monotonic-nn 0.2.0rc2

Monotonic Neural Networks

Constrained Monotonic Neural Networks

This Python library implements Monotonic Dense Layer as described in Davor Runje, Sharath M. Shankaranarayana, “Constrained Monotonic Neural Networks” [PDF].

If you use this library, please cite:

@inproceedings{runje2023,
  title={Constrained Monotonic Neural Networks},
  author={Davor Runje and Sharath M. Shankaranarayana},
  booktitle={Proceedings of the 40th {International Conference on Machine Learning}},
  year={2023}
}

This package contains an implementation of our Monotonic Dense Layer MonoDense (Constrained Monotonic Fully Connected Layer). Below is the figure from the paper for reference.

In the code, the variable monotonicity_indicator corresponds to t in the figure and parameters is_convex, is_concave and activation_weights are used to calculate the activation selector s as follows:

• if is_convex or is_concave is True, then the activation selector s will be (units, 0, 0) and (0, units, 0), respecively.

• if both is_convex or is_concave is False, then the activation_weights represent ratios between $\breve{s}$, $\hat{s}$ and $\tilde{s}$, respecively. E.g. if activation_weights = (2, 2, 1) and units = 10, then

$$ (\breve{s}, \hat{s}, \tilde{s}) = (4, 4, 2) $$

Running in Google Colab

You can start this interactive tutorial in Google Colab by clicking the button below:

Install

pip install monotonic-nn

How to use

In this example, we’ll assume we have a simple dataset with three inputs values $x_1$, $x_2$ and $x_3$ sampled from the normal distribution, while the output value $y$ is calculated according to the following formula before adding Gaussian noise to it:

$y = x_1^3 + \sin\left(\frac{x_2}{2 \pi}\right) + e^{-x_3}$

<style type="text/css"> </style>

x0	x1	x2	y
0.304717	-1.039984	0.750451	0.234541
0.940565	-1.951035	-1.302180	4.199094
0.127840	-0.316243	-0.016801	0.834086
-0.853044	0.879398	0.777792	-0.093359
0.066031	1.127241	0.467509	0.780875

Now, we’ll use the MonoDense layer instead of Dense layer to build a simple monotonic network. By default, the MonoDense layer assumes the output of the layer is monotonically increasing with all inputs. This assumtion is always true for all layers except possibly the first one. For the first layer, we use monotonicity_indicator to specify which input parameters are monotonic and to specify are they increasingly or decreasingly monotonic:

• set 1 for increasingly monotonic parameter,

• set -1 for decreasingly monotonic parameter, and

• set 0 otherwise.

In our case, the monotonicity_indicator is [1, 0, -1] because $y$ is:

• monotonically increasing w.r.t. $x_1$ $\left(\frac{\partial y}{x_1} = 3 {x_1}^2 \geq 0\right)$, and

• monotonically decreasing w.r.t. $x_3$ $\left(\frac{\partial y}{x_3} = - e^{-x_2} \leq 0\right)$.

from tensorflow.keras import Sequential
from tensorflow.keras.layers import Dense, Input

from airt.keras.layers import MonoDense

model = Sequential()

model.add(Input(shape=(3,)))
monotonicity_indicator = [1, 0, -1]
model.add(
    MonoDense(128, activation="elu", monotonicity_indicator=monotonicity_indicator)
)
model.add(MonoDense(128, activation="elu"))
model.add(MonoDense(1))

model.summary()

Model: "sequential"
_________________________________________________________________
 Layer (type)                Output Shape              Param #   
=================================================================
 mono_dense (MonoDense)      (None, 128)               512       
                                                                 
 mono_dense_1 (MonoDense)    (None, 128)               16512     
                                                                 
 mono_dense_2 (MonoDense)    (None, 1)                 129       
                                                                 
=================================================================
Total params: 17,153
Trainable params: 17,153
Non-trainable params: 0
_________________________________________________________________

Now we can train the model as usual using Model.fit:

from tensorflow.keras.optimizers import Adam
from tensorflow.keras.optimizers.schedules import ExponentialDecay

lr_schedule = ExponentialDecay(
    initial_learning_rate=0.01,
    decay_steps=10_000 // 32,
    decay_rate=0.9,
)
optimizer = Adam(learning_rate=lr_schedule)
model.compile(optimizer=optimizer, loss="mse")

model.fit(
    x=x_train, y=y_train, batch_size=32, validation_data=(x_val, y_val), epochs=10
)

Epoch 1/10
313/313 [==============================] - 3s 5ms/step - loss: 9.4221 - val_loss: 6.1277
Epoch 2/10
313/313 [==============================] - 1s 4ms/step - loss: 4.6001 - val_loss: 2.7813
Epoch 3/10
313/313 [==============================] - 1s 4ms/step - loss: 1.6221 - val_loss: 2.1111
Epoch 4/10
313/313 [==============================] - 1s 4ms/step - loss: 0.9479 - val_loss: 0.2976
Epoch 5/10
313/313 [==============================] - 1s 4ms/step - loss: 0.9008 - val_loss: 0.3240
Epoch 6/10
313/313 [==============================] - 1s 4ms/step - loss: 0.5027 - val_loss: 0.1455
Epoch 7/10
313/313 [==============================] - 1s 4ms/step - loss: 0.4360 - val_loss: 0.1144
Epoch 8/10
313/313 [==============================] - 1s 4ms/step - loss: 0.4993 - val_loss: 0.1211
Epoch 9/10
313/313 [==============================] - 1s 4ms/step - loss: 0.3162 - val_loss: 1.0021
Epoch 10/10
313/313 [==============================] - 1s 4ms/step - loss: 0.2640 - val_loss: 0.2522

<keras.callbacks.History>

License

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