Bi-Tempered Logistic Loss

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Overview of the method is here: Google AI Blogpost

Also, explore the interactive visualization that demonstrates the practical properties of the Bi-Tempered logistic loss.

Bi-Tempered logistic loss is a generalized softmax cross-entropy loss function with bounded loss value per sample and a heavy-tail softmax probability function.

Bi-tempered loss generalizes (with a bias correction term):

• Zhang & Sabuncu. "Generalized cross entropy loss for training deep neural networks with noisy labels." In NeurIPS 2018.

which is recovered when 0.0 <= t1 <= 1.0 and t2 = 1.0. It also includes:

• Ding & Vishwanathan. "t-Logistic regression." In NeurIPS 2010.

for t1 = 1.0 and t2 >= 1.0.

Bi-tempered loss is equal to the softmax cross entropy loss when t1 = t2 = 1.0. For 0.0 <= t1 < 1.0 and t2 > 1.0, bi-tempered loss provides a more robust alternative to the cross entropy loss for handling label noise and outliers.

TensorFlow

A replacement for standard logistic loss function: tf.losses.softmax_cross_entropy is available here

def bi_tempered_logistic_loss(activations,
                              labels,
                              t1,
                              t2,
                              label_smoothing=0.0,
                              num_iters=5):
  """Bi-Tempered Logistic Loss with custom gradient.
  Args:
    activations: A multi-dimensional tensor with last dimension `num_classes`.
    labels: A tensor with shape and dtype as activations.
    t1: Temperature 1 (< 1.0 for boundedness).
    t2: Temperature 2 (> 1.0 for tail heaviness, < 1.0 for finite support).
    label_smoothing: Label smoothing parameter between [0, 1).
    num_iters: Number of iterations to run the method.
  Returns:
    A loss tensor.
  """

Replacements are also available for the transfer functions:

Tempered version of tf.nn.sigmoid:

def tempered_sigmoid(activations, t, num_iters=5):
  """Tempered sigmoid function.
  Args:
    activations: Activations for the positive class for binary classification.
    t: Temperature tensor > 0.0.
    num_iters: Number of iterations to run the method.
  Returns:
    A probabilities tensor.
  """

Tempered version of tf.nn.softmax:

def tempered_softmax(activations, t, num_iters=5):
  """Tempered softmax function.
  Args:
    activations: A multi-dimensional tensor with last dimension `num_classes`.
    t: Temperature tensor > 0.0.
    num_iters: Number of iterations to run the method.
  Returns:
    A probabilities tensor.
  """

Citation

When referencing Bi-Tempered loss, cite this paper:

@inproceedings{amid2019robust,
  title={Robust bi-tempered logistic loss based on bregman divergences},
  author={Amid, Ehsan and Warmuth, Manfred KK and Anil, Rohan and Koren, Tomer},
  booktitle={Advances in Neural Information Processing Systems},
  pages={15013--15022},
  year={2019}
}

Contributions

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