def _grad_neg_log_likelihood_and_fim_fn(x):
predicted_linear_response = (
fisher_scoring.compute_predicted_linear_response(
model_matrix, x))
g, h_middle = _grad_neg_log_likelihood_and_fim(
model_matrix, predicted_linear_response, response, model)
return g, model_matrix, h_middle
def _neg_log_likelihood(x):
predicted_linear_response = (
fisher_scoring.compute_predicted_linear_response(
model_matrix, x))
log_probs = model.log_prob(response, predicted_linear_response)
return -log_probs
def fit_sparse_one_step(model_matrix,
response,
model,
model_coefficients_start,
tolerance,
l1_regularizer,
l2_regularizer=None,
maximum_full_sweeps=None,
learning_rate=None,
name=None):
"""One step of (the outer loop of) the GLM fitting algorithm.
This function returns a new value of `model_coefficients`, equal to
`model_coefficients_start + model_coefficients_update`. The increment
`model_coefficients_update in R^n` is computed by a coordinate descent method,
that is, by a loop in which each iteration updates exactly one coordinate of
`model_coefficients_update`. (Some updates may leave the value of the
coordinate unchanged.)
The particular update method used is to apply an L1-based proximity operator,
"soft threshold", whose fixed point `model_coefficients_update^*` is the
desired minimum
```none
model_coefficients_update^* = argmin{
-LogLikelihood(model_coefficients_start + model_coefficients_update')
+ l1_regularizer *
||model_coefficients_start + model_coefficients_update'||_1
+ l2_regularizer *
||model_coefficients_start + model_coefficients_update'||_2**2
: model_coefficients_update' }
```
where in each iteration `model_coefficients_update'` has at most one nonzero
coordinate.
This update method preserves sparsity, i.e., tends to find sparse solutions if
`model_coefficients_start` is sparse. Additionally, the choice of step size
is based on curvature (Fisher information matrix), which significantly speeds
up convergence.
Args:
model_matrix: (Batch of) matrix-shaped, `float` `Tensor` or `SparseTensor`
where each row represents a sample's features. Has shape `[N, n]` where
`N` is the number of data samples and `n` is the number of features per
sample.
response: (Batch of) vector-shaped `Tensor` with the same dtype as
`model_matrix` where each element represents a sample's observed response
(to the corresponding row of features).
model: `tfp.glm.ExponentialFamily`-like instance, which specifies the link
function and distribution of the GLM, and thus characterizes the negative
log-likelihood which will be minimized. Must have sufficient statistic
equal to the response, that is, `T(y) = y`.
model_coefficients_start: (Batch of) vector-shaped, `float` `Tensor` with
the same dtype as `model_matrix`, representing the initial values of the
coefficients for the GLM regression. Has shape `[n]` where `model_matrix`
has shape `[N, n]`.
tolerance: scalar, `float` `Tensor` representing the convergence threshold.
The optimization step will terminate early, returning its current value of
`model_coefficients_start + model_coefficients_update`, once the following
condition is met:
`||model_coefficients_update_end - model_coefficients_update_start||_2
/ (1 + ||model_coefficients_start||_2)
< sqrt(tolerance)`,
where `model_coefficients_update_end` is the value of
`model_coefficients_update` at the end of a sweep and
`model_coefficients_update_start` is the value of
`model_coefficients_update` at the beginning of that sweep.
l1_regularizer: scalar, `float` `Tensor` representing the weight of the L1
regularization term (see equation above).
l2_regularizer: scalar, `float` `Tensor` representing the weight of the L2
regularization term (see equation above).
Default value: `None` (i.e., no L2 regularization).
maximum_full_sweeps: Python integer specifying maximum number of sweeps to
run. A "sweep" consists of an iteration of coordinate descent on each
coordinate. After this many sweeps, the algorithm will terminate even if
convergence has not been reached.
Default value: `1`.
learning_rate: scalar, `float` `Tensor` representing a multiplicative factor
used to dampen the proximal gradient descent steps.
Default value: `None` (i.e., factor is conceptually `1`).
name: Python string representing the name of the TensorFlow operation. The
default name is `"fit_sparse_one_step"`.
Returns:
model_coefficients: (Batch of) `Tensor` having the same shape and dtype as
`model_coefficients_start`, representing the updated value of
`model_coefficients`, that is, `model_coefficients_start +
model_coefficients_update`.
is_converged: scalar, `bool` `Tensor` indicating whether convergence
occurred across all batches within the specified number of sweeps.
iter: scalar, `int` `Tensor` representing the actual number of coordinate
updates made (before achieving convergence). Since each sweep consists of
`tf.size(model_coefficients_start)` iterations, the maximum number of
updates is `maximum_full_sweeps * tf.size(model_coefficients_start)`.
"""
with tf.name_scope(name or 'fit_sparse_one_step'):
predicted_linear_response = (
fisher_scoring.compute_predicted_linear_response(
model_matrix, model_coefficients_start))
g, h_middle = _grad_neg_log_likelihood_and_fim(
model_matrix, predicted_linear_response, response, model)
return tfp.optimizer.proximal_hessian_sparse_one_step(
gradient_unregularized_loss=g,
hessian_unregularized_loss_outer=model_matrix,
hessian_unregularized_loss_middle=h_middle,
x_start=model_coefficients_start,
l1_regularizer=l1_regularizer,
l2_regularizer=l2_regularizer,
maximum_full_sweeps=maximum_full_sweeps,
tolerance=tolerance,
learning_rate=learning_rate,
name=name)