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TensorFlow Constrained Optimization (TFCO)

TFCO is a library for optimizing inequality-constrained problems in TensorFlow. 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. https://arxiv.org/abs/1804.06500

which will be referred to as [CoJiSr19] throughout the remainder of this document, and:

Cotter, Gupta, Jiang, Srebro, Sridharan, Wang, Woodworth and You. "Training Well-Generalizing Classifiers for Fairness Metrics and Other Data-Dependent Constraints". https://arxiv.org/abs/1807.00028

which will be referred to as [CotterEtAl18].

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.

On a problem including proxy constraints—even a convex problem—the Lagrangian approach discussed above isn't guaranteed to work. However, a different algorithm, based on minimizing swap regret on a slightly different formulation—which we call the "proxy-Lagrangian" formulation—does work.

Shrinking

This library is designed to deal with a very flexible class of constrained problems, but this flexibility makes 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 some cases, oscillation, instead of convergence, is inevitable.

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 usually 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 issues with 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

  • constrained_minimization_problem.py: contains the ConstrainedMinimizationProblem interface, representing an inequality-constrained problem. Your own constrained optimization problems should be represented using implementations of this interface. If using the rate-based helpers, such objects can be constructed as RateMinimizationProblems.

  • candidates.py: contains two functions, find_best_candidate_distribution and find_best_candidate_index. Both of these functions are given a set of candidate solutions to a constrained optimization problem, from which the former finds the best distribution over at most m+1 candidates, and the latter heuristically finds the single best candidate. As discussed above, the set of candidates will typically be model snapshots saved periodically during optimization. Both of these functions require that scipy be installed.

    The find_best_candidate_distribution function implements the approach described in Lemma 3 of [CoJiSr19], while find_best_candidate_index implements the heuristic used for hyperparameter search in the experiments of Section 5.2.

  • Optimizing general inequality-constrained problems

    • constrained_optimizer.py: contains the ConstrainedOptimizer interface, which is similar to (but different from) tf.train.Optimizer, with the main difference being that ConstrainedOptimizers are given ConstrainedMinimizationProblems to optimize, and perform constrained optimization.

    • lagrangian_optimizer.py: contains the LagrangianOptimizer implementation, which is a ConstrainedOptimizer implementing the Lagrangian approach discussed above (with additive updates to the Lagrange multipliers). You should use this optimizer for problems without proxy constraints. It may also work for problems with proxy constraints, but we recommend using a proxy-Lagrangian optimizer, instead.

      This optimizer is most similar to Algorithm 3 in Appendix C.3 of [CoJiSr19], and is discussed in Section 3. The two differences are that it uses proxy constraints (if they're provided) in the update of the model parameters, and uses tf.train.Optimizers, instead of SGD, for the "inner" updates.

    • proxy_lagrangian_optimizer.py: contains the ProxyLagrangianOptimizer implementation, which is a ConstrainedOptimizer implementing the proxy-Lagrangian approach mentioned above. We recommend using this optimizer for problems with proxy constraints.

      The ProxyLagrangianOptimizer with multiplicative swap-regret updates is most similar to Algorithm 2 in Section 4 of [CoJiSr19], with the difference being that it uses tf.train.Optimizers, instead of SGD, for the "inner" updates.

  • Helpers for constructing rate-based optimization problems

    • subsettable_context.py: contains the rate_context function, which takes a Tensor of predictions (i.e. the output of a TensorFlow model, through which gradients can be propagated), and optionally Tensors of labels and weights, and returns an object representing a (subset of a) minibatch on which one may calculate rates.

      The related split_rate_context function takes two Tensors of predictions, labels and weights, the first for the "penalty" portion of the objective, and the second for the "constraint" portion. The purpose of splitting the context is to improve generalization performance: see [CotterEtAl18] for full details.

      The most important property of these objects is that they are subsettable: if you want to calculate a rate on e.g. only the negatively-labeled examples, or only those examples belonging to a certain protected class, then this can be accomplished via the subset method. However, you should use great caution with the subset method: if the desired subset is a very small proportion of the dataset (e.g. a protected class that's an extreme minority), then the resulting stochastic gradients will be noisy, and during training your model will converge very slowly. Instead, it is usually better (but less convenient) to create an entirely separate dataset for each rare subset (e.g. using the "filter" method of a tf.train.Dataset), and to construct each subset context directly from each such dataset.

    • binary_rates.py: contains functions for constructing rates from contexts. These rates can then be combined into more complicated expressions using python arithmetic operators, or into constraints using comparison operators.

    • operations.py: contains functions for manipulating rate expressions, including wrap_rate, which can be used to convert a Tensor into a rate object, as well as lower_bound and upper_bound, which convert lists of rates into rates representing lower- and upper-bounds on all elements of the list.

    • loss.py: contains loss functions used in constructing rates. These can be passed as parameters to the optional penalty_loss and constraint_loss functions in "binary_rates.py" (above).

    • rate_minimization_problem.py: contains the RateMinimizationProblem class, which constructs a ConstrainedMinimizationProblem (suitable for use by ConstrainedOptimizers) from a rate expression to minimize, and a list of rate constraints to impose.

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)
predictions = tf.tensordot(constant_features, weights, axes=(1, 0)) - threshold

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.

We'll now consider each of these two options in turn.

Option 1: 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.

context = tfco.rate_context(predictions, labels=constant_labels)
problem = tfco.RateMinimizationProblem(
    tfco.error_rate(context), [tfco.recall(context) >= recall_lower_bound])

Rate-construction functions (error_rate and recall, here) 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 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 combined 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.

Option 2: explicit ConstrainedMinimizationProblem

For problems that cannot be easily expressed using the rate helpers, one could instead an explicit implementation of the ConstrainedMinimizationProblem interface. The current task (minimizing the average hinge loss subject to a recall constraint) is a rate-based problem, but for illustrative reasons we will show how to create a ConstrainedMinimizationProblem for this task.

The constructor will takes three parameters: a Tensor containing the classification labels (0 or 1) for every training example, another Tensor containing the model's predictions on every training example (sometimes called the "logits"), and the lower bound on recall that will be enforced using a constraint.

As before, this implementation will contain both constraints and proxy constraints: the former represents the constraint that the true recall (defined in terms of the number of true positives) be at least recall_lower_bound, while the latter represents the same constraint, but on a hinge approximation of the recall.

class ExampleProblem(tfco.ConstrainedMinimizationProblem):

  def __init__(self, labels, predictions, recall_lower_bound):
    self._labels = labels
    self._predictions = predictions
    self._recall_lower_bound = recall_lower_bound
    # The number of positively-labeled examples.
    self._positive_count = tf.reduce_sum(self._labels)

  @property
  def objective(self):
    return tf.losses.hinge_loss(labels=self._labels, logits=self._predictions)

  @property
  def constraints(self):
    # Recall that the labels are binary (0 or 1).
    true_positives = self._labels * tf.to_float(self._predictions > 0)
    true_positive_count = tf.reduce_sum(true_positives)
    recall = true_positive_count / self._positive_count
    # The constraint is (recall >= self._recall_lower_bound), which we convert
    # to (self._recall_lower_bound - recall <= 0) because
    # ConstrainedMinimizationProblems must always provide their constraints in
    # the form (tensor <= 0).
    #
    # The result of this function should be a tensor, with each element being
    # a quantity that is constrained to be nonpositive. We only have one
    # constraint, so we return a one-element tensor.
    return self._recall_lower_bound - recall

  @property
  def proxy_constraints(self):
    # Use 1 - hinge since we're SUBTRACTING recall in the constraint function,
    # and we want the proxy constraint function to be convex. Recall that the
    # labels are binary (0 or 1).
    true_positives = self._labels * tf.minimum(1.0, self._predictions)
    true_positive_count = tf.reduce_sum(true_positives)
    recall = true_positive_count / self._positive_count
    # Please see the corresponding comment in the constraints property.
    return self._recall_lower_bound - recall

problem = ExampleProblem(
    labels=constant_labels,
    predictions=predictions,
    recall_lower_bound=recall_lower_bound,
)

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, the Lagrangian optimizer often suffices for problems without proxy constraints, but proxy-Lagrangian optimizers are recommended for problems with proxy constraints. Since this problem contains proxy constraints, we use the ProxyLagrangianOptimizer.

For this problem, the constraint is fairly easy to satisfy, so we can use the same "inner" optimizer (an AdagradOptimizer 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 ProxyLagrangianOptimizer). 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 ProxyLagrangianOptimizer'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).

with tf.Session() as session:
  optimizer = tfco.ProxyLagrangianOptimizer(
      optimizer=tf.train.AdagradOptimizer(learning_rate=1.0))
  train_op = optimizer.minimize(problem)

  session.run(tf.global_variables_initializer())
  for ii in xrange(1000):
    session.run(train_op)

  trained_weights, trained_threshold = session.run((weights, threshold))

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))

Running the above code gives 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:

with tf.Session() as session:
  optimizer = tf.train.AdagradOptimizer(learning_rate=1.0)
  # For optimizing the unconstrained problem, we just minimize the "objective"
  # portion of the minimization problem. We could instead have used the
  # "minimize_unconstrained" method of a ConstrainedOptimizer.
  train_op = optimizer.minimize(problem.objective)

  session.run(tf.global_variables_initializer())
  for ii in xrange(1000):
    session.run(train_op)

  trained_weights, trained_threshold = session.run((weights, threshold))

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/jupyter_notebooks directory contains several Jupyter notebooks illustrating how to use this library:

  • Recall_constraint.ipynb: Start here! This is the above simple example.

  • Fairness_adult.ipynb: This notebook shows how to train classifiers for fairness constraints on the UCI Adult dataset using the helpers for constructing rate-based optimization problems.

  • Minibatch_training.ipynb: This notebook describes how to solve a rate-constrained training problem using minibatches. The notebook focuses on problems where one wishes to impose a constraint on a group of examples constituting an extreme minority of the training set, and shows how one can speed up convergence by using separate streams of minibatches for each group.

  • Oscillation_compas.ipynb: This notebook illustrates the oscillation issue raised in the "skrinking" section (above): it's possible that the individual iterates won't converge when using the Lagrangian approach to training with fairness constraints, even though they do converge on average. This motivate more careful selection of solutions or the use of a stochastic classifier.

  • Post_processing.ipynb: This notebook describes how to use the shrinking procedure of [CoJiSr19], as discussed in the "shrinking" section (above), to post-process the iterates of a constrained optimizer and construct a stochastic classifier from them. For applications where a stochastic classifier is not acceptable, we show how to use a heuristic to pick the best deterministic classifier from the iterates found by the optimizer.

  • Generalization_communities.ipynb: This notebook shows how to improve fairness generalization performance on the UCI Communities and Crime dataset with the split dataset approach of [CotterEtAl18], using the split_rate_context helper.

  • Churn.ipynb: This notebook describes how to use rate constraints for low-churn classification. That is, to train for accuracy while ensuring the predictions don't differ by much compared to a baseline model.

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