Constrained Monotonic Neural Networks

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Summary

This Python library implements Constrained Monotonic Neural Networks as described in:

Davor Runje, Sharath M. Shankaranarayana, “Constrained Monotonic Neural Networks”, in Proceedings of the 40th International Conference on Machine Learning, 2023. [PDF].

Abstract

Wider adoption of neural networks in many critical domains such as finance and healthcare is being hindered by the need to explain their predictions and to impose additional constraints on them. Monotonicity constraint is one of the most requested properties in real-world scenarios and is the focus of this paper. One of the oldest ways to construct a monotonic fully connected neural network is to constrain signs on its weights. Unfortunately, this construction does not work with popular non-saturated activation functions as it can only approximate convex functions. We show this shortcoming can be fixed by constructing two additional activation functions from a typical unsaturated monotonic activation function and employing each of them on the part of neurons. Our experiments show this approach of building monotonic neural networks has better accuracy when compared to other state-of-the-art methods, while being the simplest one in the sense of having the least number of parameters, and not requiring any modifications to the learning procedure or post-learning steps. Finally, we prove it can approximate any continuous monotone function on a compact subset of $\mathbb{R}^n$.

Citation

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}
}

Python package

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

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

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:

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

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

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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