# Criterions

Criterions are helpful to train a neural network. Given an input and a target, they compute a gradient according to a given loss function.

## Criterion

This is an abstract class which declares methods defined in all criterions. This class is serializable.

### [output] forward(input, target)

Given an input and a target, compute the loss function associated to the criterion and return the result. In general input and target are Tensors, but some specific criterions might require some other type of object.

The output returned should be a scalar in general.

The state variable self.output should be updated after a call to forward().

Given an input and a target, compute the gradients of the loss function associated to the criterion and return the result. In general input, target and gradInput are Tensors, but some specific criterions might require some other type of object.

The state variable self.gradInput should be updated after a call to backward().

### State variable: output

State variable which contains the result of the last forward(input, target) call.

State variable which contains the result of the last backward(input, target) call.

## AbsCriterion

criterion = nn.AbsCriterion()


Creates a criterion that measures the mean absolute value of the element-wise difference between input x and target y:

loss(x, y)  = 1/n \sum |x_i - y_i|


If x and y are d-dimensional Tensors with a total of n elements, the sum operation still operates over all the elements, and divides by n.

The division by n can be avoided if one sets the internal variable sizeAverage to false:

criterion = nn.AbsCriterion()
criterion.sizeAverage = false


## ClassNLLCriterion

criterion = nn.ClassNLLCriterion([weights])


The negative log likelihood criterion. It is useful to train a classification problem with n classes. If provided, the optional argument weights should be a 1D Tensor assigning weight to each of the classes. This is particularly useful when you have an unbalanced training set.

The input given through a forward() is expected to contain log-probabilities of each class: input has to be a 1D Tensor of size n. Obtaining log-probabilities in a neural network is easily achieved by adding a LogSoftMax layer in the last layer of your neural network. You may use CrossEntropyCriterion instead, if you prefer not to add an extra layer to your network. This criterion expects a class index (1 to the number of class) as target when calling forward(input, target) and backward(input, target).

The loss can be described as:

loss(x, class) = -x[class]


or in the case of the weights argument it is specified as follows:

loss(x, class) = -weights[class] * x[class]


Due to the behaviour of the backend code, it is necessary to set sizeAverage to false when calculating losses in non-batch mode.

The following is a code fragment showing how to make a gradient step given an input x, a desired output y (an integer 1 to n, in this case n = 2 classes), a network mlp and a learning rate learningRate:

function gradUpdate(mlp, x, y, learningRate)
local criterion = nn.ClassNLLCriterion()
local pred = mlp:forward(x)
local err = criterion:forward(pred, y)
local t = criterion:backward(pred, y)
mlp:backward(x, t)
mlp:updateParameters(learningRate)
end


By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed for each minibatch.

## CrossEntropyCriterion

criterion = nn.CrossEntropyCriterion([weights])


This criterion combines LogSoftMax and ClassNLLCriterion in one single class.

It is useful to train a classification problem with n classes. If provided, the optional argument weights should be a 1D Tensor assigning weight to each of the classes. This is particularly useful when you have an unbalanced training set.

The input given through a forward() is expected to contain scores for each class: input has to be a 1D Tensor of size n. This criterion expect a class index (1 to the number of class) as target when calling forward(input, target) and backward(input, target).

The loss can be described as:

loss(x, class) = -log(exp(x[class]) / (\sum_j exp(x[j])))
= -x[class] + log(\sum_j exp(x[j]))


or in the case of the weights argument being specified:

loss(x, class) = weights[class] * (-x[class] + log(\sum_j exp(x[j])))


Due to the behaviour of the backend code, it is necessary to set sizeAverage to false when calculating losses in non-batch mode.

crit = nn.CrossEntropyCriterion(weights)
crit.nll.sizeAverage = false


The losses are averaged across observations for each minibatch.

## ClassSimplexCriterion

criterion = nn.ClassSimplexCriterion(nClasses)


ClassSimplexCriterion implements a criterion for classification. It learns an embedding per class, where each class' embedding is a point on an (N-1)-dimensional simplex, where N is the number of classes.

The input given through a forward() is expected to be the output of a Normalized Linear layer with no bias: - input has to be a 1D Tensor of size n for a single sample - a 2D Tensor of size batchSize x n for a mini-batch of samples

This Criterion is best used in combination with a neural network where the last layers are: - a weight-normalized bias-less Linear layer. Example source code - followed by an output normalization layer (nn.Normalize).

The loss is described in detail in the paper Scale-invariant learning and convolutional networks.

The following is a code fragment showing how to make a gradient step given an input x, a desired output y (an integer 1 to n, in this case n = 30 classes), a network mlp and a learning rate learningRate:

nInput = 10
nClasses = 30
nHidden = 100
mlp = nn.Sequential()

criterion = nn.ClassSimplexCriterion(nClasses)

pred = mlp:forward(x)
local err = criterion:forward(pred, y)
local t = criterion:backward(pred, y)
mlp:backward(x, t)
mlp:updateParameters(learningRate)
end


This criterion also provides two helper functions getPredictions(input) and getTopPrediction(input) that return the raw predictions and the top prediction index respectively, given an input sample.

## DistKLDivCriterion

criterion = nn.DistKLDivCriterion()


The Kullback–Leibler divergence criterion. KL divergence is a useful distance measure for continuous distributions and is often useful when performing direct regression over the space of (discretely sampled) continuous output distributions. As with ClassNLLCriterion, the input given through a forward() is expected to contain log-probabilities, however unlike ClassNLLCriterion, input is not restricted to a 1D or 2D vector (as the criterion is applied element-wise).

This criterion expect a target Tensor of the same size as the input Tensor when calling forward(input, target) and backward(input, target).

The loss can be described as:

loss(x, target) = 1/n \sum(target_i * (log(target_i) - x_i))


By default, the losses are averaged for each minibatch over observations as well as over dimensions. However, if the field sizeAverage is set to false, the losses are instead summed.

## BCECriterion

criterion = nn.BCECriterion([weights])


Creates a criterion that measures the Binary Cross Entropy between the target and the output:

loss(o, t) = - 1/n sum_i (t[i] * log(o[i]) + (1 - t[i]) * log(1 - o[i]))


or in the case of the weights argument being specified:

loss(o, t) = - 1/n sum_i weights[i] * (t[i] * log(o[i]) + (1 - t[i]) * log(1 - o[i]))


This is used for measuring the error of a reconstruction in for example an auto-encoder. Note that the outputs o[i] should be numbers between 0 and 1, for instance, the output of an nn.Sigmoid layer and should be interpreted as the probability of predicting t[i] = 1. Note t[i] can be either 0 or 1.

By default, the losses are averaged for each minibatch over observations as well as over dimensions. However, if the field sizeAverage is set to false, the losses are instead summed.

## MarginCriterion

criterion = nn.MarginCriterion([margin])


Creates a criterion that optimizes a two-class classification hinge loss (margin-based loss) between input x (a Tensor of dimension 1) and output y (which is a tensor containing either 1s or -1s). margin, if unspecified, is by default 1.

loss(x, y) = sum_i (max(0, margin - y[i]*x[i])) / x:nElement()


The normalization by the number of elements in the input can be disabled by setting self.sizeAverage to false.

### Example

function gradUpdate(mlp, x, y, criterion, learningRate)
local pred = mlp:forward(x)
local err = criterion:forward(pred, y)
mlp:updateParameters(learningRate)
end

mlp = nn.Sequential()

x1 = torch.rand(5)
x1_target = torch.Tensor{1}
x2 = torch.rand(5)
x2_target = torch.Tensor{-1}
criterion=nn.MarginCriterion(1)

for i = 1, 1000 do
end

print(mlp:forward(x1))
print(mlp:forward(x2))

print(criterion:forward(mlp:forward(x1), x1_target))
print(criterion:forward(mlp:forward(x2), x2_target))


gives the output:

 1.0043
[torch.Tensor of dimension 1]

-1.0061
[torch.Tensor of dimension 1]

0
0


i.e. the mlp successfully separates the two data points such that they both have a margin of 1, and hence a loss of 0.

By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed.

## SoftMarginCriterion

criterion = nn.SoftMarginCriterion()


Creates a criterion that optimizes a two-class classification logistic loss between input x (a Tensor of dimension 1) and output y (which is a tensor containing either 1s or -1s).

loss(x, y) = sum_i (log(1 + exp(-y[i]*x[i]))) / x:nElement()


The normalization by the number of elements in the input can be disabled by setting self.sizeAverage to false.

### Example

function gradUpdate(mlp, x, y, criterion, learningRate)
local pred = mlp:forward(x)
local err = criterion:forward(pred, y)
mlp:updateParameters(learningRate)
end

mlp = nn.Sequential()

x1 = torch.rand(5)
x1_target = torch.Tensor{1}
x2 = torch.rand(5)
x2_target = torch.Tensor{-1}
criterion=nn.SoftMarginCriterion(1)

for i = 1, 1000 do
end

print(mlp:forward(x1))
print(mlp:forward(x2))

print(criterion:forward(mlp:forward(x1), x1_target))
print(criterion:forward(mlp:forward(x2), x2_target))


gives the output:


0.7471
[torch.DoubleTensor of size 1]

-0.9607
[torch.DoubleTensor of size 1]

0.38781049558836
0.32399356957564



i.e. the mlp successfully separates the two data points.

By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed.

## MultiMarginCriterion

criterion = nn.MultiMarginCriterion(p, [weights], [margin])


Creates a criterion that optimizes a multi-class classification hinge loss (margin-based loss) between input x (a Tensor of dimension 1) and output y (which is a target class index, 1 <= y <= x:size(1)):

loss(x, y) = sum_i(max(0, (margin - x[y] + x[i]))^p) / x:size(1)


where i == 1 to x:size(1) and i ~= y. Note that this criterion also works with 2D inputs and 1D targets.

Optionally, you can give non-equal weighting on the classes by passing a 1D weights tensor into the constructor. The loss function then becomes:

loss(x, y) = sum_i(max(0, w[y] * (margin - x[y] + x[i]))^p) / x:size(1)


This criterion is especially useful for classification when used in conjunction with a module ending in the following output layer:

mlp = nn.Sequential()
mlp:add(nn.Euclidean(n, m)) -- outputs a vector of distances


By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed.

## MultiLabelMarginCriterion

criterion = nn.MultiLabelMarginCriterion()


Creates a criterion that optimizes a multi-class multi-classification hinge loss (margin-based loss) between input x (a 1D Tensor) and output y (which is a 1D Tensor of target class indices):

loss(x, y) = sum_ij(max(0, 1 - (x[y[j]] - x[i]))) / x:size(1)


where i == 1 to x:size(1), j == 1 to y:size(1), y[j] ~= 0, and i ~= y[j] for all i and j. Note that this criterion also works with 2D inputs and targets.

y and x must have the same size. The criterion only considers the first non zero y[j] targets. This allows for different samples to have variable amounts of target classes:

criterion = nn.MultiLabelMarginCriterion()
input = torch.randn(2, 4)
target = torch.Tensor{{1, 3, 0, 0}, {4, 0, 0, 0}} -- zero-values are ignored
criterion:forward(input, target)


## MultiLabelSoftMarginCriterion

criterion = nn.MultiLabelSoftMarginCriterion()


Creates a criterion that optimizes a multi-label one-versus-all loss based on max-entropy, between input x (a 1D Tensor) and target y (a binary 1D Tensor):

loss(x, y) = - sum_i (y[i] log( exp(x[i]) / (1 + exp(x[i]))) + (1-y[i]) log(1/(1+exp(x[i])))) / x:nElement()


where i == 1 to x:nElement(), y[i] in {0,1}. Note that this criterion also works with 2D inputs and targets.

y and x must have the same size.

## MSECriterion

criterion = nn.MSECriterion()


Creates a criterion that measures the mean squared error between n elements in the input x and output y:

loss(x, y) = 1/n \sum |x_i - y_i|^2 .


If x and y are d-dimensional Tensors with a total of n elements, the sum operation still operates over all the elements, and divides by n. The two Tensors must have the same number of elements (but their sizes might be different).

The division by n can be avoided if one sets the internal variable sizeAverage to false:

criterion = nn.MSECriterion()
criterion.sizeAverage = false


By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed.

## SpatialAutoCropMSECriterion

criterion = nn.SpatialAutoCropMSECriterion()


Creates a criterion that measures the mean squared error between the input and target, even if the target is spatially larger than the input. It achieves this by center-cropping the target to the same spatial resolution as the input, the mean squared error is then calculated between the input and this cropped target.

If the input and cropped target tensors are d-dimensional Tensors with a total of n elements, the sum operation operates over all the elements, and divides by n.

The division by n can be avoided if one sets the internal variable sizeAverage to false:

criterion = nn.SpatialAutoCropMSECriterion()
criterion.sizeAverage = false


## MultiCriterion

criterion = nn.MultiCriterion()


This returns a Criterion which is a weighted sum of other Criterion. Criterions are added using the method:

criterion:add(singleCriterion [, weight])


where weight is a scalar (default 1). Each criterion is applied to the same input and target.

Example :

input = torch.rand(2,10)
target = torch.IntTensor{1,8}
nll = nn.ClassNLLCriterion()
nll2 = nn.CrossEntropyCriterion()
output = mc:forward(input, target)


## ParallelCriterion

criterion = nn.ParallelCriterion([repeatTarget])


This returns a Criterion which is a weighted sum of other Criterion. Criterions are added using the method:

criterion:add(singleCriterion [, weight])


where weight is a scalar (default 1). The criterion expects an input and target table. Each criterion is applied to the commensurate input and target element in the tables. However, if repeatTarget=true, the target is repeatedly presented to each criterion (with a different input).

Example :

input = {torch.rand(2,10), torch.randn(2,10)}
target = {torch.IntTensor{1,8}, torch.randn(2,10)}
nll = nn.ClassNLLCriterion()
mse = nn.MSECriterion()
output = pc:forward(input, target)


## SmoothL1Criterion

criterion = nn.SmoothL1Criterion()


Creates a criterion that can be thought of as a smooth version of the AbsCriterion. It uses a squared term if the absolute element-wise error falls below 1. It is less sensitive to outliers than the MSECriterion and in some cases prevents exploding gradients (e.g. see "Fast R-CNN" paper by Ross Girshick).

                      ⎧ 0.5 * (x_i - y_i)^2, if |x_i - y_i| < 1
loss(x, y) = 1/n \sum ⎨
⎩ |x_i - y_i| - 0.5,   otherwise


If x and y are d-dimensional Tensors with a total of n elements, the sum operation still operates over all the elements, and divides by n.

The division by n can be avoided if one sets the internal variable sizeAverage to false:

criterion = nn.SmoothL1Criterion()
criterion.sizeAverage = false


By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed.

## HingeEmbeddingCriterion

criterion = nn.HingeEmbeddingCriterion([margin])


Creates a criterion that measures the loss given an input x which is a 1-dimensional vector and a label y (1 or -1). This is usually used for measuring whether two inputs are similar or dissimilar, e.g. using the L1 pairwise distance, and is typically used for learning nonlinear embeddings or semi-supervised learning.

                 ⎧ x_i,                  if y_i ==  1
loss(x, y) = 1/n ⎨
⎩ max(0, margin - x_i), if y_i == -1


If x and y are n-dimensional Tensors, the sum operation still operates over all the elements, and divides by n (this can be avoided if one sets the internal variable sizeAverage to false). The margin has a default value of 1, or can be set in the constructor.

### Example

-- imagine we have one network we are interested in, it is called "p1_mlp"

-- But we want to push examples towards or away from each other so we make another copy
-- of it called p2_mlp; this *shares* the same weights via the set command, but has its
-- own set of temporary gradient storage that's why we create it again (so that the gradients
-- of the pair don't wipe each other)
p2_mlp:get(1).weight:set(p1_mlp:get(1).weight)
p2_mlp:get(1).bias:set(p1_mlp:get(1).bias)

-- we make a parallel table that takes a pair of examples as input.
-- They both go through the same (cloned) mlp
prl = nn.ParallelTable()

-- now we define our top level network that takes this parallel table
-- and computes the pairwise distance betweem the pair of outputs
mlp = nn.Sequential()

-- and a criterion for pushing together or pulling apart pairs
crit = nn.HingeEmbeddingCriterion(1)

-- lets make two example vectors
x = torch.rand(5)
y = torch.rand(5)

-- Use a typical generic gradient update function
function gradUpdate(mlp, x, y, criterion, learningRate)
local pred = mlp:forward(x)
local err = criterion:forward(pred, y)
mlp:updateParameters(learningRate)
end

-- push the pair x and y together, notice how then the distance between them given
-- by print(mlp:forward({x, y})[1]) gets smaller
for i = 1, 10 do
gradUpdate(mlp, {x, y}, 1, crit, 0.01)
print(mlp:forward({x, y})[1])
end

-- pull apart the pair x and y, notice how then the distance between them given
-- by print(mlp:forward({x, y})[1]) gets larger

for i = 1, 10 do
gradUpdate(mlp, {x, y}, -1, crit, 0.01)
print(mlp:forward({x, y})[1])
end


By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed.

## L1HingeEmbeddingCriterion

criterion = nn.L1HingeEmbeddingCriterion([margin])


Creates a criterion that measures the loss given an input x = {x1, x2}, a table of two Tensors, and a label y (1 or -1): this is used for measuring whether two inputs are similar or dissimilar, using the L1 distance, and is typically used for learning nonlinear embeddings or semi-supervised learning.

             ⎧ ||x1 - x2||_1,                  if y ==  1
loss(x, y) = ⎨
⎩ max(0, margin - ||x1 - x2||_1), if y == -1


The margin has a default value of 1, or can be set in the constructor.

## CosineEmbeddingCriterion

criterion = nn.CosineEmbeddingCriterion([margin])


Creates a criterion that measures the loss given an input x = {x1, x2}, a table of two Tensors, and a Tensor label y with values 1 or -1. This is used for measuring whether two inputs are similar or dissimilar, using the cosine distance, and is typically used for learning nonlinear embeddings or semi-supervised learning.

margin should be a number from -1 to 1, 0 to 0.5 is suggested. Forward and Backward have to be used alternately. If margin is missing, the default value is 0.

The loss function for each sample is:

             ⎧ 1 - cos(x1, x2),              if y ==  1
loss(x, y) = ⎨
⎩ max(0, cos(x1, x2) - margin), if y == -1


For batched inputs, if the internal variable sizeAverage is equal to true, the loss function averages the loss over the batch samples; if sizeAverage is false, then the loss function sums over the batch samples. By default, sizeAverage equals to true.

By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed.

## DistanceRatioCriterion

criterion = nn.DistanceRatioCriterion(sizeAverage)


This criterion is probabilistic treatment of margin cost. The model is trained using sample triplets {Xs, Xa, Xd} where Xa is anchor sample, Xs is sample similar to anchor sample and Xd is a sample not similar to anchor sample. Let Ds be distance between embeddings of {Xs, Xa} and Dd be distance between embeddings of {Xa, Xd} then the loss is defined as follow

   loss = -log( exp(-Ds) / ( exp(-Ds) + exp(-Dd) ) )


Sample example

   torch.setdefaulttensortype("torch.FloatTensor")

require 'nn'

-- triplet : with batchSize of 32 and dimensionality 512
sample = {torch.rand(32, 512), torch.rand(32, 512), torch.rand(32, 512)}

embeddingModel = nn.Sequential()

tripleModel = nn.ParallelTable()

-- Similar sample distance w.r.t anchor sample
posDistModel = nn.Sequential()

-- Different sample distance w.r.t anchor sample
negDistModel = nn.Sequential()

-- Complete Model

-- DistanceRatioCriterion
criterion = nn.DistanceRatioCriterion(true)

-- Forward & Backward
output = model:forward(sample)
loss   = criterion:forward(output)
dLoss  = criterion:backward(output)
model:backward(sample, dLoss)


## MarginRankingCriterion

criterion = nn.MarginRankingCriterion(margin)


Creates a criterion that measures the loss given an input x = {x1, x2}, a table of two Tensors of size 1 (they contain only scalars), and a label y (1 or -1). In batch mode, x is a table of two Tensors of size batchsize, and y is a Tensor of size batchsize containing 1 or -1 for each corresponding pair of elements in the input Tensor.

If y == 1 then it assumed the first input should be ranked higher (have a larger value) than the second input, and vice-versa for y == -1.

The loss function is:

loss(x, y) = max(0, -y * (x[1] - x[2]) + margin)


For batched inputs, if the internal variable sizeAverage is equal to true, the loss function averages the loss over the batch samples; if sizeAverage is false, then the loss function sums over the batch samples. By default, sizeAverage equals to true. By default, the losses are averaged over observations for each minibatch. However, if the field sizeAverage is set to false, the losses are instead summed.

### Example

p1_mlp = nn.Linear(5, 2)
p2_mlp = p1_mlp:clone('weight', 'bias')

prl = nn.ParallelTable()

mlp1 = nn.Sequential()

mlp2 = mlp1:clone('weight', 'bias')

mlpa = nn.Sequential()
prla = nn.ParallelTable()

crit = nn.MarginRankingCriterion(0.1)

x=torch.randn(5)
y=torch.randn(5)
z=torch.randn(5)

-- Use a typical generic gradient update function
function gradUpdate(mlp, x, y, criterion, learningRate)
local pred = mlp:forward(x)
local err = criterion:forward(pred, y)
mlp:updateParameters(learningRate)
end

for i = 1, 100 do
gradUpdate(mlpa, {{x, y}, {x, z}}, 1, crit, 0.01)
if true then
o1 = mlp1:forward{x, y}[1]
o2 = mlp2:forward{x, z}[1]
o = crit:forward(mlpa:forward{{x, y}, {x, z}}, 1)
print(o1, o2, o)
end
end

print "--"

for i = 1, 100 do
gradUpdate(mlpa, {{x, y}, {x, z}}, -1, crit, 0.01)
if true then
o1 = mlp1:forward{x, y}[1]
o2 = mlp2:forward{x, z}[1]
o = crit:forward(mlpa:forward{{x, y}, {x, z}}, -1)
print(o1, o2, o)
end
end