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🍇Python和C++計(jì)算微分正反向累積

算法微分在機(jī)器學(xué)習(xí)領(lǐng)域尤為重要。例如，它允許人們?cè)谏窠?jīng)網(wǎng)絡(luò)中實(shí)現(xiàn)反向傳播，而無(wú)需手動(dòng)計(jì)算導(dǎo)數(shù)。

計(jì)算微分的基礎(chǔ)是復(fù)合函數(shù)偏導(dǎo)數(shù)鏈?zhǔn)椒▌t提供的微分分解。簡(jiǎn)單結(jié)構(gòu)如：
$$\begin{array}{rl}y& =f(g(h(x)))=f\left(g\left(h\left(w_0\right)\right)\right)=f\left(g\left(w_1\right)\right)=f\left(w_2\right)=w_3\\ w_0& =x\\ w_1& =h\left(w_0\right)\\ w_2& =g\left(w_1\right)\\ w_3& =f\left(w_2\right)=y\end{array}$$
由鏈?zhǔn)椒▌t得出：
$$\frac{?y}{?x}=\frac{?y}{?w_2}\frac{?w_2}{?w_1}\frac{?w_1}{?x}=\frac{?f\left(w_2\right)}{?w_2}\frac{?g\left(w_1\right)}{?w_1}\frac{?h\left(w_0\right)}{?x}$$
通常，存在兩種不同的計(jì)算微分模式：正向累積和反向累積。

正向累積指定從內(nèi)到外遍歷鏈?zhǔn)椒▌t（即首先計(jì)算 $?w_1/?x$，然后計(jì)算 $?w_2/?w_1$，最后計(jì)算 $?y/?w_2$ ），而反向累積是從外到內(nèi)的遍歷（首先計(jì)算 $?y/?w_2$，然后計(jì)算 $?w_2/?w_1$，最后計(jì)算 $?w_1/?x$?）。更簡(jiǎn)潔地說(shuō)，

正向累積計(jì)算遞歸關(guān)系：$\frac{?w_i}{?x}=\frac{?w_i}{?w_{i?1}}\frac{?w_{i?1}}{?x}$ 且$w_3=y$

反向累積計(jì)算遞歸關(guān)系：$\frac{?y}{?w_i}=\frac{?y}{?w_{i+1}}\frac{?w_{i+1}}{?w_i}$ 且 $w_0=x$

正向累積在一次傳遞中計(jì)算函數(shù)和導(dǎo)數(shù)（但每個(gè)僅針對(duì)一個(gè)獨(dú)立變量）。相關(guān)方法調(diào)用期望表達(dá)式 Z 相對(duì)于變量 V 導(dǎo)出。該方法返回一對(duì)已求值的函數(shù)及其導(dǎo)數(shù)。該方法遞歸遍歷表達(dá)式樹(shù)，直到到達(dá)變量。如果請(qǐng)求相對(duì)于此變量的導(dǎo)數(shù)，則其導(dǎo)數(shù)為 1，否則為 0。然后求偏函數(shù)以及偏導(dǎo)數(shù)。

偽代碼：

tuple<float,float> evaluateAndDerive(Expression Z, Variable V) {if isVariable(Z)if (Z = V) return {valueOf(Z), 1};else return {valueOf(Z), 0};else if (Z = A + B){a, a'} = evaluateAndDerive(A, V);{b, b'} = evaluateAndDerive(B, V);return {a + b, a' + b'};else if (Z = A - B){a, a'} = evaluateAndDerive(A, V);{b, b'} = evaluateAndDerive(B, V);return {a - b, a' - b'};else if (Z = A * B){a, a'} = evaluateAndDerive(A, V);{b, b'} = evaluateAndDerive(B, V);return {a * b, b * a' + a * b'};
}

Python實(shí)現(xiàn)正向累積：

class ValueAndPartial:def __init__(self, value, partial):self.value = valueself.partial = partialdef toList(self):return [self.value, self.partial]class Expression:def __add__(self, other):return Plus(self, other)def __mul__(self, other):return Multiply(self, other)class Variable(Expression):def __init__(self, value):self.value = valuedef evaluateAndDerive(self, variable):partial = 1 if self == variable else 0return ValueAndPartial(self.value, partial)class Plus(Expression):def __init__(self, expressionA, expressionB):self.expressionA = expressionAself.expressionB = expressionBdef evaluateAndDerive(self, variable):valueA, partialA = self.expressionA.evaluateAndDerive(variable).toList()valueB, partialB = self.expressionB.evaluateAndDerive(variable).toList()return ValueAndPartial(valueA + valueB, partialA + partialB)class Multiply(Expression):def __init__(self, expressionA, expressionB):self.expressionA = expressionAself.expressionB = expressionBdef evaluateAndDerive(self, variable):valueA, partialA = self.expressionA.evaluateAndDerive(variable).toList()valueB, partialB = self.expressionB.evaluateAndDerive(variable).toList()return ValueAndPartial(valueA * valueB, valueB * partialA + valueA * partialB)# Example: Finding the partials of z = x * (x + y) + y * y at (x, y) = (2, 3)
x = Variable(2)
y = Variable(3)
z = x * (x + y) + y * y
xPartial = z.evaluateAndDerive(x).partial
yPartial = z.evaluateAndDerive(y).partial
print("?z/?x =", xPartial)  # Output: ?z/?x = 7
print("?z/?y =", yPartial)  # Output: ?z/?y = 8

C++實(shí)現(xiàn)正向累積：

#include <iostream>
struct ValueAndPartial { float value, partial; };
struct Variable;
struct Expression {virtual ValueAndPartial evaluateAndDerive(Variable *variable) = 0;
};
struct Variable: public Expression {float value;Variable(float value): value(value) {}ValueAndPartial evaluateAndDerive(Variable *variable) {float partial = (this == variable) ? 1.0f : 0.0f;return {value, partial};}
};
struct Plus: public Expression {Expression *a, *b;Plus(Expression *a, Expression *b): a(a), b(b) {}ValueAndPartial evaluateAndDerive(Variable *variable) {auto [valueA, partialA] = a->evaluateAndDerive(variable);auto [valueB, partialB] = b->evaluateAndDerive(variable);return {valueA + valueB, partialA + partialB};}
};
struct Multiply: public Expression {Expression *a, *b;Multiply(Expression *a, Expression *b): a(a), b(b) {}ValueAndPartial evaluateAndDerive(Variable *variable) {auto [valueA, partialA] = a->evaluateAndDerive(variable);auto [valueB, partialB] = b->evaluateAndDerive(variable);return {valueA * valueB, valueB * partialA + valueA * partialB};}
};
int main () {// Example: Finding the partials of z = x * (x + y) + y * y at (x, y) = (2, 3)Variable x(2), y(3);Plus p1(&x, &y); Multiply m1(&x, &p1); Multiply m2(&y, &y); Plus z(&m1, &m2);float xPartial = z.evaluateAndDerive(&x).partial;float yPartial = z.evaluateAndDerive(&y).partial;std::cout << "?z/?x = " << xPartial << ", "<< "?z/?y = " << yPartial << std::endl;// Output: ?z/?x = 7, ?z/?y = 8return 0;
}

反向累積需要兩次傳遞：在正向傳遞中，首先評(píng)估函數(shù)并緩存部分結(jié)果。在反向傳遞中，計(jì)算偏導(dǎo)數(shù)并反向傳播先前導(dǎo)出的值。相應(yīng)的方法調(diào)用期望表達(dá)式 Z 被導(dǎo)出，并以父表達(dá)式的導(dǎo)出值為種子。對(duì)于頂部表達(dá)式 Z 相對(duì)于 Z 導(dǎo)出，這是 1。該方法遞歸遍歷表達(dá)式樹(shù)，直到到達(dá)變量并將當(dāng)前種子值添加到導(dǎo)數(shù)表達(dá)式。

偽代碼：

void derive(Expression Z, float seed) {if isVariable(Z)partialDerivativeOf(Z) += seed;else if (Z = A + B)derive(A, seed);derive(B, seed);else if (Z = A - B)derive(A, seed);derive(B, -seed);else if (Z = A * B)derive(A, valueOf(B) * seed);derive(B, valueOf(A) * seed);
}

Python實(shí)現(xiàn)反向累積：

class Expression:def __add__(self, other):return Plus(self, other)def __mul__(self, other):return Multiply(self, other)class Variable(Expression):def __init__(self, value):self.value = valueself.partial = 0def evaluate(self):passdef derive(self, seed):self.partial += seedclass Plus(Expression):def __init__(self, expressionA, expressionB):self.expressionA = expressionAself.expressionB = expressionBself.value = Nonedef evaluate(self):self.expressionA.evaluate()self.expressionB.evaluate()self.value = self.expressionA.value + self.expressionB.valuedef derive(self, seed):self.expressionA.derive(seed)self.expressionB.derive(seed)class Multiply(Expression):def __init__(self, expressionA, expressionB):self.expressionA = expressionAself.expressionB = expressionBself.value = Nonedef evaluate(self):self.expressionA.evaluate()self.expressionB.evaluate()self.value = self.expressionA.value * self.expressionB.valuedef derive(self, seed):self.expressionA.derive(self.expressionB.value * seed)self.expressionB.derive(self.expressionA.value * seed)# Example: Finding the partials of z = x * (x + y) + y * y at (x, y) = (2, 3)
x = Variable(2)
y = Variable(3)
z = x * (x + y) + y * y
z.evaluate()
print("z =", z.value)        # Output: z = 19
z.derive(1)
print("?z/?x =", x.partial)  # Output: ?z/?x = 7
print("?z/?y =", y.partial)  # Output: ?z/?y = 8

C++實(shí)現(xiàn)反向累積：

#include <iostream>
struct Expression {float value;virtual void evaluate() = 0;virtual void derive(float seed) = 0;
};
struct Variable: public Expression {float partial;Variable(float _value) {value = _value;partial = 0;}void evaluate() {}void derive(float seed) {partial += seed;}
};
struct Plus: public Expression {Expression *a, *b;Plus(Expression *a, Expression *b): a(a), b(b) {}void evaluate() {a->evaluate();b->evaluate();value = a->value + b->value;}void derive(float seed) {a->derive(seed);b->derive(seed);}
};
struct Multiply: public Expression {Expression *a, *b;Multiply(Expression *a, Expression *b): a(a), b(b) {}void evaluate() {a->evaluate();b->evaluate();value = a->value * b->value;}void derive(float seed) {a->derive(b->value * seed);b->derive(a->value * seed);}
};
int main () {// Example: Finding the partials of z = x * (x + y) + y * y at (x, y) = (2, 3)Variable x(2), y(3);Plus p1(&x, &y); Multiply m1(&x, &p1); Multiply m2(&y, &y); Plus z(&m1, &m2);z.evaluate();std::cout << "z = " << z.value << std::endl;// Output: z = 19z.derive(1);std::cout << "?z/?x = " << x.partial << ", "<< "?z/?y = " << y.partial << std::endl;// Output: ?z/?x = 7, ?z/?y = 8return 0;
}

👉參閱一：計(jì)算思維

👉參閱二：亞圖跨際