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

2025-06-10 23:59:22.845447: I external/local_xla/xla/tsl/cuda/cudart_stub.cc:32] Could not find cuda drivers on your machine, GPU will not be used.
2025-06-10 23:59:22.848710: I external/local_xla/xla/tsl/cuda/cudart_stub.cc:32] Could not find cuda drivers on your machine, GPU will not be used.
2025-06-10 23:59:22.857423: E external/local_xla/xla/stream_executor/cuda/cuda_fft.cc:467] Unable to register cuFFT factory: Attempting to register factory for plugin cuFFT when one has already been registered
WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
E0000 00:00:1749599962.871425    7415 cuda_dnn.cc:8579] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered
E0000 00:00:1749599962.875582    7415 cuda_blas.cc:1407] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered
W0000 00:00:1749599962.887281    7415 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1749599962.887293    7415 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1749599962.887295    7415 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1749599962.887297    7415 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
2025-06-10 23:59:22.891554: 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.

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([0.35, 0.65, 0.15, 0.75, 0.25, 0.15, 0.35, 0.55, 0.45, 0.55])

y_train[0]

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

2025-06-10 23:59:25.626997: 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: 0.1318 - loss: 2.2748 - val_accuracy: 0.2060 - val_loss: 2.2240

Epoch 2/100

40/40 - 0s - 2ms/step - accuracy: 0.2108 - loss: 2.1842 - val_accuracy: 0.2210 - val_loss: 2.1332

Epoch 3/100

40/40 - 0s - 3ms/step - accuracy: 0.2505 - loss: 2.0883 - val_accuracy: 0.2530 - val_loss: 2.0324

Epoch 4/100

40/40 - 0s - 2ms/step - accuracy: 0.2744 - loss: 1.9895 - val_accuracy: 0.3240 - val_loss: 1.9348

Epoch 5/100

40/40 - 0s - 2ms/step - accuracy: 0.3015 - loss: 1.8975 - val_accuracy: 0.3600 - val_loss: 1.8366

Epoch 6/100

40/40 - 0s - 2ms/step - accuracy: 0.3295 - loss: 1.8084 - val_accuracy: 0.3350 - val_loss: 1.7551

Epoch 7/100

40/40 - 0s - 2ms/step - accuracy: 0.3577 - loss: 1.7289 - val_accuracy: 0.3820 - val_loss: 1.6788

Epoch 8/100

40/40 - 0s - 2ms/step - accuracy: 0.3936 - loss: 1.6558 - val_accuracy: 0.4170 - val_loss: 1.6107

Epoch 9/100

40/40 - 0s - 2ms/step - accuracy: 0.4153 - loss: 1.5938 - val_accuracy: 0.4480 - val_loss: 1.5469

Epoch 10/100

40/40 - 0s - 2ms/step - accuracy: 0.4368 - loss: 1.5336 - val_accuracy: 0.5160 - val_loss: 1.4873

Epoch 11/100

40/40 - 0s - 2ms/step - accuracy: 0.4683 - loss: 1.4771 - val_accuracy: 0.5850 - val_loss: 1.4317

Epoch 12/100

40/40 - 0s - 2ms/step - accuracy: 0.4996 - loss: 1.4233 - val_accuracy: 0.5020 - val_loss: 1.3799

Epoch 13/100

40/40 - 0s - 3ms/step - accuracy: 0.5069 - loss: 1.3785 - val_accuracy: 0.5940 - val_loss: 1.3303

Epoch 14/100

40/40 - 0s - 2ms/step - accuracy: 0.5262 - loss: 1.3341 - val_accuracy: 0.6030 - val_loss: 1.2867

Epoch 15/100

40/40 - 0s - 2ms/step - accuracy: 0.5506 - loss: 1.2927 - val_accuracy: 0.6370 - val_loss: 1.2436

Epoch 16/100

40/40 - 0s - 2ms/step - accuracy: 0.5667 - loss: 1.2530 - val_accuracy: 0.6240 - val_loss: 1.2044

Epoch 17/100

40/40 - 0s - 2ms/step - accuracy: 0.5835 - loss: 1.2136 - val_accuracy: 0.7290 - val_loss: 1.1693

Epoch 18/100

40/40 - 0s - 2ms/step - accuracy: 0.6050 - loss: 1.1802 - val_accuracy: 0.7710 - val_loss: 1.1308

Epoch 19/100

40/40 - 0s - 2ms/step - accuracy: 0.6230 - loss: 1.1472 - val_accuracy: 0.6580 - val_loss: 1.1059

Epoch 20/100

40/40 - 0s - 2ms/step - accuracy: 0.6388 - loss: 1.1166 - val_accuracy: 0.7410 - val_loss: 1.0686

Epoch 21/100

40/40 - 0s - 2ms/step - accuracy: 0.6506 - loss: 1.0890 - val_accuracy: 0.7420 - val_loss: 1.0418

Epoch 22/100

40/40 - 0s - 2ms/step - accuracy: 0.6775 - loss: 1.0570 - val_accuracy: 0.7410 - val_loss: 1.0163

Epoch 23/100

40/40 - 0s - 3ms/step - accuracy: 0.6927 - loss: 1.0275 - val_accuracy: 0.7840 - val_loss: 0.9866

Epoch 24/100

40/40 - 0s - 2ms/step - accuracy: 0.7145 - loss: 0.9994 - val_accuracy: 0.7940 - val_loss: 0.9520

Epoch 25/100

40/40 - 0s - 2ms/step - accuracy: 0.7206 - loss: 0.9763 - val_accuracy: 0.8650 - val_loss: 0.9320

Epoch 26/100

40/40 - 0s - 2ms/step - accuracy: 0.7435 - loss: 0.9485 - val_accuracy: 0.8610 - val_loss: 0.9079

Epoch 27/100

40/40 - 0s - 2ms/step - accuracy: 0.7556 - loss: 0.9250 - val_accuracy: 0.8420 - val_loss: 0.8845

Epoch 28/100

40/40 - 0s - 2ms/step - accuracy: 0.7731 - loss: 0.9016 - val_accuracy: 0.8540 - val_loss: 0.8663

Epoch 29/100

40/40 - 0s - 2ms/step - accuracy: 0.7777 - loss: 0.8802 - val_accuracy: 0.8780 - val_loss: 0.8368

Epoch 30/100

40/40 - 0s - 2ms/step - accuracy: 0.8048 - loss: 0.8532 - val_accuracy: 0.9220 - val_loss: 0.8167

Epoch 31/100

40/40 - 0s - 2ms/step - accuracy: 0.8098 - loss: 0.8357 - val_accuracy: 0.9180 - val_loss: 0.7945

Epoch 32/100

40/40 - 0s - 2ms/step - accuracy: 0.8249 - loss: 0.8137 - val_accuracy: 0.9220 - val_loss: 0.7687

Epoch 33/100

40/40 - 0s - 3ms/step - accuracy: 0.8430 - loss: 0.7910 - val_accuracy: 0.9090 - val_loss: 0.7479

Epoch 34/100

40/40 - 0s - 3ms/step - accuracy: 0.8534 - loss: 0.7721 - val_accuracy: 0.9240 - val_loss: 0.7390

Epoch 35/100

40/40 - 0s - 2ms/step - accuracy: 0.8703 - loss: 0.7520 - val_accuracy: 0.9490 - val_loss: 0.7089

Epoch 36/100

40/40 - 0s - 2ms/step - accuracy: 0.8830 - loss: 0.7302 - val_accuracy: 0.9680 - val_loss: 0.6910

Epoch 37/100

40/40 - 0s - 2ms/step - accuracy: 0.8902 - loss: 0.7117 - val_accuracy: 0.9420 - val_loss: 0.6794

Epoch 38/100

40/40 - 0s - 2ms/step - accuracy: 0.9063 - loss: 0.6910 - val_accuracy: 0.9750 - val_loss: 0.6546

Epoch 39/100

40/40 - 0s - 2ms/step - accuracy: 0.9132 - loss: 0.6741 - val_accuracy: 0.9520 - val_loss: 0.6401

Epoch 40/100

40/40 - 0s - 2ms/step - accuracy: 0.9223 - loss: 0.6536 - val_accuracy: 0.9840 - val_loss: 0.6217

Epoch 41/100

40/40 - 0s - 2ms/step - accuracy: 0.9308 - loss: 0.6357 - val_accuracy: 0.9880 - val_loss: 0.6020

Epoch 42/100

40/40 - 0s - 2ms/step - accuracy: 0.9342 - loss: 0.6182 - val_accuracy: 0.9600 - val_loss: 0.5829

Epoch 43/100

40/40 - 0s - 2ms/step - accuracy: 0.9442 - loss: 0.6012 - val_accuracy: 0.9950 - val_loss: 0.5608

Epoch 44/100

40/40 - 0s - 3ms/step - accuracy: 0.9528 - loss: 0.5834 - val_accuracy: 0.9900 - val_loss: 0.5533

Epoch 45/100

40/40 - 0s - 2ms/step - accuracy: 0.9576 - loss: 0.5659 - val_accuracy: 0.9880 - val_loss: 0.5334

Epoch 46/100

40/40 - 0s - 2ms/step - accuracy: 0.9634 - loss: 0.5511 - val_accuracy: 0.9910 - val_loss: 0.5177

Epoch 47/100

40/40 - 0s - 2ms/step - accuracy: 0.9653 - loss: 0.5336 - val_accuracy: 0.9990 - val_loss: 0.4928

Epoch 48/100

40/40 - 0s - 2ms/step - accuracy: 0.9649 - loss: 0.5198 - val_accuracy: 0.9990 - val_loss: 0.4903

Epoch 49/100

40/40 - 0s - 2ms/step - accuracy: 0.9728 - loss: 0.5026 - val_accuracy: 0.9990 - val_loss: 0.4670

Epoch 50/100

40/40 - 0s - 2ms/step - accuracy: 0.9763 - loss: 0.4885 - val_accuracy: 0.9940 - val_loss: 0.4550

Epoch 51/100

40/40 - 0s - 2ms/step - accuracy: 0.9771 - loss: 0.4756 - val_accuracy: 0.9990 - val_loss: 0.4384

Epoch 52/100

40/40 - 0s - 2ms/step - accuracy: 0.9775 - loss: 0.4600 - val_accuracy: 1.0000 - val_loss: 0.4332

Epoch 53/100

40/40 - 0s - 2ms/step - accuracy: 0.9810 - loss: 0.4460 - val_accuracy: 1.0000 - val_loss: 0.4178

Epoch 54/100

40/40 - 0s - 3ms/step - accuracy: 0.9822 - loss: 0.4311 - val_accuracy: 0.9820 - val_loss: 0.4261

Epoch 55/100

40/40 - 0s - 2ms/step - accuracy: 0.9845 - loss: 0.4201 - val_accuracy: 0.9990 - val_loss: 0.3920

Epoch 56/100

40/40 - 0s - 2ms/step - accuracy: 0.9849 - loss: 0.4061 - val_accuracy: 0.9970 - val_loss: 0.3850

Epoch 57/100

40/40 - 0s - 2ms/step - accuracy: 0.9863 - loss: 0.3941 - val_accuracy: 0.9980 - val_loss: 0.3764

Epoch 58/100

40/40 - 0s - 2ms/step - accuracy: 0.9860 - loss: 0.3841 - val_accuracy: 1.0000 - val_loss: 0.3511

Epoch 59/100

40/40 - 0s - 2ms/step - accuracy: 0.9877 - loss: 0.3701 - val_accuracy: 0.9990 - val_loss: 0.3381

Epoch 60/100

40/40 - 0s - 2ms/step - accuracy: 0.9921 - loss: 0.3575 - val_accuracy: 1.0000 - val_loss: 0.3288

Epoch 61/100

40/40 - 0s - 2ms/step - accuracy: 0.9903 - loss: 0.3476 - val_accuracy: 1.0000 - val_loss: 0.3160

Epoch 62/100

40/40 - 0s - 2ms/step - accuracy: 0.9919 - loss: 0.3352 - val_accuracy: 1.0000 - val_loss: 0.3093

Epoch 63/100

40/40 - 0s - 2ms/step - accuracy: 0.9918 - loss: 0.3245 - val_accuracy: 1.0000 - val_loss: 0.2982

Epoch 64/100

40/40 - 0s - 2ms/step - accuracy: 0.9939 - loss: 0.3129 - val_accuracy: 1.0000 - val_loss: 0.2832

Epoch 65/100

40/40 - 0s - 2ms/step - accuracy: 0.9926 - loss: 0.3044 - val_accuracy: 1.0000 - val_loss: 0.2727

Epoch 66/100

40/40 - 0s - 2ms/step - accuracy: 0.9928 - loss: 0.2936 - val_accuracy: 1.0000 - val_loss: 0.2625

Epoch 67/100

40/40 - 0s - 2ms/step - accuracy: 0.9943 - loss: 0.2826 - val_accuracy: 1.0000 - val_loss: 0.2605

Epoch 68/100

40/40 - 0s - 2ms/step - accuracy: 0.9948 - loss: 0.2743 - val_accuracy: 0.9980 - val_loss: 0.2524

Epoch 69/100

40/40 - 0s - 2ms/step - accuracy: 0.9953 - loss: 0.2646 - val_accuracy: 1.0000 - val_loss: 0.2401

Epoch 70/100

40/40 - 0s - 2ms/step - accuracy: 0.9962 - loss: 0.2541 - val_accuracy: 1.0000 - val_loss: 0.2296

Epoch 71/100

40/40 - 0s - 2ms/step - accuracy: 0.9952 - loss: 0.2474 - val_accuracy: 1.0000 - val_loss: 0.2149

Epoch 72/100

40/40 - 0s - 2ms/step - accuracy: 0.9959 - loss: 0.2383 - val_accuracy: 1.0000 - val_loss: 0.2141

Epoch 73/100

40/40 - 0s - 2ms/step - accuracy: 0.9973 - loss: 0.2294 - val_accuracy: 1.0000 - val_loss: 0.2108

Epoch 74/100

40/40 - 0s - 3ms/step - accuracy: 0.9970 - loss: 0.2221 - val_accuracy: 1.0000 - val_loss: 0.1960

Epoch 75/100

40/40 - 0s - 2ms/step - accuracy: 0.9962 - loss: 0.2142 - val_accuracy: 1.0000 - val_loss: 0.1859

Epoch 76/100

40/40 - 0s - 2ms/step - accuracy: 0.9965 - loss: 0.2067 - val_accuracy: 1.0000 - val_loss: 0.1889

Epoch 77/100

40/40 - 0s - 2ms/step - accuracy: 0.9974 - loss: 0.1997 - val_accuracy: 1.0000 - val_loss: 0.1795

Epoch 78/100

40/40 - 0s - 2ms/step - accuracy: 0.9968 - loss: 0.1925 - val_accuracy: 1.0000 - val_loss: 0.1688

Epoch 79/100

40/40 - 0s - 2ms/step - accuracy: 0.9984 - loss: 0.1846 - val_accuracy: 1.0000 - val_loss: 0.1583

Epoch 80/100

40/40 - 0s - 2ms/step - accuracy: 0.9981 - loss: 0.1787 - val_accuracy: 1.0000 - val_loss: 0.1608

Epoch 81/100

40/40 - 0s - 2ms/step - accuracy: 0.9970 - loss: 0.1728 - val_accuracy: 1.0000 - val_loss: 0.1455

Epoch 82/100

40/40 - 0s - 2ms/step - accuracy: 0.9986 - loss: 0.1650 - val_accuracy: 1.0000 - val_loss: 0.1498

Epoch 83/100

40/40 - 0s - 2ms/step - accuracy: 0.9976 - loss: 0.1601 - val_accuracy: 1.0000 - val_loss: 0.1328

Epoch 84/100

40/40 - 0s - 2ms/step - accuracy: 0.9983 - loss: 0.1532 - val_accuracy: 1.0000 - val_loss: 0.1297

Epoch 85/100

40/40 - 0s - 2ms/step - accuracy: 0.9978 - loss: 0.1486 - val_accuracy: 1.0000 - val_loss: 0.1247

Epoch 86/100

40/40 - 0s - 2ms/step - accuracy: 0.9985 - loss: 0.1428 - val_accuracy: 1.0000 - val_loss: 0.1182

Epoch 87/100

40/40 - 0s - 2ms/step - accuracy: 0.9986 - loss: 0.1369 - val_accuracy: 1.0000 - val_loss: 0.1131

Epoch 88/100

40/40 - 0s - 2ms/step - accuracy: 0.9983 - loss: 0.1323 - val_accuracy: 1.0000 - val_loss: 0.1098

Epoch 89/100

40/40 - 0s - 2ms/step - accuracy: 0.9985 - loss: 0.1276 - val_accuracy: 1.0000 - val_loss: 0.1061

Epoch 90/100

40/40 - 0s - 2ms/step - accuracy: 0.9993 - loss: 0.1225 - val_accuracy: 1.0000 - val_loss: 0.0977

Epoch 91/100

40/40 - 0s - 2ms/step - accuracy: 0.9987 - loss: 0.1179 - val_accuracy: 1.0000 - val_loss: 0.0924

Epoch 92/100

40/40 - 0s - 2ms/step - accuracy: 0.9988 - loss: 0.1141 - val_accuracy: 1.0000 - val_loss: 0.0886

Epoch 93/100

40/40 - 0s - 2ms/step - accuracy: 0.9988 - loss: 0.1098 - val_accuracy: 1.0000 - val_loss: 0.0881

Epoch 94/100

40/40 - 0s - 3ms/step - accuracy: 0.9991 - loss: 0.1059 - val_accuracy: 1.0000 - val_loss: 0.0836

Epoch 95/100

40/40 - 0s - 2ms/step - accuracy: 0.9988 - loss: 0.1025 - val_accuracy: 1.0000 - val_loss: 0.0793

Epoch 96/100

40/40 - 0s - 2ms/step - accuracy: 0.9994 - loss: 0.0978 - val_accuracy: 1.0000 - val_loss: 0.0762

Epoch 97/100

40/40 - 0s - 2ms/step - accuracy: 0.9991 - loss: 0.0937 - val_accuracy: 1.0000 - val_loss: 0.0747

Epoch 98/100

40/40 - 0s - 2ms/step - accuracy: 0.9994 - loss: 0.0914 - val_accuracy: 1.0000 - val_loss: 0.0713

Epoch 99/100

40/40 - 0s - 2ms/step - accuracy: 0.9991 - loss: 0.0873 - val_accuracy: 1.0000 - val_loss: 0.0662

Epoch 100/100

40/40 - 0s - 3ms/step - accuracy: 0.9996 - loss: 0.0838 - val_accuracy: 1.0000 - val_loss: 0.0623

<keras.src.callbacks.history.History at 0x7f2a089ba190>

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#