Abductive Knowledge Induction from Raw Data
Authors: Wang-Zhou Dai
Stephen Muggleton
Venue:  (2021)
Keywords: Neural-Symbolic Model
Inductive Logic Programming
Abduction

Abstract: For many reasoning-heavy tasks, it is challenging to find an appropriate end-to-end differentiable approximation to domain-specific inference mechanisms. Neural-Symbolic (NeSy) AI divides the end-to-end pipeline into neural perception and symbolic reasoning, which can directly exploit general domain knowledge such as algorithms and logic rules. However, it suffers from the exponential computational complexity caused by the interface between the two components, where the neural model lacks direct supervision, and the symbolic model lacks accurate input facts. As a result, they usually focus on learning the neural model with a sound and complete symbolic knowledge base while avoiding a crucial problem: where does the knowledge come from? In this paper, we present Abductive Meta-Interpretive Learning ($Meta_{Abd}$), which unites abduction and induction to learn perceptual neural network and first-order logic theories simultaneously from raw data. Given the same amount of domain knowledge, we demonstrate that $Meta_{Abd}$ not only outperforms the compared end-to-end models in predictive accuracy and data efficiency but also induces logic programs that can be re-used as background knowledge in subsequent learning tasks. To the best of our knowledge, $Meta_{Abd}$ is the first system that can jointly learn neural networks and recursive first-order logic theories with predicate invention.

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ABDUCTIVE KNOWLEDGE INDUCTION FROM RAW DATA
Anonymous authors Paper under double-blind review
ABSTRACT
For many reasoning-heavy tasks, it is challenging to find appropriate end- to-end differentiable approximation to domain-specific inference mechanisms. Neural-Symboliic (NeSy) AI divides the end-to-end pipeline into neural percep- tion and symbolic reasoning, which can directly exploit general domain knowl- edge such as algorithms and logic rules. However, it suffers from the exponential computational complexity caused by the interface between the two components, where the neural model lacks direct supervision, and the symbolic model lacks accurate input facts. As result, they usually focus on learning the neural model with a sound and complete symbolic knowledge base while avoiding a crucial problem: where does the knowledge come from? In this paper, present Abduc- tive Interprettee Learning (Meta, (Metaable Abd), which unites abduction and induction learn perceptual neural network and first-order logic theories simultaneously from raw data. Given the same amount of domain knowledge, we demonstrate that Mela Abd only outperforms the compared end-to-end models in predie- tive accuracy and data efficiency but also induces logic programs that can be re- used as background knowledge in subsequent learning tasks. To the best of our knowledge, Meta Abd is the first system that can jointly learn neural networks and recursive first-order logic theories with predicate invention.
INTRODUCTION
Inductive bias, background knowledge, is essential component in machine learning. Despite the success of data-driven end-to-end deep learning in many traditional machine learning tasks, has been shown that incorporating domain knowledge is still necessary for some complex learning
problems (Dhingra et al. 2020 Grover et al. 2019 Trask ct al. 2018).
In order to leverage complex domain knowledge that is discrete and relational, end-to-end learning systems need to represent it with a differentiable module that can be embedded in the deep learning context. For example, graph neural networks (GNN) use relational graphs an external knowledge base (Zhou et al.) 2018); some works even considers more specific domain knowledge such as differentiable primitive programs (Gaunt al., 2017). However, the design of these modules is usually ad hoc. Sometimes, it is not easy to find an appropriate approximation that is suited for single-model based end-to-end learning (Glasmachers) 2017 Garcez et al. 2019).
Therefore, many researchers propose to break the end-to-end learning pipeline apart and build hybrid model that consists of smaller modules where each of them only accounts for one specific function (Glasmachers] 2017). (.17).1 A representative branch in this line of research is Syurol-Symbolic (NeSy) AT (De Raedt et al. 20200 Garcez et al., 2019) aiming to bridge System and System AI (Kahneman) 2011] Bengio 2017). i.e., neural-network-beee based relational inference. In NeSy models, the neural network extracts high-level symbols from noisy raw data and the symbolic model performs relational inference over the extracted symbols.
machine learning and symbolic-
However, the non-differentiable interface between neural and symbolic systems (i.e., the facts ex- tracted from raw data and their truth values) leads to high computational complexity in learning. For example, due to the lack of direct supervision to the neural network and reliable inputs to the symbolic model, some works have to use Markov Chain Monte Carlo (MCMC) sampling zero- order optimisation to train the model (Li et al. 2020; Dai et al. 2019), which could be inefficient in practice. Consequently, almost all hybrid models assume the existence of very strong predefined

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domain knowledge base and focus using it to train neural networks. It limits the expressive power of the sybrid-structured model and sacrifices many benefits of symbolic learning (c.g., predicate in- vention, learning recursive theories, and re-using learned models as background knowledge).
In this paper, we integrate neural networks with Inductive Logic Programming (ILP) (Muggleton & de Raedi 1994] general framework for symbolic machine learning-ts enable first-order logic
theory induction from raw data. More specifically, we present Abductive Meta- Ietta-Interrretive Learning
(Melaata) which extends the Abductive Learning (ABL) framework (Dai al. 2019 Zhou 2019)
by combining logical induction and abduction (Flach et al. 2000) with neural networks in Meta- Interpretive Learning (MIL) (Muggleton al. 2014). Meta Abd employs neural networks to extract probabilistic logic facts from raw data, and induces an abductive logic program (Kakas et al. 1992) that can efficiently infer possible truth values of the probabilistic facts to train the neural model.
On the one hand, the abductive logic program learned by Meta can largely prune the search space of the truth value assignments to the logical facts extracted by an under-trained neural model. On the other hand, the extracted probabilistic facts, although noisy, provide distribution on the possible worlds [Nilsson (986) reflecting the raw data distribution, which helps logical induction to identify the most probable hypothesis. The two systems in Mela Abd are integrated by a probabilistic model that can be optimised with Expectation Maximisation (EM).
To the best of our knowledge, Metaabd is the first system that can simultancously (1) train neural models, (2) learn recursive logic theories and (3) perform predicate invention from domains with sub-symbolic representation. In the experiments we compare Abda to the compared state-of- the-art end-to-end deep learning models on two complex learning tasks. The results show that, given the same amount of background knowledge, M outperforms the end-to-end models significantly in terms of predictive accuracy and data efficiency, and learns human interpretable models that could be re-used in subsequent learning tasks.
2 RELATED WORK
Solving "System 2" problems require the ability of relational and logical reasoning instead of "in- tuitive and unconscious thinking" (Kahneman) 2011 Bengio, 2017). Due to the complexity of this type of tasks, many researchers have tried to embed intricate background knowledge in end-to-end deep learning models. For example, (Trakk al. (2018) propose the differentiable Neural Arithmetic Logic Units (NALU) to model basic arithmetic functions (e.g., addition, multiplication, etc.) in neu- ral cells: Grover et al.] (2019) encode permutation operators with stochastic matrix and present a continuous and differentiable approximation to the sort operation: Wang et al. (2019) introduce dif- ferentiable SAT solver to enable gradient- baaeed constraint solving. However, most of these specially designed differentiable modules are ad hoc approximations to the original inference mechanisms, which can not represent the inductive bias in a general form such as formal languages.
In order to directly exploit the complex background knowledge expressed by formal languages. Statistical Relational (StarAl) and Neural Symbolic (NeSy) AI (De Raedt et al. 22020] Garcez et al.) 2019) are proposed. Some works try to approximate logical inference with continuous functions or use probabilistic logical inference to enable the end-to-end training (Cohen et al. 2020 Manhaeve et al. 2018 Donadello et al. 2017); others try combine neural networks and pure symbolic reasoning by performing combinatorial search over the truth values of the output facts of the neural model (Li al. 2020 Dai al. 2019). Because of the highly complex statistical relational inference and combinatorial search, it is difficult for them to learn first-order logic theories. Therefore, most existing StarAI and NeSy systems focus on utilising a pre-defined symbolic knowledge base to help the parameter learning of the neural model and probabilistic model.
One way to learn symbolic models is to use Inductive Logic Programming (Muggleton & de Raedt, 1994]. Some early work on combining logical abduction and induction can learn logic theories even when input data is incomplete (Flach al. 2000). Recently, ØIL was proposed for learning first-order logic theories from noisy data (Evans & Grefenstette, 2018). However, these works are designed for learning from domains. Otherwise, they need to use a fully trained neural model to extract primitive facts from raw data before symbolic learning. Machine apperception (Evans al.) 2019) unifies reasoning and perception by combining logical inference and binary neural networks in Answer Set Programming, in which logic hypotheses and parameters of neural networks are all represented by logical groundings, making the system hard to optimise. For problems involving noisy inputs like MNIST images, it still requires fully pre-trained neural net for pre-processing.
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Different to the previous work, presented Abductive Meta-Interpretive Learning AMel aims to combine symbolic and sub-symbolic learning in a mutually beneficial way, where the in- duced abductive logic program prunes the combinatorial search of the unknown labels for training the neural model; and the prohabilistic facts output by the neural model provide a distribution on the possible worlds of the symbolic domain to help logie theory induction.
ABDUCTIVE META-INTERPRETIVE LEARNING
3.1 PROBLEM FORMULATION
typical hybrid model bridging sub-symbolic and symbolic learning contains two major parts: a perception model and a reasoning model (Dai et al. 2019). The perception model maps raw inputs E whwhh are usually noisy and represented by sub-symbolic features to some primitive symbols ε Z, such as digits, objects, ground logical expressions, etc. The reasoning model takes the interpreted = as input and deduces the final output y y according to a symbolic background knowledge base B. Because the primitive symbols are uncertain and not observable from both training data and the background knowledge, we have named them as pseudo-labels of æ.
The perception model is parameterised with 0 and outputs the conditional probability Pa(=)) P(z (r, 8): the reasoning model E H is a set of firstoordrr logical clauses such that BUHU2 - y. where " means "logically entails". Our target is to learn and H simultaneously from training
data D = { yt) Y For example, if we have one example with x = A 3 and y 6, given background knowledge about adding two numbers, the hybrid model should learn perception model that recognises = = [1, 3) and induce a program to add all numbers in z recursively.
Assuming that D is i.i.d. sample from the underlying distribution of (ar, y), the objective of our learning problem can be represented as follows:
(H*,0*) "0) = arg max
II P(. H,0). (E.Y)ED
(1)
H.
where z a hidden variable in this model. Theoretically, this problem can be solved by Expectation Maximisation (EM) algorithm. However, even if we can obtain the expectation of the hidden vari- able and efficiently estimate the perception model's parameter 0 with numerical optimisation, the hypothesis H, which is first-order logic theory, is still difficult to be optimised together with 6.
We propose to solve this problem by treating like z as an extra hidden variable, which gives us:
f = arg max
II
Σ
Σ
Ply, 22,,
(D HEH SEZ
The hybrid-mode) learning problem in Equation [ can be split into two EM steps: (1) Expectation: obtain the expected value of H and z by sampling them in their discrete hypothesis space from (H, P(H, 2| x, y, 0): (2) Maximisation: estimate # by maximising the likelihood of training data with efficient numerical optimisation approaches such as gradient descent.
As one can imagine, the main challenge is to estimate the expectation the hidden variables U2, i.e., we need to search for the most probable H and given the 0 learned in the previous iteration.
Challenges This search problem is nontrivial. Sampling the values of hidden variable v results in a search space growing exponentially with the number of training examples. Even when B is sound and complete, existing hybrid models that do not learn first-order hypotheses still have to use Zero-Order Optimisation (700pt) or Markov Chain Monte Carlo (MCMC) sampling to estimate the expectation of DDai et al.) 2019 Li et al. 2020), which could be quite inefficient in practice.
Furthermore, the size and structure hypothesis space H of first-order logic programs makes the search problem even more complicated. For example, given
1 2 3 and y 6, when the perception model is accurate enough to output the most probable 2 = 2, 3). we have at least two choices for H: cumulative sum or cumulative product. When the perception model is traieet and outputs the most probable = = [2, 2, 3), then H could be a program that only multiplies the last two digits. Hence, H and z entangled and cannot be treated independently.
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Example ((x.a)): 91. 15).
Pscudo-labck (#):
Abduced constraints:
Abductive hypotheses (H):
[0,0,0]
== 15.
=(A,B) :- (A,
Abducible Primitives (II):
[3,6,0]
d 15.
I(A,B) mull(A,B).
edd( (A,BIT), (CIT]) : C- A+B.
mul:([k,BIT]) [CI1]) :- a- A+B. eq([Al.), bead(CH)]] R).
[0,3,5]
== 15.
((A add (C,)).
tail([_IT). T).
[0,5,3]
15.
X-
(A, :- od
#(A eq(A.))
[1,3,6]
Neural Probabilistic facts (p(())))):
[(A,)) :-
pa
0, 0.02). and 1, 0.39).
[7,8,0]
H- 15.
f_I(A,8) --
na
O, 0.09). nat 0.02).
[7,8,1]
E(A :- m) 1(C,)),
O, 0.07),
0.00).
[7,3,5]
15.
(_1(A,)) mult(A,)),,e(C..
Figure 1: Example of Meta Abd's abduction-induction learning. Given training examples, back- ground knowledge of abducible primitives and probabilistic facts generated by perceptual neural net, M Abd Abd learns an abductive logic program and abduces relational constraints (implemented with the CLP(Z) predicate "#="| over the input images; it then uses them to efficiently prune the search space of the most probable pseudo-labels 2 (in grey blocks) for training the neural network.
3.2 EFFICIENT HYPOTHESIS SAMPLING BY COMBINING ABDUCTION AND INDUCTION
Inspired by early works abductive logic programming (Flach et al. 2000). we propose to solve the challenges above by combining logical induction and abduction. The induction learns an abductive logic theory H based on Po (2(22r). and the abduction by H reduces the search space of z.
Abduction refers to the process of selectively inferring specific grounded facts and hypotheses that give the best explanation to observations based on background knowledge of a deductive theory. For example, if we know that H is a cumulative sum program and observe that
1 and y = 6, then we can abduce that x must satisfy the constraint 21+22+23=6, where [11, Z2, Z3 =
2
the pseudo-labels of images in z. This constraint can largely prune the search space of in which all Zi > 6 can be excluded. If the current perception model assigns very high probabilities to Z1 = 2 and Z2 3. one easily infer that Z3 1 even when the perception model has relatively low confidence about it, as this is the only solution that satisfies the constraint.
An illustrative example of combining abduction and induction is shown in Figure Briefly speak- ing, instead of directly sampling pseudo-labels z and H/ together from the huge hypothesis space, our proposed Abductive Meta-Interpretive Learning approach only samples the abductive logic program H, and then use the abduced relational constraints to prune the search space of 2. Meanwhile, the perception model outputs the likelihood of pseudo-labels with pe( (2|2z) which defines a distribution over all possible values of and helps to find the most probable H U
Formally, for each example {x. y), we re-write the likelihood in Equation 2 as follows:
Ply. HI = = P(yl B, H, P( B,))
Ply, H 2) Po(elr)
= P(yl B, H, (F)
(3)
where Pa- BH is the Bayesian prior distribution on first-order logic hypotheses, which is defined by the transitive closure of stochastic refinements o² given the background knowledge B (Muggleton cet al. 2013), where a refinement is a unit modification (c.g., adding/removing a clause or literal) to logie theory. The equations equations hold because: (1) pseudo-label is conditioned on and since it is the output of the perception model; (2) H follows the prior distribution so it only depends on B; (3) y U H independent from x given z because the relations among B, H, y and are determined by pure logical inference, where:
[1, Ply H, o, otherwise.
iU y,
(4)
!CIP(ZZ) is constraint logic programming package accessible at http://gittuh..om/rriskkk/ More im- plementation details please refer to the Appendix.
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Abductive Meta- Inta-enterrreter
prove ([], Prog, Prog, 11, Prob, Prob)
prove [Atom] As], Progl, Progl, Abds, Prcbl, Prob2) deduce (Atom),
prove (As, Prog", Prag?, Abds, Probl, Prob2) prove ( [Atoo|AA]] Progl, Progl, Abds, Prob1, Prob2) :- aollaabducible (Atom, Abd, Prob), Prob3 is Probl Prob, get.max.prob (Max), Prob3 Max, set max prob (Prob3),
prove (As, Progl, Progl, [Abd[Abds], Prob3, Prob2). prove [[tto]] As], Progl, Prog2, Abds, Probl, Prob2) meta-rule (Name, MetaSub, (Atom Rody), Order), Order, substitue (metasub (Name, MetaSub), Progl, Prog3),
prove (Body, Prog3, Prog4), prove (As, Prog4, Proq2, Abds, Probl, Prob2)
Figure 2: Prolog code for Aed It recursively proves series of atomic goals in three ways: (1) deducing them from background knowledge; (2) abducing a possible grounded expression (e.g.- relational constraint) to satisfy them (bold fonts); (3) matching them against the heads of meta- rules and form an augmented program or prove it with the current program. Finally, the abduced groundings Abd used for searching the best pseudo-labels z; the probability of Abd is used for calculating the score function in Equation
Following Bayes' rule we have P(H, 2|, & P(y. H. H, 0). Therefore, we can sample the most probable H U the expectation step according Equation 3 as follows:
1. Sample an abductive first-order logic hypothesis H P.. B):
2. Use H U B and y abduce possible pseudo-labels 2, which guaranteed satisfy U BU y and exclude the values of = such that P(y B, H, 2) = 0;
3. According to Equation 3 and tor each sampled U calculate its score by:
(core(H,2) = Po- (H (3 Pa(la))
(5)
4. Return the H U with the highest score continue the maximisation step.
By learning an abductive logic theory H, the search space of pseudo-label be largely pruned thanks to the sparsity of the probabilistic distribution structured by U HUZty
3.3 THE Metaadd IMPLEMENTATION
We implement the above abduction-induution algorithm with Abductive Meta-Interpretive Learn- ing (AetaaAb)), whose codes are shown Figure 2 extends the general meta-interpreter of MIL (Muggleton et al. 2014) by including an abduction procedure (bold fonts in Figure 2 that can abduce relational constraints on pseudo-labels for pruning the search space.
Interpeiteeee Learning (MIL) is form of ILP (Muggleton & de Raedi, 1994). It learns first- order logic programs with second-order meta- meta-interpreter which is composed of a definite first- order background knowledge B and meta-rules M. B contains the primitive predicates for con- structing first-order hypotheses H: M is second-order clauses with existentially quantified predicate variables and universally quantified first-order variables that shape the structure of the hypothesis space H. Briefly speaking, MIL attempts to prove the training examples and saves the resulting programs for successful proofs.. However, MIL can only learn first-order logic programs from pure symbolic domains, where the examples are deterministic and noise-free.
By combining abduction and induction, Meta Abd learn abductive logic programs from noisy domains where the distribution on possible worlds (Nilsson) 1986) is given by a set of probabilistic facts. A possible world truth value assignment to the probabilistic logic facts. For the exam- ple in Figure [ each combination of the possible pseudo-labels of the three input images forms a
"The abduction he naturally accelerated by parallel computing, more details in the Appendix.
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Table Domain knowledge used by the compared models.
Domain Knowledge
End-to-end Models
Meta And
Recurrence
LSTM & RNN
Prolog s list operations
Arithmetic functions
NAC& NALU (Trask et al. 2018
Predicates add, mult and eq
Permutation
Permutation matrix eort (Grvvr el al. 2019
Prolog's i permutation
Sorting
scrt operator (Grover et al. 2019)
Predicate s (learned from sub-task)
possible world, whose probability distribution is defined by the probability values output by the per- ceptual neural net. As shown in the figure, given the abducible primitives as background knowledge Abd can construct the hypotheses H while abducing the relational constraints on 2.
After an abductive hypothesis H has been sampled, the search for will be done by logical abduc- tion. Finally, the score of H U z will be calculated by Equation [ where (2(2)) the output of the perception model, which in this work is implemented with a neural network yoo that outputs:
Pa(2|x) = 80f to
Meanwhile, we define the prior distribution on H by following Hocquette Muggleton (0088
6
Pp B) = (sT c 2)
where C(H) is the complexity of the learned program, e.g., the size of H.
4 EXPERIMENTS
This section describes the experiments which apply Mela Abd learn first-order logic programs from images of handwritten digits in two scenarios: (1) cumulative sum/product and (2) sorting. The experiments aim address the following two questions:
1. Can the abduction-induction strategy of Abd learn first-order logic programs and train perceptual neural networks jointly?
2. Given the same type and amount of background knowledge shown in Table I is hybrid modelling, which directly leverages the background knowledge in symbolic form, better than end-to-end learnin??
4.1 LEARNING CUMULATIVE SUM AND PRODUCT FROM IMAGES
Following the setting of Trask et al. 2018), the inputs of the two tasks are series of randomly chosen MNIST digits; the numerical outputs are the sum and product of the digits, respec- tively. The lengths of training sequences are 2-5. To verify if the learned models can extrapolate to longer inputs, we also include test examples with length 10 (both tasks), 15 (in the cumulative product task) and 100 (in the cumulative sum task). In the cumulative product experiments, when the randomly generated sequence is long enough, it will be very likely to contain a and makes the final outputs equal to O. So the extrapolation examples with length 15 only contain digits from 1 to The dataset contains 3000 and 1000 examples for training and validation, respectively; the test data of each length has 10,000 examples. Since the end-to-end models usually require more training data due to the model complexity, we also did experiment with 10,000 training examples for them.
Materials
Methods We compare Meta with four end-to-end learning baselines, including RNN, LSTM and LSTMs attached to Neural Accumulators(NACC) and Neural Arithmetic Logic Units (NALU Trask et al.) 2018). The performance is measured by classification accuracy (Acc.) on length-ond inputs, mean average error (MAE) in sum tasks, and mean average error on logarithm (log MAE) of the outputs in product tasks whose error grows exponentially with sequence length.
A convnet processes the input images to the recurrent networks, as Trask et al. (018)) described; it also serves as the perception model of Mela And to output the probabilistic facts. As shown in Table T all models are aware of the same amount of background knowledge: the end-to-end models use LSTM RNN handle recurring inputs and use NACs and NALUs to encode basic arithmetic functions, while Meta Abd can exploit them explicitly as primitive predicates in the Prolog language. Note that Meta Abd uses the same background knowledge for both sum and product tasks. Each experiment is carried out five times, and the average of the results are reported.
We use the implementation of NAC and NAIU J from http://gghattmmmkeeiiiit/a//LLL

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Table 2: Results on the MNIST cumulative sum/product tasks.
MNIST cumulative sum
MNIST cumulative product
Acc.
MAE
Ace.
log MAE
Seq Len
5
100
5
10
15
LSTM
9.80%
15,3008
44,3082
449.8304
9.80%
11.1037
19.5594
21.6346
RNN-Relu
10.32%
12,3664
41.4368
446,9737
9.80%
10.7635
19. 19.8029
21.8928
LSTM-NAC
7.012%
6,0531
29.8749
435.4106
0.00%
9.6164
20.9943
17.9787
LSTM-NAC10r
8.85%
1.9013
21.4870
424.2194
10.50%
9.3785
20.8712
17.2158
LSTM-NALU
0.00%
6,2233
32.7772
438.3457
0.00%
9.6154
20.9961
17.9487
LSTM-NALU
0.00%
6.1041
31.2402
436,8040
0.00%
8.9741
20.9966
18.0257
Meta
95.27% %
0,5100
1.2994
6,5867
97.73%
0.3340
0.4951
2.3735
LSTM-NAC, shot CNN
49.83%
0.8737
21.1724
426.0690
0.00%
6.0190
13.4729
17.9787
LSTM-NALU, CNN
0.00%
6.0070
30.2110
435.7494
0.00%
9.6176
20.9298
18.1792
Mela
Lobot H-shor
CNN
98.11%
0.2610
0.6813
4.7090
97. 97.94%
0.3492
0,4920
2,4521
Cumulative Sum: (add ,C
Results Our experimental results shown in Table 2 the learned first-order logic theories are shown in Fig- ure Bal End-to-end models that do exploit any back- ground knowledge (LSTM and RNN) perform worst on these tasks. For NALU and NAC, even though they can exploit background knowledge by using the specially de- signed differentiable neural modules, the performance is still significantly worse than Mbdaaba given the same amount of training data or even more.
=(A,B):-oq(AA3
Cumulative Product: fAA :mmlltA, (C,B).
=(A,B))-oq(AAB)
Bogosort:
f(A (A
=(A) BA,,
e :ttai(((AAA)) (B)
8.1 a.1(4,B):-nn.prre (6), ta, B)
Although Aeda achieves the best result among the compared methods, we observe that its EM learning sometimes converges
(a) Programs learned by Meta Abd
saddle points local optima in the cumulative sum task. This phenomenon hap- pens less in the other task, because of the distribution P(H, 2 B, I, 0) of learning the cumulative product function much sparser compared to cumulative sum. Therefore, we also carry out extra experiments with 1- shot pre-trained convnets, which are trained by randomly sampling one example in each class of MNIST data. Al-
26 02s
#
+
0.77s
0.0
0.5
1.0
1.5
2.0
Number of Prolog inferences
X10
(b) Comparison sampling and H
Figure 3: Learned programs and the time efficiency abduction.
though the pre-trained convnet weak (Ace. 20~35%), it provides good initialisation for the EM algorithm and improves the learning performance.
Figure 3b shows the time efficiency of Aed' abduction-induction strategy on one batch of examples in the cumulative sum task. "z + H means first samples pseudo-labels and then learn И with ILP; "H + z" means first sample an abductive hypothesis H and then use H to abduce z. The unit of x-axis is the average number of Prolog inferences, the number the end of each bar is the average inference time in seconds. Evidently, the abduction leads to a substantial improvement in the number of Prolog inferences and significantly reduces the search complexity.
4.2 LEARNING ROGOSORT FROM IMAGES
Materials Following the setting from Grover et al. (2019), the input of this task is sequence of randomly chosen MNIST images of distinct numbers; the output the correct ranking (from large to small) of the digits. For example, when
5 7. 3 f ([5, 9, 3, 8 then the output should be y = [3, 1, 4, 5, 2). The training dataset contains 3000 training and 1000 validation examples; the test dataset has 10,000 examples. The training examples contain five images, and we test the learned models on image sequences with lengths 3, 5 and Experiments are repeated five times, and the average of results are reported.
Methods In this task, we compare Meta Abd to NeuralSor Grover et al. 2019), which imple- ments a differentiable relaxation of sorting operator. Given input list of scalars, it generates stochastic permutation matrix by applying the pre-defined deterministic or stochastic sort oper- ator on the inputs, i.c., NeuralSort can be regarded as differentiable approximation bogosort
WWe use the implementation of NeuralSort from http://ihhhbbccmmmmmmoggpphhhnnh

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Table 3: Average accuracy of the bogosort task. First value is the rate of correct permutations; second value is the rate of correct individual element ranks.
Seq Len
3
Deterministic NeuralSort
95,49% (96.82%)
88.26% (94.32%)
80.51% (92.38%)
Stochastic NeuralSort
95.37% (96.749) 74%)
87.46% (94.03%)
78.50% (91.85%)
Meta
96.33% (97.22%)
91.75% (95.24%)
87.42% (93.58%
(permutation sort). Although for Meta is easy to include stronger background knowledge for learning more efficient sorting algorithms like quicksort (Cropper & Muggleton 2019), in order to make a fair comparison, we adapt the same background knowledge as NeuralSort logic rules and learn bogosort. We did not compare to other baselines such as LSTM/RNN with different activation layers because they are weaker than NeuralSort this task (Grover et al. 2019).
The background knowledge of permutation in Meta is implemented with Prolog's built-in in predi- cate permut iti. Meanwhile, instead of providing the information about sorting as prior knowl- edge like the NeuralSort, we try to learn the concept of "sorted" (represented by monadic predicate s) from data sub-task, whose training set is the subset of the sorted examples within the training dataset (< 20 examples). To do this, Meta Abd uses an MLP attached to the same untrained convnet as previous experiments to produce dyadic probabilistic facts nn-peed ( if the first two items in the image sequence satisfy a dyadic relation. Please note that the attached MLP is not provided with supervision on nn-pred about whether it should learn "greater than" or pred
9 -] ), which learns
"less than". Moreover, we do not provide any prior knowledge about total ordering, only learns a dyadic partial order among the MNIST images. As we can see, the background knowl- edge used by Meta Abd is much weaker than that is used by NeuralSort. The sorting task and its sub-task are trained sequentially. In our experiments, the first five epochs of M Meta learn the sub-task, and then re-uses the learned models learn bogosort.
Results Table shows the average accuracy of the compared methods in the sorting tasks; Fig- ure [3a shows the learned programs. The performance measured by the average proportion of correct permutations and individual permutations following Grover al. (2019). Although using weaker background knowledge, Meta has a significantly better performance than NeuralSort in both interpolation (length 5) and extrapolation (length 7) experiments.
The learned program of and the dyadic neural net nn-pred are both successfully re-used in the sorting task, where the learned program of s is consulted as interpreted background knowl- edge (Cropper et al., (2020), and the neural network that generates probabilistic facts of nn-pred directly re-used and continuously trained during the learning of sorting. This experiment also demonstrates Meda ability of learning recursive logic programs and predicate invention (the invented predicate s_1 in Figure Ba).
5 CONCLUSION
In this paper, we present the Abductive Meta- InterInnerrreetve Learning (Meta Aba) approach that can train neural networks and learn recursive first-order logic theories with predicate invention simul- tancously. By combining symbolic learning with neural networks, . Meta,Ad can learn human- interpretable models directly from raw-data, and the learned neural models and logic theories can be directly re-used in subsequent learning tasks. Meda is general framework for combining sub-s sub-symbolic perception with logical induction and abduction. The perception model extracts proba- bilistic facts from sub-symbolic data; the logical induction searches for first-order abductive theories in relatively small hypothesis space; the logical abduction uses the abductive theory to prune the vast search space of the truth values of the probabilistic facts. The three parts are optimised together in well-defined probabilistic model.
In future work, we would like to apply Meta Abd on more complicated tasks that involve sub- symbolic perception and symbolic induction, such as reinforcement learning. Instead of approxi- mating logical inference with continuous and differentiable functions, Meta uses pure logical inference for reasoning and it is possible to leverage more advanced symbolic inference/optimisation techniques like Satisfiability Modulo Theories (SMT) (Barrett & Tinelli 2018) and Answer Set Pro- gramming (ASP) (Liffshitz) 2019), which are able perform large scale inference efficiently.
8

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Under review as a conference paper at ICLR 2021
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10


Reviews:
[{"review_id": "WGgOeUWU-t2", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes an EM framework (Meta_abd) for iteratively learning the parameters of a neural network (\"perception\" component), and inducing the logical rules underlying a domain (\"reasoning\" component). The rule-learning component is the existing MIL system (with some modifications that are bolded in Figure 2). The paper's ability to learn recursive theories and invent predicates is directly inherited from MIL. Hence, I find its contributions with respect to these two aspects to be incremental at best.\n\nThough interesting, I find that significant details about the Meta_abd's EM model are omitted, and this prevents one from ascertaining Meta_abd's true effectiveness (please see questions below).\n\nFurther, EM as a framework for bridging noisy inputs and symbolic reasoning has been explored by prior work (the pLogic, ExpressGNN, and pGAT systems listed below). Could the authors position Meta_abd vis-a-vis these systems, and mention them as related work? Meta_abd's \"neural probabilistic facts\" map to uncertain knowledge graph triples in these systems (the \"ground logical expressions\" mentioned in Section 3.1 of the paper), and Meta_abd's neural network's parameters map to entities' embeddings. Like Meta-abd's logical component, the logical part of these systems are used to infer the probabilities of the triples (neural probabilistic facts), which in turn are used to train an embedding model. The difference between these prior systems and Meta_abd is that Meta_abd learns its logical rules whereas the previous systems require the rules to be pre-specified. In the context of these existing systems, the contribution of Meta_abd appears to lie only in learning rules with the existing MIL system, and thus seems moderately incremental.\n\n\n* Probabilistic Logic Neural Networks for Reasoning. Qu and Tang, 2019\n* Efficient Probabilistic Logic Reasoning with Graph Neural Networks, Zhang et al., 2020\n* Probabilistic Logic Graph Attention Networks for Reasoning, Vardhan et al., 2020\n\n\n\nQUESTIONS\n\n* Page 4, Figure 1: When optimizing the \\theta parameters of the neural network in the M-step, are the labels for each digit image probabilistic? If they are, how are the probabilities derived (e.g., from conflicting pseudo-labels)? If the label per image is deterministic, how is the label chosen?\n\n* Page 5, Eqn 5, step 1: How is H *sampled*? From the paper's description, it seems that H is learned deterministically from themodified MIL (Figure 2). Do you perturb the H returned by MIL probabilistically in some way? If not, how does one justify the claim that H is sampled? (Is H the mode?)\n\n* Page 5, Eqn 5, step 2: How much does the the pruning of z with the abduced constraints help? There is no ablation study to verify the contribution of the abduced constraints.\n\n* Page 5, Eqn 5: How are the parameters and/or rules initialized? EM is particularly sensitive to initialization conditions. This is something that the paper alludes to too on page 7 paragraph 2 (\"converges to saddle points or local optima\"). How does Meta_abd ameliorate this sensitivity? Does it make multiple runs of EM, each from a different initialization? If so, how does it choose the best model? What is the \"success rate\" of these runs, or how many runs are needed to learn a good model? Are the comparison systems treated fairly in being allowed multiple runs too? The answers to these questions would help to elucidate the workings of Meta_abd.\n\n* Page 6, para 2, \"provides a good initialization ... improves the learning performance\". \nHow much does the pre-trained convent help exactly? And from what base?\n\n* Could the authors comment on Meta_abd's scalability? What're its space and time complexity? This information is particularly helpful in view that the empirical evaluation only uses fairly small datasets from a narrow domain.\n\n* Supplementary material, Page 4, Figure 10:\nHow sensitive is Meta_abd to the meta-rules? Would it be fair to say that the meta-rules are cleverly specified to provide a strong enough structural bias for Meta_abd to be able to learn the correct rules? What happens empirically if the meta-rules are removed (one at a time)? These meta-rules seem to imbue Meta_abd with an unfair advantage over the comparison systems.\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "EvjRWfDQjo8", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "I like the idea of this paper, however the paper seems to be more written to impress rather than inform.\n\nI cannot see how it \"learns ... simultaneously from raw data.\"  I think the interface is in the nn(image=value,prob) but nn() is never explained in the text. I can't work out: what are the inputs? What are the outputs? What. is the relationship between them? Similarly,  prove() used in Figure 2 is never explained; what are the argument? What is the intended interpretation?\n\nIt seems funny to have 0/1 probabilities (equation (4)). Surely there is so much noise that you can't perfectly predict the outputs.\n\nI didn't see how you overcome the exponential complexity promised in the abstract (and why do you think it is caused by the interface?).\n\nIn the experiments, why is MAE (or log MAE) a reasonable measure? What is the accuracy of the correct program? (I think there is supposed to be a correct program you are learning).  Is the correct program in the search space with a non-zero prior?\n\nIs \"Acc\" in table 2 correspond to just predicting the digit? Why is there so much variability? What is the accuracy of nn()?\n\nThe paper needs to be self contained. For example, you need to tell us that #= means equality and what Prolog's permutation  predicate is  (is it related to permute() in Figure 3?)\n\nI understand you are trying to learn the simplest logic program that can produce the output from raw images. What you did at the top-level seems right, but it is not described well enough. You need to provide enough details so that it is reproducible.", "extra_scores": {}, "extra_text": {}}, {"review_id": "C4imHvOcWH", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary.\n\nIn this paper, the authors have presented a framework that combines meta-interpretive learning and the abductive learning of neural networks. The high-level idea is to formulate a unified probabilistic interpretation of the entire algorithm so that both the inductive logic programming module and the neural network modules can be trained jointly from data. The authors have demonstrated the application of the proposed algorithm to learning arithmetic operations and sorting operations by looking at input-output mnist digits.\n\nComments.\nThe key idea of the paper has been presented clearly. The authors demonstrated two tasks: cumulative sum/product, and sorting. Both tasks require learning recursive rules, and the bogosort task requires predicate invention. These are challenging tasks for both neural networks and ILP algorithms.\n\nHowever, my major comments about the paper is that the experiment sections are relatively weak and they have definitely missed some important baseline comparisons. Concretely, taking the cumulative summation task as an example, the MetaAbd model has very strong inductive biases, because of the builtin \"add\" operation and the metarules built into the system, which strongly favors recursive rules of specific forms. However, at least the \"add\" operation was not built into other baselines.\n\nSecond, there have also been many other works trying to solve this task:\n- partial ILP (Evans & Grefenstette, 2018) and machine apperception (Evans et al.,\n2019) that can learn mnist digits with much weaker assumptions: they can even learn the \"succ\" relationship between digits.\n- Neural GPU (Kaiser and Sutskever 2015) that can learn to add multi-digit numbers without any builtin \"add\" operations.\n- Differentiable Neural Computer (https://deepmind.com/blog/article/differentiable-neural-computers)\n- Neural Programmer-Interpreters (Reed et al 2015) and its follow-ups: they support integrating human-written primitive functions (such as the \"add\" operation) with neural networks.\nThe authors are encouraged to make comparisons with these methods as well.\n\nThird, the learned logic rules are relatively simple. This makes me less convinced about the applicability of the paper. The authors have made very strong claims in the abstract/intro about \"To the best of our knowledge, MetaAbd is the first system that can jointly learn neural networks and recursive first-order logic theories with predicate invention.\" For example, partial ILP and machine apperception can do that, too. Recently, there have also been other trials on using relational neural networks for bridging perception and rule learning, such as,\n- Graph Neural Networks (https://arxiv.org/abs/1806.01261)\n- Neural Logic Machines (https://arxiv.org/abs/1904.11694)\n\nOverall, I think this paper is not matching the publication standard of ICLR.\n\nMinor:\nPlease change the latex formatting of the model name. There is currently an extra space between M and e.", "extra_scores": {}, "extra_text": {}}, {"review_id": "5CPJwzn4rM5", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "In this paper, the author proposes Meta_abd which is a hybrid model that learns a deep recognition model and FOL rules the same time. The goal of this work is to learn FOL rules from raw data such as digits presented in image patches in an end-to-end fashion. The model is evaluated with 3 induction benchmarks associated to the MINST digit dataset.\n\nPersonally, I find this paper to be difficult to read and many details in methods and experiments are missing, making it hard to understand the authors' contribution. Given its current state, I would recommend rejection. My concerns are as follows:\n\nMotivation:\n\n- Neural-symbolic integration usually refers to combining logic reasoning into deep model's decision process for better interpretability or sample-efficiency, or to use deep models to help the logic reasoning tasks such as ILP or deduction.\n\n- With that being said, I find the claim for this hybrid model to be unjustified. In Meta_abd, NN is only used for data pre-processing which is completely agnostic to the later logic component.\n\n- In fact, diff-ILP (Evans & Grefenstette, 2018) also uses NN for pre-processing MNIST digits for ILP task which I personally find to be similar to the proposed method, though I'm happy to be proven wrong.\n\nClaims: I find many of the claims in the paper to be ambiguous and lack justifications\n\n- In section 2, the author claims that differentiable ILP methods rely on fully trained NN for pre-processing -  this is untrue, for example NeuralLP is an end2end model that can be extended to the MINST benchmark with a perception module that's jointly trainable\n\n- The author also claims that most existing NeSy systems only utilize a pre-defined knowledge base. I find this claim to be confusing and the author does not discuss how the proposed method has addressed this limitation\n\t\n\nMethod: I find the method to miss many details\n\n- The author defines the learning problem with Eq1 and Eq2 in Section 3.1 and claims to it would be learned through EM. However, Eq1 and 2 do not reveal details about the method, and the exact procedure of EM is unclear to me\n\n- In section 3.3 the author claims the proposed method is an extension of MIL, but this concept is not formally introduced in the paper.\n\n- I find Figure 2 to be difficult to understand - why is Prolog used here? What's the connection of Prolog to the proposed method? It seems to suggest the proposed method is using Prolog for solving the constraints?\n\nExperiment: \n\n- The proposed method is only compared to LSTM and RNN.  The author should include some ILP baselines such as diff-ILP or NeuralLP and substitute the digit image into its ground-truth digit symbol for reference, though I personally think adding a NN pre-processing module is straightforward  as well.\n\n- 3 benchmarks are fairly small consisting of only a few predicates and rules. How does the proposed method scale with the number of predicates and the size of the grounding space?\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper addresses the difficult problem of combining ILP in a meta-interpretive framework with noisy inputs from a neural system.   The essential idea is to use MIL to "efficiently" search for constraints on the neural outputs (eg z1 + z2 + z3 = 7, or z2< z3) as well as logic programs, with a score related to program complexity as well as probability of the best constraint-satisfying neural outputs.  It is interesting work for the right audience but it's clear from the reviews that the presentation was difficult for ICLR readers, even ones with appropriate background. 

Some potential weaknesses of the approach include:

1 - it's unclear how scalable the MIL framework is - presumably the intrinsic difficultly of the search means that programs and constraint sets must be small

2 - it's unclear how general the approach is beyond the digits-as-separate-inputs setting of the two experimental studies, and its unclear how accurate the perceptual layer needs to be - MNIST obviously being an example of a case where there is little noise with a modern classifier.

3 - it's unclear how constraints can in general be used to backprop any information to the underlying neural system, and without this the joint training seems to be quite limited.

Overall the paper is judged as inappropriate for ICLR.
Decision: Reject