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TensorFlow Constrained Optimization (TFCO)

TFCO is a library for optimizing inequality-constrained problems in TensorFlow 1.14 and later (including TensorFlow 2). In the most general case, both the objective function and the constraints are represented as Tensors, giving users the maximum amount of flexibility in specifying their optimization problems. Constructing these Tensors can be cumbersome, so we also provide helper functions to make it easy to construct constrained optimization problems based on rates, i.e. proportions of the training data on which some event occurs (e.g. the error rate, true positive rate, recall, etc).

For full details, motivation, and theoretical results on the approach taken by this library, please refer to:

Cotter, Jiang and Sridharan. "Two-Player Games for Efficient Non-Convex Constrained Optimization". ALT'19, arXiv

and:

Narasimhan, Cotter and Gupta. "Optimizing Generalized Rate Metrics with Three Players". NeurIPS'19

which will be referred to as [CoJiSr19] and [NaCoGu19], respectively, throughout the remainder of this document. For more information on this library's optional "two-dataset" approach to improving generalization, please see:

Cotter, Gupta, Jiang, Srebro, Sridharan, Wang, Woodworth and You. "Training Well-Generalizing Classifiers for Fairness Metrics and Other Data-Dependent Constraints". ICML'19, arXiv

which will be referred to as [CotterEtAl19].

Proxy constraints

Imagine that we want to constrain the recall of a binary classifier to be at least 90%. Since the recall is proportional to the number of true positive classifications, which itself is a sum of indicator functions, this constraint is non-differentiable, and therefore cannot be used in a problem that will be optimized using a (stochastic) gradient-based algorithm.

For this and similar problems, TFCO supports so-called proxy constraints, which are differentiable (or sub/super-differentiable) approximations of the original constraints. For example, one could create a proxy recall function by replacing the indicator functions with sigmoids. During optimization, each proxy constraint function will be penalized, with the magnitude of the penalty being chosen to satisfy the corresponding original (non-proxy) constraint.

Rate helpers

While TFCO can optimize "low-level" constrained optimization problems represented in terms of Tensors (by creating a ConstrainedMinimizationProblem directly), one of TFCO's main goals is to make it easy to configure and optimize problems based on rates. This includes both very simple settings, e.g. maximizing precision subject to a recall constraint, and more complex, e.g. maximizing ROC AUC subject to the constraint that the maximum and minimum error rates over some particular slices of the data should be within 10% of each other. To this end, we provide high-level "rate helpers", for which proxy constraints are handled automatically, and with which one can write optimization problems in simple mathematical notation (i.e. minimize this expression subject to this list of algebraic constraints).

These helpers include a number of functions for constructing (in "binary_rates.py", "multiclass_rates.py" and "general_rates.py") and manipulating ("operations.py", and Python arithmetic operators) rates. Some of these, as described in [NaCoGu19], require introducing slack variables and extra implicit constraints to the resulting optimization problem, which, again, is handled automatically.

Shrinking

This library is designed to deal with a very flexible class of constrained problems, but this flexibility can make optimization considerably more difficult: on a non-convex problem, if one uses the "standard" approach of introducing a Lagrange multiplier for each constraint, and then jointly maximizing over the Lagrange multipliers and minimizing over the model parameters, then a stable stationary point might not even exist. Hence, in such cases, one might experience oscillation, instead of convergence.

Thankfully, it turns out that even if, over the course of optimization, no particular iterate does a good job of minimizing the objective while satisfying the constraints, the sequence of iterates, on average, usually will. This observation suggests the following approach: at training time, we'll periodically snapshot the model state during optimization; then, at evaluation time, each time we're given a new example to evaluate, we'll sample one of the saved snapshots uniformly at random, and apply it to the example. This stochastic model will generally perform well, both with respect to the objective function, and the constraints.

In fact, we can do better: it's possible to post-process the set of snapshots to find a distribution over at most m+1 snapshots, where m is the number of constraints, that will be at least as good (and will often be much better) than the (much larger) uniform distribution described above. If you're unable or unwilling to use a stochastic model at all, then you can instead use a heuristic to choose the single best snapshot.

In many cases, these issues can be ignored. However, if you experience oscillation during training, or if you want to squeeze every last drop of performance out of your model, consider using the "shrinking" procedure of [CoJiSr19], which is implemented in the "candidates.py" file.

Public contents

Convex example using proxy constraints

This is a simple example of recall-constrained optimization on simulated data: we seek a classifier that minimizes the average hinge loss while constraining recall to be at least 90%.

We'll start with the required imports—notice the definition of tfco:

import math
import numpy as np
from six.moves import xrange
import tensorflow as tf

import tensorflow_constrained_optimization as tfco

We'll next create a simple simulated dataset by sampling 1000 random 10-dimensional feature vectors from a Gaussian, finding their labels using a random "ground truth" linear model, and then adding noise by randomly flipping 200 labels.

# Create a simulated 10-dimensional training dataset consisting of 1000 labeled
# examples, of which 800 are labeled correctly and 200 are mislabeled.
num_examples = 1000
num_mislabeled_examples = 200
dimension = 10
# We will constrain the recall to be at least 90%.
recall_lower_bound = 0.9

# Create random "ground truth" parameters for a linear model.
ground_truth_weights = np.random.normal(size=dimension) / math.sqrt(dimension)
ground_truth_threshold = 0

# Generate a random set of features for each example.
features = np.random.normal(size=(num_examples, dimension)).astype(
    np.float32) / math.sqrt(dimension)
# Compute the labels from these features given the ground truth linear model.
labels = (np.matmul(features, ground_truth_weights) >
          ground_truth_threshold).astype(np.float32)
# Add noise by randomly flipping num_mislabeled_examples labels.
mislabeled_indices = np.random.choice(
    num_examples, num_mislabeled_examples, replace=False)
labels[mislabeled_indices] = 1 - labels[mislabeled_indices]

We're now ready to construct our model, and the corresponding optimization problem. We'll use a linear model of the form f(x) = w^T x - t, where w is the weights, and t is the threshold.

# Create variables containing the model parameters.
weights = tf.Variable(tf.zeros(dimension), dtype=tf.float32, name="weights")
threshold = tf.Variable(0.0, dtype=tf.float32, name="threshold")

# Create the optimization problem.
constant_labels = tf.constant(labels, dtype=tf.float32)
constant_features = tf.constant(features, dtype=tf.float32)
def predictions():
  return tf.tensordot(constant_features, weights, axes=(1, 0)) - threshold

Notice that predictions is a nullary function returning a Tensor. This is needed to support eager mode, but in graph mode, it's fine for it to simply be a Tensor. To see how this example could work in graph mode, please see the Jupyter notebook containing a more-comprehensive version of this example (Recall_constraint.ipynb).

Now that we have the output of our linear model (in the predictions variable), we can move on to constructing the optimization problem. At this point, there are two ways to proceed:

  1. We can use the rate helpers provided by the TFCO library. This is the easiest way to construct optimization problems based on rates (where a "rate" is the proportion of training examples on which some event occurs).
  2. We could instead create an implementation of the ConstrainedMinimizationProblem interface. This is the most flexible approach. In particular, it is not limited to problems expressed in terms of rates.

Here, we'll only consider the first of these options. To see how to use the second option, please refer to Recall_constraint.ipynb.

Rate helpers

The main motivation of TFCO is to make it easy to create and optimize constrained problems written in terms of linear combinations of rates, where a "rate" is the proportion of training examples on which an event occurs (e.g. the false positive rate, which is the number of negatively-labeled examples on which the model makes a positive prediction, divided by the number of negatively-labeled examples). Our current example (minimizing a hinge relaxation of the error rate subject to a recall constraint) is such a problem.

# Like the predictions, in eager mode, the labels should be a nullary function
# returning a Tensor. In graph mode, you can drop the lambda.
context = tfco.rate_context(predictions, labels=lambda: constant_labels)
problem = tfco.RateMinimizationProblem(
    tfco.error_rate(context), [tfco.recall(context) >= recall_lower_bound])

The first argument of all rate-construction helpers (error_rate and recall are the ones used here) is a "context" object, which represents what we're taking the rate of. For example, in a fairness problem, we might wish to constrain the positive_prediction_rates of two protected classes (i.e. two subsets of the data) to be similar. In that case, we would create a context representing the entire dataset, then call the context's subset method to create contexts for the two protected classes, and finally call the positive_prediction_rate helper on the two resulting contexts. Here, we only create a single context, representing the entire dataset, since we're only concerned with the error rate and recall.

In addition to the context, rate-construction helpers also take two optional named parameters—not used here—named penalty_loss and constraint_loss, of which the former is used to define the proxy constraints, and the latter the "true" constraints. These default to the hinge and zero-one losses, respectively. The consequence of this is that we will attempt to minimize the average hinge loss (a relaxation of the error rate using the penalty_loss), while constraining the true recall (using the constraint_loss) by essentially learning how much we should penalize the hinge-constrained recall (penalty_loss, again).

The RateMinimizationProblem class implements the ConstrainedMinimizationProblem interface, and is constructed from a rate expression to be minimized (the first parameter), subject to a list of rate constraints (the second). Using this class is typically more convenient and readable than constructing a ConstrainedMinimizationProblem manually: the objects returned by error_rate and recall—and all other rate-constructing and rate-combining functions—can be manipulated using python arithmetic operators (e.g. "0.5 * tfco.error_rate(context1) - tfco.true_positive_rate(context2)"), or converted into a constraint using a comparison operator.

Wrapping up

We're almost ready to train our model, but first we'll create a couple of functions to measure its performance. We're interested in two quantities: the average hinge loss (which we seek to minimize), and the recall (which we constrain).

def average_hinge_loss(labels, predictions):
  # Recall that the labels are binary (0 or 1).
  signed_labels = (labels * 2) - 1
  return np.mean(np.maximum(0.0, 1.0 - signed_labels * predictions))

def recall(labels, predictions):
  # Recall that the labels are binary (0 or 1).
  positive_count = np.sum(labels)
  true_positives = labels * (predictions > 0)
  true_positive_count = np.sum(true_positives)
  return true_positive_count / positive_count

As was mentioned earlier, a Lagrangian optimizer often suffices for problems without proxy constraints, but a proxy-Lagrangian optimizer is recommended for problems with proxy constraints. Since this problem contains proxy constraints, we use the ProxyLagrangianOptimizerV2.

For this problem, the constraint is fairly easy to satisfy, so we can use the same "inner" optimizer (an Adagrad optimizer with a learning rate of 1) for optimization of both the model parameters (weights and threshold), and the internal parameters associated with the constraints (these are the analogues of the Lagrange multipliers used by the proxy-Lagrangian formulation). For more difficult problems, it will often be necessary to use different optimizers, with different learning rates (presumably found via a hyperparameter search): to accomplish this, pass both the optimizer and constraint_optimizer parameters to ProxyLagrangianOptimizerV2's constructor.

Since this is a convex problem (both the objective and proxy constraint functions are convex), we can just take the last iterate. Periodic snapshotting, and the use of the find_best_candidate_distribution or find_best_candidate_index functions, is generally only necessary for non-convex problems (and even then, it isn't always necessary).

# ProxyLagrangianOptimizerV2 is based on tf.keras.optimizers.Optimizer.
# ProxyLagrangianOptimizerV1 (which we do not use here) would work equally well,
# but is based on the older tf.compat.v1.train.Optimizer.
optimizer = tfco.ProxyLagrangianOptimizerV2(
    optimizer=tf.keras.optimizers.Adagrad(learning_rate=1.0),
    num_constraints=problem.num_constraints)

# In addition to the model parameters (weights and threshold), we also need to
# optimize over any trainable variables associated with the problem (e.g.
# implicit slack variables and weight denominators), and those associated with
# the optimizer (the analogues of the Lagrange multipliers used by the
# proxy-Lagrangian formulation).
var_list = ([weights, threshold] + problem.trainable_variables +
            optimizer.trainable_variables())

for ii in xrange(1000):
  optimizer.minimize(problem, var_list=var_list)

trained_weights = weights.numpy()
trained_threshold = threshold.numpy()

trained_predictions = np.matmul(features, trained_weights) - trained_threshold
print("Constrained average hinge loss = %f" % average_hinge_loss(
    labels, trained_predictions))
print("Constrained recall = %f" % recall(labels, trained_predictions))

Notice that this code is intended to run in eager mode (there is no session): in Recall_constraint.ipynb, we also show how to train in graph mode. Running this code results in the following output (due to the randomness of the dataset, you'll get a different result when you run it):

Constrained average hinge loss = 0.683846
Constrained recall = 0.899791

As we hoped, the recall is extremely close to 90%—and, thanks to the fact that the optimizer uses a (hinge) proxy constraint only when needed, and the actual (zero-one) constraint whenever possible, this is the true recall, not a hinge approximation.

For comparison, let's try optimizing the same problem without the recall constraint:

optimizer = tf.keras.optimizers.Adagrad(learning_rate=1.0)
var_list = [weights, threshold]

for ii in xrange(1000):
  # For optimizing the unconstrained problem, we just minimize the "objective"
  # portion of the minimization problem.
  optimizer.minimize(problem.objective, var_list=var_list)

trained_weights = weights.numpy()
trained_threshold = threshold.numpy()

trained_predictions = np.matmul(features, trained_weights) - trained_threshold
print("Unconstrained average hinge loss = %f" % average_hinge_loss(
    labels, trained_predictions))
print("Unconstrained recall = %f" % recall(labels, trained_predictions))

This code gives the following output (again, you'll get a different answer, since the dataset is random):

Unconstrained average hinge loss = 0.612755
Unconstrained recall = 0.801670

Because there is no constraint, the unconstrained problem does a better job of minimizing the average hinge loss, but naturally doesn't approach 90% recall.

More examples

The examples directory contains several illustrations of how one can use this library: