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Fenchel-Young losses

This package implements loss functions useful for probabilistic classification. More specifically, it provides

• drop-in replacements for PyTorch loss functions
• drop-in replacements for TensorFlow loss functions
• scikit-learn compatible classifiers

The package is based on the Fenchel-Young loss framework [1,2,3].

.. image:: examples/tsallis.png :alt: Tsallis losses :align: center

Notice from the center plot that sparsemax and Tsallis are able to produce exactly zero (sparse) probabilities unlike the logistic (softmax) loss.

Supported Fenchel-Young losses

• Multinomial logistic loss
• One-vs-all logistic loss
• Sparsemax loss (sparse probabilities!)
• Tsallis losses (sparse probabilities!)

Sparse means that some classes have exactly zero probability, i.e., these classes are irrelevant.

Tsallis losses are a family of losses parametrized by a positive real value α. They recover the multinomial logistic loss with α=1 and the sparsemax loss with α=2. Values of α between 1 and 2 enable to interpolate between the two losses.

In all losses above, the ground-truth can either be a n_samples 1d-array of label integers (each label should be between 0 and n_classes-1) or a n_samples x n_classes 2d-array of label proportions (each row should sum to 1).

Examples

scikit-learn compatible classifier:

.. code-block:: python

import numpy as np from sklearn.datasets import make_classification from fyl_sklearn import FYClassifier

X, y = make_classification(n_samples=10, n_features=5, n_informative=3, n_classes=3, random_state=0) clf = FYClassifier(loss="sparsemax") clf.fit(X, y) print(clf.predict_proba(X[:3]))

Drop-in replacement for PyTorch losses:

.. code-block:: python

import torch from fyl_pytorch import SparsemaxLoss

integers between 0 and n_classes-1, shape = n_samples

y_true = torch.tensor([0, 2])

model scores, shapes = n_samples x n_classes

theta = torch.tensor([[-2.5, 1.2, 0.5], [2.2, 0.8, -1.5]]) loss = SparsemaxLoss()

loss value (caution: reversed convention compared to numpy and tensorflow)

print(loss(theta, y_true))

predictions (probabilities) are stored for convenience

print(loss.y_pred)

can also recompute them from theta

print(loss.predict(theta))

label proportions are also allowed

y_true = torch.tensor([[0.8, 0.2, 0], [0.1, 0.2, 0.7]]) print(loss(theta, y_true))

Drop-in replacement for tensorflow losses:

.. code-block:: python

import tensorflow as tf from fyl_tensorflow import sparsemax_loss, sparsemax_predict

integers between 0 and n_classes-1, shape = n_samples

y_true = tf.constant([0, 2])

model scores, shapes = n_samples x n_classes

theta = tf.constant([[-2.5, 1.2, 0.5], [2.2, 0.8, -1.5]])

loss value

print(sparsemax_loss(y_true, theta))

predictions (probabilities)

print(sparsemax_predict(theta))

label proportions are also allowed

y_true = tf.constant([[0.8, 0.2, 0], [0.1, 0.2, 0.7]]) print(sparsemax_loss(y_true, theta))

Installation

Simply copy relevant files to your project.

The TensorFlow implementation requires the installation of TensorFlow Addons <https://github.com/tensorflow/addons>_.

References

.. [1] SparseMAP: Differentiable Sparse Structured Inference. Vlad Niculae, André F. T. Martins, Mathieu Blondel, Claire Cardie. In Proc. of ICML 2018. [arXiv <https://arxiv.org/abs/1802.04223>_]

.. [2] Learning Classifiers with Fenchel-Young Losses: Generalized Entropies, Margins, and Algorithms. Mathieu Blondel, André F. T. Martins, Vlad Niculae. In Proc. of AISTATS 2019. [arXiv <https://arxiv.org/abs/1805.09717>_]

.. [3] Learning with Fenchel-Young Losses. Mathieu Blondel, André F. T. Martins, Vlad Niculae. Preprint. [arXiv <https://arxiv.org/abs/1901.02324>_]

Author

• Mathieu Blondel, 2018