Controlling Neural Networks with Rule Representations
Authors: Sungyong Seo
Sercan O Arik
Jinsung Yoon
Xiang Zhang
Kihyuk Sohn
Tomas Pfister
Venue: NeurIPS 2021 Poster (2021)
Keywords: Controlling neural networks
Incorporating rules
Unsupervised adaptation

Abstract: We propose a novel training method that integrates rules into deep learning, in a way the strengths of the rules are controllable at inference. Deep Neural Networks with Controllable Rule Representations (DeepCTRL) incorporates a rule encoder into the model coupled with a rule-based objective, enabling a shared representation for decision making. DeepCTRL is agnostic to data type and model architecture. It can be applied to any kind of rule defined for inputs and outputs. The key aspect of DeepCTRL is that it does not require retraining to adapt the rule strength -- at inference, the user can adjust it based on the desired operation point on accuracy vs. rule verification ratio. In real-world domains where incorporating rules is critical -- such as Physics, Retail and Healthcare -- we show the effectiveness of DeepCTRL in teaching rules for deep learning. DeepCTRL improves the trust and reliability of the trained models by significantly increasing their rule verification ratio, while also providing accuracy gains at downstream tasks. Additionally, DeepCTRL enables novel use cases such as hypothesis testing of the rules on data samples, and unsupervised adaptation based on shared rules between datasets.

Controlling Neural Networks with Rule Representations
Sungyong Seo, Sercan Ô. Arik, Jinsung Yoon, Xiang Zhang, Kihyuk Sohn, Tomas Pfister
Google Cloud Al Sunnyvale, CA, USA
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Abstract
We propose novel training method that integrates rules into deep learning, in way the strengths of the rules are controllable at inference. Deep Neural Networks with Controllable Rule Representations (DeEPCTRL) incorporates rule encoder into the model coupled with rule-based objective, enabling a shared representation for decision making. DEEPCTRL is agnostic data type and model architecture. It can be applied to any kind of rule defined for inputs and outputs. The key aspect of DEEPCTRL is that does not require retraining adapt the rule strength inference, the user can adjust it based on the desired operation point on accuracy vs. rule verification ratio. In real-world domains where incorporating rules is critical such as Physics, Retail and Healthcare we show the effectiveness of DEEPCTRL in teaching rules for deep learning. DEEPCTRL improves the trust and reliability of the trained models by significantly increasing their rule verification ratio, while also providing accuracy gains at downstream tasks. Additionally, DEEPCTRL enables novel use cases such as hypothesis testing of the rules on data samples, and unsupervised adaptation based on shared rules between datasets.
Introduction
Deep neural networks (DNNs) excel at numerous tasks such as image classification 28 machine translation (22 30), time series forecasting [1] [21], and tabular learning [ DNNs get more accurate as the size and coverage of training data increase [17]. While investing in high-quality and large-scale labeled data is one path, another is utilizing prior knowledge concisely referred to as 'rules': reasoning heuristies, equations, associative logic, constraints or blacklists.
most
scenarios, labeled datasets are not sufficient to teach all rules present about task [2 23 24 Let us consider an example from Physics: the task of predicting the next state in a double pendulum system, visualized in Fig. [ Although 'data-driven' black-box model, fitted with conventional supervised learning, can fit a relatively accurate mapping from the current state to next, it can easily fail to capture the canonical rule of 'energy conservation' In this work, we focus on how to teach 'rules' in effective ways so that DNNs absorb the knowledge from them in addition to learning from the data for the downstream task.
The benefits of learning from rules are multifaceted. First, rules can provide extra information for cases with minimal data supervision, improving the test accuracy. Second, the rules can improve trust and reliability of DNNs. One major bottleneck for widespread use of DNNs them being "black-box', The lack of understanding of the rationale behind their reasoning and inconsistencies of their outputs with human judgement often reduce the trust of the users 26] By incorporating rules, such inconsistencies for loan delinquency prediction can absorb all the decision heuristics used at a bank, the loan officers of the bank can rely the predictions more comfortably. Third, DNNs are sensitive slight changes
be minimized and the users' trust can be improved. For example, if a DNN
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Conventional learning
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Figure 1: Overview of DEEPCTRL. When DNN is trained only with the task-specific objective (in this example, predicting next vositions//eloccties of two objects connected in a pendulum), it may easily violate the rule (E, E.+1) that it must have followed according to the energy damping rule from physics (top graph). DEEPCTRL (with outputs shown via blue arrows) provides a controllable mechanism that enables the rule dependency to be adjusted at inference time in order to achieve an optimal behavior (middle graph) in regards to accuracy and rule verification ratio. With increased rule strength, DEEPCTRL yields operation point (bottom graph) where is satisfied for all time steps.
to the inputs that are human- -impreiipprreppibble [15 20 31. With rules, the impact of these changes can be minimized as the model search space is further constrained to reduce underspecification 10
When 'data-driven' and 'rule-driven' learning are considered jointly, a fundamental question is how to balance the contribution from each. Even when rule is known hold 100% of the time (such as the principles in natural sciences), the contribution of rule-driven learning should not be increased arbitrarily. There is an optimal trade-off that depends not only on the dataset, but also on each sample. If there are training samples that are very similar to a particular test sample, a weaker rule-driven contribution would be desirable at inference. On the other hand, if the rule is known to hold for only subset of samples (e.g. in Retail, the varying impact of a price change on different products (6). the strength of the rule-driven learning contribution should reflect that. Thus, framework where the contributions of data- and rule-driven learning can be controlled would be valuable. Ideally, such control should be enabled at inference without the need for retraining in order to minimize the computational cost, shorten the deployment time, and to adjust to different samples or changing distributions flexibly.
In this paper, we propose DEEPCTRL that enables joint learning from labeled data and rules. DEEPCTRL employs separate encoders for data and rules with the outputs combined stochastically to cover intermediate representations coupling with corresponding objectives. This representation learning is the key to controllability, as it allows increasing decceasin the rule strength gradually at inference without retraining. To convert any nin-differentiable rules into differentiable objectives, we propose novel perturbation- based method. DEEPCTRL is agnostic to the data type or the model architecture, and DEEPCTRL can be flexibly used in different tasks and with different rules. We demonstrate DEEPCTRL on important use cases from Physics, Retail, and Healthcare, and show that it: (i) improves the rule verification ratio significantly while yielding better accuracy by merely changing the rule strength at inference; (ii) enables hypotheses to be examined for each sample based on the optimal ratio of rule strength without retraining (for the first time in literature, to the best of our knowledge); and (iii) improves target task performance by changing the rule strength, a desired capability when different subsets of the data are known to satisfy rules with different strengths.
Various methods have been studied incorporate 'rules' into deep learning, consid- ering prior knowledge in wide range of applications. Posterior regularization [3] is one approach to inject rules into predictions.
Related Work:
[8], a teacher-student framework used, where the teacher network is obtained by projecting the student network to a (logic) rule-regularized subspace and the student network updated to balance between emulating the teacher's output and predicting the labels. Adversarial learning is utilized in (32), specifically for bias rules, to penalize unwanted biases. In [2]

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Figure 2: DEEPCTRL for controllable incorporation of a rule within the learning process. DEEPC- TRL introduces two passages for the input-output relationship, a data encoder and rule encoder, that produce two latent representations 2, and zæ These representations are stochastically concatenated with a control parameter a into a single representation z. z is then fed into a decision block with objectives for each representations, Crule and Ltask, again weighed by the control parameter a. is randomly sampled during training and set by users at inference to adjust the rule strength.
framework is proposed that exploits Lagrangian duality train with rules. In (24) learning with constraints is studied, via formulation over the space of confusion matrices and optimization solvers that operate through a sequence of linear minimization steps. In 27 constraints are injected via KL divergence for variational autoencoders, for controlling output diversity or disentangled latent factor representations. DEEPCTRL differentiate from the all aforementioned in how injects the rules, with the aspect that it allows controllability of the rule strength at the inference without retraining, enabled by accurate learning of the rule representations in the data manifold. This unlocks new capabilities, beyond simply improving rule verification for target accuracy.
2 Learning Jointly from Rules and Task
Let us consider the conventional approach 9 12 14 for incorporating rules via combining the training objectives for the supervised task and a term that denotes the violation of rule:
Lraek + XLrule:
(1)
where is the coefficient for the rule-based objective. There are three limitations of this approach that we aim to address: (i) À needs to be defined before learning (e.g. can be a hyperparameter guided with validation score), (ii) is not adaptable to target data inference if there is any mismatch with the training setup, and (iii) Frate needs to be differentiable with respect to learnable parameters.
DEEPCTRL modifies canonical training by creating rule representations, coupled with data repre- sentations, which is the key to enable the rule strength to be controlled at inference time. Fig. 2 overviews DEEPCTRL and Algorithm (errithhmmmdssaa describes its corresponding training process. We propose to modify the canonical training approach by introducing two passages for the input-output relationship of the task, via data encoder da and rule encoder Ór. In this way, our goal make each encoder individually learn the latent representations (zu and z,), corresponding to extracted information from the labeled data and the rule. Then, the two representations are stochastically concatenated (with the operation denoted as () to obtain single representation z. To adjust the relative contributions of data vs. rule encoding. we use random variable E [0, 1], which also couples (za. 2,) with the corresponding objectives (Ltask, Crute) (Lines 4 & 5 in Algorithm|] D. is sampled from the distribution P(a). The motivation to use random a is to encourage learning the mapping with a range of values, so that at inference, the model can yield high performance with any particular chosen value. The output of the decision block (g) is used the entire objective.
By modifying the control parameter at inference, users can control the behavior of the model to adapt it to unseen data. The strength of the rule on the output decision can be enhanced by increasing the value of o. Setting a = O would minimize the contribution of the rule at inference, but as shown in the experiments, the result can still be better than in conventional training since during the training process a wide range of a are considered. Typically, an intermediate value yields the optimal solution given specific performance, transparency and reliability goals. ensure that model shows distinct and robust behavior when to sample a more heavily at the two extremes rather than uniformly from [0. 1]. To this end, we choose to sample from Beta distribution (Beta 3)). We observe strong results with 3 0.1 in
0 or + and thus interpolates accurately later, we propose
3

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Algorithm Training process for DEEPCTRL.
Input: Training data D = { Vi) 1, N
Output: Optimized parameters
Require: Rule encoder Φ,, data encoder da. decision block o, and distribution P(a).
1: Initialize Φr. Qd and Φ
2: while not converged do
3: Get mini-batch D3 from D and as E R from P(a)
4: Get z = 24 ( (1 - Orbza) where 2, = or (D,) and 2 = (o((D)D)))))))))))
5: Get
(z)) and compute L Pa- (afrule p(1 a)) Crask) where P
Crute,0/ Ctakk0
6: Update Ør, Pd. and o from gradients Vo,L. Vo C. and VoL
7: end while
most cases and in Section 5, further study the impact of the selection of the prior for Since a Beta distribution is highly polarized, each encoder is encouraged to learn distinct representations associated with the corresponding encoder rather than mixed representations. Similar sampling ideas were also considered in [133] to effectively sample the mixing weights for representation learning.
One concern in the superposition of Ctask and Lrute is their scale differences that may cause all learnable parameters be dominated by one particular objective regardless of a, and hence become unbalanced. This is not desired behavior as it significantly limits the expressiveness of DREPCTRL and may cause convergence into single mode. E.g., if Crule much larger than Lrask, then DEEPCTRL will become rule-based model even when is close to O. To minimize such imbalanced behavior, we propose to adjust the scaling automatically. Before starting a learning process, compute the initial loss values Crate.0 and Lrask,0 on training set and introduce scale parameter p = Lrate,0/ Ltask,o. Overall, the DEEPCTRL objective function becomes:
L = El + p(1 Cissk)
(2)
DEEPCTRL is model-type agnostic we can choose and appropriate inductive bias for encoders and the decision block based on the type and task. E.g. if the input data is an image and the task is classification, the encoders can be convolutional layers to extract hidden representations associated with local spatial coherence, and the decision block can be an MLP followed by a softmax layer.
3 Integrating Rules via Input Perturbations
Algorithm requires a rale-based objective Crute that is function of (æx 9) and only differentiable with respect to the learnable parameters of the model. In some cases, it is straightforward to convert rule into a differentiable form. For example, for rule defined as r(æx Ø) S given differentiable function r(-), we can propose Lrule = max(r(z D) 0) that has a penalty with an increasing amount as the violation increases. However, there are many valuable rules that are non-iifferentiable with respect to the input or learnable parameters, and in these cases it may not be possible to define a continuous function Lrute as above. Some examples include expressive statements represented as concatenations of Boolean rules (e.g. fitted decision trees) [8), such as 'The probability of the j-th class Dj is higher when a < Kx (where a a constant and æ& is the k-th feature) or 'The number of sales is increased when price of an item is decreased.".
We introduce an input perturbation method, overviewed in Algorithm to generalize DEEPCTRL to cindiifferentiable constraints. The method is based on introducing small perturbation dæ (Line 4 in Algorithm to input features & in order to modify the original output ý and construct the rule-based constraint Lrute for it. For example, if we were to incorporate the first sample rule in the previous paragraph (concatenations of Boolean rules), we would only consider Tpp as a valid perturbed input when Øg < a and a < F..k and Úp is computed from the Tp. frute is defined as:
Lrate (2,, Pp Пy 9p.) = ReLU - -r.) 1(xx < a) 1( >a).
This perturbation-based loss function is combined with task-based loss function (Line 6 in Algo- rithm 2 to update parameters Φr, tha and Φb. The control parameter a is used for both ý and ÿp- and Ltask only considers the original output ÿ. All learnable parameters in the rule encoder, data

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Algorithm 2 DEEPCTRL via perturbation-basse integration of rules.
Input: Training data D {z; Vi) 1. .N..
Output: Optimized parameters
Require: Rule encoder Φ,, data encoder da, decision block o, and distribution P(a).
1: Initialize Φr, Qd. and Φ
2: while not converged do
3: Get mini batch D from D and aG E R from P(a)
4: Get perturbed input x.p = x + dær where a E "Db
5: Get y and Yp from and pp through Φr, Dd, Ø, and as, respectively
6:
Define Frate = Crate (z. Tp: ÿ. Up) based on a rule and Ltask = Lrast (V. 9) to compute
C CrCCule + (1 ai Ctask
7: Update Ør, Hd, and from gradient V &, V VosC, and VoL
8: end while
34
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Ranthert LOR-NE
0335
DesoCTE
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6391
7.28
(a) MAE
(b) Verification ratio
(c) Energy prediction
Figure 3: Experimental results for the double pendulum task. (Left) Task-based prediction error and (Middle) verification ratio from different models. DEEPCTRL has scale parameter which adjusts the scale mismatch between Ltask and Lrlle. The performances of LDF method are highly sensitive hyperparameter and its output t + 1, respectively. According to the energy damping rule, Pe+1 should not be larger than Ez..
reliable. (Right) Current and predicted energy at time and
encoder, and decision block are shared across the original input and the perturbed input pp. Overall, this perturbation based method expands the scope of rules we can incorporate into DEEPCTRL.
4 Experimental Results
We evaluate DEEPCTRL on machine learning use cases from Physics, Retail, and Healthcare, where utilization of rules is particularly important.
For the rule encoder (Φr), data encoder (øa), and decision block (Ø), we use MLPs with ReLU activation at intermediate layers, similarly to 16). We compare DEEPCTRL to the TASKONLY baseline, which is trained with fixed a 0, i.e. only uses data encoder (œ4) and decision block (ő) to predict next state. In addition, we include TASKONLY with rule regularization, TASK& based on Eq. I with a fixed X. We make a comparison to Lagrangian Dual Framework (LDF) 12 that enforces rules by solving constraint optimization problem.
4.1 Improved Reliability Given Known Principles
By adjusting the control parameter a, higher rule verification ratio, and thus more reliable predic- tions, can be achieved. Operating at better verification ratio could be beneficial for performance, especially if the rules are known to be (almost) always valid as in natural sciences. We demonstrate a systematic framework to encode such principles from known literature, although they may not be completely learnable from training data.
Dataset, task and the rule: Following [4], we consider the time-series data generated from double pendulum dynamies with friction, from given initial state (01,41, w2) where 0,w are angular displacement and velocity, respectively. We first generate the time-series data with high sampling frequency (200Hz) to avoid numerical errors, and then downsample to a lower sampling frequency (10Hz) to construct the training dataset. We define the task predicting the next state 21+1 of the
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double pendulum from the current state 2; = (f1e. W11, Ozz, W2)). We construct a synthetic training dataset using the analytically defined relationship between inputs and outputs, and introduce a small additive noise (e ~ (0, 0.0001)) )) to model measurement noise in the setup. We focus on teaching the rule of energy conservation law. Since the system has friction, (m))) > E( where E(x)) is the energy of given state D. We apply an additional constraint based on the law such that Crute (x. 3) = (e)U(E(Ú) - E(x)). To quantify how much the rule is learned, we evaluate the "verification ratio', defined as the ratio of samples that satisfy the rule.
Results on accuracy and rule teaching efficacy: Fig. shows how the control parameter affects the task-based metric and the rule- metric, respectively. The additionally additionlll-incorporatee rule in DEEPCTRL is hugely beneficial for obtaining more accurate predictions of the next state, 1 Compared to TASKONLY, DEEPCTRL reduces the prediction MAE significantly - the parameters are driven to learn better representations with the domain knowledge constraints. Moreover, DEFPCTRI provides much higher verification ratio than TASKONLY and the resultant energy is not only verified, but also more stable (close to that of TASK & RULE) (c).. It demonstrates that DEFPCTRL is able to incorporate the domain knowledge into data-driven model, providing reliable and robust predictions. DEEPCTRL allows to control model's behavior by adjusting a without retraining: when is close to 0 in Fig. 3a and 3b DEEPCTRL is in the task only region where its rule-based metric is degraded; by increasing a, the model's 's behavior is more dominated by the rule-based embedding 2,- L.c although the prediction error is increased, the output is more likely to follow the rule. However, DEEPCTRL is still better than TASKONLY in both metrics regardless of the value of a.
Comparison to baselines: Fig. a, b) shows comparison of DEEPCTRL, to the baselines of train- ing with rule-based constraint as a fixed regularization term. We test different E (0.01, 0.1, 1.0} and all show that the additive regularization (Eq. [T) is helpful for both aspects. The highest À provides the highest verification ratio, however, the prediction error is slightly worse than that of = 0.1. We find that the lowest prediction error of the fixed baseline is comparable to (or even larger than) that of DEEPCTRL, but the highest verification ratio of the fixed baseline is still lower than that of DEEPCTRL. The computed energy values from the fixed baseline are also similar to those from DEEPCTRL. In addition, we consider the benchmark of imposing the constrann with LDF and demonstrate two results where its hyperparameters are chosen by the lowest MAE (LDF-MAE) and highest rule verification ratio (LDF-RATIO) on validation set, respectively. We note that LDF does not allow the capability of flexibly changing the rule strength at inference as DEEPCTRL, it needs to be retrained for different hyperparameters to find such operation points. LDF-MAE yields higher MAE and lower rule verification ratio compared to others on the test set, showing lack of generalizability of the learned rule behavior. On the other hand, LDF-RATIO provides lower MAE than others. However, only 50% of outputs follow the rule which significantly lowers the reliability of the method. Overall, these results demonstrate that DEEPCTRL is competitive in MAE compared to fixed methods like LDF, while providing the flexibility of adjusting the rule strength to operate at a more favorable point in terms of rule verification ratio, and enabling the extra capabilities presented next.
Scale of the objectives: It is important that the scales of Lrute and Lrast are balanced in order to avoid all learnable parameters becoming dominated by one of the objective. With the proposed scale parameter p. the combined loss L has to be of a similar scale for the two extremes (a - 0 and + 1). Fig. salshows that this scaling enables more U-shape curve, and the verification ratio when a + 0 is close to that of TASKONLY in Fig. 3b In other words, DEEPCTRL is close to TASKONLY as a - 0 and close RULBONLY a + I.O, which is a model trained with fixed a 1.0. It is not necessary to search the scale parameter before training; instead, it is fixed at the beginning of training with the initial loss values of Crule and Lrask.
Optimizing rule strength on validation set: Users can run inference with DEEPCTRL with any value for without retraining. In Fig. we consider the scenario where users pick an optimal based on a target verification ratio. For a goal of > 90% verification ratio on the validation set, a > 0.6 is needed, and this guarantees verification ratio of 80.2% on the test set. On the other hand, if we consider optimization of λ of the fixed objective function ('Task& & Rule") benchmark for > 90% validation verification ratio, we observe a test verification ratio of 71.6%, which is 8.6% lower than DEEPCTRL. In addition, the verification ratio on the validation set can be used to determine the rule strength that minimizes error. On the validation set, minimum MAE occurs when a 0.2, which corresponds to rule verification ratio around 60 %. If we search for the a that satisfies the same rule verification ratio of 60 % on the test set, we observe that a = 0.4 is optimal, which also gives

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Figure 4: Candidate rule testing. Sales prediction error (MAE) across three groups with different correlation coefficients between ASALES and APRICES. The dashed lines point the optimal providing the lowest error.
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Figure 5: Candidate model testing. Sales prediction error (MAE) for three groups from model trained on NEG0.2. Note that the correlation coefficients in the target groups are different to that of the source group. The dashed lines point the optimal a providing the lowest error.
the lowest MAE. In other words, the rule verification ratio is robust unsupervised model selection proxy for DEEPCTRL, underlining the generalizability of the learned representations.
4.2 Examining Candidate Rules
DEEPCTRL allows 'hypothesis testing' for rules. In many applications, rules are not scientific principles but rather come from insights, in economical sociological sciences. We do not always want rules to dominate data-driven learning as they may hurt accuracy when they are not valid.
Dataset, task and the rule: We focus on the task of sales forecasting of retail goods on the M5 " While the original task is forecasting daily sales across different goods at every store, we change the task to be forecasting weekly sales since the prices of items updated weekly. For this task, we consider the economics principle as the rule [6]: price-difference and sales- -iffe-diffeeence should have negative correlation coefficient: using the perturbation- based objective described in Sec.
0.0. This inductive bias is incorporated The positive perturbation is applied to price of an input item, and arai is function of the perturbed output Üp and the original output ÿ: Crule (9, Up) = ReLU(g (gp - ÿ) Unlike the previous task, the rule of this task soft not all items known to follow the rule and their correlation coefficients are different.
ASALOS
APRICES
Experimental setting: We split items three groups: (1) NEGÔ. items with a negative price-sales correlation (r < -0.1), (2) NEG0.2 items with negative price-sales correlation (r -022), and (3) NEGO.3 items with negative correlation (r < -0.3). We train TASKONLY, LDF, and DEEPCTRL on each group to examine how beneficial the incorporated rule is. Additionally, examine candidate models by applying a model trained on NEG0.2 to unseen groups: (I) NEGO. 1, (2) PosO.2, and (3) POSO.3 where the rule is weak in the latter two groups.
Candidate rule testing: Fig. shows the weekly sales prediction results. Compared to TASKONLY, DEEPCTRL obtains lower MAE on NEG0.2 and NEGO.3, but TASKONLY outperforms DEEPCTRL when the items have weaker correlations between ASALES and APRICES. This is reasonable since the rule we impose is more dominant in NEGO.2 and NEGO.3, however, it is less informative for

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items in NEGO.1. While LDF provides lower MAE on NEG0.2, its performance is significantly degraded on NEG0.3 due to an unstable dual learning process. Furthermore, the rule-dependency can be discovered via the value of a providing the lowest error. Fig. [ clearly demonstrates that the optimal a is shifted to larger values as the correlation gets stronger. In other words, as items have stronger correlations between APRICES and ASALES, the corresponding rule is more beneficial, and thus the rule-dependency is higher. The post-hoc controllability of DEEPCTRL enables users to examine how much the candidate rules are informative for the data.
Candidate model testing: The hypothesis testing use case can be extended to model testing to evaluate appropriateness of a model for the target distribution. Fig. shows models testing capability, where items in each domain have different price-sales correlations. Overall, DEEPCTRL is superior to other methods like LDF as it learns the task and rule representations in disentangled ways, so that the negative impact of a rule more minimal via lower when the rule is not helping for the task. There are notable points from the experimental results. First, Fig. 55 shows that TASKONLY, LDF, and DEEPCTRL can be applicable to target domain (NEG0. 1) without significant performance degradation. Compared to Fig. 4a MAE from TASKONLY is increased from 0.041 to 0.05 because TASKONLY is optimized on NEG0.2. While DEEPCTRL is also degraded (0.042 to 0.045), the gap is much smaller than that of TASKONLY as we can optimize DEEPCTRL by tuning a. Second, compared to Fig. [4b] we see that the optimal & is decreased to 0.3 from 0.45, showing that rule- dependency becomes weaker compared to source domain. Third, if the model is applied to (Pos0.2 and PosO.3) where the imposed rule for the source domain is not informative at all, not only the prediction quality is significantly degraded, but also the rule-dependency is minimized (the optimal is 0). This indicates the benefit of controllability to examine how much the existing rules or trained models are appropriate for given test samples.
4.3 Adapting to Distribution Shifts using the Rule Strength
There may be shared rules across multiple datasets for the same task law must be valid for pendulums of different lengths or counts. On the other hand, some rules may be valid with different strengths among different subsets of the data e.g. in disease prediction, the likelihood of cardiovascular disease with higher blood pressure increases more for older patients than younger patients. When the task is shared but data distribution and the validity of the rule differ among different datasets, DEEPCTRL is useful for adapting to such distribution shifts by controlling o, avoiding then need for fine-tuning or training from scratch.
e.g. - the energy conservation
Dataset, task and the rule: We evaluate this aspect with a cardiovascular classification task We consider 70,000 records of patients with half having a cardiovascular disease. The target task is to predict whether the cardiovascular disease is present or not based on 11 continuous and categorical features. Given that higher systolic blood pressure is known to be strongly associated with the press " where xn press is cardiovascular disease [9), we consider the rule "Øp.: > ÿt if x > xx
systolic blood pressure of the i-th patient, and ği is the probability of having the disease. The subscript p denotes perturbations to the input and output. Fitting a decision tree, we discover that if blood pressure is higher than 129.5, 5, more than 70% patients have the disease. Based on this information, we split the patients into two groups: (1) {i {xp
129.5 = 1) (1, (-. > 129.5 ny; 0}} (called UNUSUAL) and (2) (i : {x.Phees < 1 y = 0} U {Xp
129.5 yi = 1}} (called USUAL). The majority of patients in the UNUSUAL group likely to have the disease even though their blood pressure is relatively lower, and vice versa. In other words, the rule is not very helpful since it imposes higher risk for higher blood pressure. To evaluate the performance, target datasets by mixing the patients from the two groups with different ratios (See Appendix).
create different
Target distribution performance: We first train model on the dataset from the source domain and then apply the trained model to a dataset from the target domain without any additional training. Fig. salshows that TASKONLY outperforms DEEPCTRL regardless of the value of a. This is expected as the source data do not always follow the rule and thus, incorporating the rule is not helpful or even harmful. Note that the error monotonically increases as + 1 as the unnecessary (and sometimes harmful) inductive bias gets involved more. When a trained model is transferred to the target domain, the error is higher due the difference in data distributions. However, we show that the error can be reduced by controlling a. For TARGET where the majority of patients are from USUAL, as a is increased, the rule- representation, which is helpful now, has more weight and the resultant
dataset

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Usuall of total samples: 0.30
of Usual/ of total samples: 0,77
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(a) SOURCE
(b) TARGET
of Usualr of total samples: 0.50
Usual/e of total samples: 0.40
TaskOnly
Task Rute Qu00.11
Task Rude 9m11.0)
DecOCTRL
(3063 Entropy
Qrook Eremur
0.70
TeskCiny
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DespCTRL
0.0
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144
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(c) TARGET 2
(d) TARGET 3
Figure Test cross entropy vs. rule strength for target datasets with varying USUAL/UNUSUAL ratio.
error decreases monotonically. When the ratio of patients from USUAL is decreased, the optimal is an intermediate value between 0 and These demonstrate the capability of DEEPCTRL to adapt the trained model via towards optimal behavior if the rule is beneficial for the target domain. Moreover, we can reversely interpret the amount rule dependency for a target dataset. As the optimal is close to 1.0 for TARGET 1, expect that the rule is valid for most of the patients, unlike TARGET 3 which has the optimal a around 0.5.
Extrapolating the rule strength: While is sampled between 0 and during training, the encoder outputs can continuously changed when a > or a < 0. Thus, expected that the rule dependency should get higher lower when the rule is effective for TARGET and the error is monotonically decreased until a = I when we of the error is continued as
set >
0. respectively. We find that
apply the trained model on SOURCE (Fig. (b). The decreasing increased until 1. and extends the range of a for further controllability. This observation is particularly relevant when it is necessary increase the rule dependency more and underlines how DEEPCTRL learns rule representations effectively.
Ablation Studies
Rule strength prior P(a): To ensure that DEEPCTRL yields desired behavior with a range of a values inference, learn mimic predictions with different during training provide the appropriate supervision. We choose the Beta distribution to sample because via only one parameter, allows us to sweep between various different behaviors. One straightforward idea is uniformly simulate all potential inference scenarios during training (i.c. having Beta( (I,)),, however, this yields empirically worse results. We propose that overemphasizing on edge cases (very low and very high a) is important so that the behavior for edge cases can be learned more robustly. and the interpolation between them would be accurate. Fig. 7a and 7b] shows results with different values of Beta( 8). For the pendulum task, Lrute is much larger than Ltask and it leads DEEPCTRL getting mostly affected by Lrute on Beta(1.0, 1.0) when a > 0.1, hurting the task-specific performance. We observe that there is typically intermediate 3 < optimal for the general behavior of the curves B 0.1 seems to be reasonable choice across all datasets, and the results are not sensitive when slightly higher or lower.

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9.0055
Betaxo.05.0000
D.taro.1, 0.1)
0.0M55
0.6
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Betal0.3, 0.33
4,tal0.,
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(a) MAE
(b) Verification ratio
1.0
Add tal
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Concat Ex
Despc INL.
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Cencal le exal
Add zul
Concel ix
0.0015
0.0
DeepCTRL
0.2
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(d) Verification ratio
Figure 7: (a,b) Results on a double pendulum task for various of B3 (c,d) Comparison with two non-coupled representations (CONCAT and ADD) and concatenated input (æ @ a).
Stochastically-couplee representations: We propose to combine the two representations from the encoders enable stochastic coupling with corresponding objectives (azz, ahrule and =d (1 a-)aa) Ltask). ). As an alternative, we consider the coupled representations with two conventional ways of merging: CONCAT (z 2, + zd) and ADD
Z, zd). Fig. 7c and 7 shows that these can provide very high verification ratio (as Crate dominates over Lrask), equivalent to DEEPCTRL high a, but their errors intermediate values are higher than DEEPCTRL. The proposed coupling method in DEEPCTRE enables disentangled learning of rule vs. task representations by mixing them with a. We also concatenate network architecture. Fig. [7c and 7d shows that concatenating inputs does not improve the task error or verification ratio, although the same amount of information is used. This supports the hypothesis that the coupled separation (z, and za) is more desirable for controlling corresponding representations.
[a:; ox) instead of relying on the proposed two-passage
Conclusion and Societal Impact
Learning from rules can be crucial for constructing interpretable, robust, and reliable DNNs. In this paper, we propose DEEPCTRI. new methodology incorporate rules into data-learned DNNs. Unlike existing methods, DEEPCTRL enables controllability of rule strength inference without retraining. We propose novel perturbation- into meaningful representations. Overall, DEEPCTRL is model architecture, data type and rule type agnostic. We demonstrate three uses cases of DEEPCTRL: improving reliability given known principles, examining candidate rules, and domain adaptation using the rule strength. We leave theoretical proofs for convergence of learning, extensive empirical performance analysis on other datasets, well demonstrations of other
rule encoding method to integrate arbitrary rules
like robustness, for future work.
DEFPCTRL. has many potential benefits in real-world deep learning deployments to improve their accuracy, to increase their reliability, and to enhance human-Al interaction. On the other hand, we also note the capability of DEEPCTRL in encoding rules in effective ways can have undesired outcomes if used with bad intentions to teach unethical biases.
10

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References
[1] Serean O Ank and Tomas Plister. Protoattend: iieettionbbssl prototypical learning. Journal of Machine Learning Research, 21:1-35, 2020.
[2] Serean Ömer Arik and Tomas Pfister. Tabnet: Attentive interpretable tabular learning. arXiv: (rXXvv9908.00442 2019.
[3] Vijay Arya, Rachel KE Bellamy, Pin-Yo Chen, Amit Dhurandhar, Michael Hind, Samuel C Hoffman, Stephanie Houde, Q Vera Liao, Ronny Luss, Aleksandra Mojsilovié, et al. One explanation does not fit all: A toolkit and taxonomy of ai explainability techniques. arXiv preprint (rXXv::909.03012.. 2019.
[4] Alexis Asseman, Tomasz Kornuta, and Ahmet Ozcan. Learning beyond simulated physics. In Modeling and Decision-making in the Spatiotemporal Domain Workshop, 2018.
[5] Philip G Breen, Christopher N Folcy, Tjarda Bockholt, and Simon Portegics Zwart. Newton vs the machine: solving the chaotic three-body problem using deep neural networks. arXiv preprint #rXiv:1910.072: 2019.
[6] Edgar K. Browning and J. M. Browning. Microcconomic theory and applications. 1986.
[7] Alexander D' Amour, Katherine Heller, Dan Moldovan, Ben Adlam, Babak Alipanahi, Alex Beutel, Christina Chen, Jonathan Deaton, Jacob Eisenstein, Matthew D. Hoffmman, Farhad Hormozdiari. Neil Houlsby, Shaobo Hou, Ghassen Jerfel, Alan Karthikesalingam, Mario Lucic. Yian Ma. Cory McLean, Diana Mincu, Akinori Mitani, Andrea Montanari, Zachary Nado, Vivek Natarajan, Christopher Nielson, Thomas F. Oshorne, Rajiv Raman, Kim Ramasamy, Rory Sayres, Jessica Schrouff, Martin Seneviratne, Shannon Sequeira, Harini Suresh, Victor Veitch, Max Vladymyrov, Xuezhi Wang. Kellie Wehster, Steve Yadlowsky, Taedong Yun, Xiaohua Zhai, and D. Sculley. Underspecification presents challenges for credibility in modern machine learning, 2020.
[8] Sanjeeb Dash, Oktay Gunluk, and Dennis Wei. Boolean decision rules via column generation. In Advances in Neural Information Processing Systems, volume 31, pages 4655-4665. Curran Associates, Inc., 2018.
[9] Michelangelo Diligenti, Soumali Roychowdhury, and Marco Gori. Integrating prior knowledge into deep learning. 2017 16th IEEE International Conference on Machine Learning and Applications (ICMLA), pages 920-923, 2017.
[10] Finale Doshi Velez and Been Kim. Towards a rigorous science interpretable machine learning. arXiv preprint 1702 2017.
[11] Carson Eisenach, Yagna Patel, and Dhruv Madeka. Mqtransformer: multi-horizon forecasts with context dependent and feedback- aware attention, 2020.
[12] Ferdinando Fioretto, Pascal Van Hentenryck, Terrence WK Mak, Cuong Tran, Federico Baldo, and Michele Lombardi. Lagrangian duality for constrained deep learning, 2020.
[13] K. Ganchev, Graça, Gillenwater, and Taskar. Posterior regularization structured latent variable models. J. Mach. Learn. Res., 11:2001-2049, 2010.
[14] lan Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, 2016. http: //www.deepppp deepleerningbbook org
[15] lan Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprint arXiv: 1422.5572 2014.
[16] Samuel Greydamus, Misko Dzamba, and Jason Yosinski. Hamiltonian neural networks. In Advances in Newral Information Processing Systems, pages 15379- 15389, 2019.
[17] Joel Hestness. Sharan Narang, Newsha Ardalani, Gregory Diamos, Heewoo Jun, Hassan Kianinejad, Md. Mostofa Ali Patwary. Yang Yang, and Yanqi Zhou. Deep learning scaling is predictable, empirically. 2017.
[18] Zhiting Hu, Xuezhe Ma, Zhengzhong Liu, Eduard H. Hovy, and Eric P. Xing. Harnessing deep neural networks with logic rules. CoRR, abs/1603. 10W31660306118 18, 2016.
[19] William B Kannel, Elevated systolic blood pressure as cardiovascular risk factor. The American Journal of Cardiology, (112):2112555 2000.
[20] Alexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial examples in the physical world. arXiv preprint arXiv:1607.02533 2016.

---

[21] Bryan Lim, Sercan O. Arik, Nicolas Loefff and Tomas Pfister. Temporal fusion transformers for inter- pretable multi-horizon time series forecasting, 2020.
[22] Xiaodong Liu, Kevin Duh, Liyuan Liu, and Jianfeng Gao. Very deep transformers for neural machine translation. arXiv preprint arXiv:2008.00 077772, 2020.
[23] Michael Lutter, Christian Ritter, and Jan Peters. Deep lagrangian networks: Using physics as model prior for deep learning, 2019.
[24] Harikrishna Narasimhan. I.carning with complex loss functions and constraints, In Amos Storkey and Fernando Perez-Cruz. editors. Proceedings the Twenty-First International Conference Artificial Intelligence and Statistics. volume of Proceedings of Machine Learning Research, pages 1646-1654. PMLR, 09-11 Apr 2018.
[25] Sergei Popov, Stanislav Morozov, and Artem Babenko. Neural oblivious decision ensembles for deep learning on tabular data. arXiv:1909.06312 2019.
[26] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. " why should trust you?" explaining predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pages 1135-1144, 2016.
[27] Huajie Shao, Shuochao Yao, Dachun Sun, Aston Zhang. Shengzhong Liu, Dongxin Liu, Jun Wang, and Tarek Abdelzaher. Controlvae: Controllable variational autoencoder. In International Conference on Machine Learning, pages 6665- PMLR, 2020.
[28] Christian Szegedy, Vincent Vanhoucke, Sergey loffe, Jon Shlens, and Zbigniew Wojua. Rethinking the inception architecture for computer vision. In Proceedings the IEEE conference on computer vision and pattern recognition, pages 2818 22226, 2016.
[29] Hugo Touvron, Andrea Vedaldi, Matthijs Douze, and Herve Jegou. Fixing the train-test resolution discrepancy. Advances Neural Information Processing Systems, pages 8252 8262, 2019.
[30] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention you need. In Advances neural information processing systems, pages 5998 6008, 2017.
[31] Xiaoyong Yuan,
Qile Zhu, and Xiaolin Li. Adversarial examples: Attacks and defenses deep learning. IEEE transactions on neural networks and learning systems, 30(9):2805-25 2019.
[32] Brian Hu Zhang, Blake Lemoine, Margaret Mitchell. Mitigating unwanted biases with adversarial learning. CoRR, abs/ Ses11801,7793,
07593, 2018.
[33] Hongyi Zhang. Moustapha Cisse, Yann N. Dauphin, David Lopez-Paz. mixup: Beyond empirical risk minimization. In International Conference on Learning Representations, 2018.
12


Reviews:
[{"review_id": "xmtnhShYkl", "reviewer": "Reviewer_9N6z", "summary": "This paper proposes DeepCTRL, a method for infusing domain knowledge in the form of rules into the representation learning models. The method independently encodes the input instance and the supplied rule and couples the input and rule encodings where the coupling is controlled by a parameter \\alpha. Training the model involves a task-specific loss function and a “rule-satisfaction” loss function which are again coupled by the same parameter \\alpha. Experiments on three tasks related to Physics, Forecasting, and Health domain demonstrate several benefits of the proposed approach which include (i) ability to specify the domain knowledge, (ii) More control on coupling the information from rule and data during the test time (iii) More interpretability. \nHowever, the experiments in this paper seem to be very simplistic and lack exhaustive comparisons with the prior work and thus it’s difficult to judge the practical utility of the proposed method (E.g. all the experiments involve just one rule).  ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "Pros:\n- This paper considers an important problem of injecting domain knowledge into representation learning models. \n- The proposed method allows easy adaptation of rules to new domains and more controllability in the information coming from rules vs data, during inference.\n- Experiments cover multiple domains like Physics, Health and Forecasting (however the datasets seem to be very simplistic)\n\nLimitations:\n- Lack of comparisons with the relevant baselines: In section 4.1, I believe [1] is an important baseline, which also tries to combine domain knowledge along with data for learning better models for predicting damped pendulum dynamics. The is a large body of work around learning from rules e.g. [2] and [3]. How does this work compares with these methods? I think the paper should include some more baselines since there are a lot of papers around learning from rules.\n- All the experiments deal with just one rule and the datasets seem to be very simplistic. How well does this method do when we have significantly more rules?\n\nQuestions:\n- In Figure 4a, why is Task-Only better than DeepCTRL for \\alpha=0? Should both the methods differ significantly for \\alpha=0?\n\nReferences:\n\n[1] Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting, ICLR 2021\n\n[2] Learning from Rules Generalizing Labeled Exemplars, ICLR 2020\n\n[3] Snorkel: Fast training set generation for information extraction, VLDB\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "cWOeZLw3E0K", "reviewer": "Reviewer_xjoF", "summary": "This paper proposes to integrate a combination of rule learning and data learning in the deep neural networks. The core motivation is to introduce prior knowledge and/or other learning methods and learning how to \"balance\" the pure data-driven part of the neural network with its rule learning component. The final neural network has two components: the rule-based encoder and the data-driven encoder that are combined by a parametrized convex combination. The loss function has also a counterpart that captures the traditional loss function with penalizations on the rule violations. The second part is similar to the Lagrangian approach to training neural networks subject to constraints that has been used successfully in prior (mostly cited) work. The novelty is the introduction of the rule encoder inside the neural network, as examplified by the third application. The paper also proposes an interesting perturbation approach to capture some of the rule behavior. ", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "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": "It is an interesting paper giving an interesting direction. Some of the work described is a direct consequence of prior work on incorporating constraints in the objective function. But the paper generalizes this with two ideas\n    -- the idea of integrating rule learning directly in the neural networks\n    -- the idea of local perturbation, which is quite nice\nThese ideas are an interesting avenue for research in a variety of applications.\n\nThere are some weaknesses in the paper. The paper lists a number of limitations  of prior work\n  (1)  the Lagrangian multipliers are fixed;\n  (2) the loss function for the \"constraint\" part must be differentiable\n  (3) the components cannot be balanced at inference time.\n\n(1) is not correct and is addressed in prior work. The approach here also suffers from (2). (3) is addressed by changing how to weight the rule and data encoders during inference. It is not clear what the justification is here since the network is trained with a different weight.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The paper mentions some of its limitations at the end. The paper should mention that the rule-based encoder may codify existing biases in some possible applications. \n", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2 hours"}, "extra_text": {}}, {"review_id": "PflF9_d8DI", "reviewer": "Reviewer_hguV", "summary": "This work proposed an interesting way of mixing the power of data and rules while enabling the adaptation at the inference time. This paper proposed to use perturbation of the input data points as a novel way of injection rules compared with the continuous relaxation of rules.", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "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": "This learning framework as well as the perturbation-based rule injection looks very interesting and novel to me. First of all, combining rule-based prior knowledge with learned knowledge is an interesting and promising direction for improving neural networks' generalization performance. The proposed perturbation-based rule injection is very novel to me. Most existing works inject rules using their continuous relaxation. The perturbation-based rule injection provides a new way to inject a broader range of rules that are potentially hard to relax. \nSecond, the idea of controlling the importance weights between data (learned knowledge) and rules is quite novel. Though it is simple from hindsight, it does enable the control of a trained model and provides some interpretability to the model itself (rather than a pure black box).  Moreover, the control of the weights between data and rules can help to solve out-of-sample data points (as shown in their experiments).\nOverall I think this paper deserves a score of 8.  I recommend accepting this paper.\n\n\nOne suggestion: I hope the author gives a broader literature survey for methods that try to merge the power of data and rule/symbolic knowledge.\n\nHere are some related works that I know:\n\nDeep Reasoning Networks for Unsupervised Pattern De-mixing with Constraint Reasoning\nA semantic loss function for deep learning with symbolic knowledge\n\nQuestions:\nIt looks quite counterintuitive to concatenate the two embeddings z_d and z_r while scaling them by alpha and (1-alpha). People often use this kind of “convex combination of two” on the summation operator rather than concatenation. Did you try if you use z = alpha * z_d + (1-alpha) *z_r? \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "N/A", "review_text": "", "extra_scores": {"ethical_concerns": "NO", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "EP4Krxr4XCO", "reviewer": "Reviewer_x3gr", "summary": "The paper proposes DeepCTRL, a novel training method to integrate rules into neural networks. DeepCTRL is evaluated through several applications showing how rules can be effectively incorporated and improve performance.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "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": "Overall, the idea proposed is clear and yields strong results.\n\nFig 4 makes a compelling case that DeepCTRL works and is useful. One question about Fig 4 (especially 4c): it seems to make the case that DeepCTRL does a better job of enforcing the soft rule than LDF. It is surprising then that in 5b and 5c DeepCTRL continues to outperform LDF even when imposing the rule would ostensibly hurt perfromance. Do the authors have an explanation for why this might be?\n\nThe results in Fig 3 do not clearly show that DeepCTRL outperforms the LDF-ratio baseline. In this case, LDF-Ratio seems to perform the best, even without following the energy damping rule. Even if its verification ratio is lower, this example does not clearly show that DeepCTRL is outperforming LDF-Ratio.\n\nThe ablation studies give reasonable confidence that some of the calls made in DeepCTRL's methodology are sound.\n\nOne relatively simple baseline that comes to mind is using a convex combination of two classifiers: the rule-based model and a neural network model. This should still preserve many of the proposed useful properties (e.g. selecting the relative strength of the rule-model at test time). Is there a reason the proposed model is better than this simple baseline and does not need to be compared to it?\n\n**Minor points.**\n\n- grammar in first sentece of abstract is slightly unclear\n- fig 1 caption - \"DEEPCTRL **provide** a controllable mechanism\"\n- fig 1 itself is faily confusing - what do the blue arrows represent? Why isn't the physical rule of  monotonicity obeyed by any of the curves (it seems to be even less obeyed for the bottom plot, where the rule strength is ostensibly the highest)?", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors do not seem to explicitly address limitations and societal impact.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper considers deep learning models for applications where rules are important (e.g. physics). The idea is to construct a "rule encoder" to incorporate user-specified rules. Specifically, this integration is done by perturbing the observations and enforce some user-specified rules to "regulate" between the model's outputs on the original & perturbed inputs. 

Reviewers conclude that the proposed approach is novel and the machine learning tasks considered in the paper are important. Initially they have concerns on missing comparisons with other approaches. This is partly addressed by further experimental results provided by the author feedback. 

Personally I think the paper can be an interesting presentation to the NeurIPS community. I find this paper also connected to existing works related to constrained optimisation/control and adversarial training: these works also consider regularising networks to satisfy some constraints and/or make sure the network returns similar results on perturbed inputs. It would be useful to discuss them in the camera ready.
Decision: Accept (Poster)