Regression losses

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MeanSquaredError class

keras.losses.MeanSquaredError(
    reduction="sum_over_batch_size", name="mean_squared_error"
)

Computes the mean of squares of errors between labels and predictions.

Formula:

loss = mean(square(y_true - y_pred))

Arguments

• reduction: Type of reduction to apply to the loss. In almost all cases this should be "sum_over_batch_size". Supported options are "sum", "sum_over_batch_size" or None.
• name: Optional name for the loss instance.

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MeanAbsoluteError class

keras.losses.MeanAbsoluteError(
    reduction="sum_over_batch_size", name="mean_absolute_error"
)

Computes the mean of absolute difference between labels and predictions.

Formula:

loss = mean(abs(y_true - y_pred))

Arguments

• reduction: Type of reduction to apply to the loss. In almost all cases this should be "sum_over_batch_size". Supported options are "sum", "sum_over_batch_size" or None.
• name: Optional name for the loss instance.

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MeanAbsolutePercentageError class

keras.losses.MeanAbsolutePercentageError(
    reduction="sum_over_batch_size", name="mean_absolute_percentage_error"
)

Computes the mean absolute percentage error between y_true & y_pred.

Formula:

loss = 100 * mean(abs((y_true - y_pred) / y_true))

Arguments

• reduction: Type of reduction to apply to the loss. In almost all cases this should be "sum_over_batch_size". Supported options are "sum", "sum_over_batch_size" or None.
• name: Optional name for the loss instance.

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MeanSquaredLogarithmicError class

keras.losses.MeanSquaredLogarithmicError(
    reduction="sum_over_batch_size", name="mean_squared_logarithmic_error"
)

Computes the mean squared logarithmic error between y_true & y_pred.

Formula:

loss = mean(square(log(y_true + 1) - log(y_pred + 1)))

Arguments

• reduction: Type of reduction to apply to the loss. In almost all cases this should be "sum_over_batch_size". Supported options are "sum", "sum_over_batch_size" or None.
• name: Optional name for the loss instance.

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CosineSimilarity class

keras.losses.CosineSimilarity(
    axis=-1, reduction="sum_over_batch_size", name="cosine_similarity"
)

Computes the cosine similarity between y_true & y_pred.

Note that it is a number between -1 and 1. When it is a negative number between -1 and 0, 0 indicates orthogonality and values closer to -1 indicate greater similarity. This makes it usable as a loss function in a setting where you try to maximize the proximity between predictions and targets. If either y_true or y_pred is a zero vector, cosine similarity will be 0 regardless of the proximity between predictions and targets.

Formula:

loss = -sum(l2_norm(y_true) * l2_norm(y_pred))

Arguments

• axis: The axis along which the cosine similarity is computed (the features axis). Defaults to -1.
• reduction: Type of reduction to apply to the loss. In almost all cases this should be "sum_over_batch_size". Supported options are "sum", "sum_over_batch_size" or None.
• name: Optional name for the loss instance.

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mean_squared_error function

keras.losses.mean_squared_error(y_true, y_pred)

Computes the mean squared error between labels and predictions.

Formula:

loss = mean(square(y_true - y_pred), axis=-1)

Example

>>> y_true = np.random.randint(0, 2, size=(2, 3))
>>> y_pred = np.random.random(size=(2, 3))
>>> loss = keras.losses.mean_squared_error(y_true, y_pred)

Arguments

• y_true: Ground truth values with shape = [batch_size, d0, .. dN].
• y_pred: The predicted values with shape = [batch_size, d0, .. dN].

Returns

Mean squared error values with shape = [batch_size, d0, .. dN-1].

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mean_absolute_error function

keras.losses.mean_absolute_error(y_true, y_pred)

Computes the mean absolute error between labels and predictions.

loss = mean(abs(y_true - y_pred), axis=-1)

Arguments

• y_true: Ground truth values with shape = [batch_size, d0, .. dN].
• y_pred: The predicted values with shape = [batch_size, d0, .. dN].

Returns

Mean absolute error values with shape = [batch_size, d0, .. dN-1].

Example

>>> y_true = np.random.randint(0, 2, size=(2, 3))
>>> y_pred = np.random.random(size=(2, 3))
>>> loss = keras.losses.mean_absolute_error(y_true, y_pred)

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mean_absolute_percentage_error function

keras.losses.mean_absolute_percentage_error(y_true, y_pred)

Computes the mean absolute percentage error between y_true & y_pred.

Formula:

loss = 100 * mean(abs((y_true - y_pred) / y_true), axis=-1)

Division by zero is prevented by dividing by maximum(y_true, epsilon) where epsilon = keras.backend.epsilon() (default to 1e-7).

Arguments

• y_true: Ground truth values with shape = [batch_size, d0, .. dN].
• y_pred: The predicted values with shape = [batch_size, d0, .. dN].

Returns

Mean absolute percentage error values with shape = [batch_size, d0, .. dN-1].

Example

>>> y_true = np.random.random(size=(2, 3))
>>> y_pred = np.random.random(size=(2, 3))
>>> loss = keras.losses.mean_absolute_percentage_error(y_true, y_pred)

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mean_squared_logarithmic_error function

keras.losses.mean_squared_logarithmic_error(y_true, y_pred)

Computes the mean squared logarithmic error between y_true & y_pred.

Formula:

loss = mean(square(log(y_true + 1) - log(y_pred + 1)), axis=-1)

Note that y_pred and y_true cannot be less or equal to 0. Negative values and 0 values will be replaced with keras.backend.epsilon() (default to 1e-7).

Arguments

• y_true: Ground truth values with shape = [batch_size, d0, .. dN].
• y_pred: The predicted values with shape = [batch_size, d0, .. dN].

Returns

Mean squared logarithmic error values with shape = [batch_size, d0, .. dN-1].

Example

>>> y_true = np.random.randint(0, 2, size=(2, 3))
>>> y_pred = np.random.random(size=(2, 3))
>>> loss = keras.losses.mean_squared_logarithmic_error(y_true, y_pred)

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cosine_similarity function

keras.losses.cosine_similarity(y_true, y_pred, axis=-1)

Computes the cosine similarity between labels and predictions.

Formula:

loss = -sum(l2_norm(y_true) * l2_norm(y_pred))

Note that it is a number between -1 and 1. When it is a negative number between -1 and 0, 0 indicates orthogonality and values closer to -1 indicate greater similarity. This makes it usable as a loss function in a setting where you try to maximize the proximity between predictions and targets. If either y_true or y_pred is a zero vector, cosine similarity will be 0 regardless of the proximity between predictions and targets.

Arguments

• y_true: Tensor of true targets.
• y_pred: Tensor of predicted targets.
• axis: Axis along which to determine similarity. Defaults to -1.

Returns

Cosine similarity tensor.

Example

>>> y_true = [[0., 1.], [1., 1.], [1., 1.]]
>>> y_pred = [[1., 0.], [1., 1.], [-1., -1.]]
>>> loss = keras.losses.cosine_similarity(y_true, y_pred, axis=-1)
[-0., -0.99999994, 0.99999994]

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Huber class

keras.losses.Huber(delta=1.0, reduction="sum_over_batch_size", name="huber_loss")

Computes the Huber loss between y_true & y_pred.

Formula:

for x in error:
    if abs(x) <= delta:
        loss.append(0.5 * x^2)
    elif abs(x) > delta:
        loss.append(delta * abs(x) - 0.5 * delta^2)

loss = mean(loss, axis=-1)

See: Huber loss.

Arguments

• delta: A float, the point where the Huber loss function changes from a quadratic to linear.
• reduction: Type of reduction to apply to loss. Options are "sum", "sum_over_batch_size" or None. Defaults to "sum_over_batch_size".
• name: Optional name for the instance.

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huber function

keras.losses.huber(y_true, y_pred, delta=1.0)

Computes Huber loss value.

Formula:

for x in error:
    if abs(x) <= delta:
        loss.append(0.5 * x^2)
    elif abs(x) > delta:
        loss.append(delta * abs(x) - 0.5 * delta^2)

loss = mean(loss, axis=-1)

See: Huber loss.

Example

>>> y_true = [[0, 1], [0, 0]]
>>> y_pred = [[0.6, 0.4], [0.4, 0.6]]
>>> loss = keras.losses.huber(y_true, y_pred)
0.155

Arguments

• y_true: tensor of true targets.
• y_pred: tensor of predicted targets.
• delta: A float, the point where the Huber loss function changes from a quadratic to linear. Defaults to 1.0.

Returns

Tensor with one scalar loss entry per sample.

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LogCosh class

keras.losses.LogCosh(reduction="sum_over_batch_size", name="log_cosh")

Computes the logarithm of the hyperbolic cosine of the prediction error.

Formula:

error = y_pred - y_true
logcosh = mean(log((exp(error) + exp(-error))/2), axis=-1)`

where x is the error y_pred - y_true.

Arguments

• reduction: Type of reduction to apply to loss. Options are "sum", "sum_over_batch_size" or None. Defaults to "sum_over_batch_size".
• name: Optional name for the instance.

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log_cosh function

keras.losses.log_cosh(y_true, y_pred)

Logarithm of the hyperbolic cosine of the prediction error.

Formula:

loss = mean(log(cosh(y_pred - y_true)), axis=-1)

Note that log(cosh(x)) is approximately equal to (x ** 2) / 2 for small x and to abs(x) - log(2) for large x. This means that 'logcosh' works mostly like the mean squared error, but will not be so strongly affected by the occasional wildly incorrect prediction.

Example

>>> y_true = [[0., 1.], [0., 0.]]
>>> y_pred = [[1., 1.], [0., 0.]]
>>> loss = keras.losses.log_cosh(y_true, y_pred)
0.108

Arguments

• y_true: Ground truth values with shape = [batch_size, d0, .. dN].
• y_pred: The predicted values with shape = [batch_size, d0, .. dN].

Returns

Logcosh error values with shape = [batch_size, d0, .. dN-1].