Table of Content

• Quick start

• Quick tutorial on Concept Embedding Models

• Quick tutorial on Deep Concept Reasoning

• Quick tutorial on Logic Explained Networks

• Benchmark datasets

• Theory

• Authors

• Licence

Quick start

You can install torch_explain along with all its dependencies from PyPI:

pip install torch-explain

Quick tutorial on Concept Embedding Models

Using concept embeddings we can solve concept-based problems very efficiently! For this simple tutorial, let’s approach the XOR benchmark dataset:

import torch
import torch_explain as te
from torch_explain import datasets
from sklearn.metrics import accuracy_score
from sklearn.model_selection import train_test_split

x, c, y = datasets.xor(500)
x_train, x_test, c_train, c_test, y_train, y_test = train_test_split(x, c, y, test_size=0.33, random_state=42)

We just need to define a task predictor and a concept encoder using a concept embedding layer:

import torch
import torch_explain as te

embedding_size = 8
concept_encoder = torch.nn.Sequential(
    torch.nn.Linear(x.shape[1], 10),
    torch.nn.LeakyReLU(),
    te.nn.ConceptEmbedding(10, c.shape[1], embedding_size),
)
task_predictor = torch.nn.Sequential(
    torch.nn.Linear(c.shape[1]*embedding_size, 1),
)
model = torch.nn.Sequential(concept_encoder, task_predictor)

We can now train the network by optimizing the cross entropy loss on concepts and tasks:

optimizer = torch.optim.AdamW(model.parameters(), lr=0.01)
loss_form_c = torch.nn.BCELoss()
loss_form_y = torch.nn.BCEWithLogitsLoss()
model.train()
for epoch in range(501):
    optimizer.zero_grad()

    # generate concept and task predictions
    c_emb, c_pred = concept_encoder(x_train)
    y_pred = task_predictor(c_emb.reshape(len(c_emb), -1))

    # compute loss
    concept_loss = loss_form_c(c_pred, c_train)
    task_loss = loss_form_y(y_pred, y_train)
    loss = concept_loss + 0.5*task_loss

    loss.backward()
    optimizer.step()

Once trained we can check the performance of the model on the test set:

c_emb, c_pred = concept_encoder.forward(x_test)
y_pred = task_predictor(c_emb.reshape(len(c_emb), -1))

task_accuracy = accuracy_score(y_test, y_pred > 0)
concept_accuracy = accuracy_score(c_test, c_pred > 0.5)

As you can see the performance of the model is now great as the task task accuracy is around ~100%.

Quick tutorial on Deep Concept Reasoning

Using deep concept reasoning we can solve the same problem as above, but with an intrinsically interpretable model! In fact, Deep Concept Reasoners (Deep CoRes) make task predictions by means of interpretable logic rules using concept embeddings.

Using the same example as before, we can just change the task predictor using a Deep CoRe layer:

from torch_explain.nn.concepts import ConceptReasoningLayer
import torch.nn.functional as F

y_train = F.one_hot(y_train.long().ravel()).float()
y_test = F.one_hot(y_test.long().ravel()).float()

task_predictor = ConceptReasoningLayer(embedding_size, y_train.shape[1])
model = torch.nn.Sequential(concept_encoder, task_predictor)

We can now train the network by optimizing the cross entropy loss on concepts and tasks:

optimizer = torch.optim.AdamW(model.parameters(), lr=0.01)
loss_form = torch.nn.BCELoss()
model.train()
for epoch in range(501):
    optimizer.zero_grad()

    # generate concept and task predictions
    c_emb, c_pred = concept_encoder(x_train)
    y_pred = task_predictor(c_emb, c_pred)

    # compute loss
    concept_loss = loss_form(c_pred, c_train)
    task_loss = loss_form(y_pred, y_train)
    loss = concept_loss + 0.5*task_loss

    loss.backward()
    optimizer.step()

Once trained the Deep CoRe layer can explain its predictions by providing both local and global logic rules:

local_explanations = task_predictor.explain(c_emb, c_pred, 'local')
global_explanations = task_predictor.explain(c_emb, c_pred, 'global')

For global explanations, the reasoner will return a dictionary with entries such as {'class': 'y_0', 'explanation': '~c_0 & ~c_1', 'count': 94}, specifying for each logic rule, the task it is associated with and the number of samples associated with the explanation.

Quick tutorial on Logic Explained Networks

For this simple experiment, let’s solve the XOR problem (augmented with 100 dummy features):

import torch
import torch_explain as te
from torch.nn.functional import one_hot

x0 = torch.zeros((4, 100))
x_train = torch.tensor([
    [0, 0],
    [0, 1],
    [1, 0],
    [1, 1],
], dtype=torch.float)
x_train = torch.cat([x_train, x0], dim=1)
y_train = torch.tensor([0, 1, 1, 0], dtype=torch.long)
y_train_1h = one_hot(y_train).to(torch.float)

We can instantiate a simple feed-forward neural network with 3 layers using the EntropyLayer as the first one:

layers = [
    te.nn.EntropyLinear(x_train.shape[1], 10, n_classes=y_train_1h.shape[1]),
    torch.nn.LeakyReLU(),
    torch.nn.Linear(10, 4),
    torch.nn.LeakyReLU(),
    torch.nn.Linear(4, 1),
]
model = torch.nn.Sequential(*layers)

We can now train the network by optimizing the cross entropy loss and the entropy_logic_loss loss function incorporating the human prior towards simple explanations:

optimizer = torch.optim.AdamW(model.parameters(), lr=0.001)
loss_form = torch.nn.BCEWithLogitsLoss()
model.train()
for epoch in range(2001):
    optimizer.zero_grad()
    y_pred = model(x_train).squeeze(-1)
    loss = loss_form(y_pred, y_train_1h) + 0.0001 * te.nn.functional.entropy_logic_loss(model)
    loss.backward()
    optimizer.step()

Once trained we can extract first-order logic formulas describing how the network composed the input features to obtain the predictions:

from torch_explain.logic.nn import entropy
from torch.nn.functional import one_hot

y1h = one_hot(y_train)
global_explanations, local_explanations = entropy.explain_classes(model, x_train, y_train, c_threshold=0.5, y_threshold=0.)

Explanations will be logic formulas in disjunctive normal form. In this case, the explanation will be y=1 if and only if (f1 AND ~f2) OR (f2 AND ~f1) corresponding to f1 XOR f2.

The function automatically assesses the quality of logic explanations in terms of classification accuracy and rule complexity. In this case the accuracy is 100% and the complexity is 4.

Benchmark datasets

We provide a suite of 3 benchmark datasets to evaluate the performance of our models in the folder torch_explain/datasets. These 3 datasets were proposed as benchmarks for concept-based models in the paper “Concept Embedding Models: Beyond the Accuracy-Explainability Trade-Off”.

Real-world datasets can be downloaded from the links provided in the supplementary material of the paper.

Theory

Theoretical foundations can be found in the following papers.

Deep Concept Reasoning (recently accepted at ICML-23):

@article{barbiero2023interpretable,
  title={Interpretable Neural-Symbolic Concept Reasoning},
  author={Barbiero, Pietro and Ciravegna, Gabriele and Giannini, Francesco and Zarlenga, Mateo Espinosa and Magister, Lucie Charlotte and Tonda, Alberto and Lio, Pietro and Precioso, Frederic and Jamnik, Mateja and Marra, Giuseppe},
  journal={arXiv preprint arXiv:2304.14068},
  year={2023}
}

Concept Embedding Models:

@article{espinosa2022concept,
  title={Concept Embedding Models: Beyond the Accuracy-Explainability Trade-Off},
  author={Espinosa Zarlenga, Mateo and Barbiero, Pietro and Ciravegna, Gabriele and Marra, Giuseppe and Giannini, Francesco and Diligenti, Michelangelo and Shams, Zohreh and Precioso, Frederic and Melacci, Stefano and Weller, Adrian and others},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={21400--21413},
  year={2022}
}

Logic Explained Networks:

@article{ciravegna2023logic,
  title={Logic explained networks},
  author={Ciravegna, Gabriele and Barbiero, Pietro and Giannini, Francesco and Gori, Marco and Li{\'o}, Pietro and Maggini, Marco and Melacci, Stefano},
  journal={Artificial Intelligence},
  volume={314},
  pages={103822},
  year={2023},
  publisher={Elsevier}
}

Entropy-based LENs:

@inproceedings{barbiero2022entropy,
  title={Entropy-based logic explanations of neural networks},
  author={Barbiero, Pietro and Ciravegna, Gabriele and Giannini, Francesco and Li{\'o}, Pietro and Gori, Marco and Melacci, Stefano},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={36},
  number={6},
  pages={6046--6054},
  year={2022}
}

Psi network (“learning of constraints”):

@inproceedings{ciravegna2020constraint,
  title={A constraint-based approach to learning and explanation},
  author={Ciravegna, Gabriele and Giannini, Francesco and Melacci, Stefano and Maggini, Marco and Gori, Marco},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={34},
  number={04},
  pages={3658--3665},
  year={2020}
}

Learning with constraints:

@inproceedings{marra2019lyrics,
  title={LYRICS: A General Interface Layer to Integrate Logic Inference and Deep Learning},
  author={Marra, Giuseppe and Giannini, Francesco and Diligenti, Michelangelo and Gori, Marco},
  booktitle={Joint European Conference on Machine Learning and Knowledge Discovery in Databases},
  pages={283--298},
  year={2019},
  organization={Springer}
}

Constraints theory in machine learning:

@book{gori2017machine,
  title={Machine Learning: A constraint-based approach},
  author={Gori, Marco},
  year={2017},
  publisher={Morgan Kaufmann}
}

Authors

• Pietro Barbiero, University of Cambridge, UK.

• Mateo Espinosa Zarlenga, University of Cambridge, UK.

• Giuseppe Marra, Katholieke Universiteit Leuven, BE.

• Steve Azzolin, University of Trento, IT.

• Francesco Giannini, University of Florence, IT.

• Gabriele Ciravegna, University of Florence, IT.

• Dobrik Georgiev, University of Cambridge, UK.

Licence

Copyright 2020 Pietro Barbiero, Mateo Espinosa Zarlenga, Giuseppe Marra, Steve Azzolin, Francesco Giannini, Gabriele Ciravegna, and Dobrik Georgiev.

Licensed under the Apache License, Version 2.0 (the “License”); you may not use this file except in compliance with the License. You may obtain a copy of the License at: http://www.apache.org/licenses/LICENSE-2.0.

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.

See the License for the specific language governing permissions and limitations under the License.