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# Optimizers in TensorFlow Probability

## Abstract

In this colab we demonstrate how to use the various optimizers implemented in TensorFlow Probability.

## BFGS and L-BFGS Optimizers

Quasi Newton methods are a class of popular first order optimization algorithm. These methods use a positive definite approximation to the exact Hessian to find the search direction.

The Broyden-Fletcher-Goldfarb-Shanno algorithm (BFGS) is a specific implementation of this general idea. It is applicable and is the method of choice for medium sized problems where the gradient is continuous everywhere (e.g. linear regression with an $$L_2$$ penalty).

L-BFGS is a limited-memory version of BFGS that is useful for solving larger problems whose Hessian matrices cannot be computed at a reasonable cost or are not sparse. Instead of storing fully dense $$n \times n$$ approximations of Hessian matrices, they only save a few vectors of length $$n$$ that represent these approximations implicitly.

### L-BFGS on a simple quadratic function

# Fix numpy seed for reproducibility
np.random.seed(12345)

# The objective must be supplied as a function that takes a single
# (Tensor) argument and returns a tuple. The first component of the
# tuple is the value of the objective at the supplied point and the
# second value is the gradient at the supplied point. The value must
# be a scalar and the gradient must have the same shape as the
# supplied argument.

# The make_val_and_grad_fn decorator helps transforming a function
# returning the objective value into one that returns both the gradient
# and the value. It also works for both eager and graph mode.

dim = 10
minimum = np.ones([dim])
scales = np.exp(np.random.randn(dim))

return tf.reduce_sum(scales * (x - minimum) ** 2, axis=-1)

# The minimization routine also requires you to supply an initial
# starting point for the search. For this example we choose a random
# starting point.
start = np.random.randn(dim)

# Finally an optional argument called tolerance let's you choose the
# stopping point of the search. The tolerance specifies the maximum
# (supremum) norm of the gradient vector at which the algorithm terminates.
# If you don't have a specific need for higher or lower accuracy, leaving
# this parameter unspecified (and hence using the default value of 1e-8)
# should be good enough.
tolerance = 1e-10

@tf.function
return tfp.optimizer.lbfgs_minimize(
initial_position=tf.constant(start),
tolerance=tolerance)

# The optimization results contain multiple pieces of information. The most
# important fields are: 'converged' and 'position'.
# Converged is a boolean scalar tensor. As the name implies, it indicates
# whether the norm of the gradient at the final point was within tolerance.
# Position is the location of the minimum found. It is important to check
# that converged is True before using the value of the position.

print('L-BFGS Results')
print('Converged:', results.converged)
print('Location of the minimum:', results.position)
print('Number of iterations:', results.num_iterations)

Evaluation took: 0.014586 seconds
L-BFGS Results
Converged: True
Location of the minimum: [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
Number of iterations: 10


### Same problem with BFGS

@tf.function
return tfp.optimizer.bfgs_minimize(
initial_position=tf.constant(start),
tolerance=tolerance)

print('BFGS Results')
print('Converged:', results.converged)
print('Location of the minimum:', results.position)
print('Number of iterations:', results.num_iterations)

Evaluation took: 0.010468 seconds
BFGS Results
Converged: True
Location of the minimum: [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
Number of iterations: 10


## Linear Regression with L1 penalty: Prostate Cancer data

Example from the Book: The Elements of Statistical Learning, Data Mining, Inference, and Prediction by Trevor Hastie, Robert Tibshirani and Jerome Friedman.

Note this is an optimization problem with L1 penalty.

### Obtain dataset

def cache_or_download_file(cache_dir, url_base, filename):
filepath = os.path.join(cache_dir, filename)
if tf.io.gfile.exists(filepath):
return filepath
if not tf.io.gfile.exists(cache_dir):
tf.io.gfile.makedirs(cache_dir)
url = url_base + filename
urllib.request.urlretrieve(url, filepath)
return filepath

def get_prostate_dataset(cache_dir=CACHE_DIR):
url_base = 'http://web.stanford.edu/~hastie/ElemStatLearn/datasets/'
delim_whitespace=True, index_col=0)

prostate_df = get_prostate_dataset()

Downloading http://web.stanford.edu/~hastie/ElemStatLearn/datasets/prostate.data to /tmp/datasets/prostate.data.


### Problem definition

np.random.seed(12345)

feature_names = ['lcavol', 'lweight',   'age',  'lbph', 'svi', 'lcp',
'gleason', 'pgg45']

# Normalize features
scalar = preprocessing.StandardScaler()
prostate_df[feature_names] = pd.DataFrame(
scalar.fit_transform(
prostate_df[feature_names].astype('float64')))

# select training set
prostate_df_train = prostate_df[prostate_df.train == 'T']

# Select features and labels
features = prostate_df_train[feature_names]
labels =  prostate_df_train[['lpsa']]

# Create tensors
feat = tf.constant(features.values, dtype=tf.float64)
lab = tf.constant(labels.values, dtype=tf.float64)

dtype = feat.dtype

regularization = 0 # regularization parameter
dim = 8 # number of features

# We pick a random starting point for the search
start = np.random.randn(dim + 1)

def regression_loss(params):
"""Compute loss for linear regression model with L1 penalty

Args:
params: A real tensor of shape [dim + 1]. The zeroth component
is the intercept term and the rest of the components are the
beta coefficients.

Returns:
The mean square error loss including L1 penalty.
"""
params = tf.squeeze(params)
intercept, beta  = params, params[1:]
pred = tf.matmul(feat, tf.expand_dims(beta, axis=-1)) + intercept
mse_loss = tf.reduce_sum(
tf.cast(
tf.losses.mean_squared_error(y_true=lab, y_pred=pred), tf.float64))
l1_penalty = regularization * tf.reduce_sum(tf.abs(beta))
total_loss = mse_loss + l1_penalty


### Solving with L-BFGS

Fit using L-BFGS. Even though the L1 penalty introduces derivative discontinuities, in practice, L-BFGS works quite well still.

@tf.function
def l1_regression_with_lbfgs():
return tfp.optimizer.lbfgs_minimize(
initial_position=tf.constant(start),
tolerance=1e-8)

results = run(l1_regression_with_lbfgs)
minimum = results.position
fitted_intercept = minimum
fitted_beta = minimum[1:]

print('L-BFGS Results')
print('Converged:', results.converged)
print('Intercept: Fitted ({})'.format(fitted_intercept))
print('Beta:      Fitted {}'.format(fitted_beta))

Evaluation took: 0.017987 seconds
L-BFGS Results
Converged: True
Intercept: Fitted (2.3879985744556484)
Beta:      Fitted [ 0.68626215  0.28193532 -0.17030254  0.10799274  0.33634988 -0.24888523
0.11992237  0.08689026]


The Nelder Mead method is one of the most popular derivative free minimization methods. This optimizer doesn't use gradient information and makes no assumptions on the differentiability of the target function; it is therefore appropriate for non-smooth objective functions, for example optimization problems with L1 penalty.

For an optimization problem in $$n$$-dimensions it maintains a set of $$n+1$$ candidate solutions that span a non-degenerate simplex. It successively modifies the simplex based on a set of moves (reflection, expansion, shrinkage and contraction) using the function values at each of the vertices.

# Nelder mead expects an initial_vertex of shape [n + 1, 1].
initial_vertex = tf.expand_dims(tf.constant(start, dtype=dtype), axis=-1)

@tf.function
regression_loss,
initial_vertex=initial_vertex,
func_tolerance=1e-10,
position_tolerance=1e-10)

minimum = results.position.reshape([-1])
fitted_intercept = minimum
fitted_beta = minimum[1:]

print('Converged:', results.converged)
print('Intercept: Fitted ({})'.format(fitted_intercept))
print('Beta:      Fitted {}'.format(fitted_beta))

Evaluation took: 0.325643 seconds
Converged: True
Intercept: Fitted (2.387998456121595)
Beta:      Fitted [ 0.68626266  0.28193456 -0.17030291  0.10799375  0.33635132 -0.24888703
0.11992244  0.08689023]


## Logistic Regression with L2 penalty

For this example, we create a synthetic data set for classification and use the L-BFGS optimizer to fit the parameters.

np.random.seed(12345)

dim = 5  # The number of features
n_obs = 10000  # The number of observations

betas = np.random.randn(dim)  # The true beta
intercept = np.random.randn()  # The true intercept

features = np.random.randn(n_obs, dim)  # The feature matrix
probs = sp.special.expit(
np.matmul(features, np.expand_dims(betas, -1)) + intercept)

labels = sp.stats.bernoulli.rvs(probs)  # The true labels

regularization = 0.8
feat = tf.constant(features)
lab = tf.constant(labels, dtype=feat.dtype)

def negative_log_likelihood(params):
"""Negative log likelihood for logistic model with L2 penalty

Args:
params: A real tensor of shape [dim + 1]. The zeroth component
is the intercept term and the rest of the components are the
beta coefficients.

Returns:
The negative log likelihood plus the penalty term.
"""
intercept, beta  = params, params[1:]
logit = tf.matmul(feat, tf.expand_dims(beta, -1)) + intercept
log_likelihood = tf.reduce_sum(tf.nn.sigmoid_cross_entropy_with_logits(
labels=lab, logits=logit))
l2_penalty = regularization * tf.reduce_sum(beta ** 2)
total_loss = log_likelihood + l2_penalty

start = np.random.randn(dim + 1)

@tf.function
def l2_regression_with_lbfgs():
return tfp.optimizer.lbfgs_minimize(
negative_log_likelihood,
initial_position=tf.constant(start),
tolerance=1e-8)

results = run(l2_regression_with_lbfgs)
minimum = results.position
fitted_intercept = minimum
fitted_beta = minimum[1:]

print('Converged:', results.converged)
print('Intercept: Fitted ({}), Actual ({})'.format(fitted_intercept, intercept))
print('Beta:\n\tFitted {},\n\tActual {}'.format(fitted_beta, betas))

Evaluation took: 0.056751 seconds
Converged: True
Intercept: Fitted (1.4111415084244365), Actual (1.3934058329729904)
Beta:
Fitted [-0.18016612  0.53121578 -0.56420632 -0.5336374   2.00499675],
Actual [-0.20470766  0.47894334 -0.51943872 -0.5557303   1.96578057]


## Batching support

Both BFGS and L-BFGS support batched computation, for example to optimize a single function from many different starting points; or multiple parametric functions from a single point.

### Single function, multiple starting points

Himmelblau's function is a standard optimization test case. The function is given by:

$f(x, y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2$

The function has four minima located at:

• (3, 2),
• (-2.805118, 3.131312),
• (-3.779310, -3.283186),
• (3.584428, -1.848126).

All these minima may be reached from appropriate starting points.

# The function to minimize must take as input a tensor of shape [..., n]. In
# this n=2 is the size of the domain of the input and [...] are batching
# dimensions. The return value must be of shape [...], i.e. a batch of scalars
# with the objective value of the function evaluated at each input point.

def himmelblau(coord):
x, y = coord[..., 0], coord[..., 1]
return (x * x + y - 11) ** 2 + (x + y * y - 7) ** 2

starts = tf.constant([[1, 1],
[-2, 2],
[-1, -1],
[1, -2]], dtype='float64')

# The stopping_condition allows to further specify when should the search stop.
# The default, tfp.optimizer.converged_all, will proceed until all points have
# either converged or failed. There is also a tfp.optimizer.converged_any to
# stop as soon as the first point converges, or all have failed.

@tf.function
def batch_multiple_starts():
return tfp.optimizer.lbfgs_minimize(
himmelblau, initial_position=starts,
stopping_condition=tfp.optimizer.converged_all,
tolerance=1e-8)

results = run(batch_multiple_starts)
print('Converged:', results.converged)
print('Minima:', results.position)

Evaluation took: 0.019095 seconds
Converged: [ True  True  True  True]
Minima: [[ 3.          2.        ]
[-2.80511809  3.13131252]
[-3.77931025 -3.28318599]
[ 3.58442834 -1.84812653]]


### Multiple functions

For demonstration purposes, in this example we simultaneously optimize a large number of high dimensional randomly generated quadratic bowls.

np.random.seed(12345)

dim = 100
batches = 500
minimum = np.random.randn(batches, dim)
scales = np.exp(np.random.randn(batches, dim))

return tf.reduce_sum(input_tensor=scales * (x - minimum)**2, axis=-1)

# Make all starting points (1, 1, ..., 1). Note not all starting points need
# to be the same.
start = tf.ones((batches, dim), dtype='float64')

@tf.function
def batch_multiple_functions():
return tfp.optimizer.lbfgs_minimize(
stopping_condition=tfp.optimizer.converged_all,
max_iterations=100,
tolerance=1e-8)

results = run(batch_multiple_functions)
print('All converged:', np.all(results.converged))
print('Largest error:', np.max(results.position - minimum))

Evaluation took: 1.994132 seconds
All converged: True
Largest error: 4.4131473142527966e-08

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