Recursive Routing Networks: Learning to Compose Modules for … · Recursive Routing Networks: Learning to Compose Modules for Language Understanding Ignacio Cases 1, Clemens Rosenbaum2,

Recursive Routing Networks: Learning to Compose Modules forLanguage Understanding

Ignacio Cases1, Clemens Rosenbaum2, Matthew Riemer3, Atticus Geiger1,Tim Klinger3, Alex Tamkin1, Olivia Li1, Sandhini Agarwal1,

Joshua D. Greene4, Dan Jurafsky1, Christopher Potts1, Lauri Karttunen1

1Stanford University 2University of Massachusetts Amherst3IBM Research 4Harvard University

[email protected], [email protected]

Abstract

We introduce Recursive Routing Networks(RRNs), which are modular, adaptable mod-els that learn effectively in diverse environ-ments. RRNs consist of a set of functions, typ-ically organized into a grid, and a meta-learnerdecision-making component called the router.The model jointly optimizes the parametersof the functions and the meta-learner’s pol-icy for routing inputs through those functions.RRNs can be incorporated into existing ar-chitectures in a number of ways; we exploreadding them to word representation layers, re-current network hidden layers, and classifierlayers. Our evaluation task is natural languageinference (NLI). Using the MULTINLI corpus,we show that an RRN’s routing decisions re-flect the high-level genre structure of that cor-pus. To show that RRNs can learn to specializeto more fine-grained semantic distinctions, weintroduce a new corpus of NLI examples in-volving implicative predicates, and show thatthe model components become fine-tuned tothe inferential signatures that are characteris-tic of these predicates.

1 Introduction

Human cognition has an extraordinary ability tomodularize, decomposing problems and solvingthem by re-composing elements from prior solu-tions, and this ability is nowhere more evident thanin language understanding (Partee, 1984; Janssen,1997). Most machine learning architectures lackthis modularity, which limits their ability to gener-alize and leaves them susceptible to catastrophicinterference (McCloskey and Cohen, 1989) – for-getting past skills when acquiring new ones.

We propose to address this need for modularityby applying Routing Networks (Rosenbaum et al.,2017) to natural language understanding. RoutingNetworks are self-organizing networks with twocomponents (Figure 1): a set of function blocks

Router

5

Recursive Routing Network

3

2

4

1

6

Figure 1: Given a premise–hypothesis pair x, e.g., t man-aged to do s and t did s (1), the model needs to learn to pre-dict entails (6). The router (2) estimates the value of applyingeach of the available sub-functions to the input. Given thatmanage has certain semantic properties, the router may se-lect (3) a function f1 specialized to them. The module is thenapplied (4), yielding f1(x). This process repeats (5), now us-ing f1(x), selecting and applying another sub-function, andso on, until the router is confident in its prediction (6).

which can be applied to transform the input, anda router which makes decisions about which func-tion block to apply next. Here we introduce Re-cursive Routing Networks (RRNs), in which thereis a single set of composable functions, recursivelychosen by the router. RRNs can be applied todifferent components of modern language under-standing architectures with full end-to-end train-ing. The model jointly optimizes the parametersof selected sub-functions and the meta-learner’spolicy for how to route inputs through those func-tions. As a result, individual sub-functions spe-cialize to specific inputs, and paths through thegrid of sub-functions can similarly be trained toreflect specific concepts and capabilities.

RRNs share many intuitions with other modularmethods like: Neural Module Networks (Andreaset al., 2015, 2016; Hu et al., 2017), which learn toconstruct a neural network from pre-defined mod-ules; the Compositional Recursive Learner (Changet al., 2018), a closely related approach that uti-

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Routingacross examples

Weightsharing

Possibledistribution

OrthogonalizedKnowledge

CompressedKnowledge

High Interference

HighTransfer

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@L@✓i

· @L@✓j

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High Interference

HighTransfer

x

Router

Router

Router

x

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@L@f2

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@L@f3

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Modular Backpropagation Forward Decision Feedback

Figure 2: Left: RRN learning. Forward (grey background) and backward passes (white background) during the training of arollout of an RRN with three blocks (width) and limited to depth three (height). In this unrolled version, a path is the sequenceof function blocks selected by the router. Right: The dilemma of weight sharing and the transfer–interference trade-off.When examples are learned using largely separate weights, there is a thin possible distribution of gradient dot products thatlimits interference, which unfortunately also limits transfer. This is beneficial for unrelated examples, but frustrates the learningof related ones. Conversely, when examples are learned using largely shared weights, there is a wide possible distribution ofgradient dot products that allows for both high magnitude transfer and interference. This is beneficial for related examples, butmaximizes interference when examples are unrelated. With RRNs, we provide our network with an unprecedented degree ofleverage to learn to navigate this trade-off.

lizes curriculum learning to solve arithmetic andvision problems; and ModularNetworks (Kirschet al., 2018), which extend Routing Networks withan EM-like training approach.

We show how to incorporate RRNs into differentneural components (word representation layers,recurrent network hidden layers, classifier layers),and we study their application to natural languageinference (NLI), in which premise–hypothesispairs are labeled for whether the premise entails,contradicts, or is neutral with respect to the hy-pothesis. We chose this task because reasoningin natural language involves context-sensitive in-terpretation of words and sentences as well ascompositional structure. We make use of theMULTINLI corpus (Williams et al., 2018), whichincludes text from multiple genres that we expectto condition linguistic senses in complex ways.Our experiments show that RRNs learn policiesand components that reflect this genre structure,which leads to superior performance.

We also introduce a new corpus of NLI exam-ples involving implicative constructions like man-age to, be able to, and fail to (Karttunen, 1971,2012). This corpus follows the design of many re-cent NLI corpora, but with the added challengesof reasoning about implicatives, which have log-ical signatures that interact compositionally with

each other and with surrounding semantic oper-ators. Our experiments show that trained RRN

model components become fine-tuned to these sig-natures. Finally, we introduce an extension of theframework which leverages a Dispatcher for situ-ations where meta-information is not available attest time, obtaining very promising results.

2 Background and Motivation

RRNs are a natural extension of the Routing Net-works introduced by Rosenbaum et al. (2017),themselves part of a larger class of conditionalcomputation models (Bengio et al., 2015). Rout-ing Networks combine trainable modules with ameta-learner called a router, which is typicallytrained using reinforcement learning, though anyhard-decision making algorithm could be used.Given an example, the router selects a module andapplies it, yielding an activation that can be the in-put to another routing iteration. As Rosenbaumet al. observe, this can be modelled as a recursiveprocess where the router is applied repeatedly todefine complex paths through the modules. Thefinal routed output is passed to additional layers orinterpreted as the output of the network (Figure 1).

We are motivated to consider Routing Net-works, and in particular recursive variants, in or-der to address the transfer–interference trade-off

(Riemer et al., 2019). Vanilla neural networkslatently attempt to solve a very difficult problemof deciding when to orthogonalize and compressknowledge (Figure 2, right). When two examplesare learned with the same weights, there is highpotential for transfer as well as interference. Thisis good for related examples because it maximizesthe potential for transfer. However, weight shar-ing is bad for unrelated examples, as it increasesthe likelihood of interference. In contrast, whentwo examples are learned using different weights,there is low potential for transfer and interference.This is beneficial for unrelated examples, but lim-its the potential for learning about commonalitiesbetween related ones. RRNs extend vanilla neuralnetworks by granting them the leverage to navi-gate this trade-off by making global functional de-cisions at the module level. As a result, RRNsexplicitly make decisions to compress or orthog-onalize knowledge between examples by decidingwhether to share specific weights.

Thus far, hard-selection routing has not beenapplied to language domains. Zaremoodi et al.(2018) introduced a soft version of routing thatfalls within a larger class of Mixtures of Experts(MoE) models (Jacobs et al., 1991). However,MoE models differ from RRNs in two crucial ways.First, MoE models generally do not consider therecursive application of functions. The promise isthat we can compose functions to reflect the com-positional aspects of a problem. Imagine we havea sentence encoding and we want to answer a par-ticular question. We can now condition the routeron the question, so that it applies exactly the func-tions required that translate the encoding to extractthe answer. Second, MoE models do not allow fornearly the level of specialization as those basedon routing, because they do not eliminate weightsharing across modules and instead only gate thesharing. In practice, this still leads to significantinterference across tasks.

Composition of modules has been widely ex-plored for question answering, starting with An-dreas et al. (2015). Andreas et al. (2016) learn toassemble a deep neural network on-the-fly froma pre-specified inventory of neural modules usingtree-structured layouts based on linguistic anal-ysis. Chang et al. (2018) also explore routing,but with problems described in pseudo-languagerather than natural language.

3 Recursive Routing Networks

We now provide a formal presentation of RRNsand show how to incorporate them into existingneural architectures.

3.1 The Routing Paradigm

Formally, the router bases its decision on the tu-ple 〈f(x),m〉, where x is the sample, f(x) =fi ◦ · · · ◦ fk(x) is the composition of all appliedfunctions from the set of all available functionsF = {f1, . . . , fb}, and m is a vector that containsmeta-information that can be utilized by the router– for example, an embedding for its genre or se-mantic classification. The router consists of a pol-icy π(F|〈f(x),m〉) that determines which of thefunctions in F to apply for a given state 〈f(x),m〉.Once a new function fk is selected, the latest ac-tivation (and, thereby, the state) gets updated tofk ◦ f(x), and the process repeats.1

We focus on the recursive case where the setF is generally the same for each decision. Apartfrom being the more general formulation, thisform of recursivity allows more weight sharingand better compositional generalization. Whilethere is no necessary upper limit to the number ofselections, we found that limiting the number toa maximum of d makes the learning more stable.Consequently, the total number of possible routingpaths available is bd.

Decision Making One of the most impor-tant design choices for the router concerns themeta-learning algorithm. Routing is limited tohard decision-making algorithms, in particular astochastic reparameterization using the Gumbel-Softmax function (Jang et al., 2016; Maddisonet al., 2016) and reinforcement learning (RL) algo-rithms. As with other recent approaches to com-positional architectures with hard selections (Ben-gio et al., 2015; Shazeer et al., 2017; Kirsch et al.,2018), routing networks can suffer from modulecollapse, a lack of diversity in the router’s deci-sion making. This general problem, common inarchitectures that jointly train decision making andfunctions, stems from an early over-estimation ofthe value of specific functions. This leads to thesefunctions being selected and trained more thanothers, until these functions are indeed so good

1Some architectures have constraints that prevent re-peated selection of the same functions in F . In these cases,the router has to be restricted to select from only a compatiblesubset of F .

Premise Hypothesis

Encoder Encoder

Classifier

:

Softmax

Concatenation

RNN

…M1

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Softmax

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RNN

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RNN

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Encoding

Encoding

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RNN…

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RNN RNN



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Encoding

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General Architecture Routing Classifier

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Routing RNN – I2H

Routing CBOWRouting RNN – H2H

Figure 3: Top left: Our general NLI architecture. Top center: Routing the classifier. Lower left: Word representation routingof a CBOW encoder. Top right: Routing an RNN input transformation. Bottom right: Routing an RNN hidden transformation.

that others are not considered. To mitigate this,we start with a strategy proposed by Rosenbaumet al. (2017): exclusively conditioning the routeron meta-information provided by the datasets andstoring the router’s policy in a table. We found thatthis stabilizes early training. During later train-ing, we can replace the meta-information hard as-signment rule with a meta-information “guesser”.We implement this guesser with what Rosenbaumet al. call a Dispatcher, an approximation-basedsub-policy that estimates the meta-information foreach sample and passes it to the correspondingpolicies (Section 5).

Losses and Training Once the routing proce-dure terminates, the selected modules are trainedusing standard optimization techniques, such asbackpropagation with stochastic gradient descent(SGD). The decision-making algorithm is eitheralso trained with SGD (Gumbel reparameteriza-tion) or trained using reinforcement learning. Aswe focus on classification problems, our core lossfunction Lclass(y, y) is the standard cross-entropyloss. We backpropagate this loss along the func-tion path chosen by the router for input x.

Reinforcement Learning Rewards For RL

training of the router, we define the reward r forthe reinforcement learner to be the sum of the neg-ative classification loss and an additional regular-ization reward that encourages diversity in the se-lection of modules:

r = −Lclass(y, y) + rreg (1)For discussion of this reward, see Appendix A.1.

3.2 Architectures

If not otherwise mentioned, we assume that therouted modules are all fully connected layers withthe same input and output dimensions. We fur-thermore assume that the router is always select-ing from the same set of functions when definingpaths through the routing layers, as this straight-forwardly allows recursion in the relevant sense.

Routing Classifiers The simplest method weexplore involves routing the layers of a classifier.To do this, we define a fixed set of b fully con-nected layers (FC), each of the same dimension-ality, and we allow the router to choose any paththrough this set up to a maximum length d (Fig-ure 3, top center). The final activation producedby the chosen path is then densely connected toa non-routed output layer. To further model eachexample’s dependence on its class, we hard-selectthis output layer based on the meta-information la-bel m for x:

Wm (FCk1 ◦ · · · ◦ FCkd(x)) + bm (2)

Routing RNN Encoders In routing RNNs we fo-cus on their two core transformations: from thehidden state at time step tk to time step tk+1, andfrom the input to the hidden state. We have de-signed routing architectures for both of them. Inthis paper, we start with LSTMs, but the techniquesare straightforward to adapt to other cell types.

Figure 3, top right, shows an architecture wherewe route the input-to-hidden (I2H) transformation.Similarly, Figure 3, bottom right, shows an ar-chitecture where we route the hidden-to-hidden

(H2H) transformation. While these transforma-tions are often designed as single fully connectedlayers, we allow a recursive application of d stepsfrom a selection of b modules to be applied. Theselections for the transformations f, i, o, c are tiedto be the same. The corresponding transformation(in form of their weight matrices Wf/i/o/c for I2H

and Uf/i/o/c for H2H) becomes:

K = FCk ◦ · · · ◦ FCm(xt),

∀K ∈ {Wf ,Wi,Wo,Wc, Uf , Ui, Uo, Uc}(3)

Routing CBOW Encoders We also experimentwith routing a continuous bag of words (CBOW)encoding. As routing after the main addition is justrouting the classifier, we instead add a word-leveltransformation before the addition (Figure 3, bot-tom left). This transformation can again be routedrecursively (with up to d steps through bmodules).The entire CBOW model can be defined as (withw1, . . . , wt as premise or hypothesis):

CBOWR(w1, . . . , wt) =∑ti=1 FCk1◦ · · · ◦ FCkd(wi)

(4)

Routing Transformers The Transformersmodel (Vaswani et al., 2017) was recently pre-trained as a language model (Radford et al.,2018) achieving impressive results on severalNLI datasets. Since the corpus for pretrainingis not available, we instead use the parameter-files distributed by the authors.2 The encodingconsists of twelve Transformer-blocks. Eachblock consists of a convolutional attention layerfollowed by two convolutional layers (along withseveral dropout and layer-norm layers). Thisallows routing at different levels of granularity,of which we investigate two: routing entireblocks and routing the attention step within eachblock. As we use pre-defined parameters, wecannot apply the blocks or the attention layersrecursively. Furthermore, we have to add routingin the fine-tuning phase of using Transformersby creating b copies of each routed module (thedepth is necessarily d = 12). When fine-tuning,the router then diversifies the initially identicalmodules.

4 Experiments

We now evaluate the routing architectures intro-duced in the previous section, along with min-imally different non-routing baselines. Our ex-

2https://github.com/openai/finetune-transformer-lm

periments focus on NLI, using the MULTINLI cor-pus (Williams et al., 2018) and a new EnglishCorpus of Implicatives, the Stanford Corpus ofImplicatives (SCI). In both, each example is apremise/hypothesis pair labeled with one of en-tails, contradicts, or permits. What is specialabout MULTINLI and SCI in the current contextis that the examples also have meta-informationthat can be used to guide the router: a genre la-bel for MULTINLI and an implicative signature forSCI. We expect our routing models to leverage thisinformation during policy and parameter learning.

For all models (except Transformers), we adoptthe architecture in Figure 3, top left, in which thepremise and hypothesis are processed separately,and the final representations of each are concate-nated and fed into the classifier layers. We ex-plore two methods for input processing: (i) pre-trained word representations (GloVe; Penningtonet al. 2014) and (ii) pretrained contextualized rep-resentations (ELMo; Peters et al. 2018). For allnon-routed models, we provide explicit access tothe genre label via special keywords put in front ofeach sentence before encoding. The routed modelsuse this label as meta-information.

Unless otherwise specified, we used embeddingand hidden dimensions of 300. The classifier con-sists of three fully connected layers with input andoutput dimensions of 600 (also when routed). Thefinal layer projects from 600 to the output dimen-sion, 3. The classifier nonlinearities are ReLUs.We train the modules using Adam (Kingma andBa, 2014) with a learning rate of 1e−3 and therouter using SGD with a learning rate of 3e−4 (foradditional details, see Appendix A.5). For Trans-formers, all hyperparameters are determined bythe published parameter files.

In experiments with different decision mak-ing algorithms (see Appendix A.5), we foundthat the Gumbel-Softmax reparameterization per-formed 10% worse on average, with much highervariance. QLearning was consistently as good asother more complex RL algorithms, so we reportonly our QLearning experiments. As Routing Net-works have more parameters than their non-routedcounterparts, we ran experiments with larger non-routed networks. We found that this did not affectperformance, only resulting in more overfitting.

https://github.com/openai/finetune-transformer-lm

MULTINLI SCIEncoding Embedding Routing match joint disjoint mismatch nested∗

CBOW

Glove None 61.94±0.13 57.26±0.18 55.68±0.47 53.41±0.86 50.98±0.92

Elmo None 59.12±0.59 58.28±0.67 57.99±0.90 56.30±1.28

Glove Classifier 51.99±0.17 62.33±0.80 61.17±0.28 51.55±1.61

Elmo Classifier 55.36±0.35 64.29±0.99 61.60±0.61 55.50±0.87

Glove WP 64.38±0.24 74.95±0.64 73.91±0.54 68.69±0.93

Glove WP+D 65.84±0.12 75.56±0.77 74.87±0.49 71.08±0.52 75.43±0.29

Elmo WP 61.28±0.25 65.91±4.19 61.67±1.30 57.34±0.52

RNN

Glove None 66.53±0.21 67.04±2.36 63.52±2.00 59.32±2.87 57.69±2.75

Elmo None 66.13±0.19 65.76±2.16 61.89±2.41 59.02±1.92

Glove Classifier 65.60±0.33 71.02±0.85 67.82±0.81 60.58±1.35

Elmo Classifier 59.49±0.55 68.98±2.99 67.23±2.40 52.96±1.82

Glove I2H 47.79±0.47 53.25±0.28 56.57±1.83 53.99±0.55

Elmo I2H 49.12±1.01 56.13±1.04 56.23±0.17 53.36±0.26

Glove H2H 62.34±0.25 54.89±0.93 53.08±0.57

Elmo H2H 64.04±0.19 59.34±4.41 56.57±1.83 57.61±0.55

Transformers (pretrained)

None 76.12 88.12 88.07 85.64Classifier 76.50 86.05 86.68 73.45Attention 76.63 87.91 88.45 85.95Block 75.87 88.22 87.88 85.33

Table 1: Results for MULTINLI and SCI with different baselines and their routed versions. We report average accuracy withconfidence intervals over five runs with different seeds. For Transformers, we found that finetuning was highly volatile. Wetherefore report test results from the best-of-5 train models. ∗All results for nested SCI were computed by fine-tuning the samenetwork previously trained on joint. ‘WP’ stands for Word Projection, ‘+D’ for Dispatching, ‘I2H’ for Input-to-Hidden routing,and ‘H2H’ for Hidden-to-Hidden routing. Italics mark scores whose confidence intervals overlap with the best scores.

gov

tra fic sla swi

governmenttravelfiction

slateswitchboard

0

1

2

3

Figure 4: Path (module selection) overlap for MULTINLI be-tween genres with the CBOW GloVe WP model. The diag-onal represents the number of function blocks applied for agenre. A maximum of three means that two genres would berouted through the exact same functions.

4.1 Experiments with MULTINLI

The MULTINLI corpus contains 392,702 trainingexamples, 10K dev examples, and 10K test exam-ples. The examples come from 5 genres: fiction,government reports, the Slate website, the Switch-board corpus (Godfrey and Holliman, 1997), andBerlitz travel guides. We treat these genre labelsas meta-information for the model (Section 3.1).

Our MULTINLI results are given in Table 1, andthe learning dynamics in Figure 5. The best modelcombines the Transformer base model with rout-ing in the attentional layers. Our methods for rout-ing the RNN seem to be less successful, but word-representation routing offers clear benefits with

the CBOW base model. Interestingly, the baseline(non-routed) models perform at the same level asthe very similar models without genre labels eval-uated by Williams et al. (2018). It seems that thesemodels are not able to take advantage of the meta-information. In contrast, RRNs seem to providethe space needed to condition linguistic senses onthese labels.

Our hypothesis is that routing will not onlylead to better performance on diverse tasks likeMULTINLI, but also that the paths – i.e., the se-quence of functions selected by the router – fol-lowed by the network will reflect high-level taskstructure. Figure 4 suggests that this is the casefor MULTINLI. Here we show the degree of path-overlap for all pairs of genres. As we might ex-pect, government (the 9/11 report), Slate (cur-rent affairs), and travel cluster together, as dis-tinct from the two more isolated genres (Switch-board; spoken language) and fiction (mostly fromthe 20th century).

4.2 Experiments with Implicatives

Karttunen (1971) discovered that implicative con-structions, such as manage and waste chance (e.g.They wasted their chance to win), have signatures,which characterize the inferences they support inpositive and negative contexts. This makes them

the order compelled him to appear as a witness entails he appeared as a witnesswe have missed an opportunity to examine the art market today contradicts we have examined the art market todayMr Odinga had not been forced to change his plans permits Mr Odinga had changed his plans

Table 2: Examples from SCI randomly chosen from the validation set. Each row contains a triplet formed by a premise (leftcolumn), a hypothesis (right column), and a label specifying one of the three possible relations (entails, contradicts, permits)holding between premise and hypothesis. The last row contains an example of a probabilistic implicative (see the main text).

particularly informative for NLI predictions.

For instance, the positive sentence Joan man-aged to solve the problem entails Joan solved theproblem, and the negative sentence Joan didn’tmanage to solve the problem contradicts Joansolved the problem, so we say that the verb man-age has the signature +|- (MacCartney and Man-ning, 2009; Karttunen, 2016). In contrast, wastechance has the opposite signature, since theywasted the chance to befriend him contradicts theybefriended him, and they didn’t waste the chanceto befriend him entails they befriended him. Thereare seven implicative signatures: six were pre-viously known (Karttunen, 1971, 2012), and wefound an additional one (+|+; e.g. take no timeto). See Appendix C for additional details.

Signatures are compositional: when two ormore implicative constructions are composed ina sentence, they create a nested implicative con-struction whose signature is determined by the sig-natures of the individual verbs (Nairn et al., 2006).For example, John managed to remember to getthe keys entails John got the keys, where the nestedimplicative manage to remember has the overallsignature +|-. We also see a more limited form ofcompositionality inside phrasal implicatures; theirsignatures are often largely determined by the lex-ical semantic family of their constituent words.Therefore, signatures make implicatives ideal forevaluating different degrees of compositional gen-eralization with RRNs, as they provide valuablemeta-information (Section 3.1).

Statistic SCI

Validated pairs 41.15%Unanimous gold label 94.93%Gold matches author 98.61%Gold does not match author 1.39%No gold label 0

Fleiss κ coefficient 0.95entails 0.32contradicts 0.30permits 0.38

Table 3: SCI statistics. Top: Percentage of validated pairsand basic agreement. Bottom: Fleiss κ coefficients and pro-portion of all assignments made to the corresponding label.

4.2.1 A Corpus of Implicative ConstructionsOur SCI dataset contains ≈10K premise–hypothesis pairs. All seven signature types arerepresented, in addition to pragmatic signatures,which have inferential biases (see Appendix C).

We provide SCI3 in three versions for all sin-gle and phrasal implicatives.4 In the joint ver-sion, the underlying distribution of implicatives isshared across train, validation, and test splits. Inthe disjoint version, a different subset of implica-tives is used in train from those used in validationand test. This allows us to test generalization tounseen constructions. Although disjoint with re-spect to constructions, the constructions are care-fully distributed so that all the underlying signa-tures and most lexical items are represented in allsplits. The lexical items that make up implicativeconstructions overlap between the splits. For ex-ample, take vow appears only in training and val-idation, while make vow only appears in test. Thelast version is mismatch, where different subsetsof the signatures are present in training/validationand test (see appendix C for more details).

Data collection for SCI proceeded as follows:we collected at least 12, and most often 20, seedpremises from examples found in Google Booksand on the web. For each example, six hypothe-ses were created by expert annotators.5 These sixexamples were constructed in two steps: first, thepremise was taken to be the seed sentence, andthree hypotheses were created to exhaust the la-bel space in relation to the premise. Second, thepremise was taken to be the negation of the seedsentence, and a different set of three hypotheseswas created to also exhaust the label space withrespect to this negated sentence. For example, oneof the seeds for manage was I managed to see whoit was. From this, annotators produced by hand a

3https://nlp.stanford.edu/projects/sci/4We provide nested constructions as a separate extension,

with the exception of a few nested implicatives that were usedin the development version of this corpus, for which the re-sults are presented here. In the final version of the corpus,single and phrasal implicatives will be separated from nestedimplicatives. See the Appendix for more details.

5Native speakers of English trained in semantics at theundergraduate level, with expertise in implicatives.

https://nlp.stanford.edu/projects/sci/

0 5 10 15 20 25 30epochs

54

56

58

60

62

64

66ac

cura

cy in

%MultiNLI

CBOW - Glove - NoneCBOW - Glove - WRLRNN - Glove - CLRNN - Glove - None

Figure 5: Learning on MULTINLI with GloVe inputs.

+|-

-|+ N-|+

+|.5

N+|-

-|.9

o|o

+|.3

-|.5

.9|-

+|+

-|.7

+|.8

+|.4

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N-|++|.5N+|--|.9o|o

+|.3-|.5.9|-

+|+-|.7

+|.8+|.4+|.9o|+.4|- 0

1

2

3

Figure 6: The path-overlap between different signatures onSCI, using the CBOW GloVe WP model, for b = 4, d = 3.N+|− and N−|+ are nested signatures.

negated seed, I did not manage to see who it was.Then three hypotheses were generated for each ofthese two premises. Finally, each example was la-beled by the author of the hypothesis examples.

Validation was performed on a randomly se-lected subset of constructions. Two additionalvotes were cast by different linguists, resulting inthree label votes for each example in the subset.The gold label was then defined to be the majorityclass from this set of votes. The Fleiss κ showshigh inter-annotator agreement (Table 3).

4.2.2 Implicatives Results and AnalysisFor each example in SCI, we use its associated im-plicative signature as the meta-information label.This serves as a subtler kind of semantic infor-mation than genre. The learning dynamics of dif-ferent models are shown in Figure 7. As Table 1shows, RRNs lead to considerable gains in perfor-

0 20 40 60 80 100epochs

40

50

60

70

accu

racy

in %

CI

CBOW - Glove - NoneRNN - Glove - NoneRNN - Glove - CLCBOW - Glove - WRL

Figure 7: Learning on SCI (joint) with GloVe inputs.

mance over most benchmarks. Recursive routingof the classifier helps consistently with all mod-els. Even simple word-level routing yields majorimprovements for CBOW. As with MULTINLI, webelieve that the modules allow the conditioning oflinguistic senses. Routing at the word-level for se-quential models seems suboptimal here. The train-ing accuracies (> 99.5%) suggest that the problemis overfitting. Intuitively, routing can assign toomany exclusive parameters to each class; and asthe samples within these classes are similar to be-gin with, remembering them can be easier for thenetwork than for non-routed networks.

We expect examples with similar signatures tobe routed along similar paths. Figure 6 sum-marizes the path-overlap between different signa-tures. Some similar paths involve reversals of po-larity, which calls for further scrutinity. However,many of these similarities make intuitive sense.For example, +|.4 and +|.5 are highly similar,as are -|.7 and -|.9. In addition, the isolationof the unusual signature +|+ (limited to just twoconstructions) seems expected, as does the affin-ity of the nested signatures N+|- and N-|+ totheir unnested counterparts. However, as shown inTable A3, some signatures only contain very fewsamples, resulting in highly noisy routing paths.

4.3 Navigating the Transfer–InterferenceTrade-off

We now seek to provide intuitive explanationfor how RRNs help in navigating the transfer–interference trade-off. We have characterized thetrade-off in terms of controlling weight sharingbetween examples. While baseline networks canlearn to navigate this trade-off if they are opti-mized for the appropriate objective (Riemer et al.,

2019), in general these models do not have avail-able supervision on how to do this and so optimizegreedily for the current example. As such, earlyexperiments with the baseline, non-routed modelsshow that examples with neutral signatures, suchas o|o and o|+, are the first to be learned. Wenoticed that baseline sequence models learned toclassify take vow much earlier in the training pro-cess, as the word vow is lexicalized through train-ing on make vow examples. However, its per-formance decreases over time as take is lexical-ized through training on examples of other im-plicatives with different signatures, such as takechance. This behavior is the characteristic out-come of catastrophic interference: learning takechance results in decreased performance on takevow as the two examples interfere. This is a par-ticularly revealing instance as these two phrasalverbs are similar on the surface, but have quitedifferent semantic properties, as reflected by theirsignatures. However, given that RRNs are able toroute them differently (Figure 6), interference isless likely. It is also possible that routing helpswith the transfer related to similar examples, anavenue we want to explore in the future.

5 Prediction without Meta-Information

We have shown that high-quality meta-information can be extremely useful. Unfor-tunately, it is often not available at test-time. Tocompensate, we evaluated an RRN extension inwhich an additional neural network module istrained to assign examples to meta-informationclasses (genres for MULTINLI; signatures for SCI).When we trained this model jointly, we found thatit was unstable and did not perform well. How-ever, if this model is introduced after RRN trainingon examples with known meta-information, thenthe results are extremely promising. We call thisDispatcher Training (‘+D’), borrowing similarterminology from Rosenbaum et al. (2017). Ta-ble 1 includes an initial evaluation of this variantwith a CBOW base model and WP routing. As wecan see, accuracy actually increases by a smallamount over WP alone.

Additionally, having +D allows the network togeneralize better to unseen examples and unseenpatterns. Consider the relative performance dropfor CBOW WP+D from the full joint SCI dataset(75.56%) to disjoint (−0.69%) and from disjoint(74.87%) to mismatch (−3.79%), and compare

this to the plain WP version: full (74.95%) todisjoint (−1.04%), and disjoint (73.91%) to mis-match (−5.22%)

6 Conclusion

This paper introduced Recursive Routing Net-works and showed how to incorporate them intoa variety of different neural architectures; we ex-plored a range of possibilities for this, and thetechniques generalize to other options straightfor-wardly. Our evaluations focused on NLI. Weshowed in particular that our RRNs can effectivelyleverage the meta-information in the MULTINLI

corpus and in our new corpus focused on im-plicatives; not only do RRNs use this informa-tion to achieve superior accuracy, but they alsolearn sub-structure that reflects this high-level in-formation, and our Dispatcher variant extends theframework to situations where the relevant meta-information is not available for testing. We believeexploring more powerful variants of dispatchingis an interesting avenue for future work, as ispretraining routing models on language modeltasks using large corpora. It is our hope thatthese lessons extend to other richly compositional,context-sensitive language understanding tasks.

Acknowledgements

We thank George Supaniratisai, Arun Chaganty,Kenny Xu and Abi See for valuable discussions,and the anonymous reviewers for their useful sug-gestions. Clemens Rosenbaum was a recipientof an IBM PhD Fellowship while working onthis publication. We acknowledge the Office ofthe Vice Provost for Undergraduate Education atStanford for the summer internships for AtticusGeiger, Olivia Li and Sandhini Agarwal. Thisresearch is based in part upon work supportedby the Stanford Data Science Initiative, by theNSF under Grant No. BCS-1456077, by the NSFAward IIS-1514268, and by the Air Force ResearchLaboratory and DARPA under agreement num-ber FA8750-18-2-0126. The U.S. Governmentis authorized to reproduce and distribute reprintsfor Governmental purposes notwithstanding anycopyright notation thereon. The views and con-clusions contained herein are those of the authorsand should not be interpreted as necessarily repre-senting the official policies or endorsements, eitherexpressed or implied, of the Air Force ResearchLaboratory and DARPA or the U.S. Government.

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Appendices

A Architecture Details

A.1 The reward design

Rosenbaum et al. (2017) found that it was helpfulin incentivizing different routing strategies to adda regularization reward to each action made. Thisreward is constructed by computing the averagecount of choosing a particular module, multipliedby a scaling factor. We also use a similar reward.In particular, we compute it for module mk andaction a as follows:

rc(mk) =

{(1− α) ∗ r′c(mk) + α if k = a

r′c(m), otherwiseAs we want this reward to sum to 1 over all mod-ules, we normalize. Additionally, we want to scalethis reward to stay constant even for larger routingdepths. The result is therefore scaled as such:

rc =rc

d∑rc

A.2 Losses

To make the backpropagation step to train the net-work more efficient, we combine the lossesLtotal (x, y) = Lclass(x, y) + LRL(x, r) (5)

where LRL(x, r) is the loss defined by the chosenRL algorithm.

A.3 Routing RNNs

We modify LSTMs as originally defined byHochreiter and Schmidhuber (1997):

ft = σg(Wfxt + Ufht−1 + bf ) (6)

it = σg(Wixt + Uiht−1 + bi) (7)

ot = σg(Woxt + Uoht−1 + bo) (8)

ct = ft ◦ ct−1 + it◦σc(Wcxt + Ucht−1 + bc)

(9)

ht = ot ◦ σh(ct) (10)

However, analogous architectures can be imple-mented for GRUs or different LSTMs. Here, ei-ther architecture has two core transformations –one from the hidden state at time step tk to timestep tk+1, and one from the input (in our case,word embeddings) to the hidden state. We havedesigned architectures that route either.

The resulting equations (6), (7), and (8) for theinput-to-hidden routing model illustrated in Fig-ure 3, top right, become (with g being a placeholder

for f, i, o):gt =σg(FCg,kd ◦ FCg,kd−1

◦ · · · FCg,kdxt

+ Ught−1 + bg)(11)

Similarly, the equations (6), (7), and (8) for thehidden-to-hidden routing model in Figure 3, bot-tom right, become (with g being a placeholder forf, i, o):gt =σg(Wgxt + FCg,kd ◦ FCg,kd−1

◦ · · ·FCg,kdht−1 + bg)

(12)

A.4 Routing Transformers

The following describes the two different ap-proaches for routing transformers. When routingentire transformer blocks (TB), we have:

Fkt = {TBt,1, TBt,2, . . . , TBt,b} (13)However, as the routing functions for routingthe attention layers are interleaved with the otherblock-layers (BL), the basic routing chain changesto:fR(x) =fk1 ◦ BL ◦ ... ◦ BL ◦ fkd(x) ◦ BL (14)

and the routing module sets are (with AL as atten-tion layers):

Fkt = {ALt,1, ALt,2, ..., ALt,b} (15)

A.5 Hyperparameter Search

We found that training the router with Adamcan become unstable and stay at random base-line performance. In contrast, we found that us-ing Adam for the modules increased performanceby around 1%. When training a routing utiliz-ing the Gumbel-Softmax reparameterization, wetested Adam and SGD exclusively, and settled forAdam. We found the module learning rate of 1e−3and the router learning rate of 3e−4 by perform-ing sweeps over the learning rate from 1e−2 to1e−4 for the module training and from 1e−2 to1e−5 for the router training, roughly by decre-ments of factor 3. In general, we found a branchingsize of b = 3 to work best (although performancefor b = 5 was nearly identical), after sweeps forb ∈ {2, 3, 5, 10, 15, 20}. As we also specify themaximum recursion depth, we generally allow upto d = 3 recursive steps. Higher ds add to train-ing time, but neither increase nor decrease perfor-mance significantly, as the router (nearly) neverlearns to apply more than three transformationsanyway. Figure A1 shows the effects of the valueof the decision making algorithm, the value of theregularization reward, and of the routing width.Most notably, Gumbel-Softmax routing performs

(a) Effect of the decision-making algo-rithm.

(b) Effect of the regularization rewardvalue.

(c) Effect of the routing width.

Figure A1: Encoder Routing Strategies.

on average 10% worse than RL algorithms, withvariance that is nearly an order of magnitude larger.Also, larger routing width has similar expected per-formance but increases the variance. The value ofthe regularization reward has little effect on ex-pected performance.

B Implicatives Experiments

Routed models that achieve good performance onSCI show a strong indication of proper learning ofentailment and contradiction, and some difficultieswith neutral cases, as indicated by the confusionmatrix in Table A1. An inspection of common er-rors of the CBOW WP model shows a number ofcases related to the probabilistic component of thesignature. For example, Jerry got a chance to pitchwas marked as contradicting Jerry did not manageto pitch, when the gold label is indeed permits.In this case, if the premise is true then there is asmall chance that the author would regard the hy-pothesis as false and therefore predict contradictsas the model did. Other errors involve coreference,such as the prediction of entailment between youtook the time to review my presentation and youreviewed his presentation.

Relation entailment contradiction neutral Totals

entailment 35 2 10 47contradiction 2 33 5 40neutral 7 6 28 41

Totals: 44 41 43 128

Table A1: Confusion Matrix.

C Implicatives Corpus

Implicative constructions yield a judgment aboutthe truth of their complement clause under at leastone polarity. Some are simple verbs like manage

and fail, while some are phrasal constructions likemeet one’s duty and waste a chance.

C.1 Signatures

The signature of an implicative indicates the re-lation between the matrix clause and complementclause. The symbol + indicates an entailment ofthe complement clause, the symbol − indicatesan entailment of the negation of the complementclause, and the symbol o indicates no entailmentrelation to the complement clause. A signatureconsists of a left symbol, which indicates the en-tailment relation to the complement clause whenthe matrix clause is in a positive environment, andthe right symbol, which indicates the entailmentrelation to the complement clause when the ma-trix clause is in a negative environment.

For example, in the notation of MacCartney andManning (2009) and Karttunen (2016), the signa-ture of manage is +|-. This means a sentencewith manage entails its complement in positivecontexts, and contradicts its complement in nega-tive contexts. As such, Joan didn’t manage to solvethe problem contradicts Joan solved the problem.

Fail and lack foresight have the implicative sig-nature -|+. NP failed/lacked the foresight to VPcontradicts NP VPed and entails NP didn’t VP. Thisis negative entailment under positive polarity (-),positive entailment under negative polarity (+).

Two-way implicatives like manage and fail yieldan entailment under positive and negative polarity.One-way implicatives such as force (+|o), prevent(-|o), be able (o|-), and hesitate (o|+) yield anentailment, + or -, only under one polarity and noentailment (o) under the other.6

6In addition to the entailment properties that we focuson in this study, most implicative constructions also haveanother component of meaning, sometimes called presuppo-sition (Karttunen, 1971; Baglini and Francez, 2016), some-

+

−

−

+

+to eat breakfastto forget (-|+)

manage (+|-)

did not (-|+)

Kim

Figure A2: Computation of the inference Kim ate breakfastentailed by the premise Kim did not manage to forget to eatbreakfast.

There are seven known implicative signatures,six of them documented earlier (Karttunen, 1971,2012). In the course of collecting data for thisstudy, we came across a new implicative construc-tion that had been missed in all previous studies.We code take no time and waste no time as +|+because, in the vast majority of cases we lookedat, the presence or absence of negation makes nodifference. I wasted no time to jump on board andI didn’t waste no time to jump on board both meanthat I jumped on board quickly.

C.2 Pragmatic ImplicativesIn cases where there is no logical entailment, one-way implicatives may give rise to an invited infer-ence (Geis and Zwicky, 1971). I was able to speakbut just didn’t know what to say is not a contradic-tion. But if the speaker leaves out the but-clause,she probably means that not only was she able tospeak, but she in fact did speak. There is no invitedinference in cases like Bill Clinton had a chanceto kill Bin Laden before the September 11 attacks,because it is common knowledge that the attemptto do so failed. Insofar as these cases depend onan encyclopedic knowledge about the world, as op-posed to a dictionary/lexical knowledge, we use theterminology pragmatic implicatives for this type ofverbs.

The likelihood of invited inferences variesgreatly depending on the construction. They arevery likely with be able, less likely with havechance, and very unlikely with hesitate. To bet-

times conventional implicature (Karttunen and Peters, 1979).For example, Marco failed to show up for work implicatesthat Marco tried or was expected to show up for work. TheNatLog system (MacCartney and Manning, 2009) computesentailments of implicatives but no presuppositions. No cur-rent NLP system does. The difference between entailmentand presupposition/conventional implicature can be madewith the question Did Marco fail to show up for work? Thisquestion asks whether Marco showed up for work or not;fail indicates that the speaker thinks that Marco tried or wasexpected to come to work.

ter match actual usage, we assign to be able notthe semantically correct o|- signature but .9|-,reflecting our estimation that there is a .9 proba-bility that the author is using the construction as ifit were a two-way +|- implicative. We code havechance as .4|- and hesitate as o|+.

C.3 Nested Implicatives

Implicative constructions can be nested inside ofother implicative constructions, as in Stan man-aged to fail to propose to Carol. The meaning ofnested implicatives is compositional; the signatureof a nested implicative can be determined algorith-mically by the signatures of the single implicativesthat compose it (Nairn et al. 2006; see Figure A2).Since manage has the signature +|- and fail is-|+, the nested constructions manage to fail andfail to manage are both -|+. Thus, the exampleabove entails that Stan did not propose to Carol,as does Stan failed to manage to propose to Carol.The entailments are the same, but the implicationsassociated with manage and fail differ dependingon the nesting.

Some implicative constructions – e.g., manageand fail, remember, and forget – can be nested inone another. In many cases, only one order makessense. It is easy to find examples of happen to turnout – e.g., it happened to turn out to be a success –but a search for turn out to happen sentences doesnot turn up any. One can even find triply nestedimplicative constructions, such as don’t forget toremember not to forget to watch the finale of series2 tonight, but they are rare.

The single-word, phrasal implicatives, and (afew) nested implicatives used in our data set arelisted in Table A3. The release version of SCI

will contain additional doubly-nested construc-tions (the complete list is in Table A4), and willhave additional splits.

C.4 Corpus Versions and Extension

The three different versions of the dataset de-scribed in the article are joint, disjoint, and mis-match. The disjoint test set contains the implica-tives listed in Table A2, and thus the implicativeslisted there are not contained in train and valida-tion.

The mismatch version of the SCI corpus containsall signatures in validation and test, but only a re-stricted set for train. Table A5 contains the subsetsavailable on each split.

In addition to the versions, we included all thenested constructions as a separate extension to thedataset using the same format. This expansion canthen be easily used as a test set for generalizationto longer sequences.

Figure A3 show the distribution of lengths,counted in word tokens, for premises and hypothe-ses. Figure A4 shows the distribution of lengths forboth premises and hypotheses.

Construction Sign Examples

break promise -|+ 90exploit opportunity +|- 120fail obligation -|+ 60fulfill promise +|- 90grab occasion +|- 120make vow o|o 91manage +|- 90meet duty +|- 90miss occasion -|+ 120neglect -|+ 91refrain -|+ 120require +|.3 120seize opportunity +|- 120take opportunity +|- 91use occasion +|- 120waste chance -|+ 91waste no time +|+ 76waste opportunity -|+ 91

Constructions: 18 Total 1791

Table A2: Disjoint dataset in test set in SCI.

10 20 30 40 50

0

200

400

600

800

1000# words: Example lengths (prem+hyp)

Figure A3: Distribution of sentence lengths in number ofwords for all verbs in SCI.

5 10 15 20 25 30

0

200

400

600

800

1000

1200

1400

1600

# words: Premise and hypothesis lengths

Figure A4: Distribution of lengths for premises (blue) andhypothesis (orange) in SCI.

Construction Sign Examples Construction Sign Examples

be forced +|- 128 keep promise +|- 90be kept from -|.9 132 lack foresight -|+ 91be made +|.5 126 lose opportunity -|+ 90be obstructed from -|.9 126 make +|.5 126be prevented -|.7 87 make promise o|o 91be required +|.3 120 make vow o|o 91be stopped -|.5 120 manage +|- 90bother +|- 91 manage to miss opportunity -|+ 120breach contract -|+ 120 meet duty +|- 90breach promise -|+ 120 meet obligation +|- 90break pledge -|+ 73 meet promise +|- 88break promise -|+ 90 miss chance -|+ 90bring oneself +|- 120 miss occasion -|+ 120cause +|.5 120 miss opportunity -|+ 91coerce +|.9 120 neglect -|+ 91compel +|.4 120 neglect duty -|+ 120convince +|.8 120 neglect occasion -|+ 120dare +|- 91 obey order +|- 91disobey order -|+ 87 overlook opportunity -|+ 120exploit occasion +|- 120 prevent -|.7 60exploit opportunity +|- 120 proceed +|- 120fail -|+ 91 refrain -|+ 120fail obligation -|+ 60 refuse to take advantage -|+ 114fail to take advantage -|+ 120 remember +|- 91follow order +|- 90 require +|.3 120force oneself +|- 120 seize occasion +|- 120forget -|+ 90 seize opportunity +|- 120fulfill promise +|- 90 stop -|.5 120get chance .9|- 90 succeed +|- 120get +|- 119 take advantage of opportunity +|- 120grab occasion +|- 120 take chance +|- 91grab opportunity +|- 120 take no time +|+ 76happen +|- 92 take occasion +|- 120have chance .4|- 89 take opportunity +|- 91have courage +|- 91 take time +|- 88have +|.5 120 take vow o|o 91have foresight +|- 90 turn out +|- 153have gall +|- 90 use occasion +|- 120have time .9|- 90 use opportunity +|- 120have wherewithal +|- 90 venture +|- 120hazard +|- 120 waste chance -|+ 91hesitate o|+ 91 waste money +|- 91ignore duty -|+ 120 waste no time +|+ 76ignore opportunity +|- 120 waste occasion -|+ 120jump on occasion +|- 120 waste opportunity -|+ 91keep from -|.9 132 waste time +|- 91


Table A3: Implicatives in joint SCI.

Construction Sign Examples

be able to bring oneself +|- 160fail to have courage -|+ 66fail to have means -|+ 36fail to have wherewithal -|+ 18fail to manage -|+ 90fail to meet obligation -|+ 54fail to obey order -|+ 51fail to take advantage -|+ 120forget to remember -|+ 48manage to fail -|+ 54manage to forget -|+ 18manage to have gall +|- 18manage to have power +|- 18manage to have strength +|- 54manage to have wherewithal +|- 18manage to miss opportunity -|+ 166manage to remember +|- 53manage to turn out +|- 36refuse to take advantage -|+ 114remember to forget -|+ 359remember to take time +|- 54take advantage of opportunity +|- 120


Table A4: Nested Implicatives collected to appear in therelease version of SCI.

Split Signatures

train -|+, +|-, o|-, o|o, +|.5, .4|-, -|.7dev/test -|+, +|-, o|-, o|o, +|.3, .4|-, +|.4, +|.5, -|.5, -|.7, +|.8, +|.9, -|.9, .9|-

Table A5: Mismatch Splits in SCI.

Documents

Recursive Routing Networks: Learning to Compose Modules for … · Recursive Routing Networks: Learning to Compose Modules for Language Understanding Ignacio Cases 1, Clemens Rosenbaum2,