4
$\begingroup$
# Neural Network Architecture 

no_hid_layers = 1
hid = 3
no_out = 1

# Xavier Ininitialization of weights w

w1 = np.random.randn(hid, n+1)*np.sqrt(2/(hid+n+1))
w2 = np.random.randn(no_out, hid+1)*np.sqrt(2/(no_out+hid+1))

# Sigmoid Activation Function
def g(x):
    sig = 1/(1+np.exp(-x))
    return sig

def frwrd_prop(X, w1, w2):
    z2 = w1 @ X.T
    z2 = norm(z2, axis=0)
    a2 = np.insert(g(z2), 0, 1, axis=0)
    h = g((w2@a2))
    return (h,a2)

# Calculating Cost and Gradient

def Cost(X, y, w1, w2, lmbda=0):
    # Initializing Cost J and Gradients dw
    J = 0
    dw1 = np.zeros(w1.shape)
    dw2 = np.zeros(w2.shape)
    # Forward Propagation to calculate the value of the output
    h, a2 = frwrd_prop(X, w1, w2)
    # Calculate the Cost Function J 
    J = -(np.sum(y.T*np.log(h) + (1-y).T*np.log(1-h)) - lmbda/2*(np.sum(np.sum(w1[:,1:].T@w1[:,1:])) + np.sum(w2[:,1:].T@w2[:,1:])))/m
    # Applying Back Propagation to calculate the Gradients dw
    D3 = h-y
    D2 = (w2.T@D3)*a2*(1-a2)
    dw1[:,0] = (D2[1:]@X)[:,0]/m
    dw2[:,0] = ([email protected])[:,0]/m
    dw1[:, 1:] = ((D2[1:]@X)[:,1:] + lmbda*w1[:,1:])/m
    dw2[:, 1:] = (([email protected])[:,1:] + lmbda*w2[:,1:])/m
    # Gradient clipping
    if(abs(np.linalg.norm(dw1))>4.5):
        dw1 = dw1*4.5/(np.linalg.norm(dw1))
    if(abs(np.linalg.norm(dw2))>4.5):
        dw1 = dw1*4.5/(np.linalg.norm(dw2))
    return (J, dw1, dw2)

# Adam's Optimization technique for training w 

def Train(w1, w2, maxIter=50):
    # Algorithm
    a, b1, b2, e = 0.001, 0.9, 0.999, 10**(-8)
    V1 = np.zeros(w1.shape)
    V2 = np.zeros(w2.shape)
    S1 = np.zeros(w1.shape)
    S2 = np.zeros(w2.shape)
    for i in range(maxIter):
        J, dw1, dw2 = Cost(X, y, w1, w2)
        V1 = b1*V1 + (1-b1)*dw1
        S1 = b2*S1 + (1-b2)*(dw1**2)
        V2 = b1*V2 + (1-b1)*dw2
        S2 = b2*S2 + (1-b2)*(dw2**2)
        if i!=0: 
            V1 = V1/(1-b1**i)
            S1 = S1/(1-b2**i)
            V2 = V2/(1-b1**i)
            S2 = S2/(1-b2**i)
        w1 = w1 - a*V1/(np.sqrt(S1)+e)*dw1
        w2 = w2 - a*V2/(np.sqrt(S2)+e)*dw2
        print("\t\t\tIteration : ", i+1, " \tCost : ", J)
    return (w1, w2)

# Training Neural Network     

w1, w2 = Train(w1,w2)

I'm using Adam's Optimization to converge Gradient Descent to a global minima but the cost is becoming stagnant (not changing) after around 15 iterations(the number is not fixed). The initial cost due to random initialization of weights is changing very minutely before becoming constant. And this is giving training accuracy from 45% to 70% for different runs of the exact same code. Can you help me with the reason behind this?

$\endgroup$
3
  • 2
    $\begingroup$ Welcome to SE.DataScience! Adam and similar optimizers (Nestrov, Nadam, etc.) are all converging to a local minimum, no global optimum is guaranteed. This high variability could be due to (1) too much parameters, (2) too few training samples, (3) bugs in implementation, etc.. As you see, there are many causes for this symptom. You better provide an executable code with all the imports for a fast assessment. $\endgroup$
    – Esmailian
    Commented Apr 7, 2019 at 16:33
  • $\begingroup$ @Esmailian Hello and Thank you. Is there any way to prevent the gradient from falling into local minima? I think Geoffrey Hinton produced a paper on that but I'm not sure which one. And if that's not possible how to resolve the issue? Besides few training examples or more features is an issue when overfitting but low training accuracy seems to be an issue of underfitting and doesn't the training accuracy be more for less number of features because the weights will adjust more accurately if there's less training example? P.S. I'm writing this in python and have only imported Pandas and NumPy. $\endgroup$
    – Arka Patra
    Commented Apr 7, 2019 at 17:52
  • 1
    $\begingroup$ Is there any way to prevent the gradient from falling into local minima? No. One optimizer may perform better, but all fall into local minima. The high instability of accuracy cannot be attributed to over- or under-fitting surely yet. Please place a code that can be executed with no modification. $\endgroup$
    – Esmailian
    Commented Apr 7, 2019 at 17:58

1 Answer 1

2
$\begingroup$

All stochastic gradient descent (SGD) optimizers, including Adam, have randomization built and have no guarantees of reaching a global minima. The randomization is a result of training on a sub-sample of the data at each step. There are no guarantees of reaching a global minima because gradient descent optimizers are first-order, iterative optimization techniques.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.