The Global Objectives library provides TensorFlow loss functions that optimize directly for a variety of objectives including AUC, recall at precision, and more. The global objectives losses can be used as drop-in replacements for TensorFlow's standard multilabel loss functions: tf.nn.sigmoid_cross_entropy_with_logits and tf.losses.sigmoid_cross_entropy.

Many machine learning classification models are optimized for classification accuracy, when the real objective the user cares about is different and can be precision at a fixed recall, precision-recall AUC, ROC AUC or similar metrics. These are referred to as "global objectives" because they depend on how the model classifies the dataset as a whole and do not decouple across data points as accuracy does.

Because these objectives are combinatorial, discontinuous, and essentially intractable to optimize directly, the functions in this library approximate their corresponding objectives. This approximation approach follows the same pattern as optimizing for accuracy, where a surrogate objective such as cross-entropy or the hinge loss is used as an upper bound on the error rate.

For a full example of how to use the loss functions in practice, see loss_layers_example.py.

Briefly, global objective losses can be used to replace tf.nn.sigmoid_cross_entropy_with_logits by providing the relevant additional arguments. For example,

tf.nn.sigmoid_cross_entropy_with_logits(labels=labels, logits=logits)

could be replaced with

global_objectives.recall_at_precision_loss(
    labels=labels,
    logits=logits,
    target_precision=0.95)[0]

Just as minimizing the cross-entropy loss will maximize accuracy, the loss functions in loss_layers.py were written so that minimizing the loss will maximize the corresponding objective.

The global objective losses have two return values -- the loss tensor and additional quantities for debugging and customization -- which is why the first value is used above. For more information, see Visualization & Debugging.

Binary classification problems can be represented as a multi-class problem with two classes, or as a multi-label problem with one label. (Recall that multiclass problems have mutually exclusive classes, e.g. 'cat xor dog', and multilabel have classes which are not mutually exclusive, e.g. an image can contain a cat, a dog, both, or neither.) The softmax loss (tf.nn.softmax_cross_entropy_with_logits) is used for multi-class problems, while the sigmoid loss (tf.nn.sigmoid_cross_entropy_with_logits) is used for multi-label problems.

A multiclass label format for binary classification might represent positives with the label [1, 0] and negatives with the label [0, 1], while the multilbel format for the same problem would use [1] and [0], respectively.

All global objectives loss functions assume that the multilabel format is used. Accordingly, if your current loss function is softmax, the labels will have to be reformatted for the loss to work properly.

Global objectives losses (except for roc_auc_loss) use internal variables called dual variables or Lagrange multipliers to enforce the desired constraint (e.g. if optimzing for recall at precision, the constraint is on precision).

These dual variables are created and initialized internally by the loss functions, and are updated during training by the same optimizer used for the model's other variables. To initialize the dual variables to a particular value, use the lambdas_initializer argument. The dual variables can be found under the key lambdas in the other_outputs dictionary returned by the losses.

The following arguments are common to all loss functions in the library, and are either required or very important.

• labels: Corresponds directly to the labels argument of tf.nn.sigmoid_cross_entropy_with_logits.
• logits: Corresponds directly to the logits argument of tf.nn.sigmoid_cross_entropy_with_logits.
• dual_rate_factor: A floating point value which controls the step size for the Lagrange multipliers. Setting this value less than 1.0 will cause the constraint to be enforced more gradually and will result in more stable training.

In addition, the objectives with a single constraint (e.g. recall_at_precision_loss) have an argument (e.g. target_precision) used to specify the value of the constraint. The optional precision_range argument to precision_recall_auc_loss is used to specify the range of precision values over which to optimize the AUC, and defaults to the interval [0, 1].

Optional arguments:

• weights: A tensor which acts as coefficients for the loss. If a weight of x is provided for a datapoint and that datapoint is a true (false) positive (negative), it will be counted as x true (false) positives (negatives). Defaults to 1.0.
• label_priors: A tensor specifying the fraction of positive datapoints for each label. If not provided, it will be computed inside the loss function.
• surrogate_type: Either 'xent' or 'hinge', specifying which upper bound should be used for indicator functions.
• lambdas_initializer: An initializer for the dual variables (Lagrange multipliers). See also the Dual Variables section.
• num_anchors (precision_recall_auc_loss only): The number of grid points used when approximating the AUC as a Riemann sum.

While the functional form of the global objectives losses allow them to be easily substituted in place of sigmoid_cross_entropy_with_logits, model hyperparameters such as learning rate, weight decay, etc. may need to be fine-tuned to the new loss. Fortunately, the amount of hyperparameter re-tuning is usually minor.

The most important hyperparameters to modify are the learning rate and dual_rate_factor (see the section on Loss Function Arguments, above).

The global objectives losses return two values. The first is a tensor representing the numerical value of the loss, which can be passed to an optimizer. The second is a dictionary of tensors created by the loss function which are not necessary for optimization but useful in debugging. These vary depending on the loss function, but usually include lambdas (the Lagrange multipliers) as well as the lower bound on true positives and upper bound on false positives.

When visualizing the loss during training, note that the global objectives losses differ from standard losses in some important ways:

• The global losses may be negative. This is because the value returned by the loss includes terms involving the Lagrange multipliers, which may be negative.
• The global losses may not decrease over the course of training. To enforce the constraints in the objective, the loss changes over time and may increase.

For more details, see the Global Objectives paper.

• Mariano Schain
• Elad Eban
• Alan Mackey