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

2025-04-30 17:08:21.120303: I external/local_xla/xla/tsl/cuda/cudart_stub.cc:32] Could not find cuda drivers on your machine, GPU will not be used.
2025-04-30 17:08:21.123669: I external/local_xla/xla/tsl/cuda/cudart_stub.cc:32] Could not find cuda drivers on your machine, GPU will not be used.
2025-04-30 17:08:21.132532: 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 00:00:1746032901.147675    7262 cuda_dnn.cc:8579] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered
E0000 00:00:1746032901.151962    7262 cuda_blas.cc:1407] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered
W0000 00:00:1746032901.163803    7262 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1746032901.163821    7262 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1746032901.163823    7262 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1746032901.163824    7262 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
2025-04-30 17:08:21.167929: 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([0.55, 0.45, 0.65, 0.45, 0.95, 0.25, 0.85, 0.55, 0.85, 0.35])

y_train[0]

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

2025-04-30 17:08:23.957954: 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 - 17ms/step - accuracy: 0.1315 - loss: 2.2887 - val_accuracy: 0.1980 - val_loss: 2.2504

Epoch 2/100

40/40 - 0s - 3ms/step - accuracy: 0.2110 - loss: 2.2150 - val_accuracy: 0.2490 - val_loss: 2.1667

Epoch 3/100

40/40 - 0s - 3ms/step - accuracy: 0.2509 - loss: 2.1225 - val_accuracy: 0.2730 - val_loss: 2.0637

Epoch 4/100

40/40 - 0s - 3ms/step - accuracy: 0.2768 - loss: 2.0209 - val_accuracy: 0.2680 - val_loss: 1.9587

Epoch 5/100

40/40 - 0s - 3ms/step - accuracy: 0.2978 - loss: 1.9208 - val_accuracy: 0.3200 - val_loss: 1.8569

Epoch 6/100

40/40 - 0s - 3ms/step - accuracy: 0.3268 - loss: 1.8293 - val_accuracy: 0.3860 - val_loss: 1.7715

Epoch 7/100

40/40 - 0s - 3ms/step - accuracy: 0.3564 - loss: 1.7450 - val_accuracy: 0.4040 - val_loss: 1.6872

Epoch 8/100

40/40 - 0s - 3ms/step - accuracy: 0.3901 - loss: 1.6678 - val_accuracy: 0.4520 - val_loss: 1.6168

Epoch 9/100

40/40 - 0s - 3ms/step - accuracy: 0.4212 - loss: 1.5990 - val_accuracy: 0.4810 - val_loss: 1.5470

Epoch 10/100

40/40 - 0s - 3ms/step - accuracy: 0.4402 - loss: 1.5390 - val_accuracy: 0.5410 - val_loss: 1.4879

Epoch 11/100

40/40 - 0s - 3ms/step - accuracy: 0.4725 - loss: 1.4799 - val_accuracy: 0.5220 - val_loss: 1.4299

Epoch 12/100

40/40 - 0s - 3ms/step - accuracy: 0.4913 - loss: 1.4252 - val_accuracy: 0.5610 - val_loss: 1.3791

Epoch 13/100

40/40 - 0s - 3ms/step - accuracy: 0.5191 - loss: 1.3737 - val_accuracy: 0.5350 - val_loss: 1.3364

Epoch 14/100

40/40 - 0s - 3ms/step - accuracy: 0.5347 - loss: 1.3346 - val_accuracy: 0.5730 - val_loss: 1.2944

Epoch 15/100

40/40 - 0s - 3ms/step - accuracy: 0.5494 - loss: 1.2912 - val_accuracy: 0.5300 - val_loss: 1.2496

Epoch 16/100

40/40 - 0s - 3ms/step - accuracy: 0.5599 - loss: 1.2519 - val_accuracy: 0.7010 - val_loss: 1.2075

Epoch 17/100

40/40 - 0s - 3ms/step - accuracy: 0.5882 - loss: 1.2122 - val_accuracy: 0.6640 - val_loss: 1.1705

Epoch 18/100

40/40 - 0s - 3ms/step - accuracy: 0.6069 - loss: 1.1787 - val_accuracy: 0.7010 - val_loss: 1.1349

Epoch 19/100

40/40 - 0s - 3ms/step - accuracy: 0.6264 - loss: 1.1452 - val_accuracy: 0.7340 - val_loss: 1.1004

Epoch 20/100

40/40 - 0s - 3ms/step - accuracy: 0.6460 - loss: 1.1095 - val_accuracy: 0.7720 - val_loss: 1.0711

Epoch 21/100

40/40 - 0s - 3ms/step - accuracy: 0.6551 - loss: 1.0792 - val_accuracy: 0.8030 - val_loss: 1.0399

Epoch 22/100

40/40 - 0s - 3ms/step - accuracy: 0.6837 - loss: 1.0487 - val_accuracy: 0.7980 - val_loss: 1.0080

Epoch 23/100

40/40 - 0s - 3ms/step - accuracy: 0.6900 - loss: 1.0240 - val_accuracy: 0.8100 - val_loss: 0.9849

Epoch 24/100

40/40 - 0s - 3ms/step - accuracy: 0.7095 - loss: 0.9953 - val_accuracy: 0.8000 - val_loss: 0.9591

Epoch 25/100

40/40 - 0s - 3ms/step - accuracy: 0.7198 - loss: 0.9694 - val_accuracy: 0.8140 - val_loss: 0.9414

Epoch 26/100

40/40 - 0s - 3ms/step - accuracy: 0.7425 - loss: 0.9435 - val_accuracy: 0.8460 - val_loss: 0.9106

Epoch 27/100

40/40 - 0s - 3ms/step - accuracy: 0.7569 - loss: 0.9206 - val_accuracy: 0.7070 - val_loss: 0.8950

Epoch 28/100

40/40 - 0s - 3ms/step - accuracy: 0.7693 - loss: 0.8973 - val_accuracy: 0.8230 - val_loss: 0.8615

Epoch 29/100

40/40 - 0s - 3ms/step - accuracy: 0.7861 - loss: 0.8728 - val_accuracy: 0.8820 - val_loss: 0.8361

Epoch 30/100

40/40 - 0s - 3ms/step - accuracy: 0.8063 - loss: 0.8505 - val_accuracy: 0.8810 - val_loss: 0.8186

Epoch 31/100

40/40 - 0s - 3ms/step - accuracy: 0.8169 - loss: 0.8268 - val_accuracy: 0.9260 - val_loss: 0.7877

Epoch 32/100

40/40 - 0s - 3ms/step - accuracy: 0.8263 - loss: 0.8081 - val_accuracy: 0.9030 - val_loss: 0.7712

Epoch 33/100

40/40 - 0s - 3ms/step - accuracy: 0.8420 - loss: 0.7856 - val_accuracy: 0.9330 - val_loss: 0.7463

Epoch 34/100

40/40 - 0s - 3ms/step - accuracy: 0.8531 - loss: 0.7640 - val_accuracy: 0.9160 - val_loss: 0.7294

Epoch 35/100

40/40 - 0s - 3ms/step - accuracy: 0.8669 - loss: 0.7445 - val_accuracy: 0.8990 - val_loss: 0.7153

Epoch 36/100

40/40 - 0s - 3ms/step - accuracy: 0.8781 - loss: 0.7238 - val_accuracy: 0.9080 - val_loss: 0.6972

Epoch 37/100

40/40 - 0s - 3ms/step - accuracy: 0.8932 - loss: 0.7035 - val_accuracy: 0.9580 - val_loss: 0.6702

Epoch 38/100

40/40 - 0s - 3ms/step - accuracy: 0.9012 - loss: 0.6865 - val_accuracy: 0.9360 - val_loss: 0.6538

Epoch 39/100

40/40 - 0s - 3ms/step - accuracy: 0.9099 - loss: 0.6643 - val_accuracy: 0.9540 - val_loss: 0.6333

Epoch 40/100

40/40 - 0s - 3ms/step - accuracy: 0.9140 - loss: 0.6497 - val_accuracy: 0.9420 - val_loss: 0.6152

Epoch 41/100

40/40 - 0s - 3ms/step - accuracy: 0.9300 - loss: 0.6295 - val_accuracy: 0.9670 - val_loss: 0.6045

Epoch 42/100

40/40 - 0s - 3ms/step - accuracy: 0.9354 - loss: 0.6135 - val_accuracy: 0.9940 - val_loss: 0.5818

Epoch 43/100

40/40 - 0s - 3ms/step - accuracy: 0.9440 - loss: 0.5963 - val_accuracy: 0.9780 - val_loss: 0.5677

Epoch 44/100

40/40 - 0s - 3ms/step - accuracy: 0.9468 - loss: 0.5803 - val_accuracy: 0.9920 - val_loss: 0.5546

Epoch 45/100

40/40 - 0s - 3ms/step - accuracy: 0.9561 - loss: 0.5630 - val_accuracy: 0.9810 - val_loss: 0.5333

Epoch 46/100

40/40 - 0s - 3ms/step - accuracy: 0.9599 - loss: 0.5460 - val_accuracy: 0.9770 - val_loss: 0.5234

Epoch 47/100

40/40 - 0s - 3ms/step - accuracy: 0.9636 - loss: 0.5285 - val_accuracy: 0.9990 - val_loss: 0.4998

Epoch 48/100

40/40 - 0s - 3ms/step - accuracy: 0.9670 - loss: 0.5162 - val_accuracy: 1.0000 - val_loss: 0.4827

Epoch 49/100

40/40 - 0s - 3ms/step - accuracy: 0.9721 - loss: 0.4987 - val_accuracy: 0.9980 - val_loss: 0.4721

Epoch 50/100

40/40 - 0s - 3ms/step - accuracy: 0.9749 - loss: 0.4843 - val_accuracy: 1.0000 - val_loss: 0.4502

Epoch 51/100

40/40 - 0s - 3ms/step - accuracy: 0.9783 - loss: 0.4686 - val_accuracy: 0.9990 - val_loss: 0.4538

Epoch 52/100

40/40 - 0s - 3ms/step - accuracy: 0.9813 - loss: 0.4524 - val_accuracy: 1.0000 - val_loss: 0.4201

Epoch 53/100

40/40 - 0s - 3ms/step - accuracy: 0.9818 - loss: 0.4400 - val_accuracy: 0.9990 - val_loss: 0.4150

Epoch 54/100

40/40 - 0s - 3ms/step - accuracy: 0.9840 - loss: 0.4274 - val_accuracy: 1.0000 - val_loss: 0.3976

Epoch 55/100

40/40 - 0s - 3ms/step - accuracy: 0.9883 - loss: 0.4133 - val_accuracy: 0.9950 - val_loss: 0.3850

Epoch 56/100

40/40 - 0s - 3ms/step - accuracy: 0.9877 - loss: 0.4005 - val_accuracy: 1.0000 - val_loss: 0.3762

Epoch 57/100

40/40 - 0s - 3ms/step - accuracy: 0.9880 - loss: 0.3889 - val_accuracy: 1.0000 - val_loss: 0.3640

Epoch 58/100

40/40 - 0s - 3ms/step - accuracy: 0.9898 - loss: 0.3752 - val_accuracy: 1.0000 - val_loss: 0.3507

Epoch 59/100

40/40 - 0s - 3ms/step - accuracy: 0.9920 - loss: 0.3628 - val_accuracy: 1.0000 - val_loss: 0.3318

Epoch 60/100

40/40 - 0s - 3ms/step - accuracy: 0.9927 - loss: 0.3501 - val_accuracy: 1.0000 - val_loss: 0.3266

Epoch 61/100

40/40 - 0s - 3ms/step - accuracy: 0.9926 - loss: 0.3395 - val_accuracy: 1.0000 - val_loss: 0.3124

Epoch 62/100

40/40 - 0s - 3ms/step - accuracy: 0.9927 - loss: 0.3287 - val_accuracy: 1.0000 - val_loss: 0.3049

Epoch 63/100

40/40 - 0s - 3ms/step - accuracy: 0.9928 - loss: 0.3178 - val_accuracy: 1.0000 - val_loss: 0.2879

Epoch 64/100

40/40 - 0s - 3ms/step - accuracy: 0.9941 - loss: 0.3080 - val_accuracy: 1.0000 - val_loss: 0.2822

Epoch 65/100

40/40 - 0s - 3ms/step - accuracy: 0.9947 - loss: 0.2987 - val_accuracy: 1.0000 - val_loss: 0.2769

Epoch 66/100

40/40 - 0s - 3ms/step - accuracy: 0.9948 - loss: 0.2880 - val_accuracy: 1.0000 - val_loss: 0.2579

Epoch 67/100

40/40 - 0s - 3ms/step - accuracy: 0.9959 - loss: 0.2770 - val_accuracy: 1.0000 - val_loss: 0.2484

Epoch 68/100

40/40 - 0s - 3ms/step - accuracy: 0.9961 - loss: 0.2689 - val_accuracy: 1.0000 - val_loss: 0.2403

Epoch 69/100

40/40 - 0s - 3ms/step - accuracy: 0.9969 - loss: 0.2582 - val_accuracy: 1.0000 - val_loss: 0.2312

Epoch 70/100

40/40 - 0s - 3ms/step - accuracy: 0.9970 - loss: 0.2497 - val_accuracy: 1.0000 - val_loss: 0.2256

Epoch 71/100

40/40 - 0s - 3ms/step - accuracy: 0.9969 - loss: 0.2424 - val_accuracy: 1.0000 - val_loss: 0.2193

Epoch 72/100

40/40 - 0s - 3ms/step - accuracy: 0.9974 - loss: 0.2343 - val_accuracy: 1.0000 - val_loss: 0.2129

Epoch 73/100

40/40 - 0s - 3ms/step - accuracy: 0.9971 - loss: 0.2254 - val_accuracy: 1.0000 - val_loss: 0.1980

Epoch 74/100

40/40 - 0s - 3ms/step - accuracy: 0.9977 - loss: 0.2170 - val_accuracy: 1.0000 - val_loss: 0.1953

Epoch 75/100

40/40 - 0s - 3ms/step - accuracy: 0.9983 - loss: 0.2100 - val_accuracy: 1.0000 - val_loss: 0.1876

Epoch 76/100

40/40 - 0s - 3ms/step - accuracy: 0.9980 - loss: 0.2029 - val_accuracy: 1.0000 - val_loss: 0.1825

Epoch 77/100

40/40 - 0s - 3ms/step - accuracy: 0.9974 - loss: 0.1963 - val_accuracy: 1.0000 - val_loss: 0.1720

Epoch 78/100

40/40 - 0s - 3ms/step - accuracy: 0.9974 - loss: 0.1894 - val_accuracy: 1.0000 - val_loss: 0.1628

Epoch 79/100

40/40 - 0s - 3ms/step - accuracy: 0.9977 - loss: 0.1826 - val_accuracy: 1.0000 - val_loss: 0.1573

Epoch 80/100

40/40 - 0s - 3ms/step - accuracy: 0.9977 - loss: 0.1770 - val_accuracy: 1.0000 - val_loss: 0.1492

Epoch 81/100

40/40 - 0s - 3ms/step - accuracy: 0.9992 - loss: 0.1700 - val_accuracy: 1.0000 - val_loss: 0.1462

Epoch 82/100

40/40 - 0s - 3ms/step - accuracy: 0.9990 - loss: 0.1629 - val_accuracy: 1.0000 - val_loss: 0.1393

Epoch 83/100

40/40 - 0s - 3ms/step - accuracy: 0.9985 - loss: 0.1565 - val_accuracy: 1.0000 - val_loss: 0.1321

Epoch 84/100

40/40 - 0s - 3ms/step - accuracy: 0.9985 - loss: 0.1509 - val_accuracy: 1.0000 - val_loss: 0.1289

Epoch 85/100

40/40 - 0s - 3ms/step - accuracy: 0.9986 - loss: 0.1452 - val_accuracy: 1.0000 - val_loss: 0.1259

Epoch 86/100

40/40 - 0s - 3ms/step - accuracy: 0.9990 - loss: 0.1388 - val_accuracy: 1.0000 - val_loss: 0.1197

Epoch 87/100

40/40 - 0s - 3ms/step - accuracy: 0.9990 - loss: 0.1340 - val_accuracy: 1.0000 - val_loss: 0.1106

Epoch 88/100

40/40 - 0s - 3ms/step - accuracy: 0.9991 - loss: 0.1285 - val_accuracy: 1.0000 - val_loss: 0.1057

Epoch 89/100

40/40 - 0s - 3ms/step - accuracy: 0.9992 - loss: 0.1247 - val_accuracy: 1.0000 - val_loss: 0.1028

Epoch 90/100

40/40 - 0s - 3ms/step - accuracy: 0.9992 - loss: 0.1198 - val_accuracy: 1.0000 - val_loss: 0.1012

Epoch 91/100

40/40 - 0s - 3ms/step - accuracy: 0.9992 - loss: 0.1147 - val_accuracy: 1.0000 - val_loss: 0.0973

Epoch 92/100

40/40 - 0s - 3ms/step - accuracy: 0.9995 - loss: 0.1104 - val_accuracy: 1.0000 - val_loss: 0.0961

Epoch 93/100

40/40 - 0s - 3ms/step - accuracy: 0.9989 - loss: 0.1063 - val_accuracy: 1.0000 - val_loss: 0.0865

Epoch 94/100

40/40 - 0s - 3ms/step - accuracy: 0.9995 - loss: 0.1020 - val_accuracy: 1.0000 - val_loss: 0.0820

Epoch 95/100

40/40 - 0s - 3ms/step - accuracy: 0.9997 - loss: 0.0985 - val_accuracy: 1.0000 - val_loss: 0.0760

Epoch 96/100

40/40 - 0s - 3ms/step - accuracy: 0.9996 - loss: 0.0937 - val_accuracy: 1.0000 - val_loss: 0.0778

Epoch 97/100

40/40 - 0s - 3ms/step - accuracy: 0.9996 - loss: 0.0909 - val_accuracy: 1.0000 - val_loss: 0.0830

Epoch 98/100

40/40 - 0s - 3ms/step - accuracy: 0.9994 - loss: 0.0879 - val_accuracy: 1.0000 - val_loss: 0.0676

Epoch 99/100

40/40 - 0s - 3ms/step - accuracy: 0.9991 - loss: 0.0840 - val_accuracy: 1.0000 - val_loss: 0.0674

Epoch 100/100

40/40 - 0s - 3ms/step - accuracy: 0.9990 - loss: 0.0817 - val_accuracy: 1.0000 - val_loss: 0.0621

<keras.src.callbacks.history.History at 0x7fbbe1aefb90>

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#