = y_samples.mean(axis=0)
ax.plot(x_grid, y_mean, linewidth=1.5, label="Prediction")
ax.set_xlim(x_lim)
ax.set_ylim(y_lim)
#ax.legend(loc=4, borderpad=0.5, handlelength=4, fancybox=False, edgecolor="k")
all_figures = {}
inputs_grid = jnp.linspace(inputs.min(), inputs.max(), len(inputs))
intermediate_steps = [10, 20, 30, 200]
for step in intermediate_steps:
print('ntraining =', step)
fig, ax = plt.subplots()
plot_mlp_prediction(
apply_fn, (inputs[:step], emissions[:step]), inputs_grid, w_means[step - 1], w_covs[step - 1], ax, key=step
)
ax.set_title(f"Step={step}")
all_figures[f"ekf_mlp_step_{step}"] = fig
ntraining = 10
ntraining = 20
ntraining = 30
ntraining = 200
