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Safe modification of autodiff

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This commit is contained in:
Alexander Nozik 2019-05-05 09:39:51 +03:00
parent 765097cbbe
commit 6f9b704aa7
3 changed files with 100 additions and 78 deletions
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2
build.gradle.kts
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@ -1,4 +1,4 @@
val kmathVersion by extra("0.1.2-dev-3")
val kmathVersion by extra("0.1.2-dev-4")
allprojects {
repositories {

80
kmath-core/src/commonMain/kotlin/scientifik/kmath/operations/AutoDiff.kt
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@ -12,12 +12,12 @@ import kotlin.math.sqrt
* Differentiable variable with value and derivative of differentiation ([deriv]) result
* with respect to this variable.
*/
data class ValueWithDeriv(var x: Double) {
open class Variable(val x: Double) {
constructor(x: Number) : this(x.toDouble())
}
//TODO move set accessor inside AutoDiffField
var d: Double = 0.0
internal set
class DerivationResult(x: Double, val deriv: Map<Variable, Double>): Variable(x) {
fun deriv(variable: Variable) = deriv[variable] ?: 0.0
}
/**
@ -27,48 +27,53 @@ data class ValueWithDeriv(var x: Double) {
*
* Example:
* ```
* val x = ValueWithDeriv(2) // define variable(s) and their values
* val x = Variable(2) // define variable(s) and their values
* val y = deriv { sqr(x) + 5 * x + 3 } // write formulate in deriv context
* assertEquals(17.0, y.x) // the value of result (y)
* assertEquals(9.0, x.d) // dy/dx
* ```
*/
fun deriv(body: AutoDiffField.() -> ValueWithDeriv): ValueWithDeriv =
AutoDiffFieldImpl().run {
fun deriv(body: AutoDiffField.() -> Variable): DerivationResult =
AutoDiffContext().run {
val result = body()
result.d = 1.0 // computing derivative w.r.t result
runBackwardPass()
result
DerivationResult(result.x, derivatives)
}
abstract class AutoDiffField : Field<ValueWithDeriv> {
abstract class AutoDiffField : Field<Variable> {
/**
* Performs update of derivative after the rest of the formula in the back-pass.
*
* For example, implementation of `sin` function is:
*
* ```
* fun AD.sin(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(sin(x.x)) { z -> // call derive with function result
* fun AD.sin(x: Variable): Variable = derive(Variable(sin(x.x)) { z -> // call derive with function result
* x.d += z.d * cos(x.x) // update derivative using chain rule and derivative of the function
* }
* ```
*/
abstract fun <R> derive(value: R, block: (R) -> Unit): R
/**
* A variable accessing inner state of derivatives. Use only in extensions
*/
abstract var Variable.d: Double
// Overloads for Double constants
operator fun Number.plus(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() + that.x)) { z ->
operator fun Number.plus(that: Variable): Variable = derive(Variable(this.toDouble() + that.x)) { z ->
that.d += z.d
}
operator fun ValueWithDeriv.plus(b: Number): ValueWithDeriv = b.plus(this)
operator fun Variable.plus(b: Number): Variable = b.plus(this)
operator fun Number.minus(that: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(this.toDouble() - that.x)) { z ->
operator fun Number.minus(that: Variable): Variable = derive(Variable(this.toDouble() - that.x)) { z ->
that.d -= z.d
}
operator fun ValueWithDeriv.minus(that: Number): ValueWithDeriv = derive(ValueWithDeriv(this.x - that.toDouble())) { z ->
operator fun Variable.minus(that: Number): Variable = derive(Variable(this.x - that.toDouble())) { z ->
this.d += z.d
}
}
@ -76,12 +81,19 @@ abstract class AutoDiffField : Field<ValueWithDeriv> {
/**
* Automatic Differentiation context class.
*/
private class AutoDiffFieldImpl : AutoDiffField() {
private class AutoDiffContext : AutoDiffField() {
// this stack contains pairs of blocks and values to apply them to
private var stack = arrayOfNulls<Any?>(8)
private var sp = 0
internal val derivatives = HashMap<Variable, Double>()
override var Variable.d: Double
get() = derivatives[this] ?: 0.0
set(value) {
derivatives[this] = value
}
@Suppress("UNCHECKED_CAST")
override fun <R> derive(value: R, block: (R) -> Unit): R {
@ -104,71 +116,71 @@ private class AutoDiffFieldImpl : AutoDiffField() {
// Basic math (+, -, *, /)
override fun add(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
derive(ValueWithDeriv(a.x + b.x)) { z ->
override fun add(a: Variable, b: Variable): Variable =
derive(Variable(a.x + b.x)) { z ->
a.d += z.d
b.d += z.d
}
override fun multiply(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
derive(ValueWithDeriv(a.x * b.x)) { z ->
override fun multiply(a: Variable, b: Variable): Variable =
derive(Variable(a.x * b.x)) { z ->
a.d += z.d * b.x
b.d += z.d * a.x
}
override fun divide(a: ValueWithDeriv, b: ValueWithDeriv): ValueWithDeriv =
derive(ValueWithDeriv(a.x / b.x)) { z ->
override fun divide(a: Variable, b: Variable): Variable =
derive(Variable(a.x / b.x)) { z ->
a.d += z.d / b.x
b.d -= z.d * a.x / (b.x * b.x)
}
override fun multiply(a: ValueWithDeriv, k: Number): ValueWithDeriv =
derive(ValueWithDeriv(k.toDouble() * a.x)) { z ->
override fun multiply(a: Variable, k: Number): Variable =
derive(Variable(k.toDouble() * a.x)) { z ->
a.d += z.d * k.toDouble()
}
override val zero: ValueWithDeriv get() = ValueWithDeriv(0.0)
override val one: ValueWithDeriv get() = ValueWithDeriv(1.0)
override val zero: Variable get() = Variable(0.0)
override val one: Variable get() = Variable(1.0)
}
// Extensions for differentiation of various basic mathematical functions
// x ^ 2
fun AutoDiffField.sqr(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(x.x * x.x)) { z ->
fun AutoDiffField.sqr(x: Variable): Variable = derive(Variable(x.x * x.x)) { z ->
x.d += z.d * 2 * x.x
}
// x ^ 1/2
fun AutoDiffField.sqrt(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(sqrt(x.x))) { z ->
fun AutoDiffField.sqrt(x: Variable): Variable = derive(Variable(sqrt(x.x))) { z ->
x.d += z.d * 0.5 / z.x
}
// x ^ y (const)
fun AutoDiffField.pow(x: ValueWithDeriv, y: Double): ValueWithDeriv = derive(ValueWithDeriv(x.x.pow(y))) { z ->
fun AutoDiffField.pow(x: Variable, y: Double): Variable = derive(Variable(x.x.pow(y))) { z ->
x.d += z.d * y * x.x.pow(y - 1)
}
fun AutoDiffField.pow(x: ValueWithDeriv, y: Int): ValueWithDeriv = pow(x, y.toDouble())
fun AutoDiffField.pow(x: Variable, y: Int): Variable = pow(x, y.toDouble())
// exp(x)
fun AutoDiffField.exp(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.exp(x.x))) { z ->
fun AutoDiffField.exp(x: Variable): Variable = derive(Variable(kotlin.math.exp(x.x))) { z ->
x.d += z.d * z.x
}
// ln(x)
fun AutoDiffField.ln(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.ln(x.x))) { z ->
fun AutoDiffField.ln(x: Variable): Variable = derive(Variable(kotlin.math.ln(x.x))) { z ->
x.d += z.d / x.x
}
// x ^ y (any)
fun AutoDiffField.pow(x: ValueWithDeriv, y: ValueWithDeriv): ValueWithDeriv = exp(y * ln(x))
fun AutoDiffField.pow(x: Variable, y: Variable): Variable = exp(y * ln(x))
// sin(x)
fun AutoDiffField.sin(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.sin(x.x))) { z ->
fun AutoDiffField.sin(x: Variable): Variable = derive(Variable(kotlin.math.sin(x.x))) { z ->
x.d += z.d * kotlin.math.cos(x.x)
}
// cos(x)
fun AutoDiffField.cos(x: ValueWithDeriv): ValueWithDeriv = derive(ValueWithDeriv(kotlin.math.cos(x.x))) { z ->
fun AutoDiffField.cos(x: Variable): Variable = derive(Variable(kotlin.math.cos(x.x))) { z ->
x.d -= z.d * kotlin.math.sin(x.x)
}

96
kmath-core/src/commonTest/kotlin/scientifik/kmath/operations/AutoDiffTest.kt
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@ -7,145 +7,155 @@ import kotlin.test.assertEquals
class AutoDiffTest {
@Test
fun testPlusX2() {
val x = ValueWithDeriv(3) // diff w.r.t this x at 3
val x = Variable(3) // diff w.r.t this x at 3
val y = deriv { x + x }
assertEquals(6.0, y.x) // y = x + x = 6
assertEquals(2.0, x.d) // dy/dx = 2
assertEquals(2.0, y.deriv(x)) // dy/dx = 2
}
@Test
fun testPlus() {
// two variables
val x = ValueWithDeriv(2)
val y = ValueWithDeriv(3)
val x = Variable(2)
val y = Variable(3)
val z = deriv { x + y }
assertEquals(5.0, z.x) // z = x + y = 5
assertEquals(1.0, x.d) // dz/dx = 1
assertEquals(1.0, y.d) // dz/dy = 1
assertEquals(1.0, z.deriv(x)) // dz/dx = 1
assertEquals(1.0, z.deriv(y)) // dz/dy = 1
}
@Test
fun testMinus() {
// two variables
val x = ValueWithDeriv(7)
val y = ValueWithDeriv(3)
val x = Variable(7)
val y = Variable(3)
val z = deriv { x - y }
assertEquals(4.0, z.x) // z = x - y = 4
assertEquals(1.0, x.d) // dz/dx = 1
assertEquals(-1.0, y.d) // dz/dy = -1
assertEquals(1.0, z.deriv(x)) // dz/dx = 1
assertEquals(-1.0, z.deriv(y)) // dz/dy = -1
}
@Test
fun testMulX2() {
val x = ValueWithDeriv(3) // diff w.r.t this x at 3
val x = Variable(3) // diff w.r.t this x at 3
val y = deriv { x * x }
assertEquals(9.0, y.x) // y = x * x = 9
assertEquals(6.0, x.d) // dy/dx = 2 * x = 7
assertEquals(6.0, y.deriv(x)) // dy/dx = 2 * x = 7
}
@Test
fun testSqr() {
val x = ValueWithDeriv(3)
val x = Variable(3)
val y = deriv { sqr(x) }
assertEquals(9.0, y.x) // y = x ^ 2 = 9
assertEquals(6.0, x.d) // dy/dx = 2 * x = 7
assertEquals(6.0, y.deriv(x)) // dy/dx = 2 * x = 7
}
@Test
fun testSqrSqr() {
val x = ValueWithDeriv(2)
val x = Variable(2)
val y = deriv { sqr(sqr(x)) }
assertEquals(16.0, y.x) // y = x ^ 4 = 16
assertEquals(32.0, x.d) // dy/dx = 4 * x^3 = 32
assertEquals(32.0, y.deriv(x)) // dy/dx = 4 * x^3 = 32
}
@Test
fun testX3() {
val x = ValueWithDeriv(2) // diff w.r.t this x at 2
val x = Variable(2) // diff w.r.t this x at 2
val y = deriv { x * x * x }
assertEquals(8.0, y.x) // y = x * x * x = 8
assertEquals(12.0, x.d) // dy/dx = 3 * x * x = 12
assertEquals(12.0, y.deriv(x)) // dy/dx = 3 * x * x = 12
}
@Test
fun testDiv() {
val x = ValueWithDeriv(5)
val y = ValueWithDeriv(2)
val x = Variable(5)
val y = Variable(2)
val z = deriv { x / y }
assertEquals(2.5, z.x) // z = x / y = 2.5
assertEquals(0.5, x.d) // dz/dx = 1 / y = 0.5
assertEquals(-1.25, y.d) // dz/dy = -x / y^2 = -1.25
assertEquals(0.5, z.deriv(x)) // dz/dx = 1 / y = 0.5
assertEquals(-1.25, z.deriv(y)) // dz/dy = -x / y^2 = -1.25
}
@Test
fun testPow3() {
val x = ValueWithDeriv(2) // diff w.r.t this x at 2
val x = Variable(2) // diff w.r.t this x at 2
val y = deriv { pow(x, 3) }
assertEquals(8.0, y.x) // y = x ^ 3 = 8
assertEquals(12.0, x.d) // dy/dx = 3 * x ^ 2 = 12
assertEquals(12.0, y.deriv(x)) // dy/dx = 3 * x ^ 2 = 12
}
@Test
fun testPowFull() {
val x = ValueWithDeriv(2)
val y = ValueWithDeriv(3)
val x = Variable(2)
val y = Variable(3)
val z = deriv { pow(x, y) }
assertApprox(8.0, z.x) // z = x ^ y = 8
assertApprox(12.0, x.d) // dz/dx = y * x ^ (y - 1) = 12
assertApprox(8.0 * kotlin.math.ln(2.0), y.d) // dz/dy = x ^ y * ln(x)
assertApprox(12.0, z.deriv(x)) // dz/dx = y * x ^ (y - 1) = 12
assertApprox(8.0 * kotlin.math.ln(2.0), z.deriv(y)) // dz/dy = x ^ y * ln(x)
}
@Test
fun testFromPaper() {
val x = ValueWithDeriv(3)
val x = Variable(3)
val y = deriv { 2 * x + x * x * x }
assertEquals(33.0, y.x) // y = 2 * x + x * x * x = 33
assertEquals(29.0, x.d) // dy/dx = 2 + 3 * x * x = 29
assertEquals(29.0, y.deriv(x)) // dy/dx = 2 + 3 * x * x = 29
}
@Test
fun testInnerVariable() {
val x = Variable(1)
val y = deriv {
Variable(1) * x
}
assertEquals(1.0, y.x) // y = x ^ n = 1
assertEquals(1.0, y.deriv(x)) // dy/dx = n * x ^ (n - 1) = n - 1
}
@Test
fun testLongChain() {
val n = 10_000
val x = ValueWithDeriv(1)
val x = Variable(1)
val y = deriv {
var pow = ValueWithDeriv(1)
for (i in 1..n) pow *= x
pow
var res = Variable(1)
for (i in 1..n) res *= x
res
}
assertEquals(1.0, y.x) // y = x ^ n = 1
assertEquals(n.toDouble(), x.d) // dy/dx = n * x ^ (n - 1) = n - 1
assertEquals(n.toDouble(), y.deriv(x)) // dy/dx = n * x ^ (n - 1) = n - 1
}
@Test
fun testExample() {
val x = ValueWithDeriv(2)
val x = Variable(2)
val y = deriv { sqr(x) + 5 * x + 3 }
assertEquals(17.0, y.x) // the value of result (y)
assertEquals(9.0, x.d) // dy/dx
assertEquals(9.0, y.deriv(x)) // dy/dx
}
@Test
fun testSqrt() {
val x = ValueWithDeriv(16)
val x = Variable(16)
val y = deriv { sqrt(x) }
assertEquals(4.0, y.x) // y = x ^ 1/2 = 4
assertEquals(1.0 / 8, x.d) // dy/dx = 1/2 / x ^ 1/4 = 1/8
assertEquals(1.0 / 8, y.deriv(x)) // dy/dx = 1/2 / x ^ 1/4 = 1/8
}
@Test
fun testSin() {
val x = ValueWithDeriv(PI / 6)
val x = Variable(PI / 6)
val y = deriv { sin(x) }
assertApprox(0.5, y.x) // y = sin(PI/6) = 0.5
assertApprox(kotlin.math.sqrt(3.0) / 2, x.d) // dy/dx = cos(PI/6) = sqrt(3)/2
assertApprox(kotlin.math.sqrt(3.0) / 2, y.deriv(x)) // dy/dx = cos(PI/6) = sqrt(3)/2
}
@Test
fun testCos() {
val x = ValueWithDeriv(PI / 6)
val x = Variable(PI / 6)
val y = deriv { cos(x) }
assertApprox(kotlin.math.sqrt(3.0) / 2, y.x) // y = cos(PI/6) = sqrt(3)/2
assertApprox(-0.5, x.d) // dy/dx = -sin(PI/6) = -0.5
assertApprox(-0.5, y.deriv(x)) // dy/dx = -sin(PI/6) = -0.5
}
private fun assertApprox(a: Double, b: Double) {