Keras and the Last Number Problem

Keras and the Last Number Problem#

Let’s see if we can do better than our simple hidden layer NN with the last number problem.

import numpy as np
import keras
from keras.utils import to_categorical

2025-12-04 16:04:30.392591: I external/local_xla/xla/tsl/cuda/cudart_stub.cc:31] Could not find cuda drivers on your machine, GPU will not be used.
2025-12-04 16:04:30.438277: I tensorflow/core/platform/cpu_feature_guard.cc:210] This TensorFlow binary is optimized to use available CPU instructions in performance-critical operations.
To enable the following instructions: AVX2 FMA, in other operations, rebuild TensorFlow with the appropriate compiler flags.

2025-12-04 16:04:31.965408: I external/local_xla/xla/tsl/cuda/cudart_stub.cc:31] Could not find cuda drivers on your machine, GPU will not be used.

We’ll use the same data class

class ModelDataCategorical:
    """this is the model data for our "last number" training set.  We
    produce input of length N, consisting of numbers 0-9 and store
    the result in a 10-element array as categorical data.

    """
    def __init__(self, N=10):
        self.N = N
        
        # our model input data
        self.x = np.random.randint(0, high=10, size=N)
        self.x_scaled = self.x / 10 + 0.05
        
        # our scaled model output data
        self.y = np.array([self.x[-1]])
        self.y_scaled = np.zeros(10) + 0.01
        self.y_scaled[self.x[-1]] = 0.99
        
    def interpret_result(self, out):
        """take the network output and return the number we predict"""
        return np.argmax(out)

For Keras, we need to pack the scaled data (both input and output) into arrays. We’ll use the Keras to_categorical() to make the data categorical.

Let’s make both a training set and a test set

x_train = []
y_train = []
for _ in range(10000):
    m = ModelDataCategorical()
    x_train.append(m.x_scaled)
    y_train.append(m.y)

x_train = np.asarray(x_train)
y_train = to_categorical(y_train, 10)

x_test = []
y_test = []
for _ in range(1000):
    m = ModelDataCategorical()
    x_test.append(m.x_scaled)
    y_test.append(m.y)

x_test = np.asarray(x_test)
y_test = to_categorical(y_test, 10)

Check to make sure the data looks like we expect:

x_train[0]

array([0.75, 0.35, 0.55, 0.55, 0.35, 0.05, 0.95, 0.15, 0.05, 0.85])

y_train[0]

array([0., 0., 0., 0., 0., 0., 0., 0., 1., 0.])

Creating the network#

Now let’s build our network. We’ll use just a single hidden layer, but instead of the sigmoid used before, we’ll use RELU and the softmax activations.

from keras.models import Sequential
from keras.layers import Input, Dense, Dropout, Activation
from keras.optimizers import RMSprop

model = Sequential()
model.add(Input((10,)))
model.add(Dense(100, activation="relu"))
model.add(Dropout(0.1))
model.add(Dense(10, activation="softmax"))

2025-12-04 16:04:33.205343: E external/local_xla/xla/stream_executor/cuda/cuda_platform.cc:51] failed call to cuInit: INTERNAL: CUDA error: Failed call to cuInit: UNKNOWN ERROR (303)

rms = RMSprop()
model.compile(loss='categorical_crossentropy',
              optimizer=rms, metrics=['accuracy'])

model.summary()

Model: "sequential"

┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━┓
┃ Layer (type)                    ┃ Output Shape           ┃       Param # ┃
┡━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━┩
│ dense (Dense)                   │ (None, 100)            │         1,100 │
├─────────────────────────────────┼────────────────────────┼───────────────┤
│ dropout (Dropout)               │ (None, 100)            │             0 │
├─────────────────────────────────┼────────────────────────┼───────────────┤
│ dense_1 (Dense)                 │ (None, 10)             │         1,010 │
└─────────────────────────────────┴────────────────────────┴───────────────┘

 Total params: 2,110 (8.24 KB)

 Trainable params: 2,110 (8.24 KB)

 Non-trainable params: 0 (0.00 B)

Now we have ~ 2k parameters to fit.

Training#

Now we can train and test each epoch to see how we do

epochs = 100
batch_size = 256
model.fit(x_train, y_train, epochs=epochs, batch_size=batch_size,
          validation_data=(x_test, y_test), verbose=2)

Epoch 1/100

40/40 - 1s - 16ms/step - accuracy: 0.1731 - loss: 2.2571 - val_accuracy: 0.2100 - val_loss: 2.1967

Epoch 2/100

40/40 - 0s - 2ms/step - accuracy: 0.2418 - loss: 2.1488 - val_accuracy: 0.2240 - val_loss: 2.0932

Epoch 3/100

40/40 - 0s - 2ms/step - accuracy: 0.2619 - loss: 2.0475 - val_accuracy: 0.2560 - val_loss: 1.9885

Epoch 4/100

40/40 - 0s - 2ms/step - accuracy: 0.2886 - loss: 1.9456 - val_accuracy: 0.2990 - val_loss: 1.8905

Epoch 5/100

40/40 - 0s - 2ms/step - accuracy: 0.3201 - loss: 1.8474 - val_accuracy: 0.3440 - val_loss: 1.7918

Epoch 6/100

40/40 - 0s - 2ms/step - accuracy: 0.3472 - loss: 1.7596 - val_accuracy: 0.3940 - val_loss: 1.7122

Epoch 7/100

40/40 - 0s - 2ms/step - accuracy: 0.3685 - loss: 1.6815 - val_accuracy: 0.3730 - val_loss: 1.6382

Epoch 8/100

40/40 - 0s - 2ms/step - accuracy: 0.3991 - loss: 1.6088 - val_accuracy: 0.4360 - val_loss: 1.5709

Epoch 9/100

40/40 - 0s - 2ms/step - accuracy: 0.4296 - loss: 1.5477 - val_accuracy: 0.4000 - val_loss: 1.5137

Epoch 10/100

40/40 - 0s - 2ms/step - accuracy: 0.4451 - loss: 1.4899 - val_accuracy: 0.5360 - val_loss: 1.4505

Epoch 11/100

40/40 - 0s - 2ms/step - accuracy: 0.4775 - loss: 1.4338 - val_accuracy: 0.4640 - val_loss: 1.3946

Epoch 12/100

40/40 - 0s - 2ms/step - accuracy: 0.4933 - loss: 1.3857 - val_accuracy: 0.4700 - val_loss: 1.3509

Epoch 13/100

40/40 - 0s - 3ms/step - accuracy: 0.5146 - loss: 1.3386 - val_accuracy: 0.5470 - val_loss: 1.3088

Epoch 14/100

40/40 - 0s - 2ms/step - accuracy: 0.5350 - loss: 1.2955 - val_accuracy: 0.6390 - val_loss: 1.2636

Epoch 15/100

40/40 - 0s - 2ms/step - accuracy: 0.5602 - loss: 1.2530 - val_accuracy: 0.6070 - val_loss: 1.2184

Epoch 16/100

40/40 - 0s - 2ms/step - accuracy: 0.5785 - loss: 1.2177 - val_accuracy: 0.6780 - val_loss: 1.1820

Epoch 17/100

40/40 - 0s - 2ms/step - accuracy: 0.5882 - loss: 1.1822 - val_accuracy: 0.6310 - val_loss: 1.1499

Epoch 18/100

40/40 - 0s - 2ms/step - accuracy: 0.6166 - loss: 1.1467 - val_accuracy: 0.6350 - val_loss: 1.1232

Epoch 19/100

40/40 - 0s - 2ms/step - accuracy: 0.6242 - loss: 1.1178 - val_accuracy: 0.6760 - val_loss: 1.0873

Epoch 20/100

40/40 - 0s - 2ms/step - accuracy: 0.6400 - loss: 1.0852 - val_accuracy: 0.7060 - val_loss: 1.0532

Epoch 21/100

40/40 - 0s - 2ms/step - accuracy: 0.6592 - loss: 1.0549 - val_accuracy: 0.6780 - val_loss: 1.0283

Epoch 22/100

40/40 - 0s - 2ms/step - accuracy: 0.6810 - loss: 1.0249 - val_accuracy: 0.7660 - val_loss: 0.9913

Epoch 23/100

40/40 - 0s - 2ms/step - accuracy: 0.7029 - loss: 0.9977 - val_accuracy: 0.7710 - val_loss: 0.9677

Epoch 24/100

40/40 - 0s - 2ms/step - accuracy: 0.7136 - loss: 0.9714 - val_accuracy: 0.8040 - val_loss: 0.9371

Epoch 25/100

40/40 - 0s - 2ms/step - accuracy: 0.7306 - loss: 0.9448 - val_accuracy: 0.7970 - val_loss: 0.9187

Epoch 26/100

40/40 - 0s - 2ms/step - accuracy: 0.7445 - loss: 0.9221 - val_accuracy: 0.8420 - val_loss: 0.8912

Epoch 27/100

40/40 - 0s - 2ms/step - accuracy: 0.7614 - loss: 0.9013 - val_accuracy: 0.8670 - val_loss: 0.8645

Epoch 28/100

40/40 - 0s - 2ms/step - accuracy: 0.7809 - loss: 0.8726 - val_accuracy: 0.8630 - val_loss: 0.8494

Epoch 29/100

40/40 - 0s - 2ms/step - accuracy: 0.7900 - loss: 0.8547 - val_accuracy: 0.9030 - val_loss: 0.8198

Epoch 30/100

40/40 - 0s - 2ms/step - accuracy: 0.8072 - loss: 0.8313 - val_accuracy: 0.8900 - val_loss: 0.7972

Epoch 31/100

40/40 - 0s - 2ms/step - accuracy: 0.8196 - loss: 0.8091 - val_accuracy: 0.9010 - val_loss: 0.7831

Epoch 32/100

40/40 - 0s - 2ms/step - accuracy: 0.8346 - loss: 0.7901 - val_accuracy: 0.9020 - val_loss: 0.7601

Epoch 33/100

40/40 - 0s - 2ms/step - accuracy: 0.8530 - loss: 0.7673 - val_accuracy: 0.9340 - val_loss: 0.7339

Epoch 34/100

40/40 - 0s - 2ms/step - accuracy: 0.8647 - loss: 0.7453 - val_accuracy: 0.8650 - val_loss: 0.7199

Epoch 35/100

40/40 - 0s - 2ms/step - accuracy: 0.8688 - loss: 0.7284 - val_accuracy: 0.9490 - val_loss: 0.6976

Epoch 36/100

40/40 - 0s - 2ms/step - accuracy: 0.8894 - loss: 0.7059 - val_accuracy: 0.9320 - val_loss: 0.6811

Epoch 37/100

40/40 - 0s - 2ms/step - accuracy: 0.8938 - loss: 0.6892 - val_accuracy: 0.9590 - val_loss: 0.6554

Epoch 38/100

40/40 - 0s - 2ms/step - accuracy: 0.9116 - loss: 0.6702 - val_accuracy: 0.9570 - val_loss: 0.6402

Epoch 39/100

40/40 - 0s - 2ms/step - accuracy: 0.9198 - loss: 0.6507 - val_accuracy: 0.9440 - val_loss: 0.6265

Epoch 40/100

40/40 - 0s - 2ms/step - accuracy: 0.9270 - loss: 0.6336 - val_accuracy: 0.9690 - val_loss: 0.6030

Epoch 41/100

40/40 - 0s - 2ms/step - accuracy: 0.9299 - loss: 0.6163 - val_accuracy: 0.9630 - val_loss: 0.5907

Epoch 42/100

40/40 - 0s - 2ms/step - accuracy: 0.9424 - loss: 0.5964 - val_accuracy: 0.9750 - val_loss: 0.5695

Epoch 43/100

40/40 - 0s - 2ms/step - accuracy: 0.9471 - loss: 0.5807 - val_accuracy: 0.9930 - val_loss: 0.5441

Epoch 44/100

40/40 - 0s - 3ms/step - accuracy: 0.9559 - loss: 0.5610 - val_accuracy: 0.9890 - val_loss: 0.5320

Epoch 45/100

40/40 - 0s - 2ms/step - accuracy: 0.9614 - loss: 0.5453 - val_accuracy: 0.9830 - val_loss: 0.5116

Epoch 46/100

40/40 - 0s - 2ms/step - accuracy: 0.9649 - loss: 0.5285 - val_accuracy: 0.9970 - val_loss: 0.4952

Epoch 47/100

40/40 - 0s - 2ms/step - accuracy: 0.9706 - loss: 0.5121 - val_accuracy: 0.9970 - val_loss: 0.4903

Epoch 48/100

40/40 - 0s - 2ms/step - accuracy: 0.9738 - loss: 0.4984 - val_accuracy: 0.9950 - val_loss: 0.4642

Epoch 49/100

40/40 - 0s - 2ms/step - accuracy: 0.9765 - loss: 0.4825 - val_accuracy: 0.9970 - val_loss: 0.4530

Epoch 50/100

40/40 - 0s - 2ms/step - accuracy: 0.9774 - loss: 0.4664 - val_accuracy: 0.9940 - val_loss: 0.4434

Epoch 51/100

40/40 - 0s - 2ms/step - accuracy: 0.9810 - loss: 0.4535 - val_accuracy: 0.9950 - val_loss: 0.4287

Epoch 52/100

40/40 - 0s - 2ms/step - accuracy: 0.9825 - loss: 0.4393 - val_accuracy: 0.9920 - val_loss: 0.4119

Epoch 53/100

40/40 - 0s - 2ms/step - accuracy: 0.9821 - loss: 0.4250 - val_accuracy: 0.9980 - val_loss: 0.3999

Epoch 54/100

40/40 - 0s - 2ms/step - accuracy: 0.9847 - loss: 0.4129 - val_accuracy: 0.9980 - val_loss: 0.3876

Epoch 55/100

40/40 - 0s - 2ms/step - accuracy: 0.9865 - loss: 0.3973 - val_accuracy: 0.9900 - val_loss: 0.3736

Epoch 56/100

40/40 - 0s - 2ms/step - accuracy: 0.9856 - loss: 0.3855 - val_accuracy: 0.9960 - val_loss: 0.3595

Epoch 57/100

40/40 - 0s - 2ms/step - accuracy: 0.9883 - loss: 0.3726 - val_accuracy: 1.0000 - val_loss: 0.3496

Epoch 58/100

40/40 - 0s - 2ms/step - accuracy: 0.9909 - loss: 0.3609 - val_accuracy: 1.0000 - val_loss: 0.3335

Epoch 59/100

40/40 - 0s - 2ms/step - accuracy: 0.9916 - loss: 0.3485 - val_accuracy: 1.0000 - val_loss: 0.3189

Epoch 60/100

40/40 - 0s - 2ms/step - accuracy: 0.9926 - loss: 0.3371 - val_accuracy: 1.0000 - val_loss: 0.3114

Epoch 61/100

40/40 - 0s - 2ms/step - accuracy: 0.9919 - loss: 0.3276 - val_accuracy: 1.0000 - val_loss: 0.3041

Epoch 62/100

40/40 - 0s - 2ms/step - accuracy: 0.9930 - loss: 0.3141 - val_accuracy: 1.0000 - val_loss: 0.2882

Epoch 63/100

40/40 - 0s - 2ms/step - accuracy: 0.9930 - loss: 0.3046 - val_accuracy: 0.9990 - val_loss: 0.2870

Epoch 64/100

40/40 - 0s - 2ms/step - accuracy: 0.9937 - loss: 0.2948 - val_accuracy: 1.0000 - val_loss: 0.2657

Epoch 65/100

40/40 - 0s - 2ms/step - accuracy: 0.9955 - loss: 0.2841 - val_accuracy: 1.0000 - val_loss: 0.2555

Epoch 66/100

40/40 - 0s - 2ms/step - accuracy: 0.9952 - loss: 0.2742 - val_accuracy: 1.0000 - val_loss: 0.2495

Epoch 67/100

40/40 - 0s - 2ms/step - accuracy: 0.9956 - loss: 0.2646 - val_accuracy: 1.0000 - val_loss: 0.2375

Epoch 68/100

40/40 - 0s - 2ms/step - accuracy: 0.9963 - loss: 0.2543 - val_accuracy: 1.0000 - val_loss: 0.2281

Epoch 69/100

40/40 - 0s - 2ms/step - accuracy: 0.9975 - loss: 0.2448 - val_accuracy: 1.0000 - val_loss: 0.2154

Epoch 70/100

40/40 - 0s - 2ms/step - accuracy: 0.9971 - loss: 0.2368 - val_accuracy: 1.0000 - val_loss: 0.2101

Epoch 71/100

40/40 - 0s - 2ms/step - accuracy: 0.9974 - loss: 0.2279 - val_accuracy: 1.0000 - val_loss: 0.2047

Epoch 72/100

40/40 - 0s - 2ms/step - accuracy: 0.9974 - loss: 0.2198 - val_accuracy: 1.0000 - val_loss: 0.1980

Epoch 73/100

40/40 - 0s - 2ms/step - accuracy: 0.9976 - loss: 0.2116 - val_accuracy: 1.0000 - val_loss: 0.1921

Epoch 74/100

40/40 - 0s - 2ms/step - accuracy: 0.9983 - loss: 0.2038 - val_accuracy: 1.0000 - val_loss: 0.1824

Epoch 75/100

40/40 - 0s - 2ms/step - accuracy: 0.9975 - loss: 0.1966 - val_accuracy: 1.0000 - val_loss: 0.1699

Epoch 76/100

40/40 - 0s - 2ms/step - accuracy: 0.9974 - loss: 0.1892 - val_accuracy: 1.0000 - val_loss: 0.1648

Epoch 77/100

40/40 - 0s - 2ms/step - accuracy: 0.9979 - loss: 0.1818 - val_accuracy: 1.0000 - val_loss: 0.1631

Epoch 78/100

40/40 - 0s - 3ms/step - accuracy: 0.9982 - loss: 0.1764 - val_accuracy: 1.0000 - val_loss: 0.1524

Epoch 79/100

40/40 - 0s - 2ms/step - accuracy: 0.9982 - loss: 0.1682 - val_accuracy: 1.0000 - val_loss: 0.1472

Epoch 80/100

40/40 - 0s - 2ms/step - accuracy: 0.9988 - loss: 0.1618 - val_accuracy: 1.0000 - val_loss: 0.1390

Epoch 81/100

40/40 - 0s - 2ms/step - accuracy: 0.9990 - loss: 0.1548 - val_accuracy: 1.0000 - val_loss: 0.1350

Epoch 82/100

40/40 - 0s - 2ms/step - accuracy: 0.9983 - loss: 0.1499 - val_accuracy: 1.0000 - val_loss: 0.1359

Epoch 83/100

40/40 - 0s - 2ms/step - accuracy: 0.9994 - loss: 0.1423 - val_accuracy: 1.0000 - val_loss: 0.1201

Epoch 84/100

40/40 - 0s - 2ms/step - accuracy: 0.9986 - loss: 0.1381 - val_accuracy: 1.0000 - val_loss: 0.1157

Epoch 85/100

40/40 - 0s - 3ms/step - accuracy: 0.9994 - loss: 0.1321 - val_accuracy: 1.0000 - val_loss: 0.1086

Epoch 86/100

40/40 - 0s - 2ms/step - accuracy: 0.9994 - loss: 0.1267 - val_accuracy: 1.0000 - val_loss: 0.1058

Epoch 87/100

40/40 - 0s - 2ms/step - accuracy: 0.9992 - loss: 0.1219 - val_accuracy: 1.0000 - val_loss: 0.1005

Epoch 88/100

40/40 - 0s - 2ms/step - accuracy: 0.9993 - loss: 0.1165 - val_accuracy: 1.0000 - val_loss: 0.0945

Epoch 89/100

40/40 - 0s - 2ms/step - accuracy: 0.9993 - loss: 0.1121 - val_accuracy: 1.0000 - val_loss: 0.0939

Epoch 90/100

40/40 - 0s - 2ms/step - accuracy: 0.9989 - loss: 0.1085 - val_accuracy: 1.0000 - val_loss: 0.0858

Epoch 91/100

40/40 - 0s - 2ms/step - accuracy: 0.9995 - loss: 0.1033 - val_accuracy: 1.0000 - val_loss: 0.0818

Epoch 92/100

40/40 - 0s - 2ms/step - accuracy: 0.9996 - loss: 0.0984 - val_accuracy: 1.0000 - val_loss: 0.0781

Epoch 93/100

40/40 - 0s - 2ms/step - accuracy: 0.9994 - loss: 0.0955 - val_accuracy: 1.0000 - val_loss: 0.0761

Epoch 94/100

40/40 - 0s - 2ms/step - accuracy: 0.9993 - loss: 0.0916 - val_accuracy: 1.0000 - val_loss: 0.0715

Epoch 95/100

40/40 - 0s - 2ms/step - accuracy: 0.9992 - loss: 0.0884 - val_accuracy: 1.0000 - val_loss: 0.0689

Epoch 96/100

40/40 - 0s - 2ms/step - accuracy: 0.9996 - loss: 0.0841 - val_accuracy: 1.0000 - val_loss: 0.0648

Epoch 97/100

40/40 - 0s - 2ms/step - accuracy: 0.9995 - loss: 0.0817 - val_accuracy: 1.0000 - val_loss: 0.0638

Epoch 98/100

40/40 - 0s - 2ms/step - accuracy: 0.9992 - loss: 0.0781 - val_accuracy: 1.0000 - val_loss: 0.0632

Epoch 99/100

40/40 - 0s - 2ms/step - accuracy: 0.9994 - loss: 0.0759 - val_accuracy: 1.0000 - val_loss: 0.0561

Epoch 100/100

40/40 - 0s - 2ms/step - accuracy: 0.9998 - loss: 0.0713 - val_accuracy: 1.0000 - val_loss: 0.0578

<keras.src.callbacks.history.History at 0x7feec66fbad0>

As we see, the network is essentially perfect now.

Keras and the Last Number Problem

Contents

Keras and the Last Number Problem#

Creating the network#

Training#