# cntk.losses package¶

Loss functions.

binary_cross_entropy(output, target, name='')[source]

Computes the binary cross entropy (aka logistic loss) between the output and target.

Parameters: output – the computed posterior probability for a variable to be 1 from the network (typ. a sigmoid) target – ground-truth label, 0 or 1 name (str, optional) – the name of the Function instance in the network Function
cosine_distance(x, y, name='')[source]

Computes the cosine distance between x and y:

Example

>>> a = np.asarray([-1, -1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1]).reshape(3,2,2)
>>> b = np.asarray([1, 1, -1, 1, 1, -1, 1, -1, -1, -1, -1, 1]).reshape(3,2,2)
>>> x = C.sequence.input_variable(shape=(2,))
>>> y = C.sequence.input_variable(shape=(2,))
>>> np.round(C.cosine_distance(x,y).eval({x:a,y:b}),5)
array([[-1.,  1.],
[ 1.,  0.],
[ 0., -1.]], dtype=float32)

Parameters: x – numpy array or any Function that outputs a tensor name (str, optional) – the name of the Function instance in the network Function
cosine_distance_with_negative_samples(x, y, shift, num_negative_samples, name='')[source]

Given minibatches for x and y, this function computes for each element in x the cosine distance between it and the corresponding y and additionally the cosine distance between x and some other elements of y (referred to a negative samples). The x and y pairs are samples often derived from embeddings of textual data, though the function can be used for any form of numeric encodings. When using this function to compute textual similarity, x represents search query term embedding and y represents a document embedding. The negative samples are formed on the fly by shifting the right side (y). The shift indicates how many samples in y one should shift while forming each negative sample pair. It is often chosen to be 1. As the name suggests num_negative_samples indicates how many negative samples one would want to generate.

Example

>>> qry = np.asarray([1., 1., 0., 0., 0., 1., 1., 0., 0., 0., 1., 1.], dtype=np.float32).reshape(3, 1, 4)
>>> doc = np.asarray([1., 1., 0., 0., 0., 1., 1., 0., 0., 0., 1., 1.], dtype=np.float32).reshape(3, 1, 4)
>>> x = C.sequence.input_variable(shape=(4,))
>>> y = C.sequence.input_variable(shape=(4,))
>>> model = C.cosine_distance_with_negative_samples(x, y, shift=1, num_negative_samples=2)
>>> np.round(model.eval({x: qry, y: doc}), decimals=4)
array([[[ 1. ,  0.5,  0. ]],

[[ 1. ,  0.5,  0.5]],

[[ 1. ,  0. ,  0.5]]], dtype=float32)

Parameters: x – numpy array or any Function that outputs a tensor y – numpy array or any Function that outputs a tensor shift – non-zero positive integer representing number of shift to generate a negative sample num_negative_samples – number of negative samples to generate, a non-zero positive integer name (str, optional) – the name of the Function instance in the network Function
cross_entropy_with_softmax(output_vector, target_vector, axis=-1, name='')[source]

This operation computes the cross entropy between the target_vector and the softmax of the output_vector. The elements of target_vector have to be non-negative and should sum to 1. The output_vector can contain any values. The function will internally compute the softmax of the output_vector. Concretely,

$$\mathrm{softmax}(x)=\left[\frac{\exp(x_1)}{\sum_i\exp(x_i)}\quad\frac{\exp(x_1)}{\sum_i\exp(x_i)}\quad\ldots\quad\frac{\exp(x_1)}{\sum_i\exp(x_i)}\right]$$

$$\mathrm{cross\_entropy\_with\_softmax}(o, t) = -\sum_{i} t_i \log(\mathrm{softmax}(o)_i)$$

with the understanding that the implementation can use equivalent formulas for efficiency and numerical stability.

Example

>>> C.cross_entropy_with_softmax([[1., 1., 1., 50.]], [[0., 0., 0., 1.]]).eval()
array([[ 0.]], dtype=float32)

>>> C.cross_entropy_with_softmax([[1., 2., 3., 4.]], [[0.35, 0.15, 0.05, 0.45]]).eval()
array([[ 1.84019]], dtype=float32)

Parameters: output_vector – the unscaled computed output values from the network target_vector – usually it is one-hot vector where the hot bit corresponds to the label index. But it can be any probability distribution over the labels. axis (int or Axis, optional) – if given, cross entropy will be computed along this axis name (str, optional) – the name of the Function instance in the network Function
fmeasure(output, target, beta=1)[source]

This operation computes the f-measure between the output and target. If beta is set as one, its called the f1-scorce or dice similarity coefficient. f1-scorce is monotonic in jaccard distance.

f-measure = (1 + beta ** 2) * precision * recall / (beta ** 2 * precision + recall)

This loss function is frequently used in semantic segmentation of images. Works with imbalanced classes, for balanced classes you should prefer cross_entropy instead. This operation works with both binary and multiclass classification.

Parameters: output – the output values from the network target – it is usually a one-hot vector where the hot bit corresponds to the label index beta – greater than one weights recall higher than precision, less than one for the opposite. chosen values are 0.5, 1 or 2. (Commonly) – Function
hierarchical_softmax_layer(input_var, label_index, label_dim, label_classes=None)[source]

A two layers hierarchical softmax function:

Parameters: input_var – Variable with shape: [#,*](dim_x) label_index – index of label’s category: [#,*](1) label_dim – number of the label categories label_classes – number of classes of the label categories the probability of the given label [#,*](1) class_probs: the probability of all the label classes [#,*](label_classes) all_probs: the probability of all label classes output_prob
lambda_rank(output, gain, group, name='')[source]

Groups samples according to group, sorts them within each group based on output and computes the Normalized Discounted Cumulative Gain (NDCG) at infinity for each group. Concretely, the Discounted Cumulative Gain (DCG) at infinity is:

$$\mathrm{DCG_{\infty}}()=\sum_{i=0}^{\infty} \frac{gain_{(i)}}{\log(i+2)}$$

where $$gain_{(i)}$$ means the gain of the $$i$$-th ranked sample.

The NDCG is just the DCG divided by the maximum achievable DCG (obtained by placing the samples with the largest gain at the top of the ranking).

Samples in the same group must appear in order of decreasing gain.

It returns 1 minus the average NDCG across all the groups in the minibatch multiplied by 100 times the number of samples in the minibatch.

In the backward direction it back-propagates LambdaRank gradients.

Example

>>> group = C.input_variable((1,))
>>> gain  = C.input_variable((1,))
>>> g = np.array([1, 1, 2, 2], dtype=np.float32).reshape(4,1)
>>> s = np.array([1, 2, 3, 4], dtype=np.float32).reshape(4,1)
>>> n = np.array([7, 1, 3, 1], dtype=np.float32).reshape(4,1)
>>> f = C.lambda_rank(score, gain, group)
>>> np.round(f.grad({score:s, gain:n, group: g}, wrt=[score]),4)
array([[-0.2121],

[ 0.2121],

[-0.1486],

[ 0.1486]], dtype=float32)

Parameters: output – score of each sample gain – gain of each sample group – group of each sample name (str, optional) – the name of the Function instance in the network Function
lattice_sequence_with_softmax(label, prediction, loglikelihood, lattice, symListPath, phonePath, stateListPath, transProbPath, latticeConfigPath='LatticeNode.config', hSmoothingWeight=0.95, frameDropThresh=1e-10, doReferenceAlign=False, seqGammarUsesMBR=False, seqGammarAMF=14.0, seqGammarLMF=14.0, seqGammarBMMIFactor=0.0, seqGammarWordPen=0.0, name='')[source]
nce_loss(weights, biases, inputs, labels, noise_distribution, num_samples=32, allow_duplicates=True, seed=auto_select, name='')[source]

Computes the noise contrastive estimation loss. This implementation mostly follows Chris Dyer’s notes [1]. At a high level, this layer draws num_samples random labels from noise_distribution and then forms num_samples+1 binary classification problems where the true label is considered a positive example and the random labels are considered negative examples. The negatives are shared among all the examples in the minibatch. This operation only computes the logits for the labels in the minibatch and the random labels drawn from noise_distribution. The gradients will be sparse if the labels are sparse.

The noise_distribution is read once and certain quantities are precomputed based on it. This operation will need to be reinstantiated if the noise_distribution changes.

Shape inference for the weights is currently not supported when inputs are placeholders. Either a concrete input must be used or the weights must be provided without any inferred dimensions.

Example

>>> import scipy
>>> # dimensions of input, number of noise labels, batch size, number of classes
>>> xdim = 10
>>> samples = 32
>>> batch = 4
>>> classes = 100
>>> # some variables; typically x will be the output of a layer
>>> x = C.input_variable(xdim)
>>> y = C.input_variable(classes, is_sparse=True)
>>> # dummy data
>>> x0 = np.arange(batch * xdim, dtype=np.float32).reshape((batch, xdim))/(batch * xdim)
>>> data = np.ones(batch, dtype=np.float32)
>>> indices = list(range(10, 10*batch+1, 10))
>>> indptr = list(range(batch + 1))
>>> y0 = scipy.sparse.csr_matrix((data, indices, indptr), shape=(batch, classes))
>>> # a dummy noise distribution
>>> q = np.arange(classes, dtype=np.float32) + 1 # normalization not necessary
>>> # the parameters
>>> b = C.parameter((classes, 1), init=-np.log(classes))
>>> W = C.parameter((classes, C.InferredDimension), init=C.glorot_uniform(seed=98052))
>>> # the loss
>>> loss = C.nce_loss(W, b, x, y, q, seed=98052)
>>> # evaluate the loss at our dummy data
>>> np.round(loss.eval({x:x0, y:y0}), decimals=3)
array([ 2.385,  3.035,  3.886,  3.868], dtype=float32)
>>> # after training, use the logits for predictions
>>> logits = C.times_transpose(x, W) + C.reshape(b, -1)

Parameters: weights – parameter (or variable in general) containing the weights with which inputs will be multiplied. Its shape must be (number of classes, dimension of input) biases – parameter (or variable in general) containing the biases that will be added to the product of weights and inputs. Its shape must be (number of classes, 1) inputs – vector of inputs to this layer. Multiplying by the weights and adding the biases gives the logits. labels – a one-hot vector with the ground-truth labels. noise_distribution – a constant vector with dimension equal to the number of classes. The entries must be positive numbers but do not have to sum to 1. random labels will be drawn according to the normalized distribution. num_samples – number of random labels that will be drawn from the noise_distribution. allow_duplicates – boolean. If True (default), the random labels can contain duplicates. Compared to allow_duplicates=False it is faster but the quality of the approximations is slightly worse for the same number of samples. seed – random seed. The default value selects a unique random seed. name (str, optional) – the name of the Function instance in the network Function

squared_error(output, target, name='')[source]

This operation computes the sum of the squared difference between elements in the two input matrices. The result is a scalar (i.e., one by one matrix). This is often used as a training criterion.

Example

>>> i1 = C.input_variable((1,2))
>>> i2 = C.input_variable((1,2))
>>> C.squared_error(i1,i2).eval({i1:np.asarray([[[2., 1.]]], dtype=np.float32), i2:np.asarray([[[4., 6.]]], dtype=np.float32)})
array([ 29.], dtype=float32)

>>> C.squared_error(i1,i2).eval({i1:np.asarray([[[1., 2.]]], dtype=np.float32), i2:np.asarray([[[1., 2.]]], dtype=np.float32)})
array([ 0.], dtype=float32)

Parameters: output – the output values from the network target – it is usually a one-hot vector where the hot bit corresponds to the label index name (str, optional) – the name of the Function instance in the network Function
weighted_binary_cross_entropy(output, target, weight, name='')[source]

This operation computes the weighted binary cross entropy (aka logistic loss) between the output and target.

Parameters: output – the computed posterior probability from the network target – ground-truth label, 0 or 1 weight – weight of each example name (str, optional) – the name of the Function instance in the network Function