التمايز المخصص

import Glibc

func sillyExp(_ x: Float) -> Float {
    let 𝑒 = Float(M_E)
    print("Taking 𝑒(\(𝑒)) to the power of \(x)!")
    return pow(𝑒, x)
}

@derivative(of: sillyExp)
func sillyDerivative(_ x: Float) -> (value: Float, pullback: (Float) -> Float) {
    let y = sillyExp(x)
    return (value: y, pullback: { v in v * y })
}

print("exp(3) =", sillyExp(3))
print("𝛁exp(3) =", gradient(of: sillyExp)(3))

Taking 𝑒(2.7182817) to the power of 3.0!
exp(3) = 20.085535
Taking 𝑒(2.7182817) to the power of 3.0!
𝛁exp(3) = 20.085535

let x: Float = 2.0
let y: Float = 3.0
let xyGradient = gradient(at: x, y) { x, y in
    sin(sin(sin(x))) + withoutDerivative(at: cos(cos(cos(y))))
}
print(xyGradient)

(-0.18009877, 0.0)

var x: Float = 30
let xGradient = gradient(at: x) { x -> Float in
    // Print the partial derivative with respect to the result of `sin(x)`.
    let a = sin(x).withDerivative { print("∂+/∂sin = \($0)") } 
    // Force the partial derivative with respect to `x` to be `0.5`.
    let b = log(x.withDerivative { (dx: inout Float) in
        print("∂log/∂x = \(dx), but rewritten to 0.5");
        dx = 0.5
    })
    return a + b
}
print(xGradient)

∂log/∂x = 0.033333335, but rewritten to 0.5
∂+/∂sin = 1.0
0.65425146

import TensorFlow

struct MLP: Layer {
    var layer1 = Dense<Float>(inputSize: 2, outputSize: 10, activation: relu)
    var layer2 = Dense<Float>(inputSize: 10, outputSize: 1, activation: relu)

    @differentiable
    func callAsFunction(_ input: Tensor<Float>) -> Tensor<Float> {
        let h0 = layer1(input).withDerivative { print("∂L/∂layer1 =", $0) }
        return layer2(h0)
    }
}

var classifier = MLP()
let optimizer = SGD(for: classifier, learningRate: 0.02)

let x: Tensor<Float> = [[0, 0], [0, 1], [1, 0], [1, 1]]
let y: Tensor<Float> = [0, 1, 1, 0]

for _ in 0..<10 {
    let 𝛁model = gradient(at: classifier) { classifier -> Tensor<Float> in
        let ŷ = classifier(x).withDerivative { print("∂L/∂ŷ =", $0) }
        let loss = (ŷ - y).squared().mean()
        print("Loss: \(loss)")
        return loss
    }
    optimizer.update(&classifier, along: 𝛁model)
}

Loss: 0.45304087
∂L/∂ŷ = [[     -0.25],
 [     -0.25],
 [-0.2143442],
 [-0.1791575]]
∂L/∂layer1 = [[         0.0,          0.0,          0.0,          0.0,          0.0,          0.0,
           0.0,          0.0,          0.0,          0.0],
 [         0.0,          0.0,          0.0,          0.0,          0.0,          0.0,
           0.0,          0.0,          0.0,          0.0],
 [-0.046330024,  -0.07919147, -0.077494234,  -0.07907715,   0.14447221,  -0.07965051,
     0.0873662, -0.016764779,    0.1293755,  0.027867926],
 [-0.038724493, -0.066191405,   -0.0647728,  -0.06609586,   0.12075568,  -0.06657509,
    0.07302418, -0.014012676,  0.108137235,  0.023293132]]
Loss: 0.43502235
∂L/∂ŷ = [[-0.24459878],
 [-0.24358931],
 [-0.19911093],
 [-0.16190395]]
∂L/∂layer1 = [[-0.053103957,  -0.09203638,   -0.0885385,  -0.09065656,   0.16429774, -0.090893134,
    0.09901551, -0.019131118,   0.14763679,   0.03180147],
 [-0.052884795,  -0.09165655,   -0.0881731,  -0.09028242,   0.16361968,  -0.09051801,
    0.09860687, -0.019052165,   0.14702748,  0.031670224],
 [-0.043228254, -0.074920446, -0.072073065, -0.073797226,   0.13374342,   -0.0739898,
    0.08060167, -0.015573319,   0.12018088,  0.025887374],
 [-0.035150383, -0.060920395, -0.058605086,  -0.06000707,   0.10875137, -0.060163658,
    0.06553999, -0.012663202,   0.09772321,  0.021049915]]
Loss: 0.40576553
∂L/∂ŷ = [[-0.23289952],
 [-0.22639728],
 [-0.17728773],
 [-0.13724682]]
∂L/∂layer1 = [[-0.050774142,  -0.08952092, -0.084402055, -0.086720824,   0.15596299, -0.086545676,
    0.09358021,  -0.01821607,    0.1403872,  0.030280393],
 [-0.049356595,  -0.08702162,  -0.08204567,   -0.0842997,    0.1516087,  -0.08412944,
    0.09096757, -0.017707502,   0.13646778,  0.029435005],
 [ -0.03865028,   -0.0681451,  -0.06424852,  -0.06601361,   0.11872211,  -0.06588028,
   0.071235105, -0.013866433,  0.106865525,  0.023050034],
 [-0.029921012, -0.052754343, -0.049737815,  -0.05110426,    0.0919084, -0.051001046,
   0.055146467, -0.010734662,   0.08272966,  0.017844122]]
Loss: 0.38182113
∂L/∂ŷ = [[ -0.22214013],
 [ -0.21068493],
 [ -0.15761846],
 [-0.115079075]]
∂L/∂layer1 = [[-0.048611242,  -0.08700116,  -0.08059354,  -0.08307868,   0.14837542,  -0.08254748,
    0.08869235, -0.017374532,   0.13374089,  0.028881513],
 [ -0.04610448,  -0.08251473,  -0.07643753, -0.078794524,   0.14072408, -0.078290716,
    0.08411872, -0.016478572,    0.1268442,  0.027392166],
 [ -0.03449187, -0.061731257,  -0.05718476,  -0.05894808,  0.105279066, -0.058571167,
    0.06293123, -0.012328016,    0.0948952,  0.020492738],
 [-0.025182918, -0.045070708, -0.041751258,  -0.04303868,   0.07686547,  -0.04276349,
   0.045946825, -0.009000828,   0.06928409,  0.014961987]]
Loss: 0.36222494
∂L/∂ŷ = [[ -0.2122466],
 [-0.19632757],
 [-0.13990551],
 [-0.09517485]]
∂L/∂layer1 = [[ -0.046605036,   -0.08450727,  -0.077087075,   -0.07970615,    0.14145951,  -0.078871034,
     0.08428629,  -0.016600717,    0.12764633,   0.027595207],
 [ -0.043109544,   -0.07816901,   -0.07130535,   -0.07372799,    0.13084969,  -0.072955504,
    0.077964604, -0.0153556205,    0.11807254,   0.025525497],
 [ -0.030720405,   -0.05570423,  -0.050813094,  -0.052539498,    0.09324514,   -0.05198902,
    0.055558562,   -0.01094261,    0.08413999,   0.018189792],
 [ -0.020898461,  -0.037894443,  -0.034567107,   -0.03574154,    0.06343276,   -0.03536706,
     0.03779535,  -0.007444033,   0.057238705,   0.012374142]]
Loss: 0.34618416
∂L/∂ŷ = [[-0.20314947],
 [ -0.1832107],
 [-0.12396976],
 [-0.07732913]]
∂L/∂layer1 = [[  -0.04474547,  -0.082062505,   -0.07385858,   -0.07658187,    0.13514856,   -0.07549053,
     0.08030583,   -0.01588919,   0.122056164,   0.026412444],
 [  -0.04035378,   -0.07400821,   -0.06660949,    -0.0690655,   0.121883966,   -0.06808127,
     0.07242396,  -0.014329694,    0.11007657,    0.02382011],
 [  -0.02730544,  -0.050077755,    -0.0450714,  -0.046733256,    0.08247295,   -0.04606728,
    0.049005765,  -0.009696207,   0.074483454,   0.016117908],
 [ -0.017032426,  -0.031237207,  -0.028114373,  -0.029150996,    0.05144449,  -0.028735576,
    0.030568527, -0.0060482426,   0.046460852,  0.0100539345]]
Loss: 0.33304712
∂L/∂ŷ = [[ -0.19478384],
 [  -0.1712287],
 [ -0.10964805],
 [-0.061354905]]
∂L/∂layer1 = [[ -0.04302273,  -0.07968434,  -0.07088566,   -0.0736866,   0.12938349, -0.072381854,
   0.076702625, -0.015234879,   0.11692673,  0.025324788],
 [ -0.03782001, -0.070048146, -0.062313486,  -0.06477571,   0.11373719,  -0.06362875,
      0.067427, -0.013392531,   0.10278683,  0.022262271],
 [-0.024218429,  -0.04485604, -0.039903075, -0.041479785,   0.07283277, -0.040745318,
    0.04317757, -0.008576044,    0.0658206,  0.014255873],
 [-0.013551718, -0.025099747, -0.022328254, -0.023210522,  0.040754467,  -0.02279954,
   0.024160538,  -0.00479883,   0.03683072,  0.007977048]]
Loss: 0.32227832
∂L/∂ŷ = [[  -0.187089],
 [-0.16028392],
 [-0.09679102],
 [-0.04708069]]
∂L/∂layer1 = [[ -0.041427277,   -0.07738533,   -0.06814741,  -0.071002685,   0.124111414,   -0.06952245,
     0.07343468, -0.0146330325,    0.11221778,   0.024324344],
 [  -0.03549181,  -0.066297986,   -0.05838363,  -0.060829815,    0.10632942,  -0.059561655,
    0.062913366,  -0.012536493,    0.09613983,   0.020839289],
 [  -0.02143252,   -0.04003552,  -0.035256255,  -0.036733437,   0.064209394,  -0.035967633,
     0.03799164,  -0.007570441,   0.058056183,   0.012584269],
 [ -0.010425118,   -0.01947391,  -0.017149203,  -0.017867727,   0.031232467,  -0.017495228,
    0.018479737, -0.0036823824,   0.028239448,   0.006121188]]
Loss: 0.3134383
∂L/∂ŷ = [[ -0.18000817],
 [ -0.15028599],
 [ -0.08526195],
 [-0.034349076]]
∂L/∂layer1 = [[ -0.039949864,   -0.07517394,  -0.065624304,   -0.06851376,   0.119284846,    -0.0668912,
     0.07046529,  -0.014079211,     0.1078921,   0.023403734],
 [ -0.033353515,   -0.06276154,  -0.054788698,   -0.05720106,    0.09958904,   -0.05584641,
     0.05883036,  -0.011754512,   0.090077415,   0.019539408],
 [ -0.018922493,  -0.035606585,  -0.031083344,  -0.032451954,   0.056499984,   -0.03168342,
    0.033376306, -0.0066687027,   0.051103737,   0.011085318],
 [-0.0076232147,  -0.014344656, -0.0125223985,  -0.013073765,    0.02276188,  -0.012764148,
    0.013446154, -0.0026865886,   0.020587921,  0.0044658897]]
Loss: 0.30616698
∂L/∂ŷ = [[ -0.17348853],
 [ -0.14115131],
 [-0.074935496],
 [-0.023015507]]
∂L/∂layer1 = [[ -0.038581613,   -0.07305531,  -0.063298136,   -0.06620461,    0.11486097,  -0.064468496,
    0.067762226,  -0.013569281,   0.103915446,   0.022556083],
 [ -0.031390235,  -0.059438244,  -0.051499747,  -0.053864464,   0.093451574,  -0.052451957,
    0.055131756,  -0.011040049,    0.08454623,   0.018351763],
 [  -0.01666469,  -0.031555034,  -0.027340584,  -0.028595984,    0.04961229,    -0.0278461,
    0.029268773, -0.0058610262,   0.044884555,   0.009742727],
 [-0.0051183524,  -0.009691737,   -0.00839732,  -0.008782901,   0.015237799,  -0.008552584,
     0.00898954, -0.0018001414,   0.013785734,  0.0029923574]]

/// Given a differentiable function, returns the same differentiable function except when
/// derivatives of this function are being computed. In that case, values in the original function needed
/// for computing the derivatives will be recomputed, instead of being captured by the differential or pullback.
///
/// - Parameter body: The body of the differentiable function.
/// - Returns: The same differentiable function whose derivatives, when computed, will recompute
///   some values from the original function.
func makeRecomputedInGradient<T: Differentiable, U: Differentiable>(
    _ original: @escaping @differentiable (T) -> U
) -> @differentiable (T) -> U {
    return differentiableFunction { x in
        (value: original(x), pullback: { v in pullback(at: x, in: original)(v) })
    }
}

let input: Float = 10.0
print("Running original computation...")

// Differentiable multiplication with checkpointing.
let square = makeRecomputedInGradient { (x: Float) -> Float in
    print("  Computing square...")
    return x * x
}

// Differentiate `f(x) = (cos(x))^2`.
let (output, backprop) = valueWithPullback(at: input) { input -> Float in
    return square(cos(input))
}
print("Running backpropagation...")
let grad = backprop(1)
print("Gradient = \(grad)")

Running original computation...
  Computing square...
Running backpropagation...
  Computing square...
Gradient = -0.9129453

struct Model: Layer {
    var conv = Conv2D<Float>(filterShape: (5, 5, 3, 6))
    var maxPool = MaxPool2D<Float>(poolSize: (2, 2), strides: (2, 2))
    var flatten = Flatten<Float>()
    var dense = Dense<Float>(inputSize: 36 * 6, outputSize: 10)

    @differentiable
    func call(_ input: Tensor<Float>) -> Tensor<Float> {
        return input.sequenced(through: conv, maxPool, flatten, dense)
    }
}

input.sequenced(in: context, through: conv, maxPool, flatten, dense)

// Same as the previous `makeRecomputedInGradient(_:)`, except it's for binary functions.
func makeRecomputedInGradient<T: Differentiable, U: Differentiable, V: Differentiable>(
    _ original: @escaping @differentiable (T, U) -> V
) -> @differentiable (T, U) -> V {
    return differentiableFunction { x, y in
        (value: original(x, y), pullback: { v in pullback(at: x, y, in: original)(v) })
    }
}

import TensorFlow

/// A layer wrapper that makes the underlying layer's activations be discarded during application
/// and recomputed during backpropagation.
struct ActivationDiscarding<Wrapped: Layer>: Layer {
    /// The wrapped layer.
    var wrapped: Wrapped

    @differentiable
    func callAsFunction(_ input: Wrapped.Input) -> Wrapped.Output {
        let apply = makeRecomputedInGradient { (layer: Wrapped, input: Input) -> Wrapped.Output in
            print("    Applying \(Wrapped.self) layer...")
            return layer(input)
        }
        return apply(wrapped, input)
    }
}

extension Layer {
    func discardingActivations() -> ActivationDiscarding<Self> {
        return ActivationDiscarding(wrapped: self)
    }
}

var conv = Conv2D<Float>(filterShape: (5, 5, 3, 6)).discardingActivations()

struct Model: Layer {
    var conv = Conv2D<Float>(filterShape: (5, 5, 3, 6)).discardingActivations()
    var maxPool = MaxPool2D<Float>(poolSize: (2, 2), strides: (2, 2))
    var flatten = Flatten<Float>()
    var dense = Dense<Float>(inputSize: 36 * 6, outputSize: 10)

    @differentiable
    func callAsFunction(_ input: Tensor<Float>) -> Tensor<Float> {
        return input.sequenced(through: conv, maxPool, flatten, dense)
    }
}

// Use random training data.
let x = Tensor<Float>(randomNormal: [10, 16, 16, 3])
let y = Tensor<Int32>(rangeFrom: 0, to: 10, stride: 1)

var model = Model()
let opt = SGD(for: model)

for i in 1...5 {
    print("Starting training step \(i)")
    print("  Running original computation...")
    let (logits, backprop) = model.appliedForBackpropagation(to: x)
    let (loss, dL_dŷ) = valueWithGradient(at: logits) { logits in
        softmaxCrossEntropy(logits: logits, labels: y)
    }
    print("  Loss: \(loss)")
    print("  Running backpropagation...")
    let (dL_dθ, _) = backprop(dL_dŷ)

    opt.update(&model, along: dL_dθ)
}

Starting training step 1
  Running original computation...
    Applying Conv2D<Float> layer...
  Loss: 2.6726463
  Running backpropagation...
    Applying Conv2D<Float> layer...
Starting training step 2
  Running original computation...
    Applying Conv2D<Float> layer...
  Loss: 2.3370266
  Running backpropagation...
    Applying Conv2D<Float> layer...
Starting training step 3
  Running original computation...
    Applying Conv2D<Float> layer...
  Loss: 2.0828948
  Running backpropagation...
    Applying Conv2D<Float> layer...
Starting training step 4
  Running original computation...
    Applying Conv2D<Float> layer...
  Loss: 1.8765408
  Running backpropagation...
    Applying Conv2D<Float> layer...
Starting training step 5
  Running original computation...
    Applying Conv2D<Float> layer...
  Loss: 1.701678
  Running backpropagation...
    Applying Conv2D<Float> layer...