Loss functions are one of the most important parts of a machine learning algorithm; by telling the algorithm what it got right or wrong, they essentially define what it is learning. A loss function is a scalar value, where – in general – a higher value means the model is more wrong.

When training recommenders, we often don’t care about the absolute score of the items being recommended as much as their rank relative to one another. However, few loss functions actually optimize for this.

In this post, we investigate a loss function which does optimize for rank – WARP loss. We also implement it in PyTorch, a machine learning framework. Along the way, we take the hood off PyTorch, and look at how it implements neural network layers. PyTorch implements a  tool called automatic differentiation to keep track of gradients – we also take a look at how this works.

Weighted Approximate-Rank Pairwise loss
WARP loss was first introduced in 2011 (paper), not for recommender systems but for image annotation. It was used to assign to an image the correct label from a very large sample of possible labels. Originally, the motivation for developing this loss – which in particular, has a novel sampling technique – was one of memory efficiency. However, the sampling technique also has additional benefits which make it well suited to training a recommender system.

So how does WARP loss work?
At a high level, WARP loss will randomly sample output labels of a model, until it finds a pair which it knows are wrongly labelled, and will then only apply an update to these two incorrectly labelled examples.

Consider the following example: I have developed a recommender system to recommend one of the following 5 candy bars. I’ve input a customer’s customer journey through my recommender, and it has generated an output vector, which assigns to each candy bar a probability that this customer will purchase it. To train my recommender, I also have a target vector, which describes the customer’s actual behavior using 1s if the customer purchased a specific candy bar, and 0 if they did not:

Highlighted in red is the candy bar the customer actually bought (note that for simplicity, I’m only considering a single purchase, but this loss extends to the case where the customer has made multiple purchases). This is known as the correct label; let’s label it x_3^+ for clarity (where the subscript describes which element in the vector, and the superscript highlights that this is the purchased item).

I’m now going to randomly sample the other labels, until I find one which the model assigned a higher probability of purchase to the customer (or I run out of labels to sample). I’ll know that this randomly sampled label is wrongly labelled, because I know that the Milky Way bar should have the highest probability – since this is the one the customer actually bought!

For instance, if the first random sample I look at is the Mars bar:

Now, I have two variables: my correct label, x_3^+, and my sampled label, which I’ll call a sampled negative label, x_5^- (negative because since the customer didn’t buy it).

In this case, my model was correct; 0.59 > 0.17 (or x_3^+ > x_5^-) so my model correctly ranked the Milky Way higher than the Mars bar. When this happens, I sample another label – and I’ll keep doing this until I find a case where the model was wrong.

Say the second random sample I take is of the Kit Kat (which becomes the sampled negative label, x_2^-).

In this case, 0.59 < 0.63 (or x_3^+ < x_2^-). My model was wrong here, since it thought the customer would be more likely to buy the Kit Kat. To tell my model to correct this, x_3^+ and x_2^- are the two examples I’ll therefore use for the WARP loss, where the loss is the difference between the two values.

(1)   \begin{equation*} \textnormal{Loss} = (x_2^{-} - x_3^{+}) \end{equation*}

We’re not done yet! In addition to this pair, I want to have an idea of how well my model did in general; was the Milky Way bar ranked near the top of all the candy bars? Or did the model do poorly, and stick it near the bottom?
To avoid having to look at all the examples (remember; efficiency!), I can keep track of this while I do the random sampling. If it takes me lots of random samples to find an example where my model was wrong, then I can assume it did pretty well. On the other hand, if the first random sample I look at had a higher score than my correct label, then I can assume it did pretty poorly.

I therefore multiply my loss by the following function:

(2)   \begin{equation*} ln(\frac{X-1}{N}) \end{equation*}

where X is the total number of labels (5, in this case) and N is the number of samples needed to find an example where the model was wrong (2, in this case – the Mars bar, and the Kit Kat).

This makes sense; as I have to take more samples (and N gets larger), it indicates my model is more correct, so I want my loss to be small. I also take the natural logarithm of this function, just to prevent the loss from exploding if N gets small (and since X is generally large).

So now, my loss function looks like this:

(3)   \begin{equation*} \textnormal{Loss} = ln(\frac{X-1}{N})(x_2^{-} - x_3^{+}) \end{equation*}

There’s one final step. In general, it’s not enough for my model to rank x^+ above x^-; to make sure I definitely am recommending the right items, I introduce a margin of error. So not only does my model need to rank x+^ above x^-, it needs to do so by some margin, which I can define. Adding this last step, I get my final loss function:

(4)   \begin{equation*} \textnormal{Loss} = ln(\frac{X-1}{N})(\textnormal{margin} + x_2^{-} - x_3^{+}) \end{equation*}

It’s interesting to note that the loss only depends on these two examples which I have sampled (and so only weights for those two examples will be updated). Nothing is going to be done about the fact that the Twix bar was also ranked higher than the Milky Way, or the fact that Snickers got a 0.35 chance of being bought even though the customer didn’t buy it (so in the best model, it should have a 0). The model will only learn that the Milky Way bar should be ranked above the Kit Kat.

For a recommender, this is much more desirable than a model which learns that it should output 1s for all positive examples and 0 for all negative examples, because often for recommenders, a 0 does not mean a negative interaction. Just because the customer didn’t buy a Twix, it doesn’t mean they didn’t want to buy it – many other factors could have contributed to their not purchasing it, most notably (considering the case where there are not 5 but 500 products to recommend) that they just didn’t see it.

Therefore, having a model which learns to rank items it knows are positive above others can yield a better outcome than a model which learns to rank a few items as 1s, and everything else as a 0. Losses which adopt this approach are generally known as ranking losses.

Implementing WARP loss in PyTorch
Although public implementations of WARP loss do exist (notably in Mendeley’s mrec, and lysts’s lightFM), at Canopy we were interested in implementing WARP loss not for matrix factorisation (as it is used in mrec and lightFM), but to train neural networks.

We’ve been using the PyTorch framework to experiment with different network architectures, but found it lacked an equivalent to WARP loss (in particular, we were looking to avoid the problem of learning negative examples, instead of positives).

In implementing our own WARP loss function, we got to open the hood on exactly how PyTorch implements loss functions, and also take a closer look at automatic differentiation (autodiff), PyTorch’s approach to calculating gradients to update neural networks.

How are losses implemented in PyTorch?
First, a quick overview of how PyTorch works: PyTorch is a python library which allows tensors (PyTorch’s equivalent of numpy nd-arrays) to be manipulated. However, unlike numpy, you can also wrap tensors in Variables(); these allow you to keep track of the gradients of the tensors as you manipulate them, using a technique called automatic differentiation.

PyTorch treats losses as an additional layer of the neural network, so that when I am writing a loss ‘layer’, its actually an nn.Module class (the same class as any layer in PyTorch’s neural networks). nn.Module classes consists of a constructor ( __init__ ), and a single (required) method: forward(), which describes the operations to execute on the Variables.

This is sufficient to write a loss function; using automatic differentiation, PyTorch can calculate the gradients of the Variables as a result of the forward() method for free, allowing backpropagation to occur. However, you can go a step further and define a function to calculate those gradients yourself. This has two advantages: you are sure the gradients are being calculated correctly, and it can be more efficient.

To implement a function to calculate gradients of the Variables, the approach is a little different; instead of writing an nn.Module class, you actually implement an autograd.Function class, which is then ‘applied’ in the nn.Module class (this is actually what happens in the example above; F.relu written as an autograd.Function, and then applied using nn.Module).

It helped me a lot to work through a very simple example – the Linear function used as an example in the PyTorch docs – before implementing a WARP function, so I’ll do the same here.

This function has three inputs, and returns a single output; it’s a very straightforward transformation:

(5)   \begin{equation*} \textnormal{output} = \textnormal{weight}\times \textnormal{input} + \textnormal{bias} \end{equation*}

or, more succinctly

(6)   \begin{equation*} Y = W\cdot X + b \end{equation*}

This is implemented in the Function’s forward method ( .mm() is PyTorch syntax for a dot product, and the bias term is optional, but forward() otherwise easy to read).

Now, for the backward method (warning – some math ahead).
The backward() method has for input grad_output – it will receive as many of these inputs as there were outputs from the forward() method, so in this case, 1.

This grad_output represents the gradient of the layer output with respect to the entire computational graph (in this – and most -cases, a neural network with a loss function). So for our toy one-layer network,

(7)   \begin{equation*} \textnormal{grad output} = \frac{\partial F}{\partial Y} \end{equation*}

The backward() method should then take these grad_outputs, and return the gradient of the function with respect to each input. This particular backward() method therefore has 3 outputs; the gradient of the function with respect to X, with respect to W and with respect to b.
Let’s consider just the first output of backward(), grad_input (where input = X). What I want to return is

(8)   \begin{equation*} \textnormal{grad input} = \frac{\partial F}{\partial X} \end{equation*}

I can calculate this using the chain rule:

(9)   \begin{equation*} \frac{\partial F}{\partial X} = \frac{\partial F}{\partial Y}\cdot \frac{\partial Y}{\partial X} \end{equation*}

Since I know that Y=W\cdot X + b, I can easily calculate

(10)   \begin{equation*} \frac{\partial Y}{\partial X} = W \end{equation*}

Substituting this into my expression for grad_input,

(11)   \begin{equation*} \textnormal{grad input} = \frac{\partial F}{\partial X} = \frac{\partial F}{\partial Y}\cdot W \end{equation*}

As expected, in the backward() function of LinearFunction, grad_input is the dot product (.mm()) of grad_output and the weight matrix!
Note: in this example, where F(x) = Y, \frac{\partial F}{\partial Y} may seem to be trivially 1. However, in practice, F defines not the single layer, but the entire neural network (including the loss function). These partial derivatives allow me to find how this layer’s weights affect the loss function, even if there are a bunch of layers after this one! For a loss function, \frac{\partial F}{\partial Y} is indeed 1. However, if there are layers after this Linear one, then the \frac{\partial F}{\partial X} output from the previous layer will become the \frac{\partial F}{\partial Y} input of this layer during backpropagation.

As an aside, this is a really cool illustration of how autodiff works, and in particular how it combines both numerical and symbolic differentiation:
Since I only care about the gradient matrix (or more precisely, the Jacobian matrix), the backward() method gets as input some matrix, and returns some matrix; it is blind to whatever operations happened before it received the matrix, and to whatever operations happen afterwards. This is akin to numerical differentiation, and is far more memory efficient than trying to save a whole mathematical expression and then analytically differentiate it, which is what happens in symbolic differentiation (especially when you consider how massive these expressions can get for neural networks).

However, the actual operations that happen within the backward() method are analytically motivated, and are therefore exact (none of numerical differentiation’s rounding or truncation errors). This is similar to symbolic differentiation.

Implementing WARP loss in PyTorch
To implement the WARP function, I need to implement a forward() and a backward() method.
forward()
The algorithm for the forward() method works exactly as described at the beginning of this page, with a few additions for efficiency. The locations of the selected positive and negative samples are stored in matrices of 1s and 0s of the same shape as the input matrix, so that an element-by-element multiplication of these matrices with the input matrix zeroes out all non selected elements.

To clarify what this means, lets return to the candy bar example from above. As a reminder, here were the ultimately selected samples:

My negative example is the Kit Kat, and my positive example is the Milky Way, so my positive indices and negative indices ‘matrices’ (vectors, since there is only one sample) would look like this:

so that, multiplying this with the original input,

I also add all of the loss multiples, ln(\frac{X-1}{N}), into a vector, L. This allows me to calculate my loss in a single calculation, instead of adding to it with every loop:
loss = L * (1-torch.sum(positive_indices*input, dim = 1) + torch.sum( negative_indices*input, dim = 1))
Note that the margin here is 1. The positive_indices and negative_indices matrices are also useful for the backwards() method.
backward()

The backward() method needs to return grad_input, the gradient with respect to the input matrix, \frac{\partial F}{\partial X}.
Since random sampling is a non-differentiable operation, the rank multiplier ln(\frac{X-1}{N}) is considered constant for the purpose of differentiation. The way I approach this is to imagine that WARP is simply a sampling technique which defines the loss function, and therefore a unique loss function is created for every training example the recommender sees – each with unique (‘randomly chosen’) samples, and weights (the rank multiplier).
In this case, applying the same notation as for the linear function,

(12)   \begin{equation*} Y = F(X) = L(\textnormal{margin} + X^- - X^+) \end{equation*}

where

(13)   \begin{equation*} X^+ = \textnormal{positive indices} \cdot X \end{equation*}

(14)   \begin{equation*} X^- = \textnormal{negative indices} \cdot X \end{equation*}

Then, given grad_output = \frac{\partial F}{\partial Y}, I want to find grad_input, \frac{\partial F}{\partial X}. As before, this can be achieved by chain rule,

(15)   \begin{equation*} \frac{\partial F}{\partial X} = \frac{\partial F}{\partial Y}\cdot \frac{\partial Y}{\partial X} \end{equation*}

where

(16)   \begin{equation*} \frac{\partial Y}{\partial X} = L(\textnormal{margin} + \textnormal{positive indices} - \textnormal{negative indices}) \end{equation*}

Substituting this into the expression for grad_input yields the backward() method:

(17)   \begin{equation*} \frac{\partial F}{\partial X} = \frac{\partial F}{\partial Y}\cdot L(\textnormal{margin} + \textnormal{positive indices} - \textnormal{negative indices}) \end{equation*}

or, in code:

import torch
import torch.nn as nn
from torch.autograd import Variable, Function

import random

class WARP(Function): 
    '''
    autograd function of WARP loss
    '''
    @staticmethod
    def forward(ctx, input, target, max_num_trials = None):
        
        batch_size = target.size()[0]
        if max_num_trials is None: 
            max_num_trials = target.size()[1] - 1
        
        positive_indices = torch.zeros(input.size())
        negative_indices = torch.zeros(input.size())
        L = torch.zeros(input.size()[0])
        
        all_labels_idx = np.arange(target.size()[1])
        
        Y = float(target.size()[1])
        J = torch.nonzero(target)

        for i in range(batch_size): 
            
            msk = np.ones(target.size()[1], dtype = bool)
            
            # Find the positive label for this example
            j = J[i, 1]
            positive_indices[i, j] = 1
            msk[j] = False
            
            # initialize the sample_score_margin
            sample_score_margin = -1
            num_trials = 0
            
            neg_labels_idx = all_labels_idx[msk]

            while ((sample_score_margin < 0) and (num_trials < max_num_trials)):
                 
                #randomly sample a negative label
                neg_idx = random.sample(neg_labels_idx, 1)[0]
                msk[neg_idx] = False
                neg_labels_idx = all_labels_idx[msk]
                
                num_trials += 1
                # calculate the score margin 
                sample_score_margin = 1 + input[i, neg_idx] - input[i, j] 
            
            if sample_score_margin < 0:
                # checks if no violating examples have been found 
                continue
            else: 
                loss_weight = np.log(math.floor((Y-1)/(num_trials)))
                L[i] = loss_weight
                negative_indices[i, neg_idx] = 1
                
        loss = L * (1-torch.sum(positive_indices*input, dim = 1) + torch.sum(negative_indices*input, dim = 1))
        
        ctx.save_for_backward(input, target)
        ctx.L = L
        ctx.positive_indices = positive_indices
        ctx.negative_indices = negative_indices
        
        return torch.sum(loss , dim = 0, keepdim = True)

    # This function has only a single output, so it gets only one gradient 
    @staticmethod
    def backward(ctx, grad_output):
        input, target = ctx.saved_variables
        L = Variable(torch.unsqueeze(ctx.L, 1), requires_grad = False)

        positive_indices = Variable(ctx.positive_indices, requires_grad = False) 
        negative_indices = Variable(ctx.negative_indices, requires_grad = False)
        grad_input = grad_output*L*(negative_indices - positive_indices)

        return grad_input, None, None  

class WARPLoss(nn.Module): 
    def __init__(self, max_num_trials = None): 
        super(WARPLoss, self).__init__()
        self.max_num_trials = max_num_trials
        
    def forward(self, input, target): 
        return WARP.apply(input, target, self.max_num_trials)

Conclusion
To conclude, WARP loss is a powerful loss function, which can be applied to deep neural networks to provide recommendations based on customer journeys. In particular, its approach of optimizing for ranking instead of absolute outputs makes it particularly well suited to recommenders, and in addition to its current applications in matrix factorisation recommenders, it can also be applied to neural network – based recommenders.