Dual Numbers and Automatic Differentiation

Dual Numbers and Automatic Differentiation

This literate essay develops an implementation of a type called Differential. A Differential is a generalization of a type called a "dual number", and the glowing, pulsing core of the SICMUtils implementation of forward-mode automatic differentiation.

As we'll discuss, passing these numbers as arguments to some function \(f\) built out of the sicmutils.generic operators allows us to build up the derivative of \(f\) in parallel to our evaluation of \(f\). Complex programs are built out of simple pieces that we know how to evaluate; we can build up derivatives of entire programs in a similar way, building them out of the derivatives of the smaller pieces of those programs.

(ns sicmutils.differential
  "This namespace contains an implementation of [[Differential]], a generalized
  dual number type that forms the basis for the forward-mode automatic
  differentiation implementation in sicmutils.

  See `sicmutils.calculus.derivative` for a fleshed-out derivative
  implementation using [[Differential]]."
  (:refer-clojure :rename {compare core-compare}
                  #?@(:cljs [:exclude [compare]]))
  (:require [clojure.string :refer [join]]
            [sicmutils.function :as f]
            [sicmutils.generic :as g]
            [sicmutils.util :as u]
            [sicmutils.util.stream :as us]
            [sicmutils.util.vector-set :as uv]
            [sicmutils.value :as v])
  #?(:clj
     (:import (clojure.lang AFn IFn))))

Forward-Mode Automatic Differentiation

For many scientific computing applications, it's valuable be able to generate a "derivative" of a function; given some wiggle in the inputs, how much wobble does the output produce?

we know how to take derivatives of many of the generic functions exposed by SICMUtils, like +, *, sin and friends. It turns out that we can take the derivatives of large, complicated functions by combining the derivatives of these smaller functions using the chain rule as a clever bookkeeping device.

The technique of evaluating a function and its derivative in parallel is called "forward-mode Automatic Differentiation". The SICMUtils wiki has more information on the history of this technique, and links to the many other implementations you'll find in different languages. See the cljdocs Automatic Differentiation page for "how do I use this?"-style questions.

NOTE: The other flavor of automatic differentiation (AD) is "reverse-mode AD". See sicmutils.tape for an implementation of this style, coming soon!

Dual Numbers and AD

Our goal is to build up derivatives of complex functions out of the derivatives of small pieces. A dual number is a relatively simple piece of machinery that will help us accomplish this goal.

A dual number is a pair of numbers of the form

\[
a+b\varepsilon
\]

where \(a\) and \(b\) are real numbers, and \(\varepsilon\) is an abstract thing, with the property that \(\varepsilon^2 = 0\).

NOTE: This might remind you of the definition of a complex number of the form \(a + bi\), where \(i\) is also a new thing with the property that \(i^2 = -1\). You are very wise! The bigger idea lurking here is the "generalized complex number", and of course mathematicians have pushed this very far. You might explore the "Split-complex numbers" too, which arise when you set \(i^2 = 1\).

Why are dual numbers useful (in SICMUtils)? If you pass \(a+b\varepsilon\) in to a function \(f\), the result is a dual number \(f(a) + Df(a) b \varepsilon\); you get both the function and its derivative evaluated at the same time!

To see why, look at what happens when you pass a dual number into the Taylor series expansion of some arbitrary function \(f\). As a reminder, the Taylor series expansion of \(f\) around some point \(a\) is:

\[
f(x) = f(a)+\frac{Df(a)}{1!}(x-a)+\frac{D^2f(a)}{2!}(x-a)^{2}+\frac{D^3f(a)}{3!}(x-a)^{3}+\cdots
\]

NOTE: See this nice overview of Taylor series expansion by Andrew Chamberlain if you want to understand this idea and why we can approximate (smooth) functions this way.

If you evaluate the expansion of \(f(x)\) around \(a\) with a dual number argument whose first component is \(a\) – take \(x=a+b\varepsilon\), for example – watch how the expansion simplifies:

\[
f(a+b\varepsilon) = f(a)+\frac{Df(a)}{1!}(b\varepsilon)+\frac{D^2f(a)}{2!}(b\varepsilon)^2+\cdots
\]

Since \(\varepsilon^2=0\) we can ignore all terms beyond the first two:

\[
f(a+b\varepsilon) = f(a)+ (Df(a)b)\varepsilon
\]

NOTE: See lift-1 for an implementation of this idea.

Interesting! This justifies our claim above: applying a function to some dual number \(a+\varepsilon\) returns a new dual number, where

  • the first component is \(f(a)\), the normal function evaluation
  • the second component is \(Df(a)\), the derivative.

If we do this twice, the second component of the returned dual number beautifully recreates the Chain Rule:

\begin{aligned}
g(f(a+\varepsilon)) & = g(f(a) + Df(a)\varepsilon) \
& = g(f(a)) + (Dg(f(a)))(Df(a))\varepsilon
\end{aligned}

Terminology Change!

A "dual number" is a very general idea. Because we're interested in dual numbers as a bookkeeping device for derivatives, we're going to specialize our terminology. From now on, we'll rename \(a\) and \(b\) to \(x\) and \(x'\). Given a dual number of the form \(x+x'\varepsilon\): we'll refer to:

  • \(x\) as the "primal" part of the dual number
  • \(x'\) as the "tangent" part
  • \(\varepsilon\) as the "tag"

NOTE: "primal" means \(x\) is tracking the "primal", or "primary", part of the computation. "tangent" is a synonym for "derivative". "tag" is going to make more sense shortly, when we start talking about mixing together multiple \(\varepsilon_1\), \(\varepsilon_2\) from different computations.

Binary Functions

What about functions of more than one variable? We can use the same approach by leaning on the multivariable Taylor series expansion. Take \(f(x, y)\) as a binary example. If we pass dual numbers in to the taylor series expansion of \(f\), the \(\varepsilon\) multiplication rule will erase all higher-order terms, leaving us with:

\[
f(x+x'\varepsilon, y+y'\varepsilon) = f(x,y) + \partial_1 f(x,y)x' + \partial_2 f(x,y) y'
\]

NOTE: See lift-2 for an implementation of this idea.

This expansion generalizes for n-ary functions; every new argument \(x_n + x'_n\varepsilon\) contributes \(\partial_n f(...)x'_n\) to the result.

We can check this with the simple cases of addition, subtraction and multiplication.

The real parts of a dual number add commutatively, so we can rearrange the components of a sum to get a new dual number:

\[
(x+x'\varepsilon)+(y+y'\varepsilon) == (x+y)+(x'+y')\varepsilon
\]

This matches the sum rule of differentiation, since the partials of \(x + y\) with respect to either \(x\) or \(y\) both equal 1.

Subtraction is almost identical and agrees with the subtraction rule:

\[
(x+x'\varepsilon)-(y+y'\varepsilon) == (x-y)+(x'-y')\varepsilon
\]

Multiplying out the components of two dual numbers again gives us a new dual number, whose tangent component agrees with the product rule:

\begin{aligned}
(x+ x'\varepsilon)*(y+y'\epsilon) &= xy+(xy')\varepsilon+(x'y)\varepsilon+(x'y')\epsilon^2 \
&= xy+(xy'+y'x)\varepsilon
\end{aligned}

Stare at these smaller derivations and convince yourself that they agree with the Taylor series expansion method for binary functions.

The upshot is that, armed with these techniques, we can implement a higher-order derivative function (almost!) as simply as this:

(defn derivative [f]
  (fn [x]
    (extract-tangent
     (f (make-dual x 1)))))

As long as f is built out of functions that know how to apply themselves to dual numbers, this will all Just Work.

Multiple Variables, Nesting

All of the examples above are about first-order derivatives. Taking higher-order derivatives is, in theory, straightforward:

(derivative
 (derivative f))

But this guess hits one of many subtle problems with the implementation of forward-mode AD. The double-call to derivative will expand out to this:

(fn [x]
  (letfn [(inner-d [x]
            (extract-tangent
             (f (make-dual x 1))))]
    (extract-tangent
     (inner-d
      (make-dual x 1)))))

the x received by inner-d will ALREADY be a dual number \(x+\varepsilon\)! This will cause two immediate problems:

  • (make-dual x 1) will return \((x+\varepsilon)+\varepsilon = x+2\varepsilon\), which is not what we we want

  • The extract-tangent call inside inner-d will return the Df(x) component of the dual number… which, remember, is no longer a dual number! So the SECOND call to extract-tangent have nothing to extract, and can only sensibly return 0.

The problem here is called "perturbation confusion", and is covered beautifully in "Confusion of Tagged Perturbations in Forward Automatic Differentiation of Higher-Order Functions", by Manzyuk et pal. (2019).

The solution is to introduce a new \(\varepsilon\) for every level, and allow different \(\varepsilon\) instances to multiply without annihalating. Each \(\varepsilon\) is called a "tag". Differential (implemented below) is a generalized dual number that can track many tags at once, allowing nested derivatives like the one described above to work.

This implies that extract-tangent needs to take a tag, to determine which tangent to extract:

(defn derivative [f]
  (let [tag (fresh-tag)]
    (fn [x]
      (-> (f (make-dual x 1 tag))
          (extract-tangent tag)))))

This is close to the final form you'll find in derivative.

What Return Values are Allowed?

Before we discuss the implementation of dual numbers (called Differential), lift-1, lift-2 and the rest of the machinery that makes this all possible; what sorts of objects is f allowed to return?

The dual number approach is beautiful because we can bring to bear all sorts of operations in Clojure that never even see dual numbers. For example, square-and-cube called with a dual number returns a PAIR of dual numbers:

(defn square-and-cube [x]
  (let [x2 (g/square x)
        x3 (g/cube x)]
    [x2 x3]))

Vectors don't care what they hold! We want the derivative of square-and-cube to also return a vector, whose entries represent the derivative of that entry with respect to the function's input.

But this implies that extract-tangent from the example above needs to know how to handle vectors and other collections; in the case of a vector v by returning (mapv extract-tangent v)

The dual number implementation is called Differential; the way that Differential instances interact with container types in SICMUtils makes it easy for these captures to occur all over. Whenever we multiply a Differential by a structure, a function, a vector, any of those things, our implementation of the SICMUtils generics pushes the Differential inside those objects, rather than forming a Differential with, for example, a vector in the primal and tangent parts.

Functions… interesting. what about higher-order functions?

(defn offset-fn
  "Returns a function that takes a single-argument function `g`, and returns a new
  function like `g` that offsets its input by `offset`."
  [offset]
  (fn [g]
    (fn [x]
      (g (+ x offset)))))

(derivative offset-fn) here returns a function! Ideally we'd like the returned function to act exactly like:

(derivative
 (fn [offset] (g (+ x offset))))

for some known g and x, but with the ability to store (derivative offset-fn) and call it later with many different g.

We might accomplish this by composing extract-tangent with the returned function, so that the extraction happens later, when the function's called.

NOTE: The real implementation is more subtle! See the sicmutils.calculus.derivative namespace for the actual implementation of IPerturbed for functions and multimethods.

All of this suggests that we need to make extract-tangent an open function that other folks can extend for other container-like types (functors, specifically).

The IPerturbed protocol accomplishes this, along with two other functions that we'll use later:

(defprotocol IPerturbed
  (perturbed? [this]
    "Returns true if the supplied object has some known non-zero tangent to be
    extracted via [[extract-tangent]], false otherwise. (Return `false` by
    default if you can't detect a perturbation.)")

  (replace-tag [this old-tag new-tag]
    "If `this` is perturbed, Returns a similar object with the perturbation
    modified by replacing any appearance of `old-tag` with `new-tag`. Else,
    return `this`.")

  (extract-tangent [this tag]
    "If `this` is perturbed, return the tangent component paired with the
    supplied tag. Else, returns `([[sicmutils.value/zero-like]] this)`."))

replace-tag exists to handle subtle bugs that can arise in the case of functional return values. See the "Amazing Bug" sections in sicmutils.calculus.derivative-test for detailed examples on how this might bite you.

The default implementations are straightforward, and match the docstrings:

(extend-protocol IPerturbed
  #?(:clj Object :cljs default)
  (perturbed? [_] false)
  (replace-tag [this _ _] this)
  (extract-tangent [this _] (v/zero-like this)))

Differential Implementation

We now have a template for how to implement derivative. What's left? We need a dual number type that we can build and split back out into primal and tangent components, given some tag. We'll call this type a Differential.

A Differential is a generalized dual number with a single primal component, and potentially many tagged terms. Identical tags cancel to 0 when multiplied, but are allowed to multiply by each other:

\[
a + b\varepsilon_1 + c\varepsilon_2 + d\varepsilon_1 \varepsilon_2 + \cdots
\]

Alternatively, you can view a Differential as a dual number with a specific tag, that's able to hold dual numbers with some other tag in its primal and tangent slots. You can turn a Differential into a dual number by specifying one of its tags. Here are the primal and tangent components for \(\varepsilon_2\):

\[
(a + b\varepsilon_1) + (c + d\varepsilon_1)\varepsilon_2
\]

And for \(\varepsilon_1\):

\[
(a + c\varepsilon_2) + (b + d \varepsilon_2) \varepsilon_1
\]

A differential term is implemented as a pair whose first element is a set of tags and whose second is the coefficient.

(def ^:private tags first)
(def ^:private coefficient peek)

The set of tags is implemented as a "vector set", from sicmutils.util.vector-set. This is a sorted set data structure, backed by a Clojure vector. vector sets provide cheap "max" and "min" operations, and acceptable union, intersection and difference performance.

(defn- make-term
  "Returns a [[Differential]] term with the supplied vector-set of `tags` paired
  with coefficient `coef`.

  `tags` defaults to [[uv/empty-set]]"
  ([coef] [uv/empty-set coef])
  ([tags coef] [tags coef]))

Since the only use of a tag is to distinguish each unnamed \(\varepsilon_n\), we'll assign a new, unique positive integer for each new tag:

(let [next-tag (atom 0)]
  (defn fresh-tag
    "Returns a new, unique tag for use inside of a [[Differential]] term list."
    []
    (swap! next-tag inc)))

(defn- tag-in-term?
  "Return true if `t` is in the tag-set of the supplied `term`, false otherwise."
  [term t]
  (uv/contains? (tags term) t))

Term List Algebra

The discussion above about Taylor series expansions revealed that for single variable functions, we can pass a dual number into any function whose derivative we already know:

\[
f(a+b\varepsilon) = f(a) + (Df(a)b)\varepsilon
\]

Because we can split a Differential into a primal and tangent component with respect to some tag, we can reuse this result. We'll default to splitting Differential instances by the highest-index tag:

\begin{aligned}
f(a &+ b\varepsilon_1 + c\varepsilon_2 + d\varepsilon_1 \varepsilon_2) \
&= f((a + b\varepsilon_1)+(c + d\varepsilon_1)\varepsilon_2) \
&= f(a + b\varepsilon_1)+Df(a + b\varepsilon_1)(c + d\varepsilon_1)\varepsilon_2 \
\end{aligned}

Note that \(f\) and \(Df\) both received a dual number! One more expansion, this time in \(\varepsilon_1\), completes the evaluation (and makes abundantly clear why we want the computer doing this, not pencil-and-paper):

\begin{aligned}
f(a &+ b\varepsilon_1)+Df(a+b\varepsilon_1)(c+d\varepsilon_1)\varepsilon_2 \
&= (f(a)+Df(a)b\varepsilon_1)+(Df(a)+D^2f(a)b\varepsilon_1)(c + d\varepsilon_1)\varepsilon_2 \
&= f(a)+(Df(a)b+D^2f(a)bc)\varepsilon_1+Df(a)c\varepsilon_2+Df(a)d\varepsilon_1\varepsilon_2
\end{aligned}

The only operations we need to implement between lists of terms are addition and multiplication.

Addition and Multiplication

To efficiently add two Differential instances (represented as vectors of terms), we keep all terms in sorted order, sorted first by the length of each tag list (the "order" of the differential term), and secondarily by the values of the tags inside the tag list.

NOTE: Clojure vectors already implement this ordering properly, so we can use clojure.core/compare to determine an ordering on a tag list.

(defn- terms:+
  "Returns the sum of the two supplied sequences of differential terms; any terms
  in the result with a zero coefficient will be removed.

  Each input must be sequence of `[tag-set, coefficient]` pairs, sorted by
  `tag-set`."
  [xs ys]
  (loop [xs xs, ys ys, result []]
    (cond (empty? xs) (into result ys)
          (empty? ys) (into result xs)
          :else (let [[x-tags x-coef :as x] (first xs)
                      [y-tags y-coef :as y] (first ys)
                      compare-flag (core-compare x-tags y-tags)]
                  (cond
                    If the terms have the same tag set, add the coefficients
                    together. Include the term in the result only if the new
                    coefficient is non-zero.
                    (zero? compare-flag)
                    (let [sum (g/+ x-coef y-coef)]
                      (recur (rest xs)
                             (rest ys)
                             (if (v/zero? sum)
                               result
                               (conj result (make-term x-tags sum)))))

                    Else, pass the smaller term on unchanged and proceed.
                    (neg? compare-flag)
                    (recur (rest xs) ys (conj result x))

                    :else
                    (recur xs (rest ys) (conj result y)))))))

Because we've decided to store terms as a vector, we can multiply two vectors of terms by:

  • taking the cartesian product of both term lists
  • discarding all pairs of terms that share any tag (since \(\varepsilon^2=0\))
  • multiplying the coefficients of all remaining pairs and union-ing their tag lists
  • grouping and adding any new terms with the SAME list of tags, and filtering out zeros

This final step is required by a number of different operations later, so we break it out into its own collect-terms function:

(defn- collect-terms
  "Build a term list up of pairs of tags => coefficients by grouping together and
  summing coefficients paired with the same term list.

  The result will be sorted by term list, and contain no duplicate term lists."
  [tags->coefs]
  (let [terms (for [[tags tags-coefs] (group-by tags tags->coefs)
                    :let [c (transduce (map coefficient) g/+ tags-coefs)]
                    :when (not (v/zero? c))]
                [tags c])]
    (into [] (sort-by tags terms))))

terms:* implements the first three steps, and calls collect-terms on the resulting sequence:

(defn- terms:*
  "Returns a vector of non-zero [[Differential]] terms that represent the product
  of the differential term lists `xs` and `ys`."
  [xs ys]
  (collect-terms
   (for [[x-tags x-coef] xs
         [y-tags y-coef] ys
         :when (empty? (uv/intersection x-tags y-tags))]
     (make-term (uv/union x-tags y-tags)
                (g/* x-coef y-coef)))))

Differential Type Implementation

Armed with our term list arithmetic operations, we can finally implement our Differential type and implement a number of important Clojure and SICMUtils protocols.

A Differential will respond to v/kind with ::differential. Because we want Differential instances to work in any place that real numbers or symbolic argument work, let's make ::differential derive from ::v/scalar:

(derive ::differential ::v/scalar)

Now the actual type. The terms field is a term-list vector that will remain (contractually!) sorted by its list of tags.

(declare d:apply compare equiv from-terms one?)

(deftype Differential [terms]
  A [[Differential]] as implemented can act as a chain-rule accounting device
  for all sorts of types, not just numbers. A [[Differential]] is
  only [[v/numerical?]] if its coefficients are numerical.
  v/Numerical
  (numerical? [_]
    (v/numerical? (coefficient (first terms))))

  IPerturbed
  (perturbed? [_] true)

  ;; There are 3 cases to consider when replacing some tag in a term, annotated
  ;; below:
  (replace-tag [_ oldtag newtag]
    (letfn [(process [term]
              (let [tagv (tags term)]
                (if-not (uv/contains? tagv oldtag)
                  if the term doesn't contain the old tag, ignore it.
                  [term]
                  (if (uv/contains? tagv newtag)
                    if the term _already contains_ the new tag
                    $\varepsilon_{new}$, then replacing $\varepsilon_1$
                    with a new instance of $\varepsilon_2$ would cause a
                    clash. Since $\varepsilon_2^2=0$, the term should be
                    removed.
                    []
                    else, perform the replacement.
                    [(make-term (-> tagv
                                    (uv/disj oldtag)
                                    (uv/conj newtag))
                                (coefficient term))]))))]
      (from-terms
       (mapcat process terms))))

  ;; To extract the tangent (with respect to `tag`) from a differential, return
  ;; all terms that contain the tag (with the tag removed!) This can create
  ;; duplicate terms, so use [[from-terms]] to massage the result into
  ;; well-formedness again.
  (extract-tangent [_ tag]
    (from-terms
     (mapcat (fn [term]
               (let [tagv (tags term)]
                 (if (uv/contains? tagv tag)
                   [(make-term (uv/disj tagv tag)
                               (coefficient term))]
                   [])))
             terms)))

  v/Value
  (zero? [this]
    (every? (comp v/zero? coefficient) terms))
  (one? [this] (one? this))
  (identity? [this] (one? this))
  (zero-like [_] 0)
  (one-like [_] 1)
  (identity-like [_] 1)
  (freeze [_] `[~'Differential ~@terms])
  (exact? [_] false)

  (kind [_] ::differential)

  Object
  ;; Comparing [[Differential]] objects using `equals` defaults to [[equiv]],
  ;; which compares instances only using their non-tagged ('finite') components.
  ;; If you want to compare two instances using their full term lists,
  ;; See [[eq]].
  #?(:clj (equals [a b] (equiv a b)))
  (toString [_] (str "D[" (join " " (map #(join " → " %) terms)) "]"))

  ;; Because a [[Differential]] is an accounting device that augments other
  ;; operations with the ability to carry around derivatives, it's possible that
  ;; the coefficient slots could be occupied by function objects. If so, then it
  ;; becomes possible to "apply" a [[Differential]] to some arguments (apply
  ;; each coefficient to the arguments).

  ;; TODO the arity, if anyone cares to ask, might be better implemented as the
  ;; joint arity of all coefficients; but my guess here is that the tangent
  ;; terms all have to be tracking derivatives of the primal term, which have to
  ;; have the same arity by definition.
  f/IArity
  (arity [_]
    (f/arity (coefficient (first terms))))

  #?@(:clj
      ;; This one is slightly subtle. To participate in control flow operations,
      ;; like comparison with both [[Differential]] and non-[[Differential]]
      ;; numbers, [[Differential]] instances should compare using ONLY their
      ;; non-tagged ("finite") terms. This means that comparison will totally
      ;; ignore any difference in tags.
      [Comparable
       (compareTo [a b] (compare a b))

       IFn
       (invoke [this]
               (d:apply this []))
       (invoke [this a]
               (d:apply this [a]))
       (invoke [this a b]
               (d:apply this [a b]))
       (invoke [this a b c]
               (d:apply this [a b c]))
       (invoke [this a b c d]
               (d:apply this [a b c d]))
       (invoke [this a b c d e]
               (d:apply this [a b c d e]))
       (invoke [this a b c d e f]
               (d:apply this [a b c d e f]))
       (invoke [this a b c d e f g]
               (d:apply this [a b c d e f g]))
       (invoke [this a b c d e f g h]
               (d:apply this [a b c d e f g h]))
       (invoke [this a b c d e f g h i]
               (d:apply this [a b c d e f g h i]))
       (invoke [this a b c d e f g h i j]
               (d:apply this [a b c d e f g h i j]))
       (invoke [this a b c d e f g h i j k]
               (d:apply this [a b c d e f g h i j k]))
       (invoke [this a b c d e f g h i j k l]
               (d:apply this [a b c d e f g h i j k l]))
       (invoke [this a b c d e f g h i j k l m]
               (d:apply this [a b c d e f g h i j k l m]))
       (invoke [this a b c d e f g h i j k l m n]
               (d:apply this [a b c d e f g h i j k l m n]))
       (invoke [this a b c d e f g h i j k l m n o]
               (d:apply this [a b c d e f g h i j k l m n o]))
       (invoke [this a b c d e f g h i j k l m n o p]
               (d:apply this [a b c d e f g h i j k l m n o p]))
       (invoke [this a b c d e f g h i j k l m n o p q]
               (d:apply this [a b c d e f g h i j k l m n o p q]))
       (invoke [this a b c d e f g h i j k l m n o p q r]
               (d:apply this [a b c d e f g h i j k l m n o p q r]))
       (invoke [this a b c d e f g h i j k l m n o p q r s]
               (d:apply this [a b c d e f g h i j k l m n o p q r s]))
       (invoke [this a b c d e f g h i j k l m n o p q r s t]
               (d:apply this [a b c d e f g h i j k l m n o p q r s t]))
       (applyTo [this xs] (AFn/applyToHelper this xs))]

      :cljs
      [IEquiv
       (-equiv [a b] (equiv a b))

       IComparable
       (-compare [a b]  (compare a b))

       IPrintWithWriter
       (-pr-writer [x writer _]
                   (write-all writer (.toString x)))

       IFn
       (-invoke [this]
                (d:apply this []))
       (-invoke [this a]
                (d:apply this [a]))
       (-invoke [this a b]
                (d:apply this [a b]))
       (-invoke [this a b c]
                (d:apply this [a b c]))
       (-invoke [this a b c d]
                (d:apply this [a b c d]))
       (-invoke [this a b c d e]
                (d:apply this [a b c d e]))
       (-invoke [this a b c d e f]
                (d:apply this [a b c d e f]))
       (-invoke [this a b c d e f g]
                (d:apply this [a b c d e f g]))
       (-invoke [this a b c d e f g h]
                (d:apply this [a b c d e f g h]))
       (-invoke [this a b c d e f g h i]
                (d:apply this [a b c d e f g h i]))
       (-invoke [this a b c d e f g h i j]
                (d:apply this [a b c d e f g h i j]))
       (-invoke [this a b c d e f g h i j k]
                (d:apply this [a b c d e f g h i j k]))
       (-invoke [this a b c d e f g h i j k l]
                (d:apply this [a b c d e f g h i j k l]))
       (-invoke [this a b c d e f g h i j k l m]
                (d:apply this [a b c d e f g h i j k l m]))
       (-invoke [this a b c d e f g h i j k l m n]
                (d:apply this [a b c d e f g h i j k l m n]))
       (-invoke [this a b c d e f g h i j k l m n o]
                (d:apply this [a b c d e f g h i j k l m n o]))
       (-invoke [this a b c d e f g h i j k l m n o p]
                (d:apply this [a b c d e f g h i j k l m n o p]))
       (-invoke [this a b c d e f g h i j k l m n o p q]
                (d:apply this [a b c d e f g h i j k l m n o p q]))
       (-invoke [this a b c d e f g h i j k l m n o p q r]
                (d:apply this [a b c d e f g h i j k l m n o p q r]))
       (-invoke [this a b c d e f g h i j k l m n o p q r s]
                (d:apply this [a b c d e f g h i j k l m n o p q r s]))
       (-invoke [this a b c d e f g h i j k l m n o p q r s t]
                (d:apply this [a b c d e f g h i j k l m n o p q r s t]))
       (-invoke [this a b c d e f g h i j k l m n o p q r s t rest]
                (d:apply this (concat [a b c d e f g h i j k l m n o p q r s t] rest)))]))

#?(:clj
   (defmethod print-method Differential
     [^Differential s ^java.io.Writer w]
     (.write w (.toString s))))

Accessor Methods

(defn differential?
  "Returns true if the supplied object is an instance of `Differential`, false
  otherwise."
  [dx]
  (instance? Differential dx))
(defn- bare-terms
  "Returns the `-terms` field of the supplied `Differential` object. Errors if any
  other type is supplied."
  [dx]
  {:pre [(differential? dx)]}
  (.-terms ^Differential dx))

Constructors

Because a Differential is really a wrapper around the idea of a generalized dual number represented as a term-list, we need to be able to get to and from the term list format from other types, not just Differential instances.

(defn- ->terms
  "Returns a vector of terms that represent the supplied [[Differential]]; any
  term with a [[v/zero?]] coefficient will be filtered out before return.

  If you pass a non-[[Differential]], [[->terms]] will return a singleton term
  list (or `[]` if the argument was zero)."
  [dx]
  (cond (differential? dx)
        (filterv (fn [term]
                   (not (v/zero? (coefficient term))))
                 (bare-terms dx))

        (v/zero? dx) []
        :else        [(make-term dx)]))
(defn- terms->differential
  "Returns a differential instance generated from a vector of terms. This method
  will do some mild cleanup, or canonicalization:

  - any empty term list will return 0
  - a singleton term list with no tags will return its coefficient

  NOTE this method assumes that the input is properly sorted, and contains no
  zero coefficients."
  [terms]
  {:pre [(vector? terms)]}
  (cond (empty? terms) 0

        (and (= (count terms) 1)
             (empty? (tags (first terms))))
        (coefficient (first terms))

        :else (->Differential terms)))
(defn from-terms
  "Accepts a sequence of terms (pairs of [tag-list, coefficient]), and returns:

  - a well-formed [[Differential]] instance, if the terms resolve to a
    differential with non-zero infinitesimal terms
  - the original input otherwise

  Duplicate (by tag list) terms are allowed; their coefficients will be summed
  together and removed if they sum to zero."
  [tags->coefs]
  (terms->differential
   (collect-terms tags->coefs)))

Differential API

This next section lifts slightly-augmented versions of terms:+ and terms:* up to operate on Differential instances. These work just as before, but handle wrapping and unwrapping the term list.

(defn d:+
  "Returns an object representing the sum of the two objects `dx` and `dy`. This
  works by summing the coefficients of all terms with the same list of tags.

    Works with non-[[Differential]] instances on either or both sides, and returns
  a [[Differential]] only if it contains any non-zero tangent components."
  [dx dy]
  (terms->differential
   (terms:+ (->terms dx)
            (->terms dy))))
(defn d:*
  "Returns an object representing the product of the two objects `dx` and `dy`.

  This works by multiplying out all terms:

  $$(dx1 + dx2 + dx3 + ...)(dy1 + dy2 + dy3...)$$

  and then collecting any duplicate terms by summing their coefficients.

  Works with non-[[Differential]] instances on either or both sides, and returns
  a [[Differential]] only if it contains any non-zero tangent components."
  [dx dy]
  (terms->differential
   (terms:* (->terms dx)
            (->terms dy))))
(defn- d:apply
  "Accepts a [[Differential]] and a sequence of `args`, interprets each
  coefficient as a function and returns a new [[Differential]] generated by
  applying the coefficient to `args`."
  [diff args]
  (terms->differential
   (into [] (mapcat (fn [term]
                      (let [result (apply (coefficient term) args)]
                        (if (v/zero? result)
                          []
                          [(make-term (tags term) result)]))))
         (->terms diff))))

Finally, the function we've been waiting for! bundle allows you to augment some non-Differential thing with a tag and push it through the generic arithmetic system, where it will accumulate the derivative of your original input (tagged with tag.)

(defn bundle
  "Generate a new [[Differential]] object with the supplied `primal` and `tangent`
  components, and the supplied internal `tag` that this [[Differential]] will
  carry around to prevent perturbation confusion.

  If the `tangent` component is `0`, acts as identity on `primal`. `tangent`
  defaults to 1.

  `tag` defaults to a side-effecting call to [[fresh-tag]]; you can retrieve
  this unknown tag by calling [[max-order-tag]]."
  ([primal]
   (bundle primal 1 (fresh-tag)))
  ([primal tag]
   (bundle primal 1 tag))
  ([primal tangent tag]
   (let [term (make-term (uv/make [tag]) tangent)]
     (d:+ primal (->Differential [term])))))

Differential Parts API

These functions give higher-level access to the components of a Differential you're typically interested in.

(defn max-order-tag
  "Given one or more well-formed [[Differential]] objects, returns the
  maximum ('highest order') tag found in the highest-order term of any of
  the [[Differential]] instances.

  If there is NO maximal tag (ie, if you provide [[Differential]] instances with
  no non-zero tangent parts, or all non-[[Differential]]s), returns nil."
  ([dx]
   (when (differential? dx)
     (let [last-term   (peek (->terms dx))
           highest-tag (peek (tags last-term))]
       highest-tag)))
  ([dx & dxs]
   (letfn [(max-termv [dx]
             (if-let [max-order (max-order-tag dx)]
               [max-order]
               []))]
     (when-let [orders (seq (mapcat max-termv (cons dx dxs)))]
       (apply max orders)))))

A reminder: the primal-part of a Differential is all terms except for terms containing max-order-tag, and tangent-part is a Differential built out of the remaining terms, all of which contain that tag.

(defn primal-part
  "Returns a [[Differential]] containing only the terms of `dx` that do NOT
  contain the supplied `tag`, ie, the primal component of `dx` with respect to
  `tag`.

  If no tag is supplied, defaults to `([[max-order-tag]] dx)`.

  NOTE: every [[Differential]] can be factored into a dual number of the form

      primal + (tangent * tag)

  For each tag in any of its terms. [[primal-part]] returns this first piece,
  potentially simplified into a non-[[Differential]] if the primal part contains
  no other tags."
  ([dx] (primal-part dx (max-order-tag dx)))
  ([dx tag]
   (if (differential? dx)
     (let [sans-tag? #(not (tag-in-term? % tag))]
       (->> (->terms dx)
            (filterv sans-tag?)
            (terms->differential)))
     dx)))
(defn tangent-part
  "Returns a [[Differential]] containing only the terms of `dx` that contain the
  supplied `tag`, ie, the tangent component of `dx` with respect to `tag`.

  If no tag is supplied, defaults to `([[max-order-tag]] dx)`.

  NOTE: Every [[Differential]] can be factored into a dual number of the form

      primal + (tangent * tag)

  For each tag in any of its terms. [[tangent-part]] returns a [[Differential]]
  representing `(tangent * tag)`, or 0 if `dx` contains no terms with the
  supplied `tag`.

  NOTE: the 2-arity case is similar to `(extract-tangent dx tag)`; the only
  difference is that `extract-tangent` drops the `dx` tag from all terms in the
  returned value. Call `extract-tangent` if you want to drop `tag`."
  ([dx] (tangent-part dx (max-order-tag dx)))
  ([dx tag]
   (if (differential? dx)
     (->> (->terms dx)
          (filterv #(tag-in-term? % tag))
          (terms->differential))
     0)))
(defn primal-tangent-pair
  "Returns a pair of the primal and tangent components of the supplied `dx`, with
  respect to the supplied `tag`. See the docs for [[primal-part]]
  and [[tangent-part]] for more details.

  [[primal-tangent-pair]] is equivalent to

  `[([[primal-part]] dx tag) ([[tangent-part]] dx tag)]`

  but slightly more efficient if you need both."
  ([dx] (primal-tangent-pair dx (max-order-tag dx)))
  ([dx tag]
   (if-not (differential? dx)
     [dx 0]
     (let [[tangent-terms primal-terms]
           (us/separatev #(tag-in-term? % tag)
                         (->terms dx))]
       [(terms->differential primal-terms)
        (terms->differential tangent-terms)]))))
(defn finite-term
  "Returns the term of the supplied [[Differential]] `dx` that has no tags
  attached to it, `0` otherwise.

  [[Differential]] instances with many can be decomposed many times
  into [[primal-part]] and [[tangent-part]]. Repeated calls
  to [[primal-part]] (with different tags!) will eventually yield a
  non-[[Differential]] value. If you know you want this, [[finite-term]] will
  get you there in one shot.

  NOTE that this will only work with a well-formed [[Differential]], ie, an
  instance with all terms sorted by their list of tags."
  [dx]
  (if (differential? dx)
    (let [[head] (bare-terms dx)
          ts     (tags head)]
      (if (= [] ts)
        (coefficient head)
        0))
    dx))

Comparison, Control Flow

Functions like =, < and friends don't have derivatives; instead, they're used for control flow inside of Clojure functions. To play nicely with these functions, the Differential API exposes a number of methods for comparing numbers on ONLY their finite parts.

Why? If x is a Differential instance, (< x 10) needs to return true whenever a non-Differential x would return true. To make this work, these operations look only at the finite-part.

HOWEVER! v/one? and v/zero? are examples of SICMUtils functions that are used to skip operations that we want to happen, like multiplication.

(g/* x y) will return y if (v/one? x) is true… but to propagate the derivative through we need this multiplication to occur. The compromise is:

  • v/one? and v/zero? return true only when ALL tangent-part=s are zero and the =finite-part is either v/one? or v/zero? respectively
  • eq and compare-full similarly looks at every component in the Differential supplied to both sides

while:

  • equiv and compare only examine the finite-part of either side.
(defn one?
  "Returns true if the supplied instance has a [[finite-part]] that responds true
  to [[sicmutils.value/one?]], and zero coefficients on any of its tangent
  components; false otherwise.

  NOTE: This means that [[one?]] will not do what you expect as a conditional
  inside some function. If you want to branch inside some function you're taking
  the derivative of, prefer `(= 1 dx)`. This will only look at
  the [[finite-part]] and ignore the values of the tangent parts."
  [dx]
  (let [[p t] (primal-tangent-pair dx)]
    (and (v/one? p)
         (v/zero? t))))
(defn eq
  "For non-differentials, this is identical to [[clojure.core/=]].
  For [[Differential]] instances, equality acts on tangent components too.

  If you want to ignore the tangent components, use [[equiv]]."
  ([_] true)
  ([a b]
   (= (->terms a)
      (->terms b)))
  ([a b & more]
   (reduce eq (eq a b) more)))
(defn compare-full
  "Comparator that compares [[Differential]] instances with each other or
  non-differentials using all tangent terms each instance. Matches the response
  of [[eq]].

  Acts as [[clojure.core/compare]] for non-differentials."
  [a b]
  (core-compare
   (->terms a)
   (->terms b)))
(defn equiv
  "Returns true if all of the supplied objects have equal [[finite-part]]s, false
  otherwise.

  Use [[equiv]] if you want to compare non-differentials with
  [[Differential]]s and ignore all tangent components. If you _do_ want to take
  the tangent components into account, prefer [[eq]]."
  ([_] true)
  ([a b]
   (= (finite-term a)
      (finite-term b)))
  ([a b & more]
   (reduce equiv (equiv a b) more)))
(defn compare
  "Comparator that compares [[Differential]] instances with each other or
  non-differentials using only the [[finite-part]] of each instance. Matches the
  response of [[equiv]].

  Acts as [[clojure.core/compare]] for non-differentials."
  [a b]
  (core-compare
   (finite-term a)
   (finite-term b)))

Chain Rule and Lifted Functions

Finally, we come to the heart of it! lift-1 and lift-2 "lift", or augment, unary or binary functions with the ability to handle Differential instances in addition to whatever other types they previously supported.

These functions are implementations of the single and multivariable Taylor series expansion methods discussed at the beginning of the namespace.

There is yet another subtlety here, noted in the docstrings below. lift-1 and lift-2 really are able to lift functions like clojure.core/+ that can't accept Differentials. But the first-order derivatives that you have to supply do have to be able to take Differential instances.

This is because the tangent-part of Differential might still be a Differential, and for Df to handle this we need to be able to take the second-order derivative.

Magically this will all Just Work if you pass an already-lifted function, or a function built out of already-lifted components, as df:dx or df:dy.

(defn- lift-1
  "Given:

  - some unary function `f`
  - a function `df:dx` that computes the derivative of `f` with respect to its
    single argument

  Returns a new unary function that operates on both the original type of `f`
  and [[Differential]] instances.

  NOTE: `df:dx` has to ALREADY be able to handle [[Differential]] instances. The
  best way to accomplish this is by building `df:dx` out of already-lifted
  functions, and declaring them by forward reference if you need to."
  [f df:dx]
  (fn call [x]
    (if-not (differential? x)
      (f x)
      (let [[px tx] (primal-tangent-pair x)
            fx      (call px)]
        (if (and (v/number? tx) (v/zero? tx))
          fx
          (d:+ fx (d:* (df:dx px) tx)))))))
(defn- lift-2
  "Given:

  - some binary function `f`
  - a function `df:dx` that computes the derivative of `f` with respect to its
    single argument
  - a function `df:dy`, similar to `df:dx` for the second arg

  Returns a new binary function that operates on both the original type of `f`
  and [[Differential]] instances.

  NOTE: `df:dx` and `df:dy` have to ALREADY be able to handle [[Differential]]
  instances. The best way to accomplish this is by building `df:dx` and `df:dy`
  out of already-lifted functions, and declaring them by forward reference if
  you need to."
  [f df:dx df:dy]
  (fn call [x y]
    (if-not (or (differential? x)
                (differential? y))
      (f x y)
      (let [tag     (max-order-tag x y)
            [xe dx] (primal-tangent-pair x tag)
            [ye dy] (primal-tangent-pair y tag)
            a (call xe ye)
            b (if (and (v/number? dx) (v/zero? dx))
                a
                (d:+ a (d:* dx (df:dx xe ye))))]
        (if (and (v/number? dy) (v/zero? dy))
          b
          (d:+ b (d:* (df:dy xe ye) dy)))))))
(defn- lift-n
  "Given:

  - some function `f` that can handle 0, 1 or 2 arguments
  - `df:dx`, a fn that returns the derivative wrt the single arg in the unary case
  - `df:dx1` and `df:dx2`, fns that return the derivative with respect to the
    first and second args in the binary case

  Returns a new any-arity function that operates on both the original type of
  `f` and [[Differential]] instances.

  NOTE: The n-ary case of `f` is populated by nested calls to the binary case.
  That means that this is NOT an appropriate lifting method for an n-ary
  function that isn't built out of associative binary calls. If you need this
  ability, please file an issue at the [sicmutils issue
  tracker](https://github.com/sicmutils/sicmutils/issues)."
  [f df:dx df:dx1 df:dx2]
  (let [f1 (lift-1 f df:dx)
        f2 (lift-2 f df:dx1 df:dx2)]
    (fn call
      ([] (f))
      ([x] (f1 x))
      ([x y] (f2 x y))
      ([x y & more]
       (reduce call (call x y) more)))))

Derivatives of Differentials

One more treat before we augment the generic arithmetic system. The derivative operation is linear, so:

\[
D(x+x'\varepsilon) = D(x)+D(x')\varepsilon
\]

This implementation is valid because the coefficients of a Differential can be functions.

(defmethod g/partial-derivative [::differential v/seqtype] [a selectors]
  (let [tag (max-order-tag a)
        px  (primal-part a tag)
        tx  (extract-tangent a tag)]
    (d:+ (g/partial-derivative px selectors)
         (d:* (g/partial-derivative tx selectors)
              (bundle 0 1 tag)))))

Generic Method Installation

Armed with lift-1 and lift-2, we can install Differential into the SICMUtils generic arithmetic system.

Any function built out of these components will work with the D operator.

(defn- defunary
  "Given:

  - a generic unary multimethod `generic-op`
  - a corresponding single-arity lifted function `differential-op`

  installs an appropriate unary implementation of `generic-op` for
  `::differential` instances."
  [generic-op differential-op]
  (defmethod generic-op [::differential] [a] (differential-op a)))
(defn- defbinary
  "Given:

  - a generic binary multimethod `generic-op`
  - a corresponding 2-arity lifted function `differential-op`

  installs an appropriate binary implementation of `generic-op` between
  `:differential` and `::v/scalar` instances."
  [generic-op differential-op]
  (doseq [signature [[::differential ::differential]
                     [::v/scalar ::differential]
                     [::differential ::v/scalar]]]
    (defmethod generic-op signature [a b] (differential-op a b))))

And now we're off to the races. The rest of the namespace provides defunary and defbinary calls for all of the generic operations for which we know how to declare partial derivatives.

(defbinary g/add
  (lift-2 g/add
          (fn [_ _] 1)
          (fn [_ _] 1)))

(defunary g/negate
  (lift-1 g/negate (fn [_] -1)))

(defunary g/negative?
  (fn [x] (g/negative? (finite-term x))))

(defbinary g/sub
  (lift-2 g/sub
          (fn [_ _] 1)
          (fn [_ _] -1)))

(let [mul (lift-2
           g/mul
           (fn [_ y] y)
           (fn [x _] x))]
  (defbinary g/mul mul)
  (defunary g/square (fn [x] (mul x x)))
  (defunary g/cube (fn [x] (mul x (mul x x))))
  (defbinary g/dot-product mul))

(defunary g/invert
  (lift-1 g/invert
          (fn [x] (g/div -1 (g/square x)))))

(defbinary g/div
  (lift-2 g/div
          (fn [_ y] (g/div 1 y))
          (fn [x y] (g/div (g/negate x)
                          (g/square y)))))

(defunary g/abs
  (fn [x]
    (let [f (finite-term x)
          func (cond (< f 0) (lift-1 (fn [x] (g/negate x)) (fn [_] -1))
                     (> f 0) (lift-1 (fn [x] x) (fn [_] 1))
                     (= f 0) (u/illegal "Derivative of g/abs undefined at zero")
                     :else (u/illegal (str "error! derivative of g/abs at" x)))]
      (func x))))

(defunary g/sqrt
  (lift-1 g/sqrt
          (fn [x]
            (g/invert
             (g/mul (g/sqrt x) 2)))))

This first case of g/expt, where the exponent itself is non-Differential, is special-cased and slightly simpler. The second partial derivative throws, since the more general definition below should always override.

(let [power (lift-2
             g/expt
             (fn [x y]
               (g/mul y (g/expt x (g/sub y 1))))
             (fn [_ _]
               (u/illegal "can't get there from here")))]
  (defmethod g/expt [::differential ::v/scalar] [d n] (power d n)))

The remaining two cases allow for a differential exponent.

NOTE: I took this implementation-split from scmutils, but I'm not sure that it matters… if the second partial never gets called, why is this a good optimization?

(let [expt (lift-2
            g/expt
            (fn [x y]
              (g/mul y (g/expt x (g/sub y 1))))
            (fn [x y]
              (if (and (v/number? x) (v/zero? y))
                (if (v/number? y)
                  (if (not (g/negative? y))
                    0
                    (u/illegal "Derivative undefined: expt"))
                  0)
                (g/* (g/log x) (g/expt x y)))))]
  (defmethod g/expt [::differential ::differential] [d n] (expt d n))
  (defmethod g/expt [::v/scalar ::differential] [d n] (expt d n)))
(defunary g/log
  (lift-1 g/log g/invert))

(defunary g/exp
  (lift-1 g/exp g/exp))

(defunary g/sin
  (lift-1 g/sin g/cos))

(defunary g/cos
  (lift-1 g/cos
          (fn [x] (g/negate (g/sin x)))))

(defunary g/tan
  (lift-1 g/tan
          (fn [x]
            (g/invert
             (g/square (g/cos x))))))

(defunary g/asin
  (lift-1 g/asin
          (fn [x]
            (g/invert
             (g/sqrt (g/sub 1 (g/square x)))))))

(defunary g/acos
  (lift-1 g/acos
          (fn [x]
            (g/negate
             (g/invert
              (g/sqrt (g/sub 1 (g/square x))))))))

(defunary g/atan
  (lift-1 g/atan (fn [x]
                   (g/invert
                    (g/add 1 (g/square x))))))

(defbinary g/atan
  (lift-2 g/atan
          (fn [y x]
            (g/div x (g/add (g/square x)
                            (g/square y))))
          (fn [y x]
            (g/div (g/negate y)
                   (g/add (g/square x)
                          (g/square y))))))

(defunary g/sinh
  (lift-1 g/sinh g/cosh))

(defunary g/cosh
  (lift-1 g/cosh g/sinh))

(defunary g/tanh
  (lift-1 g/tanh
          (fn [x]
            (g/sub 1 (g/square (g/tanh x))))))

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