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@atry 2017-11-16T02:04:53.000000Z 字数 8053 阅读 1197

神经网络与函数式编程（六）依赖类型的类型类和多态函数

神经网络与函数式编程

在本系列的前几篇文章中，我们学习了深度学习和函数式编程的对应关系，以及如何用DeepLearning.scala创建函数式风格的神经网络。你可能会好奇，DeepLearning.scala是如何提供这些能力的。在接下来几篇文章中，我将揭示DeepLearning.scala如何实现以下这些功能的内部细节：

  1. 多态函数
  2. 反向传播
  3. 插件

在本篇文章中，我们将首先谈谈多态函数。

动机

DeepLearning.scala中内置了矩阵乘法的函数dot。dot接受两个多维数组INDArray作为参数，返回一个多维数组的计算图，比如可以这样用：

  1. val ndArray1: INDArray = ???
  2. val ndArray2: INDArray = ???
  4. val ndArrayLayer: INDArrayLayer = dot(ndArray1, ndArray2)

如果用这个dot函数实现全连接层的话，两个参数中有一个会是权重INDArrayWeight，比如：

  1. val x: INDArray = ???
  2. val w: INDArrayWeight = ???
  4. val y: INDArrayLayer = dot(x, w)

此外，通常神经网络有多层，除了第一层以外，其他层的输入都是上一层的输出，那么这种情况下，dot的两个参数中还会有一个是其他层输出的计算图INDArrayLayer，比如：

  1. val x1: INDArray = ???
  2. val w1: INDArrayWeight = ???
  3. val x2: INDArrayLayer = dot(x1, w1)
  4. val w2: INDArrayWeight = ???
  5. val y: INDArrayLayer = dot(x2, w2)

结果就是，我们需要定义一个dot函数，能支持以上所有用法，就必须能支持各种不同的参数类型。
理想情况下，两个参数都应该支持INDArray、INDArrayLayer、INDArrayWeight三个类型，排列起来有九种签名：

  1. def dot(operand0: INDArray, operand0: INDArray): INDArrayLayer
  2. def dot(operand0: INDArrayLayer, operand0: INDArray): INDArrayLayer
  3. def dot(operand0: INDArrayWeight, operand0: INDArray): INDArrayLayer
  4. def dot(operand0: INDArray, operand0: INDArrayLayer): INDArrayLayer
  5. def dot(operand0: INDArrayLayer, operand0: INDArrayLayer): INDArrayLayer
  6. def dot(operand0: INDArrayWeight, operand0: INDArrayLayer): INDArrayLayer
  7. def dot(operand0: INDArray, operand0: INDArrayWeight): INDArrayLayer
  8. def dot(operand0: INDArrayLayer, operand0: INDArrayWeight): INDArrayLayer
  9. def dot(operand0: INDArrayWeight, operand0: INDArrayWeight): INDArrayLayer

如果要重载这么多函数的话，就太过冗余了。

DeepLearning类型类

我们的做法是定义一个
DeepLearningAux模式的依赖类型的类型类，其中利用simulacrum生成繁琐的boilerplate代码：

  1. @simulacrum.typeclass
  2. trait DeepLearning[Differentiable] {
  3.   type Data
  4.   type Delta
  5.   def forward(differentiable: Differentiable): Do[Tape[Data, Delta]]
  6. }
  7. object DeepLearning {
  8.   type Aux[Differentiable, Data0, Delta0] = DeepLearning[Differentiable] {
  9.     type Data = Data0
  10.     type Delta = Delta0
  11.   }
  12. }

由于DeepLearning是个依赖类型的类型类，Data与Delta分别表示计算图的值类型与反向传播的导数类型。所以为Differentiable召唤DeepLearning实例时，可以在编译时求出Data与Delta。比如DeepLearning.scala内置插件提供了DeepLearning.Aux[INDArray, INDArray, INDArray]、DeepLearning.Aux[INDArrayLayer, INDArray, INDArray]和DeepLearning.Aux[INDArrayWeight, INDArray, INDArray]：

  1. implicit def indArrayLiteralDeepLearning: DeepLearning.Aux[INDArray, INDArray, INDArray] = ???
  2. implicit def indArrayLayerDeepLearning: DeepLearning.Aux[INDArrayLayer, INDArray, INDArray] = ???
  3. implicit def indArrayWeightDeepLearning: DeepLearning.Aux[INDArrayWeight, INDArray, INDArray] = ???

那么召唤DeepLearning[INDArray]、DeepLearning[INDArrayLayer]或DeepLearning[INDArrayWeight]都可以在编译时把Data和Delta推断为INDArray。

  1. val summonINDArrayDeepLearning = DeepLearning[INDArray]
  2. type INDArrayData = summonINDArrayDeepLearning.Data
  3. type INDArrayDelta = summonINDArrayDeepLearning.Delta
  5. val summonINDArrayLayerDeepLearning = DeepLearning[INDArrayLayer]
  6. type INDArrayLayerData = summonINDArrayLayerDeepLearning.Data
  7. type INDArrayLayerDelta = summonINDArrayLayerDeepLearning.Delta
  9. val summonINDArrayWeightDeepLearning = DeepLearning[INDArrayWeight]
  10. type INDArrayWeightData = summonINDArrayWeightDeepLearning.Data
  11. type INDArrayWeightDelta = summonINDArrayWeightDeepLearning.Delta

比如上面几行代码中，INDArrayData、INDArrayDelta、INDArrayLayerData、INDArrayLayerDelta、INDArrayWeightData、INDArrayWeightDelta都是INDArray。

而假如要召唤DeepLearning[DoubleLayer]，由于存在以下隐式值：

  1. implicit def doubleLayerDeepLearning: DeepLearning.Aux[DoubleLayer, Double, Double] = ???

那么Data和Delta就会是Double：

  1. val summonDoubleLayerDeepLearning = DeepLearning[DoubleLayer]
  2. type DoubleLayerData = summonDoubleLayerDeepLearning.Data
  3. type DoubleLayerDelta = summonDoubleLayerDeepLearning.Delta

利用DeepLearning类型类实现dot

有了DeepLearning类型类之后，我们把dot两个参数实现成泛型类型Operand0、Operand1，然后利用隐式参数DeepLearning.Aux来证明它们是可差分的多维数组。

  1. def dot[Operand0, Operand1](operand0: Operand0, operand0: Operand1)(
  2.   implicit
  3.   deeplearning0: DeepLearning.Aux[Operand0, INDArray, INDArray],
  4.   deeplearning1: DeepLearning.Aux[Operand1, INDArray, INDArray],
  5. ): INDArrayLayer = {
  6.   val do0: Do[Tape[INDArray, INDArray]] = deeplearning0.forward(operand0)
  7.   val do1: Do[Tape[INDArray, INDArray]] = deeplearning1.forward(operand1)
  8.   ???
  9. }

这样一来，deeplearning0和deeplearning1要满足DeepLearning.Aux[Operand0, INDArray, INDArray]类型的话，就只能是DeepLearning.Aux[INDArray, INDArray, INDArray]、DeepLearning.Aux[INDArrayLayer, INDArray, INDArray]或者
DeepLearning.Aux[INDArrayWeight, INDArray, INDArray]，那么也就把Operand0和Operand1限制为INDArray、INDArrayLayer或INDArrayWeight了。

由于所有的DeepLearning实例都实现了forward方法，所以dot内部可以统一把Operand0和Operand1转为Do[Tape[INDArray, INDArray]]。

这样一来，dot就可以在参数中同时支持各种多维数组类型，包括多维数组的计算图和多维数组的权重，然后统一处理了。

多态方法

尽管我们的dot可以支持以上九种签名，但有的时候还是不够。比如max函数既可以支持多维数组之间的逐元素比较，也可以用来让多维数组和标量浮点比较，以便写出max(ndArray, 0.0)实现ReLU激活函数。

理想情况下，max应该支持额外的四倍签名：

  1. def max[Operand0, Operand1](operand0: Operand0, operand0: Operand1)(
  2.   implicit
  3.   deeplearning0: DeepLearning.Aux[Operand0, INDArray, INDArray],
  4.   deeplearning1: DeepLearning.Aux[Operand1, INDArray, INDArray],
  5. ): INDArrayLayer = ???
  6. def max[Operand0, Operand1](operand0: Operand0, operand0: Operand1)(
  7.   implicit
  8.   deeplearning0: DeepLearning.Aux[Operand0, Double, Double],
  9.   deeplearning1: DeepLearning.Aux[Operand1, INDArray, INDArray],
  10. ): INDArrayLayer = ???
  11. def max[Operand0, Operand1](operand0: Operand0, operand0: Operand1)(
  12.   implicit
  13.   deeplearning0: DeepLearning.Aux[Operand0, INDArray, INDArray],
  14.   deeplearning1: DeepLearning.Aux[Operand1, Double, Double],
  15. ): INDArrayLayer = ???
  16. def max[Operand0, Operand1](operand0: Operand0, operand0: Operand1)(
  17.   implicit
  18.   deeplearning0: DeepLearning.Aux[Operand0, Double, Double],
  19.   deeplearning1: DeepLearning.Aux[Operand1, Double, Double],
  20. ): DoubleLayer = ???

不幸的是，Scala编译器不支持这样的重载定义，有个两个原因：

  1. 这四个函数的签名在类型擦除之后都一样，导致生成的Java字节码冲突。
  2. Scala编译器必须在隐式参数搜索之前确定调用哪个重载函数，而上述四个重载函数在搜索到隐式参数以前都无法确定Operand0和Operand1，也就没办法确定选用哪个重载函数了。

我们用Shapeless中Poly来解决重载问题。

我们把max定义为Poly2：

  1. object max extends Poly2

然后提供上述4个max.Case：

  1. implicit def maxDoubleDouble[Operand0, Operand1](
  2.   implicit
  3.   deepLearning0: DeepLearning.Aux[Operand0, Double, Double],
  4.   deepLearning1: DeepLearning.Aux[Operand1, Double, Double]
  5. ) = max.at[Operand0, Operand1] { (operand0, operand1) =>
  6.   ???
  7. }
  8. implicit def maxDoubleINDArray[Operand0, Operand1](
  9.   implicit
  10.   deepLearning0: DeepLearning.Aux[Operand0, Double, Double],
  11.   deepLearning1: DeepLearning.Aux[Operand1, INDArray, INDArray]
  12. ) = max.at[Operand0, Operand1] { (operand0, operand1) =>
  13.   ???
  14. }
  15. implicit def maxINDArrayDouble[Operand0, Operand1](
  16.   implicit
  17.   deepLearning0: DeepLearning.Aux[Operand0, INDArray, INDArray],
  18.   deepLearning1: DeepLearning.Aux[Operand1, Double, Double]
  19. ) = max.at[Operand0, Operand1] { (operand0, operand1) =>
  20.   ???
  21. }
  22. implicit def maxINDArrayINDArray[Operand0, Operand1](
  23.   implicit
  24.   deepLearning0: DeepLearning.Aux[Operand0, INDArray, INDArray],
  25.   deepLearning1: DeepLearning.Aux[Operand1, INDArray, INDArray]
  26. ) = max.at[Operand0, Operand1] { (operand0, operand1) =>
  27.   ???
  28. }

上面每一个Case函数根据Operand0和Operand1是普通值、计算图还是权重，又可以展开成9种Case。

最终就可以在调用max时支持四九三十六种Case，相当于36种签名。

比如：

  1. val operand0: DoubleWeight = ???
  2. val operand1: INDArrayLayer = ???
  4. max(operand0, operand1)

在搜索到隐式参数之后，函数调用等价于：

  1. max(operand0, operand1)(maxDoubleINDArray[DoubleWeight, INDArrayLayer](doubleLayerDeepLearning, indArrayLayerDeepLearning))

多态方法

除了多态函数以外，DeepLearning.scala的内置插件中还提供了一些中缀操作的多态方法，比如四则运算。这些多态方法是通过转发到shapeless.Poly2上实现的：

  1. object + extends Poly2
  2. object - extends Poly2
  3. object * extends Poly2
  4. object / extends Poly2
  6. implicit final class PolymorphicOps[Operand0](operand0: Operand0) {
  8.   def +[Operand1](operand1: Operand1)(
  9.     implicit methodCase: +.Case[Operand0, Operand1]
  10.   ): methodCase.Result = methodCase(operand0, operand1)
  12.   def -[Operand1](operand1: Operand1)(
  13.     implicit methodCase: -.Case[Operand0, Operand1]
  14.   ): methodCase.Result = methodCase(operand0, operand1)
  16.   def *[Operand1](operand1: Operand1)(
  17.     implicit methodCase: *.Case[Operand0, Operand1]
  18.   ): methodCase.Result = methodCase(operand0, operand1)
  20.   def /[Operand1](operand1: Operand1)(
  21.     implicit methodCase: /.Case[Operand0, Operand1]
  22.   ): methodCase.Result = methodCase(operand0, operand1)
  24. }

比如：

  1. implicit doubleDivINDArray[Operand0, Operand1](
  2.   implicit
  3.   deepLearning0: DeepLearning.Aux[Operand0, Double, Double],
  4.   deepLearning1: DeepLearning.Aux[Operand1, INDArray, INDArray]
  5. ) = /.at[Operand0, Operand1] { (operand0, operand1) =>
  6.   ???
  7. }
  9. val operand0: DoubleWeight = ???
  10. val operand1: INDArrayLayer = ???
  12. operand0 / operand1

在搜索到隐式参数之后，函数调用等价于：

  1. PolymorphicOps(operand0)./(operand1)(doubleDivINDArray[DoubleWeight, INDArrayLayer](doubleLayerDeepLearning, indArrayLayerDeepLearning))

结论

通过类型类DeepLearning和shapeless.Poly2，我们支持了多态函数和多态方法。用这种方式实现的多态函数和多态方法具有扩展性，只要增加新的隐式值，就能支持同名函数的新签名。

和其他功能一样，本篇文章中介绍的隐式值也是可以由插件实现。我将在本系列的下一篇文章中揭示DeepLearning.scala插件系统的内部实现细节。届时你将发现，如此强大的插件系统，其核心部分却异常简单。

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