Going Beyond Approximation: Encoding  Constraints for Explainable Multi-hop Inference via Differentiable Combinatorial Solvers
Authors: Mokanarangan Thayaparan
Marco Valentino
Andre Freitas
Venue: Submitted to ICLR 2023 (2023)
Keywords: Explainable AI
Constrained Optimization
Integer Linear Programming
Question Answering

Abstract: Integer Linear Programming (ILP) provides a viable mechanism to encode explicit and controllable assumptions about explainable multi-hop inference with natural language. However, an ILP formulation is non-differentiable and cannot be integrated into broader deep learning architectures. Recently, Thayaparan et al. (2021a) proposed a novel methodology to integrate ILP with Transformers to achieve end-to-end differentiability for complex multi-hop inference. While this hybrid framework has been demonstrated to deliver better answer and explanation selection than transformer-based and existing ILP solvers, the neuro-symbolic integration still relies on a convex relaxation of the ILP formulation, which can produce sub-optimal solutions. To improve these limitations, we propose Diff-Comb Explainer, a novel neuro-symbolic architecture based on Differentiable BlackBox Combinatorial solvers (DBCS) (Pogančić et al., 2019). Unlike existing differentiable solvers, the presented model does not require the transformation and relaxation of the explicit semantic constraints, allowing for direct and more efficient integration of ILP formulations. Diff-Comb Explainer demonstrates improved accuracy and explainability over non-differentiable solvers, Transformers and existing differentiable constraint-based multi-hop inference frameworks.

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GOING BEYOND APPROXIMATION: ENCODING CON- STRAINTS FOR EXPLAINABLE MULTI-HOP INFERENCE VIA DIFFERENTIABLE COMBINATORIAL SOLVERS
Anonymous authors Paper under double-bling blind review
ABSTRACT
Integer Linear Programming (ILP) provides a viable mechanism to encode ex- plicit and controllable assumptions about explainable multi-hop inference with natural language. However, an ILP formulation is non-difeerentiable and can- not integrated into broader deep learning architectures. Recently, Thayaparan eet al. (2022) proposed a novel methodology to integrate ILP with Transformers to achieve end-to-end differentiability for complex multi-hop inference. While this hybrid framework demonstrates to deliver better answer and explanation lection than transformer-baee and II.P solvers, the neuro-symbolic integration still relies convex relaxation of the ILP formulation, which can produce sub- optimal solutions. To improve these limitations, we propose Diff-Comb Explainer, Differentiable BlackBox Combi-
novel neuro-symbolic architecture based natorial solvers (DBCS) (Pogančić et al. 2019). Unlike existing differentiable solvers, the presented model does not require the transformation and relaxation of the explicit semantic constraints, allowing for direct and a more efficient integra- tion of IL.P formulations. Diff-Comm Explainer demonstrates improved accuracy in answer and explanation selection over non-differentiable solvers, Transformers and constraint-bad based diffcrentiable multi-hop inference frameworks.
INTRODUCTION
Given a question expressed in natural language, I..--based Multi-hop Question Answering (QA) aims to construct explanation graph of interconnected facts (i.e., natural language sentences) to support the answer (see Figure This framework provides a viable mechanism to encode explicit and controllable assumptions about the structure of the inference (Khashabi al. 2018 Khot al.) 2017 Khashabi et al.) (016). For this reason, inference based on constrained optimization is gen- erally regarded as interpretable and transparent, providing structured explanations in support of the underlying reasoning process (Thayaparan et al. 2020).
However, IIP solvers non-differentiable and cannot be integrated part of broader deep learning architecture (Paulus et al. [202]] Pogančči et al.) (019).. Moreover, these approaches are often limited by the exclusive adoption of hard-coded heuristies for the inference and cannot be optimised end-to-end on annotated corpora to achieve performance comparable to deep learning
counterparts Thayaparan al. 2022. Khashabi al. 2018).
In an attempt to combine the best of both worlds, Thayaparan et al. 2022) proposed novel neuro- symbolic framework (Diff- Explaine))))) that integrates explicit constraints with neural representations via Differentiable Convex Optimization Layers (Agrawal et al., 2019). Diff-Explainer combines constraint optimization solvers with Transformers-baseed representations, enabling end-to-end train- ing for explainable multi-hop inference. The non-differenitability of ILP solvers is alleviated by approximating the constraints using semi-detinite programming (Helmberg 20000. This approxi- mation usually requires non-trivial transformations of ILP formulations into convex optimization problems.
Since constraint-based multi-hop inference is typically framed optimal subgraph selection via binary optimization (0, 1). The semi-definite relaxation employed in Diff- Etfllaxee necessitates a continuous relaxation of the discrete variables (from (0, 1) to (0, 1)). While this process can provide

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Question:
Light rays are focused by the lens of
camero throough the process of Answer:
refraction
Explanations:
convex lens causes light to refract and concentrate to magnify distant objects
F2: convex lens is used in canera
focus means concentrate
F4: refrreciionn is a kind of process
Figure 1: Example of question, answer and explanations as graph (Xie et al. 2020 Jansen et al. 2018
tight approximations for ILP problems. this relaxation can still lead to sub-optimal solutions in practice (Yoshida 2011 Thapper & Zivný. 2018) leading to errorncous answer and explanation prediction.
To improve on these limitations, we propose Diff-Comb Explainer, a novel neuro-symbolic archi- tecture based Differentiable BlackBox Combinatorial solvers (DBCS) (Pogančié et al. (019). The proposed algorithm transforms a combinatorial optimization solver into a composable building block of neural network. DBCS achieves this by leveraging the minimisation structure of the com- binatorial optimization problem, computing a gradient of continuous interpolation to address the non-differeniaailiiy of ILP solvers. In contrast to Diff- Exflaineri (Thayaparan et al. 2022). DBCS makes it possible to compute exact solutions for the original ILP problem under consideration, ap- proximating the gradient.
Our experiments on multi-hop question answering with constraints adopted from Explana- tionLP (Thayaparan et al. 2021) yielded an improvement of 11% over non-differentiable solvers and 2.08% over Diff-Explainer. Moreover, we demonstrate that the proposed approach produces more accurate and faithful explanation-ban based inference, outperforming non-differentiable IL.P-bsed solvers, Diff- Diff-Explainer and Transformer-basee approaches.
2 RELATED WORK
Constraint-based multi-hop inference ILP has been applied for structured representa-
tion (Khashabi et al. 2016) and over semi-structured representation extracted from text Khot al.)
2017 Khashabi et al. 2018). Early approaches were unsupervised. However, recently Thayaparan et al. (021)) proposed the ExplanationLP model optimised towards answer selection via Bayesian limited to fine-tuning only nine parameters and used pre- pre-trained neural embedding. Diff-Explainer (TThayparan et al.) 2022) was the first approach to integrate con- straints into deep-learning network via Differentiable Convex Optimisation Layer (Agrawal al., 2019 by approximating ILP constraints using Semi-definite programming. (Lovász & Schrijver 1991
optimisation. ExplanationLP
Hybrid reasoning with Transformers Clark et al. (2021) proposed "soft theorem provers" op- erating over explicit theories language. This hybrid reasoning solver integrates natural language rules with transformers to perform deductive reasoning. [Saha et al.] (2020) improved top of it, enabling the answering of binary questions along with the proofs supporting the prediction. The multiProver (Saha al. 2021] evolves on top of these conceptions to produce approach that is ca- pable producing muluple proofs supporting the answer. While these hybrid reasoning approaches produce explainable and controllable inference, they assume the existence of natural language rules and have only been applied to synthetic datasets. On the other hand, our approach does not require extensive rules set and tackle complex scientific and commonsense QA.
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Drigh Construction
Subgraph Seledion
Answer Explanation Selediio
boe beladion
Calcubtte Semarts Relewanne
[CexToa] Relevaace Scocen
Ai STRANG
STRAAS
UB0S
Verseble
(P')
ge
Par
STRAAG
Cce
Aq
Toonster
keighee (4")
Difforentaato etentoon Conbinatoriss Solves
tWolghto
oegentiv
Closs
Back Propogation
Figure 2: End-to-end architectural diagram Diff-Comb Explainer. The integration of Differen- tiable Blackbox Combinatorial solvers will result in better explanation and answer prediction.
Differentiable Blackbox Combinatorial Optimisation Solver teger problem:
Given the following bounded in-
min ZEX
subject to
Ax S b,
where X E 2", I"" x are the variables, A [ei,.... Um E R is the matrix of constraint coefficients and E R.m is the bias term. The output of the solver g(c) returns the arg min of the integer problem.
Diffcrentiable Combinatorial Optimisation Solver (Pogančić et al. b are constant and the task is to find the dL/dc given global loss function L with respect to solver output x at a given point = g()). However, a small change in c rypically not going to change the optimal ILP solution resulting in the true gradient being zero.
2019) (DBCS) assumes that A,
In order to solve this problem, the approach simplifies by considering the lincarisation f of at the point A.
f = + dL (f) (x $) dr
(2)
to derive:
df(g(c)) dc dc
dL
By introducing the linearisation, the focus is now to differentiating the piecewise constant func- tion [g(()).. The approach constructs continuous interpolation of f(g()) by function fx (w)) Here the hyper-parametts À controls the trade-off between informativeness of the gradient and faithfulness to the original function.
Diff-CoMB EXPLAINER: DIFFERENTIABLE BLACKBOX COMBINATORIAL SOLVER FOR EXPLAINABLE MULTI-HOP INFERENCE
ILP-based QA is typically applied to multiple-choice question answering (Khashabi et al. 2018 Khot et al. 2017 Khashabi al. 2016 Thayaparan et al.] 202T). Given Question (Q) and the set of candidate answers C' (c1, (22 C3 .... Co) the aim is to select the correct answer Cans-
In order to achieve this, Ia.ed approaches convert question answer pairs into a list of hypothesis H = {h1, h2, has .... hnd (where h; is the concatenation of Q with c;) and typically adopt retrieval model (e.g: BM25, FAISS (Johnson et al. 2017)), to select a list of candidate explanatory facts F {f1, f2, f3. .... fa). Then construct weighted graph G (V, E, W) with edge
weights W E + IR where V = {(h,} F}. edge weight Wa of each edge Egs denote how relevant fact fi is with respect to the hypothesis h,
Given this premise, ILP-based Multi-hop QA can be defined as follows (Thayaparan et al. 2022):
Definition 3.1 (ILP- Base Multi-Hoo Hop QA). Find subset V C V. h V* and E C E such that the induced subgraph G (V*, E²) ) is connected, weight W[G = (V*, E*) == E ((e) maximal and adheres set of constraints M, designed emulate multi-hop inference. The
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hypothesis h2 with the highest subgraph weight WGG (V*, E*)] is selected to be the correct answer Cans-
As illustrated in Figure 2 Diff-Comb Explainer has 3 major parts: Graph Construction, Subgraph Selection and Answer/Explanatiin Selection. In Craph Construction, for each candidate answer oa we construct graph G = (V', E", W*) where the V = { (FF) and weights W Ei denote how relevant a fact für is with respect to the hypothesis h4 These edge weights (Wis) are calculated using weighted (#) sum of scores calculated using transformer-based (ST'rans) embeddings and lexical overlap.
de of each edge
In the Subgraph Selection step, for each Go Differentiable Blackbox Combinatorial Solver (DBCS) with constraints are applied to extract subgraph G", In this paper, we adopt the constraints proposed for ExplanationLP (Thayaparan al., 202T). ExplanationLP explicit abstraction by grouping facts into abstract and grounding facts. Abstract facts are core scientific facts that question is attempting to test, and grounding facts link concepts in the abstract facts to specific terms the question/answer. For example, in Figure [ the core scientific fact is about the nature of convex lens and how they refract light (Fi). Facts F2, F3, F4 help to connect the abstract fact to the question/answer.
Finally, in Answer/Explanation Selection the model is to predict the correct answer Cans and relevant explanations Feep- During training time, the loss is calculated based on gold answer/explanations to fine-tune the transformers (STrans, and weights (f). The rest of the section explains each of the components in detail.
3.1 GRAPHI CONSTRUCTION
In order to facilitate grounding abstract chains, the retrieved facts F are classified into grounding ...: fm such that = FA U
facts FG = {fi f? and abstract facts FA = {ff, f2, fā ,.., FG and +
Similarly to Diff-Explainer (Thayaparan et al. lexical scores, calculate the edge weights. We iet al. 2019) bi-encoder architecture to calculate the semantic relevance. The semantic relevance score from STrans is complemented with the lexical relevance score. The semantic and lexical relevance scores are calculated as follows:
2022 we use two relevance scores: semantic and
Centence-Transfoneee (STrans) (Reimers
Semantic Relevance (s): Given hypothesis h4 and fact ʃy we compute sentence vectors of h4 = STrans(hh) and f, = STrans(ff) and calculate the semantic relevance score using cosine- similarity as follows:
8iz = S(h, F3)
(4)
Lexical Relevance (I): The lexical relevance score of hypothesis h2 and f, given by the percentage of overlaps between unique terms (here, the function trm extracts the lemmatized set of unique terms from the given text):
[trm(h.) trm(f,)) maz((tr((hi) [trm(ff)))
bi L(h,, fj)
(S)
Given the above scoring function, we construct the edge weights matrix (W) as follows:
Ogglit
(G E FG (j,)) FA jE Fo, FA
Quolsh
W
Oyulzk
(6)
Ougilgh + 8
jE FG, hi
Oynilix + OpalS5k
j E FA,k= lu:
Here relevance scores are weighted by parameters which are clamped to [0, 1].
3.2 SUBGRAPH SELECTION VIA DIFFERENTIABLE BLACKBOX COMBINATORIAL SOLVERS
Given the above premises, the objective function is defined as:
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min -1( yr)
We adopt the edge variable y {0, 1)(x+1)((*+) where j7 (i + k) takes the value of 1 iff edge Ejs belongs to the subgraph and y3 takes the value of iff V2 belongs to the subgraph.
Given the above variable, the constraints defined as follows:
Answer selection constraint The candidate hypothesis should be part of the induced subgraph:
Σ
y3 =1
1 E 1
Edge and Node selection constraint If node V and Vi are selected then edges Eix and Ei will be selected. If node V is selected, then edge Egj will also be selected:
yj S jj
A (j, E
(9)
jA S YRE
A (i, E E
(10)
y 2 Yj YEk
(j.k) E
(11)
Abstract fact selection constraint
Limit the number of abstract facts selected to M:
Σ yg M
Vj E FA
(12)
3.3 ANSWER AND EXPLANATION SELECTION
The solved adjacency variable Y represents the selected edges for each candidate answer choice C;- Not all datasets provide gold explanations. Moreover, even when the gold explanations are available, they only available for the correct answer with explanations for the wrong answer.
In order to tackle these shortcomings and ensure end-to-end differentiability, we use the softmax (a) of the objective score (W'i ÿ as the probability score for each choice.
We multiply each objective score W ÿ value by the temperature hyperparameter (T) to obtain soft probability distributions 7² (where 7 = (W', 9 T)) The aim is for the correct answer Cans have the highest probability.
In order to achieve this aim, we use the cross entropy loss le as follows to calculate the answer selection loss Cans as follows:
Lans = 1a((()) 73 7"), Cans)
(13)
If gold explanations are available, we complement Lans with explanation loss Leep- We employ binary cross entropy loss between the selected explanatory facts and gold explanatory facts Feep for the explanatory loss follows:
Ceep (y f2, .... fiel. Feep)
(14)
We calculate the total loss (L) weighted by hyperparameters Aanss Aeep follows:
= Nanahans + demphezp
(15)
EMPIRICAL EVALUATION
ANSWER AND EXPLANATION SELECTION
We use the WorldTree corpus Xie al. 2020) for training the evaluation of explanation and answer selection. The 4,400 question and explanations in the WorldTree corpus are split into three different subsets: train-sel, dev-sel and test-set. We the dev-sel assess the explainability performance
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since the explanations for test-set are not publicly available. The background knowledge is consists of 5000 abstract facts from the WorldTree table store (WTree) (Xie et al. 2020) and over 100,000 is-a grounding facts from ConceptNet (Speer al. 2017).
We use the following basclines to compare against our approach for the WorldTree corpus:
Baselines:
BERT Base and BERT Large (Dellin et al. 2019): To use BERT for this task, we concatenate every hypothesis with & retrieved facts, using the separator token [SEP]. We use the Hugging- Face (Wolf et al. 2019) implementation of rrrrforSquenceClasfffctitio taking the prediction with the highest probability for the positive class the correct answer.
ExplanationI.P. Non-differentiable version of ExplanationLPP Using the constraints stated in Section 3 fine-tune the 0 parameters using Bayesian optimization and frozen STrans rep- resentations. This baseline aims to evaluate the impact of end-to-end fine-tuning over the non- differentiable solver.
3. Diff-Explainer: Diff-Explainer has already exhibited better performance over other explainable multi-hop inference approaches, including ILP-based approaches including TableILP (Khashabi itt al. 2016), TupleILP (Khot et al., 2017) and graph-based neural approach PathNet (Kundu et al. 2019). Similar to our approach, we use Explanation constraints with Diff-Explainer. We use similar hyperparameters and knowledge base used in Thhayapaann et al. (2022).
Metries The answer selection is evaluated using accuracy. For evaluation of explanation selection, we use Precision@KK In addition to Precision @ K, we introduce two new metrics to evaluate the truthfulness of the answer selection to the underlying inference. The metries are follows:
Explanatory Consistency@ K: Question/answer pair with similar explanations indicates similar underlying inference (Atanasova et al. 2022). The expectation is that similar underlying inference would produce similar explanations (Valentino et al.] 2021 2022). Given test question Qr and retrieved explanations Es we find set of Questions Qi = (Q: Qi ...) with at least K overlap
gold explanations along with the retrieved explanations E (et. Eg ..... Given this premise, Explanatory Consistency@@ @K K is defined as follows:
2
EcjE, E Eg]
(16)
ExfEE; leil
Explanatory Consistency measures out of questions/answee pairs with at least K similar gold expla- nations and how many of them share a common retrieved explanation.
The aim is measure how much percentage of the correct prediction is derived from correct inference and wrong prediction derived from wrong inference over the entire set. Let's say that the set of questions correctly answered as Aq.. wrongly answered questions AQ... set of questions with at least one correctly retrieved explanation as Aq, and set of questions where no correctly retrieved explanations AQu. Given this premise, Faithfulness is defined as follows:
Faithfulness:
|Aq.. n Aq. + IAq. Ag.
(17)
AAq. U A
higher faithfulness implies that the underlying inference process is reflected in the final answer prediction.
Table D illlutrates the explanation and answer selection performance of Diff-Comb Explainer and the baselines. We report scores for Diff-Comb Explainer trained for only the answer and optimised jointly for answer and explanation selection.
Since BERT does not provide explanations, we use facts retrieved from the fact retrieval for the best configuration (k- = 3) as explanations. We also report the scores for BERT without explanations.
We draw the following conclusions from the results obtained in Table I (The performance increase here are expressed in absolute terms):
(1) Diff-Comb Explainer improves answer selection performance over the non-differentiable solver by 9.47% with optimising only on answer selection and 10.89% with optimising on answer and explanation selection. This observation underlines the impact of the end-to-end fine-tuning frame- work. We can also observe that strong supervision with optimising explanation selection yields better performance than weak supervision with answer selection.

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Explanation Selection (dev)
Answer Selection (test)
Model
Precision
Explanatory Consistency
Faithfulness
@2
@ 1
@
@2
@1
Baselines
BERT Basc
45.43
BERT BERT Large
49.63
Fact Retrieval (FR) Only
30.19
38.49
21.42
15.69
11.64
BERT FR
52.65
58.06
LaRge FR
51.23
59.32
ExplanationIP
40.41
51.99
29.04
14. 14
11.79
71.11
62.57
Diff-Explainer
41.91
56.77
39.04
20.64
17.01
72.22
71.48
Diff-Comb Explainer
Answer selection only
45.75
61.01
49.04
29.99
18.88
73.37
72.04
Answer and explanation selection
47.57
63.23
43.33
33.36
20.71
74.47
73.46
Table 1: Comparison of explanation and answer selection of Diff-Comb Explainer against other baselines. Explanation Selection was carried out on the dev set as the test explanation was not public available.
(2) Diff-CComb Explainer outperforms the best transformer- based model by 14.14% for answer selec- tion. This increase performance demonstrates that integrating constraints with transformer-based architectures leads to better performance.
Diff-Comb Explainer outperform the best Eiflaaxp configuration (answer and explanation selection) by 0.56% even in the weak supervision setting (answer only optimization). We also outperform Explainer by 1. 1.98% the best setting.
(4) Diff- Comb Explainer is better for selecting relevant explanations over the other constraint-basee solvers. Diff-Comb Explainer outperforms the non-differentiable solver Precision @ K by 8.41 % (k =1) and 6.05% (A: =2). We also outperform Diff- Exflainee by 3.63% (k- =1) and 4.55% (k: =2). The improvement Precision
over the Fact Retrieval only (demonstrated with BERT FR) by 16.98% (k =1) and 24.74% (k 22) underlines the robustness of our approach to noise propagated by the upstream fact retrieval.
(5) Our models also exhibit higher Explanatory Consistency over the other solvers. This perfor- mance shows that the optimization model learning and applying consistent inference across dif- ferent instances. We also outperform the fact retrieval model which was also transformer-based model trained on gold explanations.
(6) Answer prediction by black-bon models like BERT do not reflect the explanation provided. This fact is indicated by the low Faithfulness score obtained by both BERT /ose/BETT Large- In contrast, the high constraint-based solver's Faithfulness scores emphasise how the underlying inference flects on the final prediction. In particular, our approach performs better than the non-differentiable models and Diff Explainer..
In summary, despite the fact that Diff-Explaier and Diff-Comb Explainer approaches use the same of constraints, model yields better performance, indicating that accurate predictions gener- ated by ILP solvers are better than approximated sub-optimal results.
4.2 QUALITATIVE ANALYSIS
Table illustrates some of the explanations extracted for ExplanationLP, Diff-Explainer and Diff- Comb Explainer. Both explanations and answer predictions in Question (1) are entirely correct for model. In this example, both ExplanationLP and Explaair have failed to retrieve any correct explanations or predict the correct answer. Both the approaches are distracted by the strong lexical overlaps with the wrong answer.

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Question (1): Which measurement is best expressed in light-years?: Correct Answer: the distance between stars the Milky Way. ExplanationI.P
Answer:
the time it takes for planets to complete their orbits. Explanations: (i) a complete revolution; orbit of planet around its star takes I; one planetary year, (ii) light-year is used for describing long distances
Diff-Expii Explainer
the time takes for planets to complete their orbits. Explanations: (i) light-year is used for describ- ing long distances, (ii) light year is a measure of the distance light travels in one year
Answer:
Diff-Comb Explainer
Answer: the distance hetween stars in the Milky Way. Explanations: (i) light years are astronomy unit used for measuring length, (ii) stars located light years apart from each other
Question (2): Which type of precipitation consists of frozen rain drops?: Correct Answer: sleet. ExplanationI.P
Answer: snow, Explanations: (i) precipitation when snow fall from clouds the Earth, (ii) snow falls Diff-Xxpliner
Answer: sleet. Explanations: (1) snow falls, (ii) precipitation is when water falls from the sky
Diff-Comb Explainer
Answer: sleet. Explanations: (i) sleet when raindrops freeze as they fall, (ii) sleet is made of ice
Question (3): Most of the mass of the atom consists of?: Correct Answer: protons and neutrons. ExplanationLP
Answer: neutrons and electrons. Explanations: (i) neutrons have more mass than an electron, (ūi neutrons have more mass than an electron
Diff-Expliner
Answer:
protons and neutrons. Explanations: the atomic mass is made of the number of protons and neutrons, (ii) precipitation when water falls from the sky
Diff-Comb Explainer
Answer:
protons and neutrons. Explanations: (i) the atomic mass is made of the number of protons and neutrons, (ii) precipitation is when water falls from the sky
Table 2:
Example of predicted answers and explanations (Only CENTRAL explanations) obtained from our model with different levels of fine-tuning.
Question (2) least explanation is correct and correct answer prediction for model. In the example provided, Diff-Explainer provides the correct answer prediction with both the retrieved facts not being explanatory. Diff-Expii Explainer arrives the correct answer prediction with no explanation addressing the correct answer.
In Question (3) both our model and Diff-Explainer provide the correct answer but with both facts not being explanations. The aforementioned qualitative (Question and 2) and quantitative measures (Explanatory Consistency@KK, Faithfulness) indicate how the underlying explanatory inference re- sults in the correct prediction; there are cases where false inference still leads to the correct answer with our model as well. In this case, the inference distracted by the strong lexical overlaps irrele- vant to the question.
However, from the qualitative analysis, we can conclude that the explainable inference that hap- pens with our model is more robust and coherent when compared to the Diff-Explainer and non- differentiable models.
4.3 KNOWLEDGE AGGREGATION WITH INCREASING DISTRACTORS
One of the key characteristics identified by Thayaparan et al. (022)) is the robustness of Diff- Explainer to distracting noise. In order to evaluate if our model also exhibits the same charac- teristics,
model for the increasing number of retrieved facts k and plotted the answer selection accuracy for WorldTree in Figure 3
As illustrated in the Figure, similar to Diff- Explainer, our approach performance remains stable with increasing distractors. We also continue to outperform Diff-Explaine across all sets of k.
BERT performance drops drastically with increasing distractors. This phenomenon is in line with existing work (Thayaparan et al. 20222 Yadav et al.) 2019a). We hypothesise that with increasing distractors, BERT overfits quickly with spurious inference correlation. On the other hand, our ap-
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Explainable
Accurncy
BERT BERTzerge
No
35.11
IR Salver Clark et al.
2016
Yes
30.36
75
TuplelLP
Yes
23.83
TableILP
2016
70
Explanatiosd..
Yes
00.21
[hhysparan et 2022
65
DGES
Partal
27.11
60
31.70
ETRK KX
2019
Partal
36.61
Acoureyy
UnsupetVised A1E
Tidav
Parial
33.87
20156
Super1520
Model Din-Explamer
20196
XutoRbee Yadav al. 2019u
Partal Yes
41.21
Explaxxl
42.95
SCAT (Large)
[Explanaionn..> (hhayparan
ct al. 2022
Diff-Comb Explainer
Yes
43.21
(Explanation.P))
Table 3: ARC challenge scores compared with other Fully or Partially explainable ap- proaches trained only the ARC dataset.
Figure 3: Accuracy for different number of retrieved facts.
proach circumvents this problem with the inductive bias provided by the constraint optimization layer.
4.4 COMPARING ANSWER SELECTION WITH ARC BASELINES
Table presents a comparison of publicly reported baselines on the ARC Challenge-Corpuu Clark et al. 2018) and our approach. These questions have proven to be challenging to answer for other L.--baee question answering and neural approaches. While models such as UnifiedQA (Khashabi
et al. 2020) and AristoBERT (Xu et al.) 2021) have demonstrated performance of 81.14 and 68 68.95, they have been trained on other guestion-answering answering datasets, including RACE (Lai et al. (017)) Moreover, despite its performance, UnifiedQA does not provide explanations supporting its infer- ence.
In Table 3 to provide a rigorous comparison, only list models that have been trained only on the ARC corpus and provides explanations supporting its inference to ensure fair comparison. Here the explainability column indicates if the model delivers explanation for the predicted answer. A subset of the models produces evidence for the answer but remains intrinsically black-box. These models have been marked Partial. As illustrated in the Table Diff-Cont Explainer outperforms the best non-differentiable constraint-solver model (Explanation LP) by 2.8%. We also outperform transformer-only model AutoRoce by 1.97%. While our improvement over Diff-Explainer is small, we still demonstrate performance improvements for answer selection. On top of performances ob- tained for explanation and answer selection with WorldTree corpus, have also established better performances than leaderboard approaches.
CONCLUSION
This paper proposed novel framework for encoding explicit and controllable assumptions as part of an end-to-end learning framework for explainable multi-hop inference using Differentiable Black- box Combinatorial Solvers (Pogančić et al.) (019). We empirically demonstrated improved answer and explanation selection performance compared with the existing differentiablee constraint-based solver for multi-hop inference (Thayaparan et al., 2022). We also demonstrated performance gain and increased robustness to noise when combining constraints with transformer-basee architectures. In this paper, we adopted the constraints ExplanationLP, but is possible to encode more complex inference constraints within the model.
Diff-Comb Explainer builds on previous work by Thayaparan et al. (0022) and investigates the combination of symbolic knowledge (expressed via constraints) with neural representations. We hope this work will encourage researchers to encode different domain semain-specific priors, leading to more robust, transparent and controllable neuro-symbolic inference models for NL.P.

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APPENDIX
The pseudo-code to train Diff-Comb Explainer end-to-end is summarized in Algorithm !
Experimental Setup:
We employ the following experimental setup:
Sentence Transformer Model: ALL-MPNET BASE-V2 (Song al., 2020).
Fact retrieval representation: ALL-MPNET- BASE-V2 trained with gold explanations of WorldTree
Corpus achieve Mean Average Precision of 40.11 the dev-set.
Fact retrieval: FAISS retrieval (Johnson et al. 2017) using pre-cached representations.
Background knowledge: 5000 abstract facts from the WorldTree table store (WTree) and over
100,000 is-a grounding facts from ConceptNet (CNet) (Speer et al. [017].
The experiments were carried out for k = {1, 3, 5, 10, 20, 30, 40, 50} and the best configuration for each model is selected.
The hyperparameters X, Xans, Aeep, T were fine-tuned for 50 epochs using the Adpative Experi-
mentation Platform!
M=2 for ExplanationLP, Diff-Explainer and Diff-Comb Explainer.
6.1 EXTERNAL TODE-BASER
Differentiable Blackbox Combinatorial Solvers Examples: https:://github.comm/sts Sentence Transformer code-base http:://wwwggiiaaa.eooo/eettttte
us- 1g-labbblackbox owr.rrfeeeeiiaaiiioooommmiiiiii
s/all-mpnet-bast base-v2
12

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Under review as a conference paper at ICLR 2023
Algorithm 1: Training Diff-Comb Explainer
Data: A, b Multi-hop Inference Constraints
Data: fw <- Graph weight Function
Data: À 4- Hyperparameter for DBCS interpolation
epoch + 0:
while epoch max_epochs do
foreach E H
G < hiet-grrphh-ccosstuutinnnii F);
1 + L(hs, F);
8 + clamp(0, 1));
- STrans((F)
h; < (Trans(h));
S S(hu, F):
W - fuls', I:: 0);
g + DBCS(-W", A, b;
r' (W.V.) ..))
end
Lans lololy) 72,
?²), Cans);
Feep available then
Leep (oPranx S2, .... Sel, Fesp): C= >anooaan + Denp
else
C= Lanni
end
update 0, STrans using AdamW optimizer by minimizing loss; epoch t epoch 1;
end
Result: Store best 0 and ST STrans
6.2 INTEGER LINEAR PROGRAMMING OPTIMIZATION
The components of the linear programming system is as follows:
Solver: Gurobi
7
The hyperparatemers used ILP constraints:
Maximum number of abstract facts (M): 2
Infrastructures used:
CPU Cores: 32
CPU Model: Intel(R) Core(TM) i7-6700 CPU 3.40GHz
Memory: 128GB
6.3 HYPERPARAMETERS
For Diff-Comb Explainer we had to fine-tune hyperparameters X, Aanes Aezp, T. We fine-tune for 50 epochs using the Adpative Experimentation Platform with seed of 42.
The bounds of the hyperparameters are as follows:
A: [100, 300]
Dempi [0.0, 1.0
Aans: [0.0. 1.0
T: [1e 2, 100
The hyperparameters adopted for our approach are as follows:
A: 152
13

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Under review as a conference paper at ICLR 2023
Nexp' 0.72
Aans: 0.99
T: 8.77
max epochs: 8
gradient accumulation steps:
learning rate: le-5
weight decay: 0.0
adam epsilon: 1e-8
warmup steps:
max grad norm: 1.0
seed: 42
The hyperparameters adopted for BERT are as follows:
gradient accumulation steps:
learning rate: le-5
weight decay: 0.0
adam epsilon: 1e-8
warmup steps:
max grad norm: 1.0
seed: 42
We fine-tuned using 4 Tesla V100 GPUs for 10 epochs in total with batch size 32 for Base and 16 for Large.
6.4 DATA
WorldTree Dataset: Data can be obtained from:
ARC- ARC-Challenge Dataset: https://laleenii.rrg/bbbtee Only used the Challenge split.
14


Reviews:
[{"review_id": "mggm__EoWPT", "reviewer": "Reviewer_eQxh", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper describes an approach to add ILP constraints into transformer-based models. The main idea is to allow for end-to-end training using a differentiable solver (DBCS) to solve the ILP. Here, DBCS produces can produce exact solutions to the ILP to approximate the gradient. The main hypothesis is that using these exact solutions to the ILP generates better explanations in multi-hop inference. \n\nSpecifically, the approach is described in the context multiple-choice question answering. Explanations are extracted in the form of weighted graphs. The weights encode the importance of a fact for a hypothesis. The subgraph that represents the explanations are selected using DBCS.\nThe transformer-based model (STrans) is used for computing embeddings needed to compute weights for the graph in the formulated ILP problem.\n\nExperiments are performed using the world tree corpus and results are reported for explanations on the dev-dataset (since test-set explanations are not available). Comparisons are made with BERT, a non-differentiable version of the approach (thus, no end-to-end training) and a recent similar approach (DiffExplainer) that uses approximate ILP solutions using transformations to a convex optimization problem. New metrics to measure consistency and faithfulness of explanations are also evaluated on all compared models.", "summary_of_the_review": "The paper proposes an interesting idea to improve explainability in Multiple-choice question answering using ILPs that give us exact solutions and can be integrated into transformers for end-to-end training. Overall, the novelty and generalizability of the model seem to be the main weaknesses.", "strength_and_weaknesses": "Strengths\n- Empirical results show improvements in explanation accuracy over existing state-of-the-art DiffExplainer\n- Using DBCS seems to be a new way to integrate symbolic constraints into transformers and the same could be applicable in other models as well.\n\nWeakness\n- Novelty seems somewhat limited since the ideas seem to be a similar to the ideas in Diffexplainer (Except the use of DBCS). I am not sure if there is only one dataset that is typically used for evaluation (since ARC does not measure explanations) to show more significant impact of the proposed method\n- In general, solving the ILP exactly may be hard (for different types of constraints), so is this approach generalizable or does it work due to the specific type of constraints formulate for the multiple-choice answering problem\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "MhPeIVKBla", "reviewer": "Reviewer_NWPP", "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": "This paper concerns the problem of question answering via multi-hop inference, where multiple separate facts need to be taken into account to answer a complex question. To allow for interpretable structured explanations of the reasoning process, this and previous work explicitly encode multi-hop inference in the constraints of an Integer Linear Program.\n\nGiven a question, a potential answer, and a set of supporting facts, all the components are encoded as the nodes of an explanation graph, on which the edge weights are computed based on lexical overlap and transformer-predicted semantic relevance. The graph is then used as the input to a discrete optimization procedure that explicitly encodes multi-hop inference in its constraints, yielding as the optimal solution a subset of the provided facts to support the potential answer. Repeating this procedure for each potential answer allows to select the most promising answer along with its supporting facts.\n \nThe main difficulty of training such an architecture end-to-end lies in the non-informative jacobian of discrete optimization algorithms. To alleviate this, previous work relies on differentiable semi-definite programming relaxations. However, these can lead to sub-optimal solutions. In contrast, the authors of this work approach the differentiability issue by relying on Differentiable BlackBox Combinatorial Solvers, a technique for computing an informative replacement for the gradient. The resulting proposed framework is called Diff-Comb Explainer.\n\nIt is empirically tested on answer and explanation selection tasks with two datasets. The proposed method is shown to outperform previous methods in terms of all relevant metrics, and is also shown to be robust to an increased number of distracting facts. Some failure cases of the presented method and its competitors are also highlighted in a qualitative analysis.", "summary_of_the_review": "While exhibiting high clarity and reasonable quality of the proposed framework, along with a good evaluation, the presented novelty is limited. I suggest to add an additional contribution, potentially further theoretical or empirical insights into the discrepancy arising from using an ILP vs. a semi-definite relaxation. ", "strength_and_weaknesses": "### Strengths:\n- The paper is well written, easy to understand, with a good presentation.\n- The demonstrated success of a hybrid approach for combining symbolic knowledge in the form of ILP constraints with neural representations opens exciting possibilities for further applications in natural language processing.\n\n### Weaknesses:\n- Technical novelty: The paper is extremely similar to Diff-Explainer [1], only the semi-definite programming relaxation of the ILP is replaced with the unrelexad ILP, and DBCS [2] is used to differentiate it. This limits the theoretical contribution to plugging one existing method into another, which is very incremental. \n- Empirical novelty: Currently the paper reports results only on experiments from [1], and additionally adds the novel consistency and faithfullness metrics. This is a reasonable empirical contribution, however, it is in my opinion not enough to outweight the limited theoretical novelty.\n- Differentiation of the combinatorial solver: Equation 13 only depends on the optimal value of the optimization problem (which is continuous), not the optimal solution (which is discrete). Therefore it is continuously differentiable w.r.t. the parameters $W$, and complicated methods like DBCS are not required here (in contrast to differentiation of Equation 14). Was this taken into account in differentiating Equation 13 or are the reported results produced by using the informative gradient replacement from DBCS? In the latter case, it would be important to report as a baseline the results of differentiating Equation 13 without the use of DBCS.\n\n### Additional questions:\n- Is there an interpretation for the performance drop from using supervision on the explanation selection in the consistency@3 column of Table 1?\n- What are the runtimes for the experiments? Solving a relxation of an ILP can be much faster than solving the true ILP, is the slight performance increase worth the additional runtime?\n\n\n[1]: M. Thayaparan, M. Valentino, D. Ferreira, J. Rozanova, and A. Freitas. Diff-Explainer: Differentiable Convex Optimization for Explainable Multi-hop Inference. In Transactions of the Association for Computational Linguistics, 2022.\n\n[2]: M. Pogancic, A. Paulus, V. Musil, G. Martius, and M. Rolinek. Differentiation of blackbox combinatorial solvers. In International Conference on Learning Representations, 2019.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "EG9RdYH6CP3", "reviewer": "Reviewer_nF5J", "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": "This paper proposes “Diff Comb Explainer”: a neuro-symbolic architecture for selecting an answer (from a given set of candidates) as well as explanations (from a corpus of facts) for a multiple-choice QA task. \nEach candidate answer is first converted into a hypothesis by concatenating the question to it, and then scoring a hypothesis is cast as a subgraph selection problem that is solved via a standard ILP for subgraph selection problem with an added constraint that hypothesis node should be a part of the subgraph. The graph contains a node for the hypothesis and k more nodes, one each for a retrieved fact from the corpus. The edges represent lexical and semantic similarities between the nodes. The facts corresponding to the nodes selected in the subgraph represent the explanations, and the objective function of the ILP is the total edge weight of the subgraph.\n\n", "summary_of_the_review": "The idea of creating a neuro-symbolic architecture by making an ILP differentiable for the task of multiple choice QA (with explanations) has already been demonstrated by Diff Explainer. This work is simply replacing the SDP relaxation with Differentiable Black Box Solver and showing slight improvement. Notably, CombOptNet, which essentially uses a black box solver for ILPs and makes it differentiable, has already been mentioned/discussed in the Diff Explainer paper, but not chosen over SDP relaxation technique to make ILP differentiable. In light of all this, I believe that the novelty of this work is somewhat limited.", "strength_and_weaknesses": "This paper demonstrates that using differentiable black-box solvers gives a slight improvement over using SDP relaxation for making ILPs differentiable.\n\nThe idea of using an ILP to solve the exact same subgraph selection problem for precisely the same task has already been explored in Diff Explainer.  The only difference is in the choice of technique to make the ILP differentiable: Diff Explainer uses an SDP relaxation to make ILP differentiable, whereas Diff Comb Explainer uses differentiable blackbox solvers, which is the same as using CombOptNet (Paulus et al 2021) with known constraints and learnable cost.  CombOptNet has been discussed in the Diff Explainer paper, and this work is just replacing  SDP relaxation with CombOptNet with known constraints (similar to blackbox differentiation).\n\n[Paulus et al 2021] CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "5pdMXnsKgK", "reviewer": "Reviewer_sAEZ", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This work proposes a differentiable model, namely Diff-Comb Explainer, that solves the integer linear programming based multi-hop QA problem. The proposed model utilizes the differentiable blackbox combinatorial solver, which solves the ILP problem in a differentiable fashion. In the experiments, the proposed method outperforms the baselines in terms of explanation and answer selection.", "summary_of_the_review": "This work focuses on improving the performance of QA tasks with explanations based on an existing approach. The proposed method relies on a unique stack of techniques and demonstrates better performance than the prior work in the experiments. That said, the impact of this work is limited. There might be some novelties for this particular line of research but I'm not familiar with the literature to be certain.", "strength_and_weaknesses": "Strength \n- The approach of generating explanations for QA tasks via graph construction is interesting\n\nWeaknesses\n- The scope of this work is to improve a specific type of approach to QA tasks and the impact could be limited\n- The paper is difficult to follow for readers unfamiliar with this line of research\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject