Recent advances in neural network modeling have lead to impressive results in fields as diverse as object, face and scene recognition (LeCun, Bengio & Hinton, 2015), reasoning (Silver et al, 2016), speech perception (Graves, Mohamed & Hinton. 2013), machine translation (Wu et al), playing computer games (Antonoglou et al, 2015) and producing art (Gatys, Ecker & Bethge. 2015). These successes have relied on a restricted set of tools (e.g., the back propagation learning algorithm or the convolutional network architecture), and principles (all learning and computations take place in links between units), and are consistent with the claim that that all forms of cognition rely on a small set of general mechanisms in which minimal innate structure needs to be included. We will call this class of connectionist models currently popular in computer science the "conventional connectionist framework"111An earlier generation of models that adopted this approach has been characterized as "eliminative connectionism" (Pinker & Prince, 1988) or “non-symbolic connectionism” (Holyoak & Hummel, 2000).. However there are still fundamental disagreements concerning the limitations of this framework, with some researchers claiming that it cannot account for a range of core cognitive capacities (e.g., Bowers, 2017; Marcus, 2018).
One of the main criticisms of the conventional connectionist approach is that it lacks the ability to represent symbolic structures and is thus unsuitable for modeling tasks requiring symbolic operations (e.g., Fodor & Pylyshyn, 1988). It is important to emphasize that the critics of this approach are not claiming that connectionist systems are in principle unable to account for symbolic processing, but rather, that models need to be augmented in order to explicitly implement symbolic computation (Holyoak & Hummel, 2000). This can involve introducing new computational mechanisms above and beyond the modification of weights between units, such the synchrony of units firing (Hummel & Biederman, 1992) or the reliance of delay lines (Davis, 2010), introducing new inductive biases specifically designed to support symbolic computation, such as graph based structures (Battaglia et al., 2018), hybrid symbolic-connectionist units (Petrov & Kokinov, 2001) or introducing new dedicated circuits for the sake of symbolic computation (Kriete, Noelle, Cohen, & O’Reilly, 2013; van der Velde, & de Kamps, 2006). We will call models that adopt some or all of these solutions the “symbolic connectionist approach”. Central to the symbolic approach is the claim that conventional connectionist models will fail on symbolic reasoning tasks that humans can perform.
Here we show that conventional connectionism systems can support at least some forms of symbolic processing when trained to output symbolic structures. We achieve this by introducing the vector approach to representing symbols (VARS) that encodes complex symbolic structures of varying complexity as a static numeric vector at the output layer. We show that VARS representations can be learned and that this enables conventional neural networks to achieve combinatorial generalization, a core capacity of symbolic processing. It is important to emphasize that we do not take our findings to rule out symbolic connectionist architectures - there may well be functional and biological pressures that lead the brain to adopt special mechanisms devoted to symbolic computation. But our findings do undermine one of the motivations for this approach - that dedicated mechanisms are necessary to support combinatorial generalization.
The structure of the paper is as follows. We start by briefly reviewing the the limited successes of conventional networks in modeling tasks that require combinatorial generalization. We then describe VARS and demonstrate how it can be used to represent symbolic structures in connectionist terms and report two simulation studies showing that neural network models trained to output VARS are able to support an impressive degree of combinatorial generalization in short term memory and visual reasoning tasks. We argue that our approach does a better job than others existing symbolic and non-symbolic models, and highlights the importance of training conventional neural networks on tasks requiring the explicit representation of symbols.
2 Review of previous studies assessing combinatorial generalization in conventional connectionist architectures.
Fodor and Pylyshyn (1988) provided an early seminal criticism of the conventional connectionist framework. They argued that a wide range of cognitive capacities rest on the fact that the mind is compositional, with a small set of context-independent elements (e.g., words, units of meaning) used to productively compose more complex representations in limitless ways. On their view, conventional connectionist models that fail to build in mechanisms to explicitly code for the compositional structure of cognition are doomed to fail in combinatorial generalization tasks in which networks are required to produce novel outputs based on novel combinations of familiar symbols. For example, after training a network on the symbols John’, ‘Mary’, ‘Loves’, and the relation ‘Loves (John, Mary)’ a conventional network would not be able to output the relation ‘Loves (Mary, John)’ in response to any query. Combinatorial generalization is at the heart of what Fodor and Pylyshyn call the systematicity and productivity of thought.
In subsequent debates surrounding this issue, a number of authors highlighted the generalization capacities of conventional connectionist models (e.g., McClelland et al., 2010) and others have highlighted their limitations (Marcus, 1998). But the conclusions one should draw with regards to Fodor and Pylyshyn’s critique are far from obvious for a number of reasons. In some cases the apparent success of a model is not relevant because the model was not actually tested in a condition that required combinatorial generalization (Botvinick & Plaut, 2006; O’Reilly, 2001). In other cases, the failure of a model is taken as a virtue as it is thought to mirror limited human performance (e.g., Thomas & McClelland, 2008). But the more basic problem in reaching any strong conclusion is that the failure of a given conventional connectionist model does not provide a demonstration that all such models will fail. Indeed, many of the early failures of conventional connectionist models that have been used to motivate symbolic models were carried out prior to the development of the more versatile modeling tools and much larger datasets used today. What is most relevant, then, is the success of current conventional networks on combinatorial generalization tasks that humans can straightforwardly perform. If these models succeed, then symbolic models cannot be motivated on the basis of computational necessity.
2.1 Combinatorial generalization in sequence learning tasks
An example of a failure to support combinatorial generalization in an earlier generation of connectionist models was reported by Bowers et al. (2009ab) in the context of modelling short-term memory. Botvinick and Plaut (2006a) had developed simple recurrent model of immediate serial recall that could correctly repeated a sequence of 6 letters approximately 50% of the time (a level of performance that matches human performance). In addition to accounting for a range of empirical phenomena, the authors had emphasized is that their model could support widespread generalization in that it could recall sequences of letters it had never been trained on. However, the model was only tested on a limited form of generalization in which each letter was trained in each position within a list. When Bowers et al. (2009a,b) excluded specific letters in specific positions during training (e.g., the letter A was trained in all positions apart from position 1) and then included them in that position at test, the model did poorly, highlighting the model’s failure in combinatorial generalization. Similar limitations in related networks were observed by other authors (van der Velde, van der Voort van der Kleij, & de Kamps, 2004; Kriete et al. , 2013).
Do these finding pose a challenge to the conventional connectionist approach to explaining human cognition? Botvinick and Plaut (2009b) defended this approach by arguing that the restricted generalization was a strength based on their claim that humans would also fail under similar training conditions. Alternatively, the conventional approach might be supported by noting that Bowers et al. (2009ab) only observed limited performance with a simple recurrent network. More recent and powerful recurrent networks that include long short term memory (LSTM) circuits (Hochreiter & Schmidhuber, 1997) might well overcome these. However, both of these lines of defence are difficult to maintain given similar findings have been reported by Lake & Baroni (2018) using state-of-the-art recurrent networks trained on tasks that humans can transparently perform. They trained LSTM models to translate a series of commands to a series of actions when the commands were composed of actions (e.g., RUN, WALK) and modifiers of the actions (LEFT, TWICE). The model was unable to perform the correct series of actions if the model had not been trained an all the relevant combinations of actions and modifiers. For instance, if the model had never been trained on LEFT-RUN it could not perform the appropriate action despite being trained on LEFT and RUN in other combinations (e.g., LEFT-WALK, TWICE-RUN).
Still, there are some successes of generalization in sequence learning tasks that appear to require some degree of combinatorial generalization. For example, Gulordava et al. (2018) trained conventional RNNs on to predict long-distance number agreement in various constructions in four different languages (e.g., predict the verb in: “The girl the boys like: IS or ARE?). The model was trained on a corpera of text in each language, and critically, succeeded at near human levels not only when tested on sentences composed of meaningful sentences (where predictions might be based on learned semantic or distributional/frequency-based information rather than abstract syntactic knowledge), but also on nonsense sentences that are grammatical but completely meaningless (motivated by the classic sentence by Chomsky: “Colorless green ideas sleep furiously”). The authors took these findings provide tentative support for the claim that RNNs can construct some abstract grammatical representations. Nevertheless, when more challenging forms of combinatorial generalization are required, current state-of-the art conventional connectionist systems continue to struggle, as detailed next.
2.2 Recent explorations of generalization using conventional neural networks in the domain of visual relational reasoning
Barrett et al. (2018) assessed the capacity of various networks to perform abstract reasoning problems analogous to the Raven-style Progressive Matrices, a well-known human IQ test. In this task an panel of images are presented that vary according to some rule such as “progression” (e.g., in a panel of 3 x 3 images in which there is an increasing number of items per image along the first two columns), and the model is trained to select the image that satisfies this rule in order to complete the third column of images (select the target image that has more items). The authors assessed various forms of generalization, including combinatorial generalization (e.g., puzzles in which the progression relation was only encountered when applied to the colour of lines and then tested tested when the progression was applied to the size of shapes). Several state-of-the-art neural network models were tested and they all performed poorly in the generalization conditions. The authors were able to improve performance somewhat by adding a “Relation Network” module specifically designed to improve relational reasoning, and more relevant to the current paper, further still by augmenting the training procedure so that the model outputted “meta-targets” that specified the relevant dimensions for correctly responding. That is, the model was trained not only to select the correct image but also the reason why the image was the correct answer. Nevertheless, the modified model with the augmentented training still performed “strikingly poorly” in the conditions that most relied on combinatorial generalization.
In a closely related paper, Hill et al. (2019) tested the ability of conventional neural networks to perform analogical reasoning on a set of visual problems. In this case, the model was presented with a “source” set of three images that shared a given relation (e.g., the number of items in each image increased by one) and a “target” set of two images along with a set of images, only one of which shared the same underlying relation (e.g., progression). The task of the model was to select the correct image. Again, the models were tested across a range of conditions, including conditions that required combinatorial generalization, such as (again) applying a familiar relation to new domains. The authors used a conventional convolutional neural network that provided input to a recurrent layer without any special mechanisms for relational reasoning and found that the type of training had a significant impact on the model’s performance. When trained in a standard manner in which the model was trained to discriminate the target from foil images that different from one another in various ways the model performed poorly in the combinatorial generalization condition. The important finding, however, was that the model did better in some (but not all) conditions that required combinatorial generation when the training foils were carefully selected so that the model was forced to learn to encode the relevant relations. The fact that performance continued to be poor in conditions requiring extrapolation highlights how difficult generalization outside the training space can be, but at the same time, the improved performance with carefully crafted training foils suggests that conventional connectionist models can be more successful in such tasks than critics often assume. The benefits of manipulating the pressure to learn to encode relations for the the sake of combinatorial generalization (e.g., Barrett et al., 2018; Hill et al., 2019) provides the motivation of our approach which we now introduce.
3 The vector approach to representing symbols
The goal of the current paper is to further explore whether out-of-the-box neural networks can solve the combinatorial generalization problem if they are pushed to learn to explicitly represent symbols. To this end, we need a way to represent symbolic structures as numeric vectors that can be used at the output of standard connectionist models. There have already been several proposals for constructing such representations, including tensor products (Smolensky, 1990) and holographically reduced representations (Plate, 1995) that are discussed (and criticized) by Hummel (2011). However, there have been no demonstrations so far that conventional neural networks can learn such representations in order to achieve combinatorial generation (but see Schlag & Schmidhuber, 2018, for a trainable recurrent neural network with a dedicated mechanism for processing tensor products). We propose a novel way of representing symbolic structures of varying complexity using numeric vectors, which we believe is more suitable for training conventional neural networks on symbolic tasks - the vector approach to representing symbols (Figure 1). Our choice to use VARS doesn’t rule out other approaches - a systematic evaluation of the alternative ways to represent symbols in connectionist terms is outside the scope of the current paper.
The main assumptions of VARS are twofold. First, we assume that the meaning of symbols (i.e. the representation of an item independent of its relation to other items) can be encoded at multiple spatial locations within a VARS vector (in either localist or distributed manner). We will refer to these locations as representational slots. Importantly, the contents of each representational slot can be addressed by assigning a distinct token to it. The idea of using tokens to address representations is analogous to the indirection principle in the model by Kriete et al (2013) and the notion of binding pools (Bowman & Wyble, 2007, Swan & Wyble, 2014). For example, if the representation of “dog” is activated in a slot addressed by token ti and the property “brown” is activated in a slot referred by token tj, then the fact that the dog is brown can be expressed as tj(ti). The allocation of symbols per slots is arbitrary which means that a symbol can be represented at any slot without affecting its interpretation (Figure 1, example A and C). The second main principle of VARS is that binding information is represented explicitly and separately from the representation of the meaning of symbols which allows to bind the arguments of predicate symbol to any other symbol, thus ensuring role-filler independence. The order of argument bindings is represented by having separate binding spaces (A1, A2, etc) - see Examples A and B in Figure 1. The complexity of symbolic structures that can be encoded using VARS is limited by two parameters: the number of addressable representational slots nslots and the maximum arity of the symbols maxarity.
4.1 Simulation 1
In this simulation we replicate a combinatorial generalization task in the context of short-term memory as developed by Kriete et al. (2013). The model was given a sequence of role-fillers pairs and then cued to recall the filler that was was associated with a given role (e.g., after encoding e sequence DOG-SUBJECT, EAT-VERB, STEAK-PATIENT the model was cued to recall the filler EAT when probed with the role VERB). In the combinatorial generalization condition, one of the fillers was never paired with a specific role during training. The authors reported that a simple recurrent network failed on this task whereas a network with an architecture specifically designed to support combinatorial generalization in this task was successful.
In order to test whether a recurrent neural network with a conventional architecture can also achieve combinatorial generalization in this task we contrasted the performance of two models (Figure 2
). Both of them included the same conventional long short term memory architecture and were trained to solve the same task, but only one was also trained to output a VARS representation of the structural information provided during the encoding phase (Figure 3). In this way, the model had to solve two task in parallel - the main task which required to output the filler corresponding to the requested role and the VARS representation of the three encoded symbols and their relationship. Performance in combinatorial generalization was measured in both tasks: in the main task this was simply checking whether the correct filler was at the output and in the VARS task we checked whether the slots have been filled and bound correctly. In order to train the model on two tasks, we defined the loss function as a sum of the error on the main task and the VARS task. More details about the simulation are provided in appendixA.
The results of the simulation clearly show that a neural network model without dedicated mechanisms for symbolic processing is much better at combinatorial generalization when pressure to represent knowledge symbolically is enforced (Table 1). Importantly, the model trained with VARS outputs not only managed to output correct VARS representations of untrained role-filler binding over 90% of the time, but performed over twice as well on the main task (74%) compared to the network without VARS (30%). Performance of the model model on the main task was comparable to the Kriete et al. (2013) model that included special circuit to support symbolic computations. This finding suggests that training a neural network model to explicitly output symbols qualitatively changes the nature of its internal representations, allowing it to solve problems it otherwise fails on.
|Combinatorial generalization accuracy (SD)|
|Model||Main Task||VARS task|
|Simulation 1||LSTM||0.30 (0.15)||n/a|
|LSTM + VARS||0.74 (0.18)||0.93 (0.10)|
|Simulation 2||Custom CNN||0.29 (0.05)||n/a|
|VGG 16||0.24 (0.05)||n/a|
|Custom CNN + VARS, no bindings||0.34 (0.11)||1.00 (0.00)|
|Custom CNN + VARS||0.99 (0.01)||0.99 (0.01)|
Combinatorial generalization mean accuracy rates. In both simulations, the models achieved combinatorial generalization only when trained to explicitly represent symbolic structures. The standard deviation is shown in parentheses.
4.2 Simulation 2
The goal of our second simulation is to confirm and further extend our finding that conventional neural architectures can achieve combinatorial generalization when pressed to learn symbolic representations. Here we assessed whether training a feed-forward convolutional neural network on VARS representations improves combinatorial generalization in a visual reasoning task.
In each trial, the model was presented with three objects and had to choose which one of them was “the odd man out”. Each object had three features: position (top, middle or bottom), shape (one out of six) and color (one out of six). In each triplet of objects, exactly two of the objects had either the same color or the same shape, but not both (Figure 4). The task of the model was to output the position of the “odd” object, i.e. the one which shared neither shape nor color with the other two. In order to test combinatorial generalization in this task, we excluded all examples in which the “odd” object was green from the training set and tested the model on these examples. In other words, during training the model never had to report the position of a green object. Green objects did appeared in the training set, but never in the “odd” role.
In order to assess the ability of neural networks to solve the ‘odd man out’ problem with or without the pressure to represent knowledge symbolically we constructed a custom convolution neural network model displayed in Figure 5
. To make sure that a failure in this task can not be attributed to the details of our custom CNN architecture, we also tested VGG16 (Simonyan, & Zisserman, 2015) - a state-of-the-art model of visual object recognition, which has been pretrained on the ImageNet dataset. The pressure to learn symbolic representation was implemented by making the custom CNN model output VARS parallel to the main task (Figure 6). Just as in Simulation 1, the loss function the of model trained on VARS was a sum of the error on the main task and the VARS task (more details about the training procedure are available in appendixB) The VARS representation contained the following symbols: X, Y, Z, different-from(X, (Y, Z)), where X, Y, Z belonged to the set (‘top’, ‘middle’ and ‘bottom’). In other words, the VARS output represented information such as “the middle object is different from the top and from the bottom one”. We also trained the custom CNN model on VARS representations without binding information (i.e. without the ‘different-from’ symbol) in order to make sure that binding is essential for combinatorial generalization. The allocation of symbols to representational slots was determined by a random 222The results of the simulations reported in this paper do not depend on whether the sequence of tokens is random or fixed. However, the arbitrary allocation of symbols to slots is a general assumption of VARS which may be decisive in other simulations. permutation of tokens which was fed to the fully connected layers of the network (Figure 6).
The results of the simulation clearly show that the pressure to learn symbolic representations enables it to solve the combinatorial generalization problem (Table 1). The inability of the model using VARS targets with no binding to account for combinatorial generalization suggests that symbolic structure is indeed what drives the model improvement in generalization and not some idiosyncratic effect of forcing the network to represent the objects at different slots in the VARS output. On the other hand, the failure of the pre-trained VGG 16 model suggests that the difficulty of combinatorial generalization in this task can’t be attributed to the relatively limited visual experience the custom CNN model was exposed to.
The two simulations above show that combinatorial generalization can be greatly improved in conventional neural networks trained on simple short-term memory and visual reasoning tasks. In order to achieve this we introduced the vector approach to representing symbols (VARS) that encodes arbitrarily complex symbolic structures as a static numeric vector in the output layer. The output VARS representations lead the models to learn internal representations that could better support combinatorial generalization, suggesting that one of the hallmarks of symbolic computation can be performed without adding new special purpose mechanisms or processes that are often claimed to be necessary.
The current findings are consistent with some recent work that also observed improved combinatorial generalization in conventional connectionist models trained in specific ways designed to improve symbolic computations. This includes training on carefully tailored training sets that force networks to induce relational representations in order to succeed (Hill et al., 2019), or including output codes that explicitly code for the relevant dimensions of the input patterns (Barrett et al., 2018). The later finding is similar to our own, although we believe using VARS is more promising as it allows to train the network on a broader variety of tasks requiring combinatorial generalization, as well as other forms of symbolic processing.
It is important to be clear what we take our contribution to be. We are not claiming that VARS are necessarily the best way to force conventional networks to learn internal representations capable of supporting combinatorial generalization. There may be other approaches that are as good or better in achieving this goal. But we are claiming that our findings provide the best evidence to date that conventional neural networks can learn to support combinatorial generalization, and this challenges one of the main motivations for introducing special mechanisms and processes in symbolic networks.
We are also not claiming that our findings rule out symbolic theories of the mind. Despite our findings and other recent research (Barrett et al., 2018; Gulordava et al., 2018; Hill et al., 2019), it is still far from clear whether mainstream connectionism can address all the concerns that Fodor and Pylyshyn (1988) raised so long ago. We look forward to more research pushing the boundaries of what neural networks can do without implementing dedicated mechanisms for symbolic processing. And even if such models are shown to support human-level generalization, this alone does not rule out symbolic models. But then, the debate will not be about whether conventional neural networks are capable of symbolic computation, but about what kind of models provide the best account of human learning and performance.
Ivan Vankov has been supported by the European Office for Aerospace Research and Development (grant FA9550-15-1-0510). Jeffrey Bower is grateful for support from the ERC Advanced Grant 741134 (M and M).
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Appendix A Techbical details of Simulation 1
The training set was constructed by using a vocabulary of 3 role and 10 filler items. Roles and fillers were encoded locally. All three roles were used within each trial. The order of roles was also varied. Thus, the total number of possible role-filler bindings to encode was 6000. In order to test combinatorial generalization, we excluded all examples in which a particular filler was assigned to a particular role (e. g. filler 1 to role 1) from training set, resulting in a 5400 training examples. The set consisted of the remaining 600 trials which contained the untrained role-filler binding.
The model architecture is shown in Figure 2
of the main paper. The input unit had 14 units when trained without VARS (10 fillers + 3 roles + 1 unit encoding the test phase) and 17 units when trained on VARS (3 additional additional units indicating the representational slot the current symbol has to be assigned to, see Figure 3). The recurrent layer contained standard LSTM units (Hochreiter & Schmidhuber, 1997), implemented using the Keras deep learning library (Chollet, 2015). No bias was used for the recurrent layer and the activation function was hyperbolic tangent. A fully connected layer with a bias connected the recurrent layer to the output of the model. The activation function of this layer was also hyperbolic tangent. The output of the model was split in two parts - 10 softmax units representing the response to the main task and 48 (3 * 10 for representing the symbols and two binding 3x3 matrices) sigmoid units for the VARS task.
The model without VARS was trained to minimize the cross entropy of the difference between the output and the target of the main task. When VARS was used, the overall error was the sum of the cross entropy of the main task and the cross entropy of the VARS task. The training proceeded in batches of 50 examples and was organized in epochs of 5000 batches. The optimization algorithm was Adam (Kingma & Ba, 2015) with a learning rate of 0.001. Both models managed to perfectly fit the examples from the training set.
The performance of the model was assessed in two ways. For the main task, we checked whether the most active unit of the output was the correct one (i.e. the active unit in the target pattern). For the VARS task, we calculated whether the positions of the n most active units in the output match the positions of the active units of the targets, where n is the number of active target units (in this simulation n was always equal to 5 - three target units encoding the symbols in the representational slots and 2 units used for binding).
The model without VARS was trained until the accuracy of the main task in the test set stopped improving for 10 epochs or until the model was trained for 50 epochs, whichever came first. The stopping criterion for the model with VARS was similar, but we used performance on the VARS task in the test set to track training progress. Each simulation was repeated 100 times. The average number of training epochs was 17 (sd = 7.1) and 31 (sd = 12.70)in the conditions without and with VARS, respectively. We also replicated 100 times the simulation with the model without VARS allowing it to run for the maximum of 50 epochs. The results were worst that the ones reported in the main paper, indicating that the amount of training is not responsible for the effect of VARS on combinatorial generalization (mean accuracy 0.27, standard deviation 0.12).
Appendix B Techbical details of Simulation 2
The training set was constructed by generating triplets of object of varying shape and color, such that exactly two of the objects had the same shape or the same color (but not both). Three distinct shapes and colors were used. The dimensions of the objects were 12x12 pixels and they were arranged one under the other in the centre of the image (see Figure 4 in the main text). The size of the images containing the objects was 48x48 pixels. The training set excluded all examples, in which the odd object (the one which was shared neither the same color or the same shape with another one) was green, which resulted in 18000 training examples and 3600 test examples.
The model architecture is shown in Figure 5
of the main paper. The dimension of the image input was 48x48x3. The custom CNN model had four convolutional layers, each having a kernel of size 3x3, rectified linear activation function and a stride of 1. The convolutional layers were followed by 2x2 max pooling transformations. At the end of the hiercharchy, there were two fully connected hidden layers with rectified linear activation functions. The output of the model consisted of three softmax units used to represent to response of the main task (i.e. determining the position of the odd object) and additional 48 sigmoid units when the model was trained to output a VARS representation parallel to the main task. When trained on VARS there an additional input vector of 4 units used to represent the allocation of symbols per slots (Figure 6). In each trial, the additional input vector was a random permutation of the numbers from 1 to 4. We also trained a second model without VARS, which was based on the VGG16 architecture (Simonyan & Zisserman, 2015). That model was already pre-trained on an object recognition task using the ImageNet dataset. We used the convolutional layers of the existing model and added two more fully connected layers on top of them, just like in the custom CNN model. Training this model affected only the fully connected layers, assuming that the network had already extracted relevant visual features from the much larger training ImageNet dataset.
The model without VARS was trained to minimize the cross entropy of the difference between the output and the target of the main task. When VARS was used, the overall error was the sum of the cross entropy of the main task and the cross entropy of the VARS task. The training proceeded in batches of 100 examples and was organized in epochs of 500 batches. The optimization algorithm was RMSprop (Tieleman & Hinton, 2012) with a learning rate of 0.0001. All models were trained to perfection on the training set.
The performance measures were the same as in Simulation 1. For the main task, we checked whether the most active unit of the output was the correct one (i.e. the active unit in the target pattern). All models were trained for 50 epochs and we reported accuracy at the end of training. The simulation was repeated 100 times for each condition (no VARS, VARS with binding, VARS without binding, VGG). We also recorded the maximal performance on the main combinatorial generalization task during training (i.e. at any epoch of training, rather than only at the last one) - the pattern of results was similar no different from the one reported in the main paper (mean accuracy of Custom CNN model: 0.37, VGG 16: 0.34, Custom CNN + VARS without binding: 0.42, Custom CNN + VARS: 0.99).
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