def derivMulPar(x: ndarray, y: ndarray, sigma: Callable) -> ndarray:
assert (x.shape == y.shape)
#in book: deriv of (x+y)/dx is 1
alpha = x * y
dx = ms.deriv(sigma, alpha) * y
dy = ms.deriv(sigma, alpha) * x
return dx, dy
def chainDeriv_3(
chain: Chain, #[f, g, h]
input_: ndarray,
diff: float = 0.001) -> ndarray:
assert len(chain) == 3
# d[f(g(h(x)))] / dx = f'(g(h(x))) * d[g(h(x))] / dx = f'(g(h(x))) * g'(h(x)) * h'(x)
return ms.deriv(chain[0], chain[1](chain[2](input_)), diff) * \
ms.deriv(chain[1], chain[2](input_), diff) * \
ms.deriv(chain[2], input_, diff)
def derivMulMatSigma(X: ndarray, # m x n
W: ndarray, # n x p
sigma: Callable) -> ndarray:
assert(X.shape[1] == W.shape[0])
N = np.dot(X, W)
dSdu = ms.deriv(sigma, N)
return np.dot(dSdu, np.transpose(W))
def chainDeriv_3_with_chainDeriv_2(
chain: Chain, #[f, g, h]
input_: ndarray,
diff: float = 0.001) -> ndarray:
assert len(chain) == 3
f = chain[0]
g_h = lambda input_: chain[1](chain[2](input_))
# d[f(g(h(x)))] / dx = f'(g(h(x))) * d[g(h(x))] / dx
return ms.deriv(f, g_h(input_), diff) * ms.chainDeriv([chain[1], chain[2]],
input_, diff)
def matmul_backward_sum(
X: ndarray, # m x n
W: ndarray, # n x p
sigma: Callable) -> ndarray:
N = np.dot(X, W) # m x p
S = sigma(N) # m x p
# would be the same calc how, but not so efficient: S = ms.matmul_forward(X, W, sigma)
dSdN = ms.deriv(sigma, N) # delta = 0.001
# dLdN = dSdN * dLdS = dSdN
dLdX = np.dot(dSdN, np.transpose(W)) # (m x p) * (p x n) = m x n
dLdW = np.dot(np.transpose(X), dSdN) # (n x m) * (m x p) = n x p
return dLdX, dLdW
assert len(chain) == 3
f = chain[0]
g_h = lambda input_: chain[1](chain[2](input_))
# d[f(g(h(x)))] / dx = f'(g(h(x))) * d[g(h(x))] / dx
return ms.deriv(f, g_h(input_), diff) * ms.chainDeriv([chain[1], chain[2]],
input_, diff)
def g(input_: ndarray) -> ndarray: # x + 3
return input_ + 3
def h(input_: ndarray) -> ndarray: # x^3 - 5
return np.power(input_, 3) - 5
def f_g_h(input_: ndarray) -> ndarray: # f(g(h(x))) = (x^3 - 5 + 3)^2 =
# (x^3 - 2)^2= x^6 - 4x^3 + 4
return np.power(input_, 6) - 4 * np.power(input_, 3) + 4
x = np.array([1, 2, 3, 4])
print("x values", x, ", diff is 0.001")
print("f_g_h(x)", f_g_h(x))
print("f(g(h(x)))", ms.parabel(g(h(x))), '\n')
print("deriv of f_g_h at x", ms.deriv(f_g_h, x))
print("chainDeriv_3 of [f, g, h] at x", chainDeriv_3([ms.parabel, g, h], x))
print("chainDeriv_3 with chainDeriv_2 of [f, g, h] at x",
chainDeriv_3_with_chainDeriv_2([ms.parabel, g, h], x))