Add backpropagation and first round of learning

diff --git a/README.md b/README.md
index 96974c990119387ee722e8f5d809536d0885233c..760002f714dd303e786e5d55989023d9a15d2ede 100644 (file)
--- a/README.md
+++ b/README.md
@@ -49,3 +49,21 @@ output vector of first image: [    0.         52766.88424917     0.
 classification of first image: 1 with confidence 52766.88424917019; real label 5
 correctly recognized images after initialization: 10.076666666666668%
 ```
+
+ - Add backpropagation algorithm and run a first training round. This is slow, as expected:
+ ```
+ $ time ./train.py
+output vector of first image: [    0.         52766.88424917     0.             0.
+ 14840.28619491 14164.62850135     0.          7011.882333
+     0.         46979.62976127]
+classification of first image: 1 with confidence 52766.88424917019; real label 5
+correctly recognized images after initialization: 10.076666666666668%
+round #0 of learning...
+./train.py:18: RuntimeWarning: overflow encountered in exp
+  return 1 / (1 + np.exp(-x))
+correctly recognized images: 14.211666666666666%
+
+real   0m37.927s
+user   1m19.103s
+sys    1m10.169s
+```

diff --git a/train.py b/train.py
index 366a5a06a2d7b7a9f436228fb4d4b873d0ed5495..fc9bb4b251add6108737281c06e6d92b961d80f4 100755 (executable)
--- a/train.py
+++ b/train.py
@@ -63,6 +63,45 @@ def classify(y):
     return np.argmax(y), np.max(y)
 
 
+def cost_grad(x, target_y, transfer=sigmoid, transfer_prime=sigmoid_prime):
+    '''Return (∂C/∂w, ∂C/∂b) for a particular input and desired output vector'''
+
+    # forward pass, remember all z vectors and activations for every layer
+    z_s, a_s = feed_forward(x, transfer)
+
+    # backward pass
+    deltas = [None] * len(weights)  # delta = dC/dz error for each layer
+    # insert the last layer error
+    deltas[-1] = transfer_prime(z_s[-1]) * 2 * (a_s[-1] - target_y)
+    for i in reversed(range(len(deltas) - 1)):
+        deltas[i] = (weights[i + 1].T @ deltas[i + 1]) * transfer_prime(z_s[i])
+
+    dw = [d @ a_s[i+1] for i, d in enumerate(deltas)]
+    db = deltas
+    return dw, db
+
+
+def label_vector(label):
+    x = np.zeros(10)
+    x[label] = 1.0
+    return x
+
+
+def backpropagate(image_batch, label_batch, eta):
+    '''Update NN with gradient descent and backpropagation to a batch of inputs
+
+    eta is the learning rate.
+    '''
+    global weights, biases
+
+    num_images = image_batch.shape[1]
+    for i in range(num_images):
+        y = label_vector(label_batch[i])
+        dws, dbs = cost_grad(image_batch[:, i], y)
+        weights = [w + eta * dw for w, dw in zip(weights, dws)]
+        biases = [b + eta * db for b, db in zip(biases, dbs)]
+
+
 def test():
     """Count percentage of test inputs which are being recognized correctly"""
 
@@ -80,3 +119,9 @@ print(f'output vector of first image: {res[1][-1]}')
 digit, conf = classify(res[1][-1])
 print(f'classification of first image: {digit} with confidence {conf}; real label {test_labels[0]}')
 print(f'correctly recognized images after initialization: {test()}%')
+
+for i in range(1):
+    print(f"round #{i} of learning...")
+    backpropagate(test_images, test_labels, 1)
+
+print(f'correctly recognized images: {test()}%')