We introduce Recursive Routing Networks ( RRN s), which are modular, adaptable models that learn effectively in diverse environments. RRN s consist of a set of functions, typically organized into a grid, and a meta-learner decision-making component called the router . The model jointly optimizes the parameters of the functions and the meta-learner's policy for routing inputs through those functions. RRN s can be incorporated into existing architectures in a number of ways; we explore adding them to word representation layers, recurrent network hidden layers, and classifier layers. Our evaluation task is natural language inference ( NLI ). Using the MULTINLI corpus, we show that an RRN 's routing decisions reflect the high-level genre structure of that corpus. To show that RRN s can learn to specialize to more fine-grained semantic distinctions, we introduce a new corpus of NLI examples involving implicative predicates, and show that the model components become fine-tuned to the inferential signatures that are characteristic of these predicates. Human cognition has an extraordinary ability to modularize, decomposing problems and solving them by re-composing elements from prior solutions, and this ability is nowhere more evident than in language understanding (Partee, 1984; Janssen, 1997). Most machine learning architectures lack this modularity, which limits their ability to generalize and leaves them susceptible to catastrophic interference (McCloskey and Cohen, 1989)  forgetting past skills when acquiring new ones. We propose to address this need for modularity by applying Routing Networks (Rosenbaum et al., 2017) to natural language understanding. Routing Networks are self-organizing networks with two components (Figure 1): a set of function blocks Router 5 Recursive Routing Network 3 2 4 1 6 Figure 1 : Given a premisehypothesis pair x , e.g., t managed to do s and t did s (1), the model needs to learn to predict entails (6). The router (2) estimates the value of applying each of the available sub-functions to the input. Given that manage has certain semantic properties, the router may select (3) a function f 1 specialized to them. The module is then applied (4), yielding f 1 ( x ) . This process repeats (5), now using f 1 ( x ) , selecting and applying another sub-function, and so on, until the router is confident in its prediction (6). which can be applied to transform the input, and a router which makes decisions about which function block to apply next. Here we introduce Recursive Routing Networks ( RRN s), in which there is a single set of composable functions, recursively chosen by the router. RRN s can be applied to different components of modern language understanding architectures with full end-to-end training. The model jointly optimizes the parameters of selected sub-functions and the meta-learner's policy for how to route inputs through those functions. As a result, individual sub-functions specialize to specific inputs, and paths through the grid of sub-functions can similarly be trained to reflect specific concepts and capabilities. RRN s share many intuitions with other modular methods like: Neural Module Networks (Andreas et al., 2015, 2016; Hu et al., 2017), which learn to construct a neural network from pre-defined modules; the Compositional Recursive Learner (Chang et al., 2018), a closely related approach that uti-x Routing across examples Weightsharing Possibledistribution OrthogonalizedKnowledge CompressedKnowledge High Interference HighTransfer  i <latexit sha1_base64="7oNlFgnMzVWHsxisJ1PN/N2kgNU=">AAAB73icbZBNS8NAEIYn9avWr6pHL8EieCqJCHosevFYwX5AG8pmO22XbjZxdyKU0D/hxYMiXv073vw3btsctPWFhYd3ZtiZN0ykMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2cao4NHstYt0NmUAqFDRIksZ1oZFEosRWOb2f11hNqI2L1QJMEg4gNlRgIzsha7S6NkFhP9MoVr+rN5a6Cn0MFctV75a9uP+ZphIq4ZMZ0fC+hIGOaBJc4LXVTgwnjYzbEjkXFIjRBNt936p5Zp+8OYm2fInfu/p7IWGTMJAptZ8RoZJZrM/O/WielwXWQCZWkhIovPhqk0qXYnR3v9oVGTnJigXEt7K4uHzHNONmISjYEf/nkVWheVH3L95eV2k0eRxFO4BTOwYcrqMEd1KEBHCQ8wyu8OY/Oi/PufCxaC04+cwx/5Hz+ACJYkAQ=</latexit> <latexit 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Forward (grey background) and backward passes (white background) during the training of a rollout of an RRN with three blocks (width) and limited to depth three (height). In this unrolled version, a path is the sequence of function blocks selected by the router. Right: The dilemma of weight sharing and the transferinterference trade-off. When examples are learned using largely separate weights, there is a thin possible distribution of gradient dot products that limits interference, which unfortunately also limits transfer. This is beneficial for unrelated examples, but frustrates the learning of related ones. Conversely, when examples are learned using largely shared weights, there is a wide possible distribution of gradient dot products that allows for both high magnitude transfer and interference. This is beneficial for related examples, but maximizes interference when examples are unrelated. With RRN s, we provide our network with an unprecedented degree of leverage to learn to navigate this trade-off. lizes curriculum learning to solve arithmetic and vision problems; and ModularNetworks (Kirsch et al., 2018), which extend Routing Networks with an EM-like training approach. We show how to incorporate RRN s into different neural components (word representation layers, recurrent network hidden layers, classifier layers), and we study their application to natural language inference ( NLI ), in which premisehypothesis pairs are labeled for whether the premise entails, contradicts, or is neutral with respect to the hypothesis. We chose this task because reasoning in natural language involves context-sensitive interpretation of words and sentences as well as compositional structure. We make use of the MULTINLI corpus (Williams et al., 2018), which includes text from multiple genres that we expect to condition linguistic senses in complex ways. Our experiments show that RRN s learn policies and components that reflect this genre structure, which leads to superior performance. We also introduce a new corpus of NLI examples involving implicative constructions like manage to , be able to , and fail to (Karttunen, 1971, 2012). This corpus follows the design of many recent NLI corpora, but with the added challenges of reasoning about implicatives, which have logical signatures that interact compositionally with each other and with surrounding semantic operators. Our experiments show that trained RRN model components become fine-tuned to these signatures. Finally, we introduce an extension of the framework which leverages a Dispatcher for situations where meta-information is not available at test time, obtaining very promising results. RRN s are a natural extension of the Routing Networks introduced by Rosenbaum et al. (2017), themselves part of a larger class of conditional computation models (Bengio et al., 2015). Routing Networks combine trainable modules with a meta-learner called a router , which is typically trained using reinforcement learning, though any hard-decision making algorithm could be used. Given an example, the router selects a module and applies it, yielding an activation that can be the input to another routing iteration. As Rosenbaum et al. observe, this can be modelled as a recursive process where the router is applied repeatedly to define complex paths through the modules. The final routed output is passed to additional layers or interpreted as the output of the network (Figure 1). We are motivated to consider Routing Networks, and in particular recursive variants, in order to address the transferinterference trade-off (Riemer et al., 2019). Vanilla neural networks latently attempt to solve a very difficult problem of deciding when to orthogonalize and compress knowledge (Figure 2, right). When two examples are learned with the same weights, there is high potential for transfer as well as interference. This is good for related examples because it maximizes the potential for transfer. However, weight sharing is bad for unrelated examples, as it increases the likelihood of interference. In contrast, when two examples are learned using different weights, there is low potential for transfer and interference. This is beneficial for unrelated examples, but limits the potential for learning about commonalities between related ones. RRN s extend vanilla neural networks by granting them the leverage to navigate this trade-off by making global functional decisions at the module level. As a result, RRN s explicitly make decisions to compress or orthogonalize knowledge between examples by deciding whether to share specific weights. Thus far, hard-selection routing has not been applied to language domains. Zaremoodi et al. (2018) introduced a soft version of routing that falls within a larger class of Mixtures of Experts ( M o E ) models (Jacobs et al., 1991). However, M o E models differ from RRN s in two crucial ways. First, M o E models generally do not consider the recursive application of functions. The promise is that we can compose functions to reflect the compositional aspects of a problem. Imagine we have a sentence encoding and we want to answer a particular question. We can now condition the router on the question, so that it applies exactly the functions required that translate the encoding to extract the answer. Second, M o E models do not allow for nearly the level of specialization as those based on routing, because they do not eliminate weight sharing across modules and instead only gate the sharing. In practice, this still leads to significant interference across tasks. Composition of modules has been widely explored for question answering, starting with Andreas et al. (2015). Andreas et al. (2016) learn to assemble a deep neural network on-the-fly from a pre-specified inventory of neural modules using tree-structured layouts based on linguistic analysis. Chang et al. (2018) also explore routing, but with problems described in pseudo-language rather than natural language. We now provide a formal presentation of RRN s and show how to incorporate them into existing neural architectures. Formally, the router bases its decision on the tuple (cid:104) f ( x ) , m (cid:105) , where x is the sample, f ( x ) = f i      f k ( x ) is the composition of all applied functions from the set of all available functions F = { f 1 , . . . , f b } , and m is a vector that contains meta-information that can be utilized by the router  for example, an embedding for its genre or semantic classification. The router consists of a policy  ( F|(cid:104) f ( x ) , m (cid:105) ) that determines which of the functions in F to apply for a given state (cid:104) f ( x ) , m (cid:105) . Once a new function f k is selected, the latest activation (and, thereby, the state) gets updated to f k  f ( x ) , and the process repeats. 1 We focus on the recursive case where the set F is generally the same for each decision. Apart from being the more general formulation, this form of recursivity allows more weight sharing and better compositional generalization. While there is no necessary upper limit to the number of selections, we found that limiting the number to a maximum of d makes the learning more stable. Consequently, the total number of possible routing paths available is b d . Decision Making One of the most important design choices for the router concerns the meta-learning algorithm. Routing is limited to hard decision-making algorithms, in particular a stochastic reparameterization using the Gumbel-Softmax function (Jang et al., 2016; Maddison et al., 2016) and reinforcement learning ( RL ) algorithms. As with other recent approaches to compositional architectures with hard selections (Ben-gio et al., 2015; Shazeer et al., 2017; Kirsch et al., 2018), routing networks can suffer from module collapse , a lack of diversity in the router's decision making. This general problem, common in architectures that jointly train decision making and functions, stems from an early over-estimation of the value of specific functions. This leads to these functions being selected and trained more than others, until these functions are indeed so good 1 Some architectures have constraints that prevent repeated selection of the same functions in F . In these cases, the router has to be restricted to select from only a compatible subset of F . that others are not considered. To mitigate this, we start with a strategy proposed by Rosenbaum et al. (2017): exclusively conditioning the router on meta-information provided by the datasets and storing the router's policy in a table. We found that this stabilizes early training. During later training, we can replace the meta-information hard assignment rule with a meta-information guesser. We implement this guesser with what Rosenbaum et al. call a Dispatcher , an approximation-based sub-policy that estimates the meta-information for each sample and passes it to the corresponding policies (Section 5). Losses and Training Once the routing procedure terminates, the selected modules are trained using standard optimization techniques, such as backpropagation with stochastic gradient descent ( SGD ). The decision-making algorithm is either also trained with SGD (Gumbel reparameterization) or trained using reinforcement learning. As we focus on classification problems, our core loss function L class ( y,  y ) is the standard cross-entropy loss. We backpropagate this loss along the function path chosen by the router for input x . Reinforcement Learning Rewards For RL training of the router, we define the reward r for the reinforcement learner to be the sum of the negative classification loss and an additional regularization reward that encourages diversity in the selection of modules: r = L class ( y,  y ) + r reg (1) For discussion of this reward, see Appendix A.1. If not otherwise mentioned, we assume that the routed modules are all fully connected layers with the same input and output dimensions. We furthermore assume that the router is always selecting from the same set of functions when defining paths through the routing layers, as this straightforwardly allows recursion in the relevant sense. Routing Classifiers The simplest method we explore involves routing the layers of a classifier. To do this, we define a fixed set of b fully connected layers ( FC ), each of the same dimensionality, and we allow the router to choose any path through this set up to a maximum length d (Fig-ure 3, top center). The final activation produced by the chosen path is then densely connected to a non-routed output layer. To further model each example's dependence on its class, we hard-select this output layer based on the meta-information label m for x : W m ( FC k 1      FC k d ( x )) + b m (2) Routing RNN Encoders In routing RNN s we focus on their two core transformations: from the hidden state at time step t k to time step t k +1 , and from the input to the hidden state. We have designed routing architectures for both of them. In this paper, we start with LSTM s, but the techniques are straightforward to adapt to other cell types. Figure 3, top right, shows an architecture where we route the input-to-hidden ( I 2 H ) transformation. Similarly, Figure 3, bottom right, shows an architecture where we route the hidden-to-hidden ( H 2 H ) transformation. While these transformations are often designed as single fully connected layers, we allow a recursive application of d steps from a selection of b modules to be applied. The selections for the transformations f, i, o, c are tied to be the same. The corresponding transformation (in form of their weight matrices W f/i/o/c for I 2 H and U f/i/o/c for H 2 H ) becomes: K = FC k      FC m ( x t ) ,  K  { W f , W i , W o , W c , U f , U i , U o , U c } (3) Routing CBOW Encoders We also experiment with routing a continuous bag of words ( CBOW ) encoding. As routing after the main addition is just routing the classifier, we instead add a word-level transformation before the addition (Figure 3, bottom left). This transformation can again be routed recursively (with up to d steps through b modules). The entire CBOW model can be defined as (with w 1 , . . . , w t as premise or hypothesis): CBOWR ( w 1 , . . . , w t ) = (cid:80) ti =1 FC k 1      FC k d ( w i ) (4) Routing Transformers The Transformers model (Vaswani et al., 2017) was recently pretrained as a language model (Radford et al., 2018) achieving impressive results on several NLI datasets. Since the corpus for pretraining is not available, we instead use the parameter-files distributed by the authors. 2 The encoding consists of twelve Transformer-blocks. Each block consists of a convolutional attention layer followed by two convolutional layers (along with several dropout and layer-norm layers). This allows routing at different levels of granularity, of which we investigate two: routing entire blocks and routing the attention step within each block. As we use pre-defined parameters, we cannot apply the blocks or the attention layers recursively. Furthermore, we have to add routing in the fine-tuning phase of using Transformers by creating b copies of each routed module (the depth is necessarily d = 12 ). When fine-tuning, the router then diversifies the initially identical modules. ex-2 https://github.com/openai/finetune-transformer-lm periments focus on NLI , using the MULTINLI corpus (Williams et al., 2018) and a new English Corpus of Implicatives, the Stanford Corpus of Implicatives ( SCI ). In both, each example is a premise/hypothesis pair labeled with one of entails , contradicts , or permits . What is special about MULTINLI and SCI in the current context is that the examples also have meta-information that can be used to guide the router: a genre label for MULTINLI and an implicative signature for SCI . We expect our routing models to leverage this information during policy and parameter learning. For all models (except Transformers), we adopt the architecture in Figure 3, top left, in which the premise and hypothesis are processed separately, and the final representations of each are concatenated and fed into the classifier layers. We explore two methods for input processing: (i) pretrained word representations (GloVe; Pennington et al. 2014) and (ii) pretrained contextualized representations (ELMo; Peters et al. 2018). For all non-routed models, we provide explicit access to the genre label via special keywords put in front of each sentence before encoding. The routed models use this label as meta-information. Unless otherwise specified, we used embedding and hidden dimensions of 300. The classifier consists of three fully connected layers with input and output dimensions of 600 (also when routed). The final layer projects from 600 to the output dimension, 3. The classifier nonlinearities are ReLUs. We train the modules using Adam (Kingma and Ba, 2014) with a learning rate of 1 e  3 and the router using SGD with a learning rate of 3 e  4 (for additional details, see Appendix A.5). For Transformers, all hyperparameters are determined by the published parameter files. In experiments with different decision making algorithms (see Appendix A.5), we found that the Gumbel-Softmax reparameterization performed 10% worse on average, with much higher variance. QLearning was consistently as good as other more complex RL algorithms, so we report only our QLearning experiments. As Routing Networks have more parameters than their non-routed counterparts, we ran experiments with larger non-routed networks. We found that this did not affect performance, only resulting in more overfitting. Table 1 : Results for MULTINLI and SCI with different baselines and their routed versions. We report average accuracy with confidence intervals over five runs with different seeds. For Transformers, we found that finetuning was highly volatile. We therefore report test results from the best-of-5 train models. All results for nested SCI were computed by fine-tuning the same network previously trained on joint. WP' stands for Word Projection, +D' for Dispatching,  I 2 H ' for Input-to-Hidden routing, and  H 2 H ' for Hidden-to-Hidden routing. Italics mark scores whose confidence intervals overlap with the best scores. Figure 4 : Path (module selection) overlap for MULTINLI between genres with the CBOW GloVe WP model. The diagonal represents the number of function blocks applied for a genre. A maximum of three means that two genres would be routed through the exact same functions. The MULTINLI corpus contains 392,702 training examples, 10K dev examples, and 10K test examples. The examples come from 5 genres: fiction, government reports, the Slate website, the Switchboard corpus (Godfrey and Holliman, 1997), and Berlitz travel guides. We treat these genre labels as meta-information for the model (Section 3.1). Our MULTINLI results are given in Table 1, and the learning dynamics in Figure 5. The best model combines the Transformer base model with routing in the attentional layers. Our methods for routing 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 as the very similar models without genre labels evaluated by Williams et al. (2018). It seems that these models are not able to take advantage of the meta-information. In contrast, RRN s seem to provide the space needed to condition linguistic senses on these labels. Our hypothesis is that routing will not only lead to better performance on diverse tasks like MULTINLI , but also that the paths  i.e., the sequence of functions selected by the router  followed by the network will reflect high-level task structure. Figure 4 suggests that this is the case for MULTINLI . Here we show the degree of path-overlap for all pairs of genres. As we might expect, government (the 9/11 report), Slate (cur-rent affairs), and travel cluster together, as distinct from the two more isolated genres (Switch-board; spoken language) and fiction (mostly from the 20th century). Karttunen (1971) discovered that implicative constructions, such as manage and waste chance (e.g. They wasted their chance to win ), have signatures , which characterize the inferences they support in positive and negative contexts. This makes them the order compelled him to appear as a witness entails he appeared as a witness we have missed an opportunity to examine the art market today contradicts we have examined the art market today Mr 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 (left column), 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). For instance, the positive sentence Joan managed to solve the problem entails Joan solved the problem , and the negative sentence Joan didn't manage to solve the problem contradicts Joan solved the problem , so we say that the verb manage has the signature +|(MacCartney and Manning, 2009; Karttunen, 2016). In contrast, waste chance has the opposite signature, since they wasted the chance to befriend him contradicts they befriended him , and they didn't waste the chance to befriend him entails they befriended him . There are seven implicative signatures: six were previously known (Karttunen, 1971, 2012), and we found an additional one ( +|+ ; e.g. take no time to ). See Appendix C for additional details. Signatures are compositional: when two or more implicative constructions are composed in a sentence, they create a nested implicative construction whose signature is determined by the signatures of the individual verbs (Nairn et al., 2006). For example, John managed to remember to get the keys entails John got the keys , where the nested implicative manage to remember has the overall signature +|. We also see a more limited form of compositionality inside phrasal implicatures; their signatures are often largely determined by the lexical semantic family of their constituent words. Therefore, signatures make implicatives ideal for evaluating different degrees of compositional generalization with RRN s, as they provide valuable meta-information (Section 3.1). Table 3 : SCI statistics. Top: Percentage of validated pairs and basic agreement. Bottom: Fleiss  coefficients and proportion of all assignments made to the corresponding label. Our SCI dataset contains  10K premise hypothesis pairs. All seven signature types are represented, in addition to pragmatic signatures, which have inferential biases (see Appendix C). We provide SCI 3 in three versions for all single and phrasal implicatives. 4 In the joint version, the underlying distribution of implicatives is shared across train, validation, and test splits. In the disjoint version, a different subset of implicatives is used in train from those used in validation and test. This allows us to test generalization to unseen constructions. Although disjoint with respect to constructions, the constructions are carefully distributed so that all the underlying signatures and most lexical items are represented in all splits. The lexical items that make up implicative constructions overlap between the splits. For example, take vow appears only in training and validation, while make vow only appears in test. The last version is mismatch , where different subsets of the signatures are present in training/validation and test (see appendix C for more details). Data collection for SCI proceeded as follows: we collected at least 12, and most often 20, seed premises from examples found in Google Books and on the web. For each example, six hypotheses were created by expert annotators. 5 These six examples were constructed in two steps: first, the premise was taken to be the seed sentence, and three hypotheses were created to exhaust the label space in relation to the premise. Second, the premise was taken to be the negation of the seed sentence, and a different set of three hypotheses was created to also exhaust the label space with respect to this negated sentence. For example, one of the seeds for manage was I managed to see who it was . From this, annotators produced by hand a 3 https://nlp.stanford.edu/projects/sci/ 4 We provide nested constructions as a separate extension, with the exception of a few nested implicatives that were used in the development version of this corpus, for which the results are presented here. In the final version of the corpus, single and phrasal implicatives will be separated from nested implicatives. See the Appendix for more details. 5 Native speakers of English trained in semantics at the undergraduate level, with expertise in implicatives. Figure 5 : Learning on MULTINLI with GloVe inputs. Figure 6 : The path-overlap between different signatures on SCI , 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 of these two premises. Finally, each example was labeled by the author of the hypothesis examples. Validation was performed on a randomly selected subset of constructions. Two additional votes were cast by different linguists, resulting in three label votes for each example in the subset. The gold label was then defined to be the majority class from this set of votes. The Fleiss  shows high inter-annotator agreement (Table 3). For each example in SCI , we use its associated implicative signature as the meta-information label. This serves as a subtler kind of semantic information than genre. The learning dynamics of different models are shown in Figure 7. As Table 1 shows, RRN s lead to considerable gains in perfor-0 20 40 60 80 100 epochs 40 50 60 70 a cc u r a c y i n % CICBOW Glove None RNN Glove None RNN Glove CL CBOW Glove WRL Figure 7 : Learning on SCI (joint) with GloVe inputs. mance over most benchmarks. Recursive routing of the classifier helps consistently with all models. Even simple word-level routing yields major improvements for CBOW . As with MULTINLI , we believe that the modules allow the conditioning of linguistic senses. Routing at the word-level for sequential models seems suboptimal here. The training accuracies ( > 99 . 5% ) suggest that the problem is overfitting. Intuitively, routing can assign too many exclusive parameters to each class; and as the samples within these classes are similar to be-gin with, remembering them can be easier for the network than for non-routed networks. We expect examples with similar signatures to be routed along similar paths. Figure 6 summarizes the path-overlap between different signatures. Some similar paths involve reversals of polarity, 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 isolation of the unusual signature +|+ (limited to just two constructions) seems expected, as does the affin-ity of the nested signatures N+|and N-|+ to their unnested counterparts. However, as shown in Table A3, some signatures only contain very few samples, resulting in highly noisy routing paths. We now seek to provide intuitive explanation for how RRN s help in navigating the transfer interference trade-off. We have characterized the trade-off in terms of controlling weight sharing between examples. While baseline networks can learn to navigate this trade-off if they are optimized for the appropriate objective (Riemer et al., 2019), in general these models do not have available supervision on how to do this and so optimize greedily for the current example. As such, early experiments with the baseline, non-routed models show that examples with neutral signatures, such as o|o and o|+ , are the first to be learned. We noticed that baseline sequence models learned to classify take vow much earlier in the training process, as the word vow is lexicalized through training on make vow examples. However, its performance decreases over time as take is lexicalized through training on examples of other implicatives with different signatures, such as take chance . This behavior is the characteristic outcome of catastrophic interference: learning take chance results in decreased performance on take vow as the two examples interfere. This is a particularly revealing instance as these two phrasal verbs are similar on the surface, but have quite different semantic properties, as reflected by their signatures. However, given that RRN s are able to route them differently (Figure 6), interference is less likely. It is also possible that routing helps with the transfer related to similar examples, an avenue we want to explore in the future. We have shown that high-quality meta-information can be extremely useful. Unfortunately, it is often not available at test-time. To compensate, we evaluated an RRN extension in which an additional neural network module is trained to assign examples to meta-information classes (genres for MULTINLI ; signatures for SCI ). When we trained this model jointly, we found that it was unstable and did not perform well. However, if this model is introduced after RRN training on examples with known meta-information, then the results are extremely promising. We call this Dispatcher Training (+D'), borrowing similar terminology from Rosenbaum et al. (2017). Table 1 includes an initial evaluation of this variant with a CBOW base model and WP routing. As we can see, accuracy actually increases by a small amount over WP alone. Additionally, having +D allows the network to generalize better to unseen examples and unseen patterns. Consider the relative performance drop for 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%) to disjoint (  1.04%), and disjoint (73.91%) to mismatch (  5.22%) 6 Conclusion This paper introduced Recursive Routing Networks and showed how to incorporate them into a variety of different neural architectures; we explored a range of possibilities for this, and the techniques generalize to other options straightforwardly. Our evaluations focused on NLI . We showed in particular that our RRN s can effectively leverage the meta-information in the MULTINLI corpus and in our new corpus focused on implicatives; not only do RRN s use this information to achieve superior accuracy, but they also learn sub-structure that reflects this high-level information, and our Dispatcher variant extends the framework to situations where the relevant meta-information is not available for testing. We believe exploring more powerful variants of dispatching is an interesting avenue for future work, as is pretraining routing models on language model tasks using large corpora. It is our hope that these lessons extend to other richly compositional, context-sensitive language understanding tasks. We thank George Supaniratisai, Arun Chaganty, Kenny Xu and Abi See for valuable discussions, and the anonymous reviewers for their useful suggestions. Clemens Rosenbaum was a recipient of an IBM PhD Fellowship while working on this publication. We acknowledge the Office of the Vice Provost for Undergraduate Education at Stanford for the summer internships for Atticus Geiger, Olivia Li and Sandhini Agarwal. This research is based in part upon work supported by the Stanford Data Science Initiative, by the NSF under Grant No. BCS-1456077, by the NSF Award IIS-1514268, and by the Air Force Research Laboratory and DARPA under agreement number FA8750-18-2-0126. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory and DARPA or the U.S. Government.
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