[Categorical search] Add documentation for score normalizer

The forthcoming score normalization algorithm, which is a tweak of
adafang@'s previous work, is complicated enough that it warrants
some documentation in the codebase. This CL adds a .md, and follow-up
CLs will implement the algorithm itself.

Bug: 1199206
Change-Id: I87a77e1579c65422c00d0f908916b56a7e35cb4a
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3189836
Commit-Queue: Tony Yeoman <[email protected]>
Reviewed-by: Rachel Wong <[email protected]>
Cr-Commit-Position: refs/heads/main@{#925647}

Author: tby

chrome/browser/ui/app_list/search/ranking/score_normalizer.md

@@ -0,0 +1,137 @@
+# Score normalizer
+
+The score normalizer is a heuristic algorithm that attempts to map a incoming
+stream of numbers drawn from some unknown distribution to a uniform
+distribution. This allows us to directly compare scores produced by different
+underlying distributions, namely, by different search providers.
+
+## Overview and terminology
+
+The input to the normalizer is a stream of scores, `x`, drawn from an
+underlying continuous distribution, `P(x)`. Its goal is to learn a function
+`f(x)` such that `f(x ~ P)` is a uniform distribution over `[0,1]`.
+
+## Reservoir sampling
+
+Reservoir sampling is a method for computing `f`. Its simplest form is:
+
+- Take the first `N` (eg. 100) scores as a 'reservoir'.
+- Define `f(x) = sum(y < x for y in reservoir) / N`
+
+In other words, we take the first `N` scores as a reservoir, and then
+normalize a new score by computing its quantile within the reservoir.
+
+We've constructed `f` by choosing a series of dividers (the reservoir scores)
+which we hope are a representative sample of `P`. Consequently, a new `x ~ P`
+will fall into any 'bucket' between two dividers with equal probability. So,
+the index of the bucket x falls into is the quantile of x within `P`. Thus,
+with some allowance for the fact that f is discrete, `f(x ~ P) ~= U(0,1)`.
+
+The drawback of this is that the reservoir needs to be quite large before
+this is effective.
+
+## Bin-entropy reservoir sampling
+
+We're going to trade off some accuracy for efficiency, and build on the core
+idea of reservoir sampling. We will store 'bins' that cover the real line, and
+try to keep the number of elements we've seen in each bin approximately equal.
+Then we can compute `f(x)` by using the bin index of `x` as a quantile.
+
+__Data structure__. We store a vector of N 'bins', each one with a count and
+a lower divider. The bottom bin always has a lower divider of negative
+infinity. `bins[i].count` roughly tracks the number of scores we've seen
+between `bins[i-1].lower_divider` and `bins[i].lower_divider`.
+
+__Algorithm overview.__ Each time a score is recorded, we increment the count
+of the appropriate bin. Then, we propose a repartition of the bins as follows:
+
+- Split the bin corresponding to the new score into two bins, divided by the
+  new score. Halve the counts of the old bin into the two new bins.
+
+- Merge together the two smallest contiguous bins, so that the total number of
+  bins stays equal.
+
+We accept this repartition if it increases the evenness across all bins, which
+we measure using entropy. For simplicity, we prevent the merged bins from
+containing either of the split bins.
+
+__Entropy.__ Given a collection of bins, its entropy is:
+
+```
+H(p) = -sum( bin.count/total * log(bin.count/total) for bin in bins )
+```
+
+Entropy has two key properties:
+1. The discrete distribution with maximum entropy is the uniform distribution.
+2. Entropy is convex.
+
+As such, we can make our bins as uniform as possible by seeking to maximize
+entropy. This is implemented by only accepting a repartition if it increases
+entropy.
+
+__Fast entropy calculations.__ Suppose we are considering a repartition that:
+ - splits bin `s` into `s_l` and `s_r`,
+ - merges bins `m_l` and `m_r` into `m`.
+
+We don't need to compute the full entropy over the old and new sets of bins,
+because the entropy contributions of bins not related to the split or merge
+cancel out. As such, we can calculate this with only six terms as follows,
+where `p(bin) = bin.count / total`.
+
+```
+H(new) - H(old) =   [ -p(s_l) log p(s_l) - p(s_r) log p(s_r) - p(m) log p(m) ]
+                  - [ -p(s) log p(s) - p(m_l) log p(m_l) - p(m_r) log p(m_r) ]
+```
+
+__Algorithm pseudocode.__ This leaves us with the following algorithm for
+an update.
+
+```
+def update(score):
+  if there aren't N bins yet:
+    insert Bin(lower_divider=score, count=1)
+    return
+
+  s = score's bin
+  s.count++
+
+  m_l, m_r = smallest contiguous pair of bins separate from s
+  m = Bin(lower_divider=m_l.lower_divider, count=m_l.count + m_r.count)
+
+  s_l = Bin(lower_divider=s.lower_divider, count=s.count/2)
+  s_r = Bin(lower_divider=score, count=s.count/2)
+
+  if H(new) > H(old):
+    replace s with s_l and s_r
+    replace m_l and m_r with m
+```
+
+## Performance
+
+We did some benchmarking under these conditions:
+ - Two tests: samples from `Beta(2,5)`, and a power-law distribution.
+ - The bin-entropy algorithm targeting 5 bins.
+ - The 'simple' algorithm from above storing `N` samples.
+ - The 'better-simple' algorithm that stores the last `N` samples and uses
+   them to decide the dividers for 5 bins.
+
+Results showed that the bin-entropy algorithm performs about as well as the
+better-simple algorithm when `N = 150`, and actually slightly outperforms it for
+`N < 150`. This may be due to the split/merge making it less sensitive to noise.
+It's difficult to compare the results of the simple algorithm because the number
+of bins is different, but the results were significantly noisier than the
+better-simple algorithm.
+
+# Ideas for improvement
+
+Here are some things that anecdotally improve performance:
+
+ - The algorithm loses information when a bin is split, because splitting the
+   counts evenly is uninformed. Tracking moments of the distribution within
+   each bin can help make this a more informed choice.
+
+ - We only perform one split/merge operation per insert, but it's possible
+   several split-merges should be done after one insert.
+
+ - Allowing the merged bins to contain one of the split bins provides a small
+   improvement in performance.