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MPP Univariate Autodiff from Roman Elizarov
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package scientifik.kmath.operations
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import kotlin.math.pow
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import kotlin.math.sqrt
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/*
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* Implementation of backward-mode automatic differentiation.
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* Initial gist by Roman Elizarov: https://gist.github.com/elizarov/1ad3a8583e88cb6ea7a0ad09bb591d3d
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*/
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/**
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* Differentiable variable with value and derivative of differentiation ([deriv]) result
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* with respect to this variable.
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*/
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data class ValueWithDeriv(var x: Double, var d: Double = 0.0) {
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constructor(x: Number) : this(x.toDouble())
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}
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/**
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* Runs differentiation and establishes [Field<ValueWithDeriv>] context inside the block of code.
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*
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* Example:
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* ```
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* val x = ValueWithDeriv(2) // define variable(s) and their values
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* val y = deriv { sqr(x) + 5 * x + 3 } // write formulate in deriv context
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* assertEquals(17.0, y.x) // the value of result (y)
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* assertEquals(9.0, x.d) // dy/dx
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* ```
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*/
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fun deriv(body: AutoDiffField.() -> ValueWithDeriv): ValueWithDeriv =
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ValueWithDerivField().run {
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val result = body()
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result.d = 1.0 // computing derivative w.r.t result
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runBackwardPass()
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result
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}
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abstract class AutoDiffField : Field<ValueWithDeriv> {
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/**
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* Performs update of derivative after the rest of the formula in the back-pass.
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*
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* For example, implementation of `sin` function is:
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*
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* ```
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* fun AD.sin(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(sin(x.x)) { z -> // call derive with function result
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* x.d += z.d * cos(x.x) // update derivative using chain rule and derivative of the function
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* }
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* ```
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*/
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abstract fun <R> derive(value: R, block: (R) -> Unit): R
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// Overloads for Double constants
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operator fun Number.plus(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() + that.x)) { z ->
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that.d += z.d
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}
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operator fun ValueWithDeriv.plus(b: Number): ValueWithDeriv = b.plus(this)
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operator fun Number.minus(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() - that.x)) { z ->
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that.d -= z.d
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}
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operator fun ValueWithDeriv.minus(that: Number): ValueWithDeriv = derive(ValueWithDeriv(this.x - that.toDouble())) { z ->
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this.d += z.d
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}
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override operator fun Number.times(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() * that.x)) { z ->
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that.d += z.d * this.toDouble()
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}
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override operator fun ValueWithDeriv.times(b: Number): ValueWithDeriv = b.times(this)
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override operator fun Number.div(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() / that.x)) { z ->
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that.d -= z.d * this.toDouble() / (that.x * that.x)
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}
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override operator fun ValueWithDeriv.div(that: Number): ValueWithDeriv = derive(ValueWithDeriv(this.x / that.toDouble())) { z ->
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this.d += z.d / that.toDouble()
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}
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}
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/**
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* Automatic Differentiation context class.
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*/
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private class ValueWithDerivField : AutoDiffField() {
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// this stack contains pairs of blocks and values to apply them to
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private var stack = arrayOfNulls<Any?>(8)
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private var sp = 0
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@Suppress("UNCHECKED_CAST")
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override fun <R> derive(value: R, block: (R) -> Unit): R {
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// save block to stack for backward pass
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if (sp >= stack.size) stack = stack.copyOf(stack.size * 2)
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stack[sp++] = block
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stack[sp++] = value
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return value
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}
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@Suppress("UNCHECKED_CAST")
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fun runBackwardPass() {
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while (sp > 0) {
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val value = stack[--sp]
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val block = stack[--sp] as (Any?) -> Unit
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block(value)
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}
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}
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// Basic math (+, -, *, /)
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override fun add(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
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derive(ValueWithDeriv(a.x + b.x)) { z ->
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a.d += z.d
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b.d += z.d
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}
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override fun multiply(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
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derive(ValueWithDeriv(a.x * b.x)) { z ->
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a.d += z.d * b.x
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b.d += z.d * a.x
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}
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override fun divide(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
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derive(ValueWithDeriv(a.x / b.x)) { z ->
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a.d += z.d / b.x
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b.d -= z.d * a.x / (b.x * b.x)
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}
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override fun multiply(a: ValueWithDeriv, k: Number): ValueWithDeriv =
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derive(ValueWithDeriv(k.toDouble() * a.x)) { z ->
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a.d += z.d * k.toDouble()
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}
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override val zero: ValueWithDeriv get() = ValueWithDeriv(0.0, 0.0)
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override val one: ValueWithDeriv get() = ValueWithDeriv(1.0, 0.0)
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}
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// Extensions for differentiation of various basic mathematical functions
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// x ^ 2
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fun AutoDiffField.sqr(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(x.x * x.x)) { z ->
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x.d += z.d * 2 * x.x
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}
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// x ^ 1/2
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fun AutoDiffField.sqrt(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(sqrt(x.x))) { z ->
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x.d += z.d * 0.5 / z.x
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}
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// x ^ y (const)
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fun AutoDiffField.pow(x: ValueWithDeriv, y: Double): ValueWithDeriv = derive(ValueWithDeriv(x.x.pow(y))) { z ->
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x.d += z.d * y * x.x.pow(y - 1)
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}
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fun AutoDiffField.pow(x: ValueWithDeriv, y: Int): ValueWithDeriv = pow(x, y.toDouble())
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// exp(x)
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fun AutoDiffField.exp(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.exp(x.x))) { z ->
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x.d += z.d * z.x
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}
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// ln(x)
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fun AutoDiffField.ln(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.ln(x.x))) { z ->
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x.d += z.d / x.x
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}
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// x ^ y (any)
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fun AutoDiffField.pow(x: ValueWithDeriv, y: ValueWithDeriv): ValueWithDeriv = exp(y * ln(x))
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// sin(x)
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fun AutoDiffField.sin(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.sin(x.x))) { z ->
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x.d += z.d * kotlin.math.cos(x.x)
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}
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// cos(x)
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fun AutoDiffField.cos(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.cos(x.x))) { z ->
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x.d -= z.d * kotlin.math.sin(x.x)
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}
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@ -0,0 +1,155 @@
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package scientifik.kmath.operations
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import kotlin.math.PI
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import kotlin.test.Test
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import kotlin.test.assertEquals
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class AutoDiffTest {
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@Test
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fun testPlusX2() {
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val x = ValueWithDeriv(3) // diff w.r.t this x at 3
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val y = deriv { x + x }
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assertEquals(6.0, y.x) // y = x + x = 6
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assertEquals(2.0, x.d) // dy/dx = 2
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}
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@Test
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fun testPlus() {
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// two variables
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val x = ValueWithDeriv(2)
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val y = ValueWithDeriv(3)
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val z = deriv { x + y }
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assertEquals(5.0, z.x) // z = x + y = 5
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assertEquals(1.0, x.d) // dz/dx = 1
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assertEquals(1.0, y.d) // dz/dy = 1
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}
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@Test
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fun testMinus() {
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// two variables
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val x = ValueWithDeriv(7)
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val y = ValueWithDeriv(3)
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val z = deriv { x - y }
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assertEquals(4.0, z.x) // z = x - y = 4
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assertEquals(1.0, x.d) // dz/dx = 1
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assertEquals(-1.0, y.d) // dz/dy = -1
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}
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@Test
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fun testMulX2() {
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val x = ValueWithDeriv(3) // diff w.r.t this x at 3
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val y = deriv { x * x }
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assertEquals(9.0, y.x) // y = x * x = 9
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assertEquals(6.0, x.d) // dy/dx = 2 * x = 7
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}
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@Test
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fun testSqr() {
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val x = ValueWithDeriv(3)
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val y = deriv { sqr(x) }
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assertEquals(9.0, y.x) // y = x ^ 2 = 9
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assertEquals(6.0, x.d) // dy/dx = 2 * x = 7
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}
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@Test
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fun testSqrSqr() {
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val x = ValueWithDeriv(2)
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val y = deriv { sqr(sqr(x)) }
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assertEquals(16.0, y.x) // y = x ^ 4 = 16
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assertEquals(32.0, x.d) // dy/dx = 4 * x^3 = 32
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}
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@Test
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fun testX3() {
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val x = ValueWithDeriv(2) // diff w.r.t this x at 2
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val y = deriv { x * x * x }
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assertEquals(8.0, y.x) // y = x * x * x = 8
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assertEquals(12.0, x.d) // dy/dx = 3 * x * x = 12
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}
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@Test
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fun testDiv() {
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val x = ValueWithDeriv(5)
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val y = ValueWithDeriv(2)
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val z = deriv { x / y }
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assertEquals(2.5, z.x) // z = x / y = 2.5
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assertEquals(0.5, x.d) // dz/dx = 1 / y = 0.5
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assertEquals(-1.25, y.d) // dz/dy = -x / y^2 = -1.25
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}
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@Test
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fun testPow3() {
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val x = ValueWithDeriv(2) // diff w.r.t this x at 2
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val y = deriv { pow(x, 3) }
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assertEquals(8.0, y.x) // y = x ^ 3 = 8
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assertEquals(12.0, x.d) // dy/dx = 3 * x ^ 2 = 12
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}
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@Test
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fun testPowFull() {
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val x = ValueWithDeriv(2)
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val y = ValueWithDeriv(3)
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val z = deriv { pow(x, y) }
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assertApprox(8.0, z.x) // z = x ^ y = 8
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assertApprox(12.0, x.d) // dz/dx = y * x ^ (y - 1) = 12
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assertApprox(8.0 * kotlin.math.ln(2.0), y.d) // dz/dy = x ^ y * ln(x)
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}
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@Test
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fun testFromPaper() {
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val x = ValueWithDeriv(3)
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val y = deriv { 2 * x + x * x * x }
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assertEquals(33.0, y.x) // y = 2 * x + x * x * x = 33
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assertEquals(29.0, x.d) // dy/dx = 2 + 3 * x * x = 29
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}
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@Test
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fun testLongChain() {
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val n = 10_000
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val x = ValueWithDeriv(1)
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val y = deriv {
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var pow = ValueWithDeriv(1)
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for (i in 1..n) pow *= x
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pow
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}
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assertEquals(1.0, y.x) // y = x ^ n = 1
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assertEquals(n.toDouble(), x.d) // dy/dx = n * x ^ (n - 1) = n - 1
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}
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@Test
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fun testExample() {
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val x = ValueWithDeriv(2)
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val y = deriv { sqr(x) + 5 * x + 3*one }
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assertEquals(17.0, y.x) // the value of result (y)
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assertEquals(9.0, x.d) // dy/dx
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}
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@Test
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fun testSqrt() {
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val x = ValueWithDeriv(16)
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val y = deriv { sqrt(x) }
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assertEquals(4.0, y.x) // y = x ^ 1/2 = 4
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assertEquals(1.0 / 8, x.d) // dy/dx = 1/2 / x ^ 1/4 = 1/8
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}
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@Test
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fun testSin() {
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val x = ValueWithDeriv(PI / 6)
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val y = deriv { sin(x) }
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assertApprox(0.5, y.x) // y = sin(PI/6) = 0.5
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assertApprox(kotlin.math.sqrt(3.0) / 2, x.d) // dy/dx = cos(PI/6) = sqrt(3)/2
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}
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@Test
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fun testCos() {
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val x = ValueWithDeriv(PI / 6)
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val y = deriv { cos(x) }
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assertApprox(kotlin.math.sqrt(3.0) / 2, y.x) // y = cos(PI/6) = sqrt(3)/2
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assertApprox(-0.5, x.d) // dy/dx = -sin(PI/6) = -0.5
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}
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private fun assertApprox(a: Double, b: Double) {
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if ((a - b) > 1e-10) assertEquals(a, b)
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}
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}
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