Online learning for an MLP using extended Kalman filtering

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.

RLS

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
../../_images/c9736dd83ca61ec18ab5dd16bea383d21fc1979a9bf5dcafb06a82d05a10f057.png ../../_images/c2aab1640914ce8af84ad50988d0e6235e18e04813c20c413821a92dc2b162b6.png ../../_images/e4f1de2ce51ed059f7bf78e97f3f7fa86e64808f1dc90e0b222b2597c8f4a23a.png ../../_images/33c136cd5e6d80d3a2c17da7feb9482a4328f1aea3bd45ea2ece631ad24e42c7.png