Neural-based classification rule learning for sequential data
Authors: Marine Collery
Philippe Bonnard
François Fages
Remy Kusters
Venue: ICLR 2023 poster (2023)
Keywords: classification rule learning
binary neural network
interpretable AI
sequential data

Abstract: Discovering interpretable patterns for classification of sequential data is of key importance for a variety of fields, ranging from genomics to fraud detection or more generally interpretable decision-making.
In this paper, we propose a novel differentiable fully interpretable method to discover both local and global patterns (i.e. catching a relative or absolute temporal dependency) for rule-based binary classification.
It consists of a convolutional binary neural network with an interpretable neural filter and a training strategy based on dynamically-enforced sparsity.
We demonstrate the validity and usefulness of the approach on synthetic datasets and on an open-source peptides dataset.
Key to this end-to-end differentiable method is that the expressive patterns used in the rules are learned alongside the rules themselves.

Published conference paper at ICLR 2023
NEURAL-BASED CLASSIFICATION RULE LEARNING FOR SEQUENTIAL DATA
Marine Collery 122g Philippe Bonnard', François Fages? & Remy Kusters IBM France Lab, Innia Saclay, 'IBM Research
ABSTRACT
Discovering interpretable patterns for classification of sequential data is of key im- portance for variety of fields, ranging from genomics fraud detection or more generally interpretable decision-making In this paper, we propose a novel differ- entiable fully interpretable method to discover both local and global patterns (i.c. catching a relative or absolute temporal dependency) for rule-basee binary classi- fication. It consists of a convolutional binary neural network with an interpretable neural filter and training strategy based dynaaicalll--efooveed sparsity. We demonstrate the validity and usefulness of the approach on synthetic datasets and on an open-source peptides dataset. Key to this end-to-end differentiable method is that the expressive patterns used in the rules are learned alongside the rules themselves.
INTRODUCTION
During the last decades, machine learning and in particular neural networks have made tremendous progress on classification tasks for a variety of fields such as healthcare, fraud detection or entertain- ment. They are able to learn from various data types ranging from images timeseries and achieve impressive classification accuracy. However, they are difficult or impossible to understand by hu- man. Recently, explaining those black-box models has attracted considerable research interest under the field of Explainable AI (XAI). However, stated by Rudin (2019). those aposteriori approaches are not the solution for high stakes pecision-making and more interest should be placed learning models that are interpretable in the first place.
Rule-base based methods are interpretable, human readable and have been widely adopted in different industrial fields with Business Rule Management Systems (BRMS). In practice however, those rules are manually written by experts. One of the reasons manually- -writtnn rule models cannot easily be replaced with learned rule models is that rule- base learning models are not able to learn as expressive rules with higher-level concepts and complex grammar (Kramer 2020). Moreover, due to the lack latent representations, rule-based learning methods underperform w.r.t. state-of-the-art neural
networks (Beck& & Fürnkranz] [202T)
Classical classification rule learning algorithms (Cohen 1995 Breiman et al. 1984 Dash et al.
Lakkaraju et al. [016 Su ct al. 2016) as well as neural-basce approaches learn rules (Qiao ct al. 2021 Kusters et al.) (2022) (or logical expressions with Riegel et al. (2020)) do not provide the
2018
grammar required to learn classification rules on sequential data. Numerous approaches for learning classification rules sequential data in the field of sequential pattern mining have been studied the past such as Eggh et al. (2015); Zhou et al.) (2013); Holat et al.] (2014) but with a different goal in mind: improve the performance of extracted patterns for a fixed rule grammar as opposed to extending the rule grammar. Another domain of research focuses training binary neural networks to obtain more computational efficient model storing, computation and evaluation efficiency (Geiger & Team 2020: Helwegen et al.) 2019). It comes with fundamental optimization challenges around weights updates and gradient computation.
this paper. we bridge three domains and introduce a binary neural network to learn classification rules on sequential data. We propose a differrntiablee rule-based classification model for sequen- tial data where the conditions composed of sequence-dependen patterus that are discovered
"c "Corresooniigg author r nee col ary8 bom

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alongside the classification task itself. More precisely. we aim at learning structure: if pattern then class = else class = 0. In particular we consider two types of patterns: lo- cal and global patterns as introduced in Aggarwal (2002) that are in practice studied independently with local and global model. A local pattern describes a subsequence a specific position in the sequence while a global pattern is invariant to the location the sequence (Fig(). The network, that we refer to Convolutional Rule Neural Network (CR2N), builds top of base rule model that comparable to rule models for tabular data presented in Qiao al. (202T); Kusters al. (2022).
of the following
The contributions of this paper are the following: We propose a convolutional binary neural net- work that learns classification rules together with the sequence-dee -ruuecee-deeenent patterns in use. ii) We present training strategy train binarized neural network while dynamically enforcing sparsity. iii) We show on synthetic and real world datasets the usefulness of our architecture with the impor- tance of the rule grammar and the validity of our training process with the importance of sparsity. The code is publicly available at athttps: /tt:://gttthi com/ I3M/cr2n
2 BASE RULE MODEL
The base rule model we invoke is composed of three consecutive layers (Fig l}. The two last layers respectively mimic logical AND and OR operators (Qiao el al., [02T] Kusters et al. (022). On top of these layers, we add an additional layer that is specific for categorical input data and corresponds to an OR operator for each categorical variable over every possible value it can take.
StackedOR layer (E nk. K)
AND layer (K, H)
OR layer (H, 1)
Ai
B
C1
Bi and D2 h2
D1
A2
Da
B2
42
h3
Ca
D3 and C3
D2
A3
h4
if (B, and Da) or (D2 and C.) then else 0
Cs
Ba
x3
Ca
h5
Da
Figure 1: Example trained base rule model architecture for the rule if (B, and D2) or (Da and Cg) then I else 0 on 3 categorical variables Fe, Eea and Ties (Ieg E {As, By, Ct, Dx)). For simplicity, the truth value of ze, B, is replaced by B, for example. Plain (dotted) lines represent activated (masked) weights. An example evaluation the model is represented with the filled neurons (neuron=1) for the binary input Te, B1,
D2 and Tea- Ag.
The AND layer takes binary features (which are atomic boolean formulae) as input and outputs to the OR layer. The output of the OR layer is mapped to the classification label y. These layers have binary weights specifying the nodes that included in the respective boolean expression (conjunction disjunction). In other words, this network implements the evaluation of a DNF and has a direct equivalence with a binary classification rule like if (A A B) C' then class 1 else class 0, where A, B and are binary input features (atoms logical terms). In this paper, we focus supervised binary classification where we predict the label y E {0. 1} given input data x.
The base rule model is illustrated in Fig and is composed of three binary neural layers.
Input neurons æ, are binarized input features of size K (æ, are one-hot encoded categorical input features of size 12).

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Hidden neurons h, are conjunctions of the input features of size H. Output neuron y, is a disjunction of the (hidden) conjunctions.
We assign to cach of boolean operations, i.c. AND and OR operations, a binary weight (W and and War respectively) that plays the role of mask filter nodes with regards their respective logical operation. For the sake of simplicity, we did not extend the model with logical NOT operation.
The disjunction operation is implemented as,
y min( Wo h, 1).
(I)
none of the neurons h are activated then y = 0, and = if at least one is.
For the conjunction operation, we use the De Morgan's law that express the conjunction with the we obtain:
OR operator A A B - -B). Combined with Eq 1
(min Wond( 1)) = min( W Wand(1 x),
(2)
StackedOR Input Layer As defined previously, the AND layer takes binary input features
imput. In this paper, we propose add an additional layer for categorical data. A categorical variable De can take one value of out of fixed number of possible values e.g. {ao..... .a3) [A, B, C, D).. Without any additional layer, it requires one-hot encoding to be provided as input the AND layer. Binary inputs 2e A and x. B are then given as input to the AND layer that can in theory represent the impossible expression F. = AAA = B i. the model has to learn the hidden categorical relationship between the one-hot encoded variables. To prevent learning distribution we already know, deepen the model with the addition a stacked architecture of OR layers as input of the AND layer as shown in Fig This structure defined by K weights, W atackk for each input categorical variables and will be referred to as the StackedOR layer with W Wstack weights.
To conclude, the base rule model is composed of StackedOR layer for categorical variables, a logical AND layer and an layer. The formal grammar that this architecture can express specified with the following production rules (see Appendix A for the full grammar):
rule if base expressionther class llse class -0
base expression conjunction conjunction hase expression predicate predicate A conjunction
conjunction
predicate
categorical expression literal
(3)
categorical expression categorical literal literal
categorical literal categorical literal categorical expression
25c
ab .... ane (or simply aj ....| a
This grammar is also limited by the model architecture: conjunction contains at most one occurrence of each predicate and the total number of conjunction(s) is bounded by the number of hidden nodes.
3 CONVOLUTIONAL RULE NEURAL NETWORK
Our main contribution is extend the base rule model for sequential data. We apply the base rule model as a ID-convolutional window of fixed length E N over sequence and retrieve outputs as input for an additional disjunctive layer which we refer to as the ConvOR layer as shown in Fig 2 The base rule model learns DNF over the window size length and the ConvOR layer indicates where along the sequence that logical expression true. the evaluation of the logical expression is true all along the sequence then it can be deseribed as a global pattern, otherwise the learned pattern represents local pattern.
The model input is now of size Ee I na and output of StackedOR layer (or input of the AND layer) is x K. Other dimensions are not impacted. For simplicity in the following, K' fixed to 1 L.. input data is composed of one categorical variable evolving sequentially. The method is still valid for K > Fig [ also still valid with change of index, k is referring the position in the window size
A natural extension for sequential data of the hase rule architecture would extend with explicit recursion of the base rule model, similar RNN. This approach tested but faced the same limitations any classical RNNs, i.c., vanishing gradients and only captures short-term dependencies.

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ConvOR layer
local
(B t-5 and (/) t-4 and 1-3)
Evaluation of base nule model on every window
Filter: Base Rule Model (B t-2 and D t-l) (D t-1 and C' at t-0)
global
B-D D- C' sequence
Figure 2: Example of a trained CR2N architecture. The base rule model is applied as ID- convolutional window over the sequence (i.e. sliding window). The resulting boolean values are given as input of the ConvOR layer which indicates through its activated weights where along the sequence the expression learned by the base model is true. The output the ConvOR layer mapped to the label of the sequence y- For local patterns, the base model expression needs to be shifted accordingly to the ConvOR layer weights. For real-domain application like fraud detec- tion, by providing meaning to B. C and D, we could have for example if "receiving a transaction amount X"B) is followed by "emitting transaction of amount X" (D) or "emitting transaction of amount (") followed by "closing the bank account"(C) then class=fraud.
With this approach, different sequence-dependent expressions can be extracted and their nature de- pends the weights of the ConvOR layer (Fig If all the weights, Woone. of the ConvOR layer are activated (i.e. equal to 1). the logical expression learned by the base model true sequence: global pattern is learned. If only some of the weight of the ConvOR layer are acti- vated, the logical expression learned by the base model is valid only in the window associated that weight: local pattern learned. The base model logical expression is modified accordingly match that shift (see example in Fig 2 with shift of 3 sequential steps).
the
The obtained weights thus translate to rule grammar with the following production rules:
rule expression
if expression then class else local pattern global pattern
(4)
We introduce 1, the position when the last observation in sequence was made. With / being our reference, in a sequence of size N- E N. before (0 & i N 1). A, B,C C and D are toy binary input possible values for our categorical variable (they cannot be activated simultaneously the same position in the sequence). With those definitions, we list below examples of different sequence-deeenddnt expressions that can be expressed with the proposed architecture (see Fig
refers to the moment the hhh observation
A local pattern is expression composed of predicates that are true at specific position i, for example at t-15. Based on Eq 3 we have:
local pattern
base expression
predicate
categorical expression at 1-i literal i-i.
A global pattern is an expression describing the presence of pattern anywhere in the sequence, for example B-D in sequence a global pattern where "_" sign refers to "followed by" and "*" correspond to any unique literal (equivalent to Ví E [0; N - 11, B at s-i-1 and D at i-i). If inputs are sequences of characters, global patterns can be compared simple regular expressions supporting the logical OR (metacharacter') ('). Based on Eq 3 we have:
global pattern + base expression conjunction predicate predicate conjunction
(6)

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Additional special cases can be pointed out such as the learning of a global pattern over an interval (e.g. B-*-D in window [t-6; 1-3J) or the learning of sequence characteristics dependant expression such as 4 I len(sequence) S 6 based on the sequence length (not shown on Fig but it corresponds a specific case where the base model has learned an always true rule). Also, is important note that base expression and conjunction in both grammars are bounded by the fixed window size I.
To ensure full equivalence between the model and rule, sequences boundaries need to be considered, especially for global patterns. All sequences padded both ends with sequence of 0 of size [- (not shown for simplicity on Fig 2 Also sequences different lengths are supported by creating model based on the maximal available sequence length M in the data and padding shorter sequences with sequence of of appropriate length. ConvOR layer input size is then M + 1.
With this one architecture we can model both local and global patterns. However for optimization reasons detailed below, we choose to differentiate the two into two distinct models: a local and a global model. The ConvOR layer weights for the global model are set and fixed to during training.
4 TRAINING STRATEGY
To overcome training challenges attributed to binarized neural networks (Geiger & Team 2020), we use latent weights and enforce sparsity dynamically. We define loss function that penalizes complex rules and the model trained via automatic differentiation (Pytorch) with Adam optimizer.
Latent weights The binary model parameters introduced above (Wands W W Wstack W Woone) are trained indirectly via the training of continuous parameter loc which is activated (binarized) by a sigmoid function (Kusters al., 2022). With such binary weights and continuous relaxation Biq [ and 2 are differentiable with nonzero derivatives (Kusters et al. 2022). As opposed to when ensured during the
using straight through estimator (Qiao al.) 2021). non-zero gradients backward pass. To overcome training limitations, we use hard concrete distribution (Qiao et al.,
2021 Louizos et al. 2018). It rescales the weights and the random variable introduced during taining prevents from obtaining local minima (Appendix BB. Weight values are in [0, 1] during training. while for testing and rule extraction, a Heaviside is applied them (2 0.5) to ensure striet binarization.
Loss function We define the loss function composed of a mean suuunn-qurrdddde error component along with a regularization term that penalize the complexity of the rule,
L Lmse AII
That regularization term 11, or penalty, evaluates the number of terminal conditions in the rule. In practice use A 10-5 For layer, of input size and output size O, the numher of terminal conditions per output corresponds to the weighted sum of the number of terminal conditions of each output of the previous layer i.e. Iayyer E, W Wayer, Iayerr,.,, a vector of size O. For the first layer, the StackedOR layer, Iateck is defined as the sum over the input dimension of the weights and we can then express the number of terminal conditions of the base rule model llaase.
Σ W stakk
II Hayer stack
Hhase Σ Wor Σ
Wand WaneeTsteck
(8)
W
For optimizing II for local patterns, have minimize the activated ConvOR layer weights. For global patterns, we want them to activated. condition could be set on the sum of ConvOR layer weights (Eq(9) to shift from one optimization problem to the other but with loss of continuity and thus differentiability (interesting values of being M + I-- 1. the ConvOR layer input size, that would correspond all ConvOR layer weights being equal to I, or M I + that would allow for 2(1- 1) weights to be 0, and corresponds to the padding required for properly accounting for sequence boundaries (Section 3)
IIcead Ihese Σ Weome come,
Igiotal Imases
II
IIgtobal Hocca
if Σ W cone 2
(9)
otherwise

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Table 1: Ground truth applied on sequences of letters (A to F) to generate synthetic unbalanced datasets 1, 2, 3 and 4 along with the distributions of the positive class. In the patterns, f refers the position when the last observation truth are generated and are referred to as the dataset number followed by the letter b (Appendix (D)
sequence was made. Balanced datasets with same ground
Ground truth
Distribution (%)
Cat t-4
14.2
A at t-6 and C. t-4
2
1.5
(A at t-6 and C at t-4) (B at t-5 and C. at t-3)
3
3.6
B-D in sequence
4
20.4
Due continuity of II' in Eq 9 we choose to have models with the same architecture for the two cases: the local and the global model respectively more relevant for their associated pattern. For the local model, all weights are trainable and II = Hostt For the global model, weights in the ConvOR layer fixed and set and
gtobbal
Enforced sparsity Sparsity of the model is crucial learn concise expressions, the model needs to generalize without observing all possible instances at training time. The first requirement for that matter is sparsity in the base rule model. In addition to the regularization term in the loss function, we propose use sparsify-during-rraiie
method (Hoefler et al. (02T) and dynamically enforce sparsity in weights from 0% to an end rate ry set to 99% in our case (Lin et al. 22020). gparsify-during-traie
method can also benefit the quality of the training in terms of convergence by correcting for approximation errors due to premature pruning in early iterations but is highly dependant on the sparsification schedule (Hoefler et al. 2021).
Every 16 iterations and for a total of training iterations, every trainable weight is pruned with a binary mask, m (of size of its associated weight and applied with Hadamard product (o)) (Lin al.] 2020 Zhu & Guptal 2017). We propose a mask based on maximum of weight magnitude loe and pruning rate (Zhu & Gupta 2017) making the assumption that it contributes to generalization (Eq 10P. This strategy
be more aggressive than state-of-art contributions (Lin et al. 2020) due
its dependency the loe maximum value. During training, model with the highest prediction accuracy on validation dataset and the highest sparsity (evaluated at each epoch) is kept.
T= Ty
m.m loccsl 2 mlxx),
W
W Wom
(10)
Additional training optimizations have been tested out such for example using binarized opti-
mizer (Helwegen et 2019 Geiger & Team (2200), adding scheduled cooling on the sigmoid
the binarized weights, alternating the training of each layer every few epochs (Qiao et al. 2021) or
using learning rate scheduler. Those techniques are not presented here but would be of interest for improving results on specific datasets.
5 EXPERIMENTS
In order to evaluate the validity and usefulness of this method, we apply it both synthetic datasets and UCI membranolytic anticancer peptides dataset (Grisoni et al. 2019 Dua & Graff] (2177).
Synthetic Datasets We propose 8 synthetic datasets based on ground truth expressions in both balanced and unbalanced distributions for discovering simple binary classification rules with local global patterns as shown in Table There are 1000 sequences of letters (A to F) of different length from 4 to 14 letters in each of them (Mean around 3). Generation is detailed in Appendix D
Besides the synthetic datasets, real-world UCI anticancer peptides dataset com- posed labeled one-letter amino acid sequences, is used (Grisoni al., (2019 Dua & Graff] 2017). The multi-classification problem is transformed into binary classification problem is the same manner as Nwegbu et al. (2022) (see Appendix C). Sequence length are from 5 to 38 letters (Mean: 17 + 5.5) and positive class distribution 79%.
Peptides Dataset

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Experimental Setting All datasets are partitioned in stratified fashion with 60% for training. 20% for validation and 20% for testing datasets and we use a batch size of 100 sequences. The hidden size in the base rule model is set to the double of the input size of the AND layer (which is the window size of the convolution). More details on experimental setting can found Appendix At each epoch (200 total), we evaluate the model against the validation dataset and keep the model with the highest accuracy and in case of equality the model with lowest penalty. For each experiment, we run the algorithm 10 times with different weights initializations. Resulting metries are averaged over these runs.
We rum the experiments with two different window sizes (3 and 6) for the CR2N convolution filter size. We compare the two versions of the architecture: the local and global models described in Section 4 and study three different dynamic pruning strategies: none, dynamic enforced sparsity from epoch=0 and from epoch=30 (arbitrary).
RESULTS
(a) w pruning-No No
(b) window=6, pruning-No No
Balanced Acmracy
1.0
200
Penalty lty
0.8
100
0.6
global
lncal
global
local
Model
Model
(c) moddl=loal, pruning=No
(d) windoo=66, model=local
Balanced Accuracy
1.0
Best Epoch
150
100
0.6
0-
No
Yes
30
Window Size
Pruning Strategy
(e) window=6, model=local
(f) window=6, model=local
Balanoed Accuracy
200
1.0
Penalty
100
0.8
0.6
No
Yes
30
No
Yes
30
Pruning Strategy
Pruning Strategy
Dataset
2
3
unbalanced
balanced
Figure 3: Representations key results obtained on the synthethic datasets. Error bars represent the standard deviation over the 10 executions with different weights initializations. Full results are available in Appendix

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Table 2: Performance metrics obtained for the different models, window size and pruning strategy on the peptides dataset, along with the standard deviations over the 10 executions with different weights initializations. (Bal. Acc.: balanced accuracy, Epoch: best epoch).
Model
Window
Pruning
Accuracy
Bal. Acc.
Penalty (II)
Epoch
No
88.9 + 5.0
75.3 + 12.7
58.7 + 35,9
105 = 60
3
Yes
85.3 1 5.3
67.3 1 14.3
19.4 1: 15.9
23 20
30
88.7 = 5.0
75.4 + 12.9
46,9 + 28.0
48 +: 22
global
No
91.2 + 1.0
81.8 2.0
220.0 +: 56.5
92 +: 82
Yes
89.5 + 3.6
77.6 9.3
132.7 93.9
33
30
91.0 + 1.3
82.0 2.8
97.3 + 60.6
71 + 19
No
90.0 + 3.7
78.7 9.7
694.3 + 269.3 3
77 + 48 25 1:
3
Yes
90.3 + 1.5
81.6 + 1.6
885.2 1: 328.7
30
89.2 + 5.2
76.3 + 13.2
674.7 + 483.9
49 + 13
local
No
92.1 +
83.5 4
3,9k 1.4k
91 + 58
Yes
88.0 1 4.7
74.6 12.4
1.0k + 1.0k
91.7 +
83.5 2.6
1.8k + 0.6k
49 10
Rule grammar and expressivity The importance of the rule madel expressivity can be seen con- cretely by comparing the different patterns the local and global models have learned for dataset 3b for example: 1. (A or B)-=-((C-2_.-(A or B or C or D or E or F) in sequence (global, no pruning. window size=6), and 2. (B at 1-5 and C at t-3) (A at 1-6 and C at 1-4) (local, pruning, window size=6). In the first case, the grammar is not appropriate to model the data (as a reminder, the global model is constrained version of the local model) as opposed to the local model that learned the perfect rule. In practice, on real data, obtained patterns, such "if (D or E or G or H or I N or T or +)=+.+++-(D or E or G or I or N or Q or S or T or or Y) in sequence" obtained for labelling -urttid-virtual' peptides, can be explored further by a domain expert. Black box approaches do not provide such insights.
is also highlighted by comparing experiments with local or global model and experiments with different window sizes. First of all, the accuracy of the local model is higher compared to the global model on balanced synthetic datasets 1, 2 and 3 (Fig 3 (a). For balanced and unbalanced dataset 4, both models achieve very high accuracies (> 95%) %). However, as shown on Fig Bb), it the cost of rule complexity for the local approach with averaged penalty values higher than 60 (and standard deviation higher than 50) compared to lower than 10 for the global model (and standard deviation lower than 5). It points out that the local model in that case requires on average at least more than 6 times more terminal conditions in the learned rule than the global model for comparable accuracies, but also that the weights initial states have huge impact the rule complexity when the rule grammar not expressive enough (with no pruning). Those results are confirmed on real-world dataset with the peptides dataset, accuracies between the local and global models especially for a window size of 6 are comparable. However there is an order of magnitude difference for the penalty, global approach being more concise. It is important to note that by architecture the global approach has less weights to train and thus a much lower maximum penalty.
Datasets 2b and 3b benefit from a bigger window size (highly expected for dataset 3/3b due to ground truth pattern size) as shown in Fig c). Accuracies also higher with window size 6 than 3 for the peptides dataset the cost of also higher penalties (Table
The more expressive the model is i.c. the more patterns can model, training limitations aside, the better for the performance. Of course any black box neural network with no such 1-1 rule mapping constrained architecture would reach 100% accuracy, but it is that mapping in particular that makes the model relevant, expressive and fully interpretable. Also, the best performances in accuracy for the peptides dataset (~~ 91%) are comparable to the best results (~ 92%) obtained from classification with single kernels when applied that same dataset in Nwegbu al. 2022). our model providing an additional inllreinterreetable property. The presented model is also flexible due to its logical equivalence and can be inputted into other logical layers for deeper architectures to extend the rule
grammar (Beck & Fürnkranz 2021). p. It can also he extended for timeseries, temporal aggregates

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culti-classification problems. Other rule grammar extensions can be inspired by Linear Temporal Logic domain and regular expression pattern mining (De Giacomo et al. 2022). However the more expressive the model is the more attention is required for training and rule complexity.
Sparsity and training strategy The importance of the model sparsity is pointed out by the exper- iments with different pruning strategies. First, looking at training scenarios, both on synthethic and peptide datasets, experiments with parsity-during during-traainng approaches reach the best model faster on average than without (lower best epoch Fig (()). Then, regarding the performance terms accuracy, we can differentiate two cases: balanced and unbalanced datascts. Training of unbalanced datasets is more affected by the aggressive dynamic pruning strategy than balanced datasets with a drop in average of around 0.2 accuracy for dataset compared to an equivalent accuracy for balanced dataset I for example (e)). The pruning strategy starting after 30 epochs is preferred in both cases. Average accuracies with pruming strategy not starting immediately (30 epochs) are comparable to the ones obtain without pruning balanced datasets. In terms of rule complexity. penalty values are lower with pruning and even lower when starting after 30 epochs in most cases (Fig
With our pruning strategy (Eq T0p. we make the assumption that lower positive loe values are asso- ciated to overfitting or redundancy by taking into account that values closer to 0, i.c. on the sigmoid slope, are more likely to shift thus less "certain'. As pointed out in early work by Prechelt| (1997). the dynamic pruning strategy helps overcome possible lower generalization ability compared fixed pruning which could explain cases of better performance (peptide dataset local model window size of 3 for example). Prechelt proposed different pruning strategy based on generalization loss characterize the amount of overfitting. When this strategy relevant in more general cases and can be applied to many different networks, our strategy is tailored for minimizing positive trainable parameter values.
Sparsity of the model is also induced via the regularization term II in the loss function L (Eqq While this method is parameterized with relative importance of sparsity for training optimization and provides an uncontrolled target sparsity, a dynamic pruning strategy is easier to control for both target sparsity and accuracy but is highly dependent on the pruning schedule (Hoefler al. 202T).
An interesting point is made by Hoefler et al. (2021) about the convolutional operator that "can be seen as a sparse version fully-connected layers'. That level of forced sparsity in our model therefore defined by the fixed window size model parameter with respect to the maximum sequence length. The ideal sparser window size would be the size of the maximum temporal hidden pattern distribution that can only be approximated with external or expert knowledge and/or tuned with trial and error.
With or without dynamic pruning strategy, for highly unbalanced datasets (2 and 3), experiments have shown that the training strategy of the model is not suitable. Indeed most of them, label everything with the majority class (50% balanced accuracy). It corresponds to the specific case of learning an empty rule (penalty=0) (Fig 3 (,c,e)). For unbalanced datasets I and 4, their best models do not reach on average the same accuracies as in their balanced versions.
Overall this training strategy is both the key and the main limitation our approach: it can provide a sharp concise rule with minimal redundancy and simplified logical expression but it is highly dependent numerous model, training and pruning parameters and is not suited as is for highly unbalanced datasets.
7 CONCLUSION
To conclude, we presented a ID-convolutional neural architecture to discover local and global pat- terns in sequential data while learning binary classification rules. This architecture is fully differ- entiable, interpretable and requires sparsity that enforeed dynamically. One main limitation is its dependence to the window size and sparsity scheduler parameters. Further work will consist in integrating this block into more complex architectures to augment the expressivity of the learned rules well extending it for multi-classification.

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AUTHOR CONTRIBUTIONS
M.C. and R.K. designed the model. R.K. encouraged M.C. to investigate the use of convolutions and supervised the findings of this work. P.B. and F.F. helped supervise the project. M.C. vali- dated the training strategy and carried out the implementation and the experiments. M.C. wrote the manuscript. All authors provided critical feedback and helped shape the manuscript.
ACKNOWLEDGMENTS
We would like to thank Shubham Gupta for helpful discussions and constructive feedback well Yusik Kim for reviewing the manuscript. This work has been partially funded by the French government as part of project PSPC AIDA 2019-PSPC-09, in the framework of the "Programme d' Innvestissseent d' Avenir".
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Francesca Grisoni, Claudia Neuhaus, Miyabi Hishinuma. Gisela Gabernet, Jan A. Hiss, Masaaki Kotera, and Gisbert Schneider. De novo design of anticancer peptides by ensemble artificial neural networks. Journal of Molecular Modeling, 12, April 2019. ISSN 0948-5023. doi: 000777/00089--19-407777
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Cheng Zhou, Boris Cule, and Bart Goethals. Itemset Based Sequence Classification. In David Hutchison, Takeo Kanade, Josef Kittler, Jon M. Kleinberg, Friedemann Mattern, John C. Mitchell, Moni Naor, Oscar Nierstrasz. C. Pandu Rangan, Bernhard Steffen, Madhu Sudan, Demetri Ter- zopoulos, Doug Tygar, Moshe Vardi, Gerhard Weikum, Camille Salinesi, Moira Norrie, and Óscar Pastor (eds.), Advanced Information Systems Engineering, volume 7908, pp. 353- 368. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. ISBN 97--3-62--3888708 978-3-642- 38709-8. doi: 10.
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A CONTEXT-FREE GRAMMAR
Context-free grammar (Chomsky 1956) (Vr, Vx, I)) where
A context free-grammar is 4-tuple G
Vr, finite set of terminals or terminal elements the language that form the alphabet of the language.
VN, disjoint from Vr, finite set non terminal elements (variables) that define a sub- language of L. We note V = Vx U Vy, the vocabulary of the grammar.
S E VN. the start symbol or variable that defines the whole sentence.
R is finite set of rules production rules of the form A + with A E Vv and V
In the following, we have 3 different types of terminal elements on the syntax level:
reserved words, distinguished with the following style reserved
signs, such as for example -, *....
other terminal elements that defined prior to the grammar, distinguished with the fol- lowing style: terminal
Production rules presented in the paper (Eq 3 Bq E 5 and Eg define grammars when associated values for Vr, VN and S.
Here is example for the base rule model grammar with production rules in Eq
Vr (if, then class else class 0, V,
Tez ob ob...... ......... Teg o . Pex 0mgg
Vy = (rule, base expression, conjunction, predicate, categorical expression, categorical literal, literal)
(1I)
S
(rule)
B HARD CONCRETE DISTRIBUTION (LoUizOS ET AL. 2018)
Parameters follows: 3 = 2/33, Ç 1.1 and -0.1.
(O,, 1), s ==o(((((u)) log(1 u) + loc)/ (3): 8* ())
+7
(12)
min(max(8,,);
(13)
C PEPTIDES DATASET
com- posed one-letter amino acid sequences (of variable length) and each sequence is labeled with its anticancer activity on breast cancer cell lines. The dataset provides classes with the following dis- tribution: 83 inactive-exp. 750 inactive-virtual, 98 moderately active and 18 very active. Sequences
UCI anticancer peptides dataset Grrssomi al. 2019) (Available on Dua & Graff 201
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lengths range from 38 letters (Mean: + 5.5). We transform this multi-classification problem into binary classification problem (as done in Nwegbu et al. (2022). Class 'inactive-virtuall is the positive class (750) and all the other are combined as the negative class (199). No other processing of the data is necessary and we leave is.
D SYNTHETIC DATASETS GENERATION
Balanced datasets are generated randomly with the same ground truth as unbalanced datasets. Then, they are upsampled until the minority class represents half of the goal dataset size and appropriate number of majority class are randomly removed.
EXPERIMENTAL SETTING
The foc-parameters for weights computation are initialized with xavier uniform initialization method (Glorot & Bengio 2010). The loss function described in Eq and depends the MSE loss and regularization coefficient
10-3, The adam optimizer is used with a fixed learning rate set to 0.1 and run consists of 200 epochs.
CPU MacBookPto18.22 (2021) with Apple MI Max chip, 10 Cores, 32 GB RAM and running macOS Monterey Version 12.4.
Experiments
F FULL EXPERIMENT RESULTS
Table 3: Performance metrics obtained for the different models, window size and pruning strategy on the synthetic datasets. Values are followed by the standard deviation over the executions with different weights initializations. (Bal Ace: balanced accuracy, Epoch: best epoch).
Dataset
Model
Window
Pruning
Bal. Acc.
Penalty
Epoch
No
50.0 0.0
0.0 + 0.0
3 +
Yes
50.0
0.0 + 0.0
3
30
50.0 + 0.0
0.0 = 0.0
2
global
No
54.9 + 3.9
9.1
109 + 64
Yes
50.0 + 0.0
0.0
1
30
50.0 0.0
0.0
1
No
100.0 0.0
1,8 = 1.5
63 + 24 47
Yes
90.0 20.0
0.8 0.4
30
85.0 + 22.9
0
41 27
local
No
0.0
5.8 5 5.6
96 50
Yes
80.9 - 23.6
1.1 1.4
26 + 18
30
95.0 + 15.0
0.9 + 0.3
79 + 43
No
70.7 1.7440
5.1 1.6
81 + 27
Yes
68.5
5.8 3.8228
47
30
71.6
5.0 + 1.3
60 + 16
global
No
76.5
37.4 20.3
108 + 49
Yes
70.2 1.244
12.5 5.5
29 + 15
1b
30
5.1
15.4 6.9
53 13
No
100.0 0.0
3.0 = 3.1
70 - 47
Yes
100.0
1.0
49 39
100.0 0.0
1.0
54
local
No
100.0 0.0
12.6 10.1
116 52
Yes
100.0 0.0
1.0
77 24
30
100.0
1.0
87 + 27

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Table 3 continued from previous page
Dataset
Model
Window
Pruning
Bal. Acc.
Penalty
Epoch
No
50.0 + 0.0
0.0 = 0.0
3 +
Yes
50.0 0.0
0.0 = 0.0
3
50.0 0.0
0.0
2
glohal
No
50.0 0.0
0.0
1
Yes
50.0 + 0.0
0.0
1
2
30
50.0 + 0.0
0.0
1
No
50.0 + 0.0
0.0 0.0
1
Yes
50.0 + 0.0
0.0
1
50.0 0.0
0.0 =
I
local
No
50.0 0.0
0.0 0.00000
1
Yes
50.0 0.0
0.0 0.0
1
30
50.0 + 0.0
0.0 = 0.0
140
No
90.5 + 0.0
3.7 # 2.1
44 + 37
Yes
85.7 9.8
2.3 0.6
39 21
90.5 0.0
2.0 0.0
+ 24
global
No
94.2 + 0.6
19.8 7.7
89 + 36
Yes
90.5 + 0.0
2.0 0.0
71 16
2b
30
91.0 1.5
12.4 13.0
61 + 26
No
2.6 97.9 2.6
2.8 = 1.2
58 42
Yes
97,4 + 2.6
1.5 0.5
43 + 28
30
98.0
1.5
54 + 29
local
No
100.0
1.45
105 + 56
Yes
100.0 0.0
0.0
30
100.0 0.0
2.4 1.2
68 + 22
No
50.0
0.0
2
Yes
50.0 +.0+0000
0.0
2
30
50.0 + 0.0
0.0
2
global
No
50.0 = 0.0
0.0
1
Yes
50.0 0.0
0.0
1
3
30
50.0 + 0.0
0.0 0.0
1+0
No
50.0 0.0
0.0
1
Yes
50.0 000+00 0.0
0.0
2
30
50.0 + 0.0
0.0 + 0.0
1
local
No
50.0 + 0.0
0.0
140
Yes
50.0 1.0
0.0
1
50.0 - 0.0
0.0
2 1
No
79.8 8.1
4.6
98 51
Yes
70.5 8.1
3.1
48 31
30
80.4 9.2
0.6
57 36
global
No
90.2 + 1.4
26.2 = 15.3
120 + 51
Yes
87.7 3.1 77.4
14.1 8.0
35 13
3b
11.5
61 10
No
89.8 - 6.3
3.9
95 64
Yes
86.2 4 8.0
6.1282
50 10420 34
30
92.5 1. 4.6
4.2 1.8
64 + 22
local
No
97.1 1.2
15.1 7.9
123 + 49
Yes
95.7 + 2.9
10.1 12.9
43 + 8
96.8 0.9
7.6 + 2.9
69 +

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Published conference paper at ICLR 2023
Table 3 continued from previous page
Dataset
Model
Window
Pruning
Bal. Acc.
Penalty
Epoch
No
100.0 + 0.0
2.2 = 0.6
42 + 28
Yes
90.0 20.0
1.6 + 0.8
28 + 14
95.0 - 15.0
1.8 0.6
42 + 25
global
No
100.0 1000000
3.4
72
Yes
21.
1.7
33 + 20
4
30
100.0 + 0.0
2.0
43 + 15
No
98.7 + 0.4
28.6 14.5
108 40
Yes
59.1 12.9
3.4
30 37
77.1 20.7
12.5 8.7
58
local
No
99.1 0.6
68.8 51.0
109 31
Yes
52.1 5.2
3.3 = 5.2
11 13
30
79.3 + 19.2
21.9 17.5
57 + 24
No
100.0 0.0
2.7 + 1.1
16 + 8
Yes
99.2 2.4
2.1 0.3
29 + 18
100.0 + 0.0
2.0 0.0
42
global
No
100.0 +
6.7 3.8
90 52
Yes
99.2 2
4.7
50
4b
30
100.0 + 0.0
69 + 25
No
100.0 - 0.0
=
76 38
Yes
99.0 2.0
29.2 = 9.4
60 + 14
30
98.7 4.0
35.9 17.3
53 14
local
No
100.0 0.0
122.6 = 100.2
107 42
Yes
88.2 7.8
103.9 + 77.3
44 16
30
98.0 +.9
79.0 55.9
68 + 12
15


Reviews:
[{"review_id": "nTysN4tASj", "reviewer": "Reviewer_JYCZ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Learning rules is difficult and often lead to low performances. From the computational point of view, it poses a problem of differentiability. This article propose to learn both the discriminant pattern and explicit decision rules.\nThe authors first insist on the cabablities associated with a 3 layer rule network. Then, they describe the convolution layer which applies rules on the discrete signal. Section 3 explains how the architecture is learnt. Then, the authors conduct experiments on synthetic and real discrete signals.", "summary_of_the_review": "The idea of the article is really interesting but the whole model description (section 3), in particular the learning process, is impossible to understand (at least for a reviewer that is not familiar with rule neural networks). For this reason, I can't accept this article.\n\n\n**AFTER REBUTAL**\n\nThe paper has been significantly improved by the authors. The procedure is much more detailed than in the original version and an anonymous repository has been made available. This encourage me to change my ratings.\n\nI am not familiar enough with the previous contribution in the field to assess reliably the novelty of the proposal.\n", "strength_and_weaknesses": "**positive**\n\nArchitecture with potential for xAI\n\nA very interesting way to learn rules with gradient descent\n\n**negative**\n\nn section 3, the authors should define the notation and matrix dimensions: I don't understand (6) and (7).\nI can't understand the regularization procedure.\n\nI am not familiar with logical networks, but I don't get how the author make the architecture differentiable\n\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "iNwFhZJ9Yt", "reviewer": "Reviewer_r7kZ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes a method based on a binary neural network to learn classification rules from sequential data. The model contains a stackedOR layer for categorical variables, a logical AND layer and an OR layer. The training strategy includes dynamically-induced sparsity.  \n", "summary_of_the_review": "The method is sound but some points should be clarified. ", "strength_and_weaknesses": "Strengths. The paper introduces a method that can be useful for real applications. The paper is well-written. \n\nWeaknesses. Comparison to state-of-the-art methods are missing. \nThe local and global patters are not so easy to understand. In general, in my opinion, Section 2.2 is not easy to follow. \n\nTwo window values (for sequences), namely, 3 and 6 were tested. It is difficult to make strong conclusions from results based on these two values only. It is also mentioned that it is a good idea to take a window size which equals to the maximum length. First, I wonder whether if such a big window is taken, the model would suffer from overfitting. Second, it is indeed interesting to take quite a big window but also include into the model some shorter windows (similarly to n-gram models). Third, windows of various lengths (performing data segmentation) would be an optimal choice.  \n\nSpeaking about interpretability: what kind of rules were learnt? \nI did not really see that the interpretable aspect of the method was validated, i.e., that some rules were extracted and considered/validated by human experts. \n\nOther remarks. In Section 3, \"the model is trained via automatic differentiation\", and it is also mentioned that the losses are differentiable. Why do you prefer automatic differentiation if the first derivative can be computed?   \n\nIn Section 3, the paragraph \"Loss function\" is not very easy to understand: why there is product of three \\Pi matrices? Is the multiplication coordinate-wise, if yes, do they have the same dimensions? What is \\Pi_{stack} and why the sum is taken over \\Pi_{stack} matrices?\n\nIt is mentioned that both balanced and unbalanced data sets are considered. How the data sets are balanced? I did not find any information on whether some undersampling or weighted loss was used. \n\nIn Table 2: what the column \"Penalty\" contains?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "Kf8YGSoFSc", "reviewer": "Reviewer_UfDD", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes a convolutional binary neural network to learn binary classification rule for sequence data. The learned rule is able to learn global/local pattern for better explainability. Furthermore, the authors propose a training strategy with latent weights and regularization term penalizing the complexity of the rule, and dynamically enforces the sparsity using sparsity-during-training method. Experiment results on 4 synthetic datasets and one real dataset are shown to exhibit the effectiveness of the proposed method.", "summary_of_the_review": "Overall I think the paper contributes a novel method for learning rule with sequence data, and recommend for acceptance. ", "strength_and_weaknesses": "Strength\n1. The paper improves on the base rule model by operating on sequence data and could learn global/local pattern\n2. The training strategy on penalizing the complexity of the rule and sparsify-during-training is useful and could benefit other rule learning algorithm  \n\nWeaknesses\n1. The proposed method does not work for unbalanced data for learning an empty rule. I am wondering if up/down-sample the positive/negative class helps\n2. The paper could be polished in terms of clarity. For example, the authors should explain the notations for grammar in equations 3-5 for better presentation. Equation 9 is also not explained with the motivation on using such form for the rate.\n3. There is no experiment results on explainability on real dataset. The paper will be stronger and better motivated with such a demo.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster