Why is there a function, but not a ? Let’s create not one, but a few of them.
People who are familiar with neural networks will likely have heard of operations named and . There is a tendency to use the shorter of the two names for both concepts, which would be confusing for this article, so we will keep them distinct. A function typically need to satisfy two properties:
- approximating the max;
- being differentiable everywhere.
In this context, by “approximating” we mean that there is an additional parameter such that, in the limit , the approximation is equal to the target function (for the mathematically inclined, we are talking about pointwise convergence). For simplicity, in the following, the dependence on will be left implicit: consider any “soft” function to have as the last argument. Here is a possible definition for :
The function is a function and each element is 1 or near 1 when the corresponding element in the argument is the maximum of its argument, 0 or near 0 otherwise. In other words, it’s an approximation of the indicator function for the maximum — a mask identifying the maximum (I will ignore the case of ties for simplicity). It also needs to be differentiable everywhere. An additional property of the , useful but perhaps not fundamental to its definition, is to be normalized to 1, meaning that the sum of the elements of is always 1.
Its typical use in the context of neural networks is to model a discrete distribution in a classification problem, owing to its elements’ range between 0 and 1 and its normalization to 1, like a multiclass equivalent of the function. Here is a possible definition:
Considering that the max is one particular quantile, and that differentiable operations are important in machine learning, let’s pose the following question: are there “soft” versions of other quantiles?
First let’s clarify what we are talking about and focus on the median, for starters. As the reader might have guessed already, these are the necessary properties of a :
- Approximating the median.
- Being differentiable everywhere.
Naturally, we can also define a function with the following properties:
- Approximates the indicator function of the median
- Is differentiable everywhere
If mathematical curiosity is not a sufficient motivation for this inquiry, let me try to suggest some possible applications of the in machine learning. If the is a soft version of a winner-take-all type operation, the is a consensus one. It reduces to the majority rule in the special case of binary classification. Consensus is very important in ensemble learning and has applications to privacy, for instance in the PATE algorithm, whereby models trained on a partition of a the training set need to mostly agree on the output — otherwise it has to be somehow obfuscated, lest it reveals too much about the training set. Of course one could take the average as consensus, but it is less robust than the median. As a pooling operation, it doesn’t seem that everywhere-differentiability is a necessity, and the max-pool operation is widely used. But there is an intuitive argument for using soft operations instead. When backpropagating through max-pool, only one unit in input to that operation (more only in case of perfect ties) receives a non-zero signal. If different units are almost tied for the max, then they may alternate, batch after batch, in receiving a non-zero backpropagation signal, which seems wasteful. In fact, a recently published paper defines a soft-max-pool operation with an adaptive parameter. This gives me more confidence that the or will also find applications.
In addition to satisfying the basic properties of the or , I considered simplicity of the definition and low computational complexity as additional goals. In our quest we will use two mathematical properties of the composition of functions: it preserves differentiability and pointwise convergence under mild assumptions. Therefore, if we find a way to express the median as composition of functions (must be continuous) and which have soft versions and then is a soft version of (but pointwise convergence doesn’t preserve continuity or differentiability, hence the limit function , , and need not be differentiable or continuous). And with this, as my probability teacher used to say, “The proof is left as an easy exercise for the student”.
The median as minmax of subsets
The median, by definition, splits a set in half: elements smaller than it and the rest (with some details to be filled in for even sized sets and in the case of ties). Therefore, if one considers all the subsets of size , where is the sample size, one of them must be the set of elements less than or equal to the median. Ignoring ties, it is also the only such set, because, if there were two distinct ones, their union would also consist of elements smaller than the median but would be larger than half the size of the set, which is a contradiction. So if we take the maximum for each such subset and consider the minimum of all such maxima, that value is the median (or a median, if multiple values are possible). Replacing the maximum and minimum operations with their soft counterparts, we have a soft median! More formally, for a sample :
has not been defined but the reader will have no trouble coming up with a definition based on and the equality . Unfortunately, the computation of this incarnation of the requires an exponential number of steps. Unless some clever simplification is possible, beyond what I could come up with, it’s not very practical. Conceptually, though, it’s very simple. As a bonus, the generalization to other quantiles is trivial.
The median as minimizer of the total absolute deviation
It is well known that the median minimizes the total absolute deviation:
Since the absolute value is not differentiable in 0, it looks like an obstacle on our quest. But we can replace it with a soft absolute value such as
One could try to find stationary points by differentiation, but it doesn’t seem to lead to a simple solution. Instead, let’s try all values in , one of which must be a median:
This approach requires steps. It is better than our first attempt and, in the context of neural networks, not any worse than the matrix-vector multiplication that dominate their computational complexity. On the other hand, compared to the complexity for the regular median, it seems quite a penalty, considering also that no such penalty is incurred to put the “soft” in .
To define the , we just need use the former as weights in a weighted sum of the sample :
To generalize to other quantiles, one can use the property that a -quantile minimizes
which can be “softened” to
and then proceed as before.
The median using ranks
The first step is defining a form of soft ranking. We need a soft version of comparing a pair of elements or, equivalently, a sign function, for instance:
This is also known as a sigmoid function, and here I picked one of the many options and scaled it to the desired range. From this, I am going to calculate a sort of rank, named , which is translated so that the median of a sample has 0 in it — this makes the following definitions simpler:
The idea is taking a sum of all the signs (positive or negative) of all the differences between one element and each of the others. For a median, the positive and negative differences should balance out. By taking the square of this quantity, we have 0, the of the median, be the smallest possible value. From there, we just need to take the :
Substituting the definition of , we obtain:
While we followed a different thought process, the result is not very different from the one in the previous section. Unfortunately, that also implies we have made no progress on the computational complexity side, which still is . One can also recognize the similarity with the trivial parallel sorting algorithm that consists of computing all the pairwise comparisons, organize them in a matrix and take the row sums to compute the ranks. This suggests that either of the last two . definitions is parallelizable along the same lines.
A definition for the can be obtained by following the same steps as in the previous section. The extension to any quantile is possible by re-centering the to be 0 for the desired quantile.
Approximating the median
If looking at parallel algorithms could have provided some inspiration, why not look at streaming algorithms as well? An approximate median algorithm for streaming settings is the remedian, whose origins can be traced back to Tukey’s ninether. The idea is to split a vector into smaller ones and take the median of those and so on recursively, until we are left with a single number. As to approximating other quantiles, the same recursive approach to any quantile other than max and min results in biased estimates. It seems that extreme quantiles admit specialized algorithms (top-k-type algorithms, with k independent of sample size, unlike, say, a quartile). It’s also possible to reduce other quantile calculations to the median by padding. For instance, if we want to calculate the first quartile of a sample of size N, we can pad the sample with elements smaller than the min, for a total of elements. The median of the padded sample is larger than elements, of which are padding and are from the original sample. For the latter number to match a first quartile we just need to set . More in general, for a quantile smaller than the median, the required padding is of size . For quantiles above the median, we can pad on the max side and follow the same calculation. So let’s focus on the median. If we take the remedian algorithm and replace any calculation of the median with the , we have soft version of the remedian, or . As a bonus, we get a streaming algorithm and one with time complexity, and parallel time complexity. The complexity is still quadratic, but it is applied only to small input vectors, so it doesn’t impact the overall cost, in the asymptotic sense.
So now what about the ? The two ingredients we need are a notion of distance from the median and the . I am sure the astute reader can take it from here. And that’s all for the theory. Stay tuned for some implementations in future posts.