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

2025-05-13 13:31:12.128631: I external/local_xla/xla/tsl/cuda/cudart_stub.cc:32] Could not find cuda drivers on your machine, GPU will not be used.
2025-05-13 13:31:12.131814: I external/local_xla/xla/tsl/cuda/cudart_stub.cc:32] Could not find cuda drivers on your machine, GPU will not be used.
2025-05-13 13:31:12.140304: 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 00:00:1747143072.154290    7387 cuda_dnn.cc:8579] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered
E0000 00:00:1747143072.158546    7387 cuda_blas.cc:1407] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered
W0000 00:00:1747143072.169986    7387 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1747143072.169997    7387 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1747143072.169999    7387 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1747143072.170001    7387 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
2025-05-13 13:31:12.174087: 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([0.95, 0.05, 0.35, 0.45, 0.35, 0.95, 0.65, 0.85, 0.35, 0.15])

y_train[0]

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

2025-05-13 13:31:14.759455: 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: 0.1452 - loss: 2.2612 - val_accuracy: 0.2200 - val_loss: 2.2045

Epoch 2/100

40/40 - 0s - 3ms/step - accuracy: 0.2134 - loss: 2.1592 - val_accuracy: 0.2500 - val_loss: 2.1104

Epoch 3/100

40/40 - 0s - 3ms/step - accuracy: 0.2621 - loss: 2.0611 - val_accuracy: 0.2760 - val_loss: 2.0036

Epoch 4/100

40/40 - 0s - 3ms/step - accuracy: 0.2881 - loss: 1.9643 - val_accuracy: 0.3200 - val_loss: 1.9099

Epoch 5/100

40/40 - 0s - 3ms/step - accuracy: 0.3182 - loss: 1.8732 - val_accuracy: 0.3250 - val_loss: 1.8200

Epoch 6/100

40/40 - 0s - 3ms/step - accuracy: 0.3448 - loss: 1.7877 - val_accuracy: 0.3330 - val_loss: 1.7377

Epoch 7/100

40/40 - 0s - 3ms/step - accuracy: 0.3738 - loss: 1.7112 - val_accuracy: 0.4100 - val_loss: 1.6659

Epoch 8/100

40/40 - 0s - 2ms/step - accuracy: 0.4096 - loss: 1.6372 - val_accuracy: 0.4300 - val_loss: 1.5951

Epoch 9/100

40/40 - 0s - 2ms/step - accuracy: 0.4215 - loss: 1.5761 - val_accuracy: 0.4790 - val_loss: 1.5325

Epoch 10/100

40/40 - 0s - 2ms/step - accuracy: 0.4547 - loss: 1.5137 - val_accuracy: 0.4500 - val_loss: 1.4779

Epoch 11/100

40/40 - 0s - 2ms/step - accuracy: 0.4714 - loss: 1.4626 - val_accuracy: 0.4720 - val_loss: 1.4305

Epoch 12/100

40/40 - 0s - 2ms/step - accuracy: 0.5063 - loss: 1.4073 - val_accuracy: 0.5490 - val_loss: 1.3743

Epoch 13/100

40/40 - 0s - 2ms/step - accuracy: 0.5292 - loss: 1.3641 - val_accuracy: 0.6310 - val_loss: 1.3264

Epoch 14/100

40/40 - 0s - 2ms/step - accuracy: 0.5520 - loss: 1.3172 - val_accuracy: 0.6530 - val_loss: 1.2794

Epoch 15/100

40/40 - 0s - 2ms/step - accuracy: 0.5774 - loss: 1.2747 - val_accuracy: 0.7000 - val_loss: 1.2467

Epoch 16/100

40/40 - 0s - 2ms/step - accuracy: 0.6002 - loss: 1.2380 - val_accuracy: 0.6200 - val_loss: 1.2057

Epoch 17/100

40/40 - 0s - 2ms/step - accuracy: 0.6111 - loss: 1.2004 - val_accuracy: 0.7140 - val_loss: 1.1654

Epoch 18/100

40/40 - 0s - 2ms/step - accuracy: 0.6391 - loss: 1.1667 - val_accuracy: 0.7340 - val_loss: 1.1350

Epoch 19/100

40/40 - 0s - 2ms/step - accuracy: 0.6576 - loss: 1.1326 - val_accuracy: 0.6840 - val_loss: 1.1117

Epoch 20/100

40/40 - 0s - 2ms/step - accuracy: 0.6653 - loss: 1.1052 - val_accuracy: 0.6250 - val_loss: 1.0750

Epoch 21/100

40/40 - 0s - 2ms/step - accuracy: 0.6772 - loss: 1.0771 - val_accuracy: 0.7740 - val_loss: 1.0406

Epoch 22/100

40/40 - 0s - 2ms/step - accuracy: 0.7009 - loss: 1.0457 - val_accuracy: 0.7760 - val_loss: 1.0145

Epoch 23/100

40/40 - 0s - 2ms/step - accuracy: 0.7148 - loss: 1.0186 - val_accuracy: 0.7180 - val_loss: 0.9933

Epoch 24/100

40/40 - 0s - 2ms/step - accuracy: 0.7188 - loss: 0.9930 - val_accuracy: 0.8550 - val_loss: 0.9640

Epoch 25/100

40/40 - 0s - 2ms/step - accuracy: 0.7397 - loss: 0.9695 - val_accuracy: 0.7810 - val_loss: 0.9491

Epoch 26/100

40/40 - 0s - 2ms/step - accuracy: 0.7536 - loss: 0.9444 - val_accuracy: 0.8700 - val_loss: 0.9138

Epoch 27/100

40/40 - 0s - 2ms/step - accuracy: 0.7682 - loss: 0.9223 - val_accuracy: 0.8170 - val_loss: 0.8914

Epoch 28/100

40/40 - 0s - 2ms/step - accuracy: 0.7752 - loss: 0.9008 - val_accuracy: 0.8690 - val_loss: 0.8679

Epoch 29/100

40/40 - 0s - 2ms/step - accuracy: 0.7924 - loss: 0.8785 - val_accuracy: 0.8820 - val_loss: 0.8460

Epoch 30/100

40/40 - 0s - 2ms/step - accuracy: 0.8079 - loss: 0.8538 - val_accuracy: 0.8460 - val_loss: 0.8230

Epoch 31/100

40/40 - 0s - 2ms/step - accuracy: 0.8088 - loss: 0.8344 - val_accuracy: 0.9020 - val_loss: 0.8008

Epoch 32/100

40/40 - 0s - 2ms/step - accuracy: 0.8224 - loss: 0.8121 - val_accuracy: 0.8640 - val_loss: 0.7856

Epoch 33/100

40/40 - 0s - 2ms/step - accuracy: 0.8376 - loss: 0.7907 - val_accuracy: 0.9010 - val_loss: 0.7588

Epoch 34/100

40/40 - 0s - 2ms/step - accuracy: 0.8442 - loss: 0.7732 - val_accuracy: 0.9170 - val_loss: 0.7438

Epoch 35/100

40/40 - 0s - 2ms/step - accuracy: 0.8526 - loss: 0.7537 - val_accuracy: 0.8740 - val_loss: 0.7342

Epoch 36/100

40/40 - 0s - 2ms/step - accuracy: 0.8558 - loss: 0.7345 - val_accuracy: 0.9040 - val_loss: 0.7063

Epoch 37/100

40/40 - 0s - 2ms/step - accuracy: 0.8658 - loss: 0.7179 - val_accuracy: 0.9310 - val_loss: 0.6880

Epoch 38/100

40/40 - 0s - 2ms/step - accuracy: 0.8681 - loss: 0.7021 - val_accuracy: 0.9170 - val_loss: 0.6764

Epoch 39/100

40/40 - 0s - 2ms/step - accuracy: 0.8829 - loss: 0.6825 - val_accuracy: 0.8950 - val_loss: 0.6573

Epoch 40/100

40/40 - 0s - 2ms/step - accuracy: 0.8847 - loss: 0.6632 - val_accuracy: 0.9370 - val_loss: 0.6351

Epoch 41/100

40/40 - 0s - 2ms/step - accuracy: 0.8927 - loss: 0.6471 - val_accuracy: 0.9670 - val_loss: 0.6127

Epoch 42/100

40/40 - 0s - 2ms/step - accuracy: 0.8998 - loss: 0.6304 - val_accuracy: 0.9590 - val_loss: 0.6024

Epoch 43/100

40/40 - 0s - 2ms/step - accuracy: 0.9051 - loss: 0.6152 - val_accuracy: 0.9580 - val_loss: 0.5850

Epoch 44/100

40/40 - 0s - 2ms/step - accuracy: 0.9169 - loss: 0.5980 - val_accuracy: 0.9470 - val_loss: 0.5724

Epoch 45/100

40/40 - 0s - 3ms/step - accuracy: 0.9175 - loss: 0.5813 - val_accuracy: 0.9490 - val_loss: 0.5584

Epoch 46/100

40/40 - 0s - 2ms/step - accuracy: 0.9213 - loss: 0.5677 - val_accuracy: 0.9800 - val_loss: 0.5342

Epoch 47/100

40/40 - 0s - 2ms/step - accuracy: 0.9275 - loss: 0.5520 - val_accuracy: 0.9650 - val_loss: 0.5253

Epoch 48/100

40/40 - 0s - 2ms/step - accuracy: 0.9303 - loss: 0.5385 - val_accuracy: 0.9730 - val_loss: 0.5141

Epoch 49/100

40/40 - 0s - 2ms/step - accuracy: 0.9403 - loss: 0.5240 - val_accuracy: 0.9740 - val_loss: 0.4963

Epoch 50/100

40/40 - 0s - 2ms/step - accuracy: 0.9457 - loss: 0.5098 - val_accuracy: 0.9900 - val_loss: 0.4862

Epoch 51/100

40/40 - 0s - 2ms/step - accuracy: 0.9474 - loss: 0.4950 - val_accuracy: 0.9790 - val_loss: 0.4706

Epoch 52/100

40/40 - 0s - 2ms/step - accuracy: 0.9536 - loss: 0.4824 - val_accuracy: 0.9590 - val_loss: 0.4611

Epoch 53/100

40/40 - 0s - 2ms/step - accuracy: 0.9570 - loss: 0.4699 - val_accuracy: 0.9950 - val_loss: 0.4468

Epoch 54/100

40/40 - 0s - 2ms/step - accuracy: 0.9610 - loss: 0.4566 - val_accuracy: 0.9840 - val_loss: 0.4335

Epoch 55/100

40/40 - 0s - 3ms/step - accuracy: 0.9651 - loss: 0.4459 - val_accuracy: 0.9990 - val_loss: 0.4169

Epoch 56/100

40/40 - 0s - 2ms/step - accuracy: 0.9662 - loss: 0.4324 - val_accuracy: 0.9810 - val_loss: 0.4204

Epoch 57/100

40/40 - 0s - 2ms/step - accuracy: 0.9702 - loss: 0.4217 - val_accuracy: 0.9820 - val_loss: 0.4072

Epoch 58/100

40/40 - 0s - 2ms/step - accuracy: 0.9728 - loss: 0.4089 - val_accuracy: 0.9990 - val_loss: 0.3815

Epoch 59/100

40/40 - 0s - 2ms/step - accuracy: 0.9744 - loss: 0.3981 - val_accuracy: 0.9990 - val_loss: 0.3701

Epoch 60/100

40/40 - 0s - 2ms/step - accuracy: 0.9762 - loss: 0.3883 - val_accuracy: 0.9990 - val_loss: 0.3638

Epoch 61/100

40/40 - 0s - 2ms/step - accuracy: 0.9787 - loss: 0.3772 - val_accuracy: 1.0000 - val_loss: 0.3487

Epoch 62/100

40/40 - 0s - 2ms/step - accuracy: 0.9809 - loss: 0.3658 - val_accuracy: 1.0000 - val_loss: 0.3356

Epoch 63/100

40/40 - 0s - 2ms/step - accuracy: 0.9816 - loss: 0.3540 - val_accuracy: 1.0000 - val_loss: 0.3298

Epoch 64/100

40/40 - 0s - 2ms/step - accuracy: 0.9831 - loss: 0.3444 - val_accuracy: 1.0000 - val_loss: 0.3216

Epoch 65/100

40/40 - 0s - 3ms/step - accuracy: 0.9836 - loss: 0.3379 - val_accuracy: 0.9990 - val_loss: 0.3165

Epoch 66/100

40/40 - 0s - 2ms/step - accuracy: 0.9855 - loss: 0.3273 - val_accuracy: 1.0000 - val_loss: 0.2992

Epoch 67/100

40/40 - 0s - 2ms/step - accuracy: 0.9893 - loss: 0.3171 - val_accuracy: 1.0000 - val_loss: 0.2898

Epoch 68/100

40/40 - 0s - 2ms/step - accuracy: 0.9902 - loss: 0.3081 - val_accuracy: 1.0000 - val_loss: 0.2809

Epoch 69/100

40/40 - 0s - 2ms/step - accuracy: 0.9901 - loss: 0.2984 - val_accuracy: 1.0000 - val_loss: 0.2729

Epoch 70/100

40/40 - 0s - 2ms/step - accuracy: 0.9898 - loss: 0.2904 - val_accuracy: 1.0000 - val_loss: 0.2700

Epoch 71/100

40/40 - 0s - 2ms/step - accuracy: 0.9911 - loss: 0.2816 - val_accuracy: 1.0000 - val_loss: 0.2642

Epoch 72/100

40/40 - 0s - 2ms/step - accuracy: 0.9924 - loss: 0.2746 - val_accuracy: 1.0000 - val_loss: 0.2481

Epoch 73/100

40/40 - 0s - 2ms/step - accuracy: 0.9934 - loss: 0.2659 - val_accuracy: 1.0000 - val_loss: 0.2422

Epoch 74/100

40/40 - 0s - 2ms/step - accuracy: 0.9921 - loss: 0.2571 - val_accuracy: 1.0000 - val_loss: 0.2314

Epoch 75/100

40/40 - 0s - 3ms/step - accuracy: 0.9920 - loss: 0.2499 - val_accuracy: 1.0000 - val_loss: 0.2265

Epoch 76/100

40/40 - 0s - 2ms/step - accuracy: 0.9956 - loss: 0.2406 - val_accuracy: 1.0000 - val_loss: 0.2173

Epoch 77/100

40/40 - 0s - 2ms/step - accuracy: 0.9951 - loss: 0.2333 - val_accuracy: 1.0000 - val_loss: 0.2149

Epoch 78/100

40/40 - 0s - 2ms/step - accuracy: 0.9954 - loss: 0.2266 - val_accuracy: 1.0000 - val_loss: 0.2065

Epoch 79/100

40/40 - 0s - 2ms/step - accuracy: 0.9958 - loss: 0.2189 - val_accuracy: 1.0000 - val_loss: 0.1948

Epoch 80/100

40/40 - 0s - 2ms/step - accuracy: 0.9954 - loss: 0.2121 - val_accuracy: 1.0000 - val_loss: 0.1899

Epoch 81/100

40/40 - 0s - 2ms/step - accuracy: 0.9956 - loss: 0.2051 - val_accuracy: 1.0000 - val_loss: 0.1885

Epoch 82/100

40/40 - 0s - 2ms/step - accuracy: 0.9963 - loss: 0.1996 - val_accuracy: 1.0000 - val_loss: 0.1725

Epoch 83/100

40/40 - 0s - 2ms/step - accuracy: 0.9966 - loss: 0.1910 - val_accuracy: 0.9990 - val_loss: 0.1876

Epoch 84/100

40/40 - 0s - 2ms/step - accuracy: 0.9968 - loss: 0.1860 - val_accuracy: 1.0000 - val_loss: 0.1642

Epoch 85/100

40/40 - 0s - 3ms/step - accuracy: 0.9972 - loss: 0.1811 - val_accuracy: 1.0000 - val_loss: 0.1637

Epoch 86/100

40/40 - 0s - 2ms/step - accuracy: 0.9965 - loss: 0.1730 - val_accuracy: 1.0000 - val_loss: 0.1520

Epoch 87/100

40/40 - 0s - 2ms/step - accuracy: 0.9974 - loss: 0.1688 - val_accuracy: 1.0000 - val_loss: 0.1492

Epoch 88/100

40/40 - 0s - 2ms/step - accuracy: 0.9972 - loss: 0.1642 - val_accuracy: 1.0000 - val_loss: 0.1437

Epoch 89/100

40/40 - 0s - 2ms/step - accuracy: 0.9985 - loss: 0.1568 - val_accuracy: 1.0000 - val_loss: 0.1380

Epoch 90/100

40/40 - 0s - 2ms/step - accuracy: 0.9981 - loss: 0.1520 - val_accuracy: 1.0000 - val_loss: 0.1278

Epoch 91/100

40/40 - 0s - 2ms/step - accuracy: 0.9980 - loss: 0.1463 - val_accuracy: 1.0000 - val_loss: 0.1240

Epoch 92/100

40/40 - 0s - 2ms/step - accuracy: 0.9978 - loss: 0.1414 - val_accuracy: 1.0000 - val_loss: 0.1173

Epoch 93/100

40/40 - 0s - 2ms/step - accuracy: 0.9989 - loss: 0.1368 - val_accuracy: 1.0000 - val_loss: 0.1138

Epoch 94/100

40/40 - 0s - 2ms/step - accuracy: 0.9987 - loss: 0.1308 - val_accuracy: 1.0000 - val_loss: 0.1108

Epoch 95/100

40/40 - 0s - 3ms/step - accuracy: 0.9985 - loss: 0.1271 - val_accuracy: 1.0000 - val_loss: 0.1067

Epoch 96/100

40/40 - 0s - 2ms/step - accuracy: 0.9987 - loss: 0.1232 - val_accuracy: 1.0000 - val_loss: 0.1056

Epoch 97/100

40/40 - 0s - 2ms/step - accuracy: 0.9988 - loss: 0.1184 - val_accuracy: 1.0000 - val_loss: 0.1028

Epoch 98/100

40/40 - 0s - 2ms/step - accuracy: 0.9991 - loss: 0.1134 - val_accuracy: 1.0000 - val_loss: 0.0926

Epoch 99/100

40/40 - 0s - 2ms/step - accuracy: 0.9987 - loss: 0.1102 - val_accuracy: 1.0000 - val_loss: 0.0895

Epoch 100/100

40/40 - 0s - 3ms/step - accuracy: 0.9990 - loss: 0.1055 - val_accuracy: 1.0000 - val_loss: 0.0847

<keras.src.callbacks.history.History at 0x7f12b3ab2690>

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#