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# The Relational Bottleneck as an Inductive Bias
for Efficient Abstraction

Taylor W. Webb  University of California, Los Angeles  Correspondence to:
<EMAIL_ADDRESS>Steven M. Frankland  Dartmouth College  Awni Altabaa
Yale University  Kamesh Krishnamurthy  Princeton University  Declan Campbell
Princeton University  Jacob Russin  Brown University  Randall O’Reilly
University of California, Davis  John Lafferty  Yale University  Jonathan D.
Cohen  Princeton University

###### Abstract

A central challenge for cognitive science is to explain how abstract concepts
are acquired from limited experience. This effort has often been framed in
terms of a dichotomy between empiricist and nativist approaches, most recently
embodied by debates concerning deep neural networks and symbolic cognitive
models. Here, we highlight a recently emerging line of work that suggests a
novel reconciliation of these approaches, by exploiting an inductive bias that
we term the relational bottleneck. We review a family of models that employ
this approach to induce abstractions in a data-efficient manner, emphasizing
their potential as candidate models for the acquisition of abstract concepts
in the human mind and brain.

## Highlights

  * •

Human learners efficiently acquire abstract concepts from limited experience.
The effort to explain this capacity has fueled debate between proponents of
symbolic and connectionist approaches, and motivated proposals for hybrid
neuro-symbolic systems.

  * •

A recently emerging approach, that we term the ‘relational bottleneck’
principle, suggests a novel way to bridge the gap. We formulate this in
information theoretic terms, and review neural network architectures that
implement this principle, displaying rapid learning of relational patterns,
and systematic generalization of those patterns to novel inputs.

  * •

The approach may help to explain a diverse set of phenomena, ranging from
cognitive development to capacity limits in cognitive function. The approach
is also consistent with findings from cognitive neuroscience, and may offer a
useful general principle for designing more powerful artificial learning
systems.

## Modeling the efficient induction of abstractions

Human cognition is characterized by a remarkable ability to transcend the
specifics of limited experience to entertain highly general, abstract ideas.
Understanding how the mind and brain accomplish this has been a central
challenge throughout the history of cognitive science, and a major
preoccupation of philosophy before that [1, 2, 3, 4]. Of particular importance
is the central role played by relations, which enable human reasoners to
abstract away from the details of individual entities and identify higher-
order patterns across distinct domains [5, 6]. The capacity to think in terms
of relations is a major component underlying the human capacity for fluid
reasoning [7, 8], and a key factor distinguishing human intelligence from that
of other species [9].

Efforts to explain this capacity have often been framed in terms of a debate
between empiricism (see Glossary), according to which concepts are primarily
acquired from experience, and nativism, according to which certain core
concepts are innate. Cognitive scientists in the empiricist tradition have for
decades explored how the abstractions associated with human cognition might
emerge through experience in neural architectures using general-purpose
learning algorithms (often termed connectionism) [10, 11, 12, 13]. This
endeavor has recently taken on new relevance, as the success of large language
models has demonstrated that it is possible, in some cases, for a human-like
capacity for abstraction to emerge given sufficient scaling of both
architecture and training data [14, 15, 16, 17]. For instance, it has recently
been shown that large language models can solve various analogy problems at a
level equal to that of college students [18]. However, the ability of these
models to perform abstract tasks (e.g., analogy) depends on exposure to a much
larger training corpus than individual humans receive in an entire lifetime
[19, 20].

An alternative approach, often associated with the nativist tradition, holds
that human cognition arises from processes akin to symbolic programs. This
approach has a long tradition in both cognitive science and AI [21, 22, 23],
due to the fact that it offers a natural explanation of the flexibility of
human cognition: processes that operate over symbols can be sensitive to their
general structure, without respect to a particular symbol’s reference [24].
Recent efforts have demonstrated that this approach is capable of inducing
abstract concepts in a data-efficient manner, mirroring the efficiency of
human concept learning [25, 26, 27, 28, 29, 30, 31]. However, a potential
limitation of this approach is that it depends on the pre-specification of
symbolic primitives. Though it remains to be seen how far this approach can be
scaled, it has thus far proven challenging to identify a set of primitives
that are expressive enough to account for the breadth and richness of human
natural concepts. It also raises the question of how the symbolic primitives
arise: are these an innate feature of the human mind, or could they too emerge
in a data-efficient manner through learning?

In this review, we highlight a recently emerging approach that suggests a
novel reconciliation of these two traditions. The central feature of this
approach is an inductive bias that we refer to as the relational bottleneck: a
constraint that biases neural network models to focus on relations between
objects rather than the attributes of individual objects. This approach
satisfies two key desiderata for models of abstract concept acquisition.
First, the approach is capable of rapid acquisition of abstract relational
concepts. Second, the approach does not require access to a set of pre-
specified primitives. This latter feature distinguishes the approach from
other so-called neuro-symbolic approaches (see Box 2 for further discussion).
In the following sections, we first provide a general characterization of this
approach, drawing on concepts from information theory, and discuss a number of
recently proposed neural network architectures that implement the approach. We
then discuss the potential of the approach for modeling human cognition,
relating it to existing cognitive theories and considering potential
mechanisms through which it might be implemented in the brain.

## The relational bottleneck

Figure 1: The relational bottleneck. An inductive bias that prioritizes the
representation of relations (e.g., ‘same’ vs. ‘different’), and discourages
the representation of the features of individual objects (e.g., the shape or
color of the objects in the images above). The result is that downstream
processing is driven primarily, or even exclusively by patterns of relations,
and can therefore systematically generalize those patterns across distinct
instances (e.g., the common ABA pattern displayed on both left and right),
even for completely novel objects. The approach is illustrated here with
same/different relations, but other relations can also be accommodated. Note
that this example is intended only to illustrate the overall goal of the
relational bottleneck framework. Figure 2 depicts neural architectures that
implement the approach.

We define the relational bottleneck as any mechanism that restricts the flow
of information from perceptual to downstream reasoning systems to consist only
of relations (see Box 1 for a formal definition). For example, given inputs
representing individual visual objects, a relational bottleneck would
constrain the representations passed to downstream reasoning processes such
that they capture only the relations between these objects (e.g., whether the
objects have the same shape), rather than the individual features of the
objects (e.g., the individual shapes). Such a representation encourages
downstream processes to identify relational patterns, such as the identity
rule in Figure 1, in a manner that is abstracted away from the details of
specific instances of those patterns, and can therefore be systematically
generalized to novel inputs. In the following section, we highlight three
recently proposed neural network architectures that instantiate this approach
in different guises, illustrating how they utilize a relational bottleneck to
induce abstract concepts in a data-efficient manner.

#### Box 1: The relational bottleneck principle

Information bottleneck theory [32] provides a normative framework for
formalizing the notion of a relational bottleneck. Consider an information
processing system that receives an input signal $X$ and aims to predict a
target signal $Y$. $X$ is processed to generate a compressed representation
$Z=f(X)$ (the ‘bottleneck’), which is then used to predict $Y$. At the heart
of information bottleneck theory is the idea of ‘minimal-sufficiency’. $Z$ is
sufficient for predicting $Y$ if it contains all the information $X$ encodes
about $Y$. That is, $I(Z;Y)=I(X;Y)$, where $I(\cdot\,;\,\cdot)$ is the mutual
information. If $Z$ is sufficient, then we write $X\to Z\to Y$, meaning that
$Y$ is conditionally independent of $X$ given the compressed representation
$Z$. $Z$ is minimal-sufficient if it is sufficient for $Y$ and does not
contain any extraneous information about $X$ which is not relevant to
predicting $Y$. That is, $I(X;Z)\leq I(X;\tilde{Z})$ for any other sufficient
compressed representation $\tilde{Z}$.

Achieving maximum compression while retaining as much relevant information as
possible is a trade-off. It is captured by the information bottleneck
objective,

$\mathrm{minimize}\ \mathcal{L}(Z)=I(X;Z)-\beta I(Z;Y).$ (1)

This objective reflects the tension between compression – which favors
discarding information as captured by the first term – and the preservation of
relevant information in $Z$, captured by the second term. The parameter
$\beta$ controls this trade-off.

While this objective is well-defined when the joint distribution $(X,Y)$ is
known, obtaining a minimal-sufficient compressed representation from data is,
in general, very challenging for the high-dimensional signals that are often
of interest. However, it may be possible to implicitly enforce a desirable
information bottleneck for a large class of tasks through architectural
inductive biases.

In particular, we hypothesize that human cognition has been fundamentally
optimized for tasks that are relational in nature. We define a ‘relational
task’ as any task for which there exists a minimal-sufficient representation
$R$ that is purely relational. Suppose the input signal represents a set of
objects,

$X=\left(x_{1},\,\ldots,\,x_{N}\right).$ (2)

A relational signal is a signal of the form,

$R=\left\\{r(x_{i},x_{j})\right\\}_{i\neq
j}=\left\\{r(x_{1},x_{2}),\,r(x_{1},x_{3}),\,\ldots,\,r(x_{N-1},x_{N})\right\\},$
(3)

where $r(x_{i},x_{j})$ is a learned relation function that satisfies certain
key relational properties (e.g., transitivity). One type of operation that
satisfies the relevant properties is inner products of the form
$\langle\phi(x_{i}),\psi(x_{j})\rangle$. Let $\mathcal{R}$ be the class of all
possible relational representations of the input signal $X$. In a relational
task, there exists $R\in\mathcal{R}$ which is sufficient for predicting $Y$
(i.e., $X\to R\to Y$).

A relational bottleneck is any mechanism that restricts the space of all
possible compressed representations to be a subset of the relational signals
$\mathcal{R}$. This gives the model a smaller space of possible compressed
representations over which it must search. This space of compressed
representations $\mathcal{R}$ is guaranteed to contain a minimal-sufficient
representation for the task and excludes many representations that encode
extraneous information about $X$, promoting efficient learning of relational
abstractions.

## The relational bottleneck in neural architectures

(a)

(b)

(c)

Figure 2: Implementing the relational bottleneck. Three neural architectures
that implement the relational bottleneck. (a) Emergent Symbol Binding Network
(ESBN) [33]. (b) Compositional Relation Network (CoRelNet) [34]. (c)
Abstractor [35]. In all cases, high-dimensional inputs (e.g., images) are
processed by a neural encoder (e.g., a convolutional network), yielding a set
of object embeddings $\mathbf{O}$. These are projected to a set of keys
$\mathbf{K}$ and queries $\mathbf{Q}$, which are then compared yielding a
relation matrix $\mathbf{R}$, in which each entry is an inner product between
a query and key. Abstract values $\mathbf{V}$ are isolated from perceptual
inputs (the core feature of the relational bottleneck), and depend only on the
relations between them.

Figure 2 (Key Figure) depicts three neural architectures that implement the
relational bottleneck through the use of architectural inductive biases. Here,
we discuss how the distinct mechanisms employed by these models implement the
same underlying principle. In particular, a common aspect of all three
architectures is the use of inner products to represent relations, which
ensures that the resulting representations are genuinely relational. In each
case, we also contrast these architectures with closely related approaches
that do not incorporate a relational bottleneck, emphasizing how this key
architectural feature gives rise to the data-efficient induction of
abstractions.

### Emergent symbol binding

We first consider the Emergent Symbol Binding Network (ESBN) (Figure 2(a))
[33]. The ESBN is a deep neural network architecture, augmented by an external
memory, that was inspired by the notion of role-filler variable binding in
cognitive models of relational reasoning [36, 37, 38]. In those models,
relational reasoning is supported by the existence of separately represented
‘roles’, which capture information about abstract variables, and ‘fillers’,
which capture information about concrete entities to which those variables are
bound. Previous work has focused on how these roles and fillers can be
dynamically bound in neural circuits. However, the role and filler
representations themselves were typically pre-specified by the modeler,
leaving open the question of how these representations might emerge from
experience.

The ESBN adopts this key idea of separate role and filler representations, but
integrates them into a larger system that can be trained end-to-end, averting
the need to pre-specify those representations. The ESBN contains three major
components: 1) a feedforward encoding pathway (‘Encoder’ in Figure 2(a)),
which generates object embeddings from high-dimensional perceptual inputs, 2)
a recurrent controller (‘Controller’ in Figure 2(a)), which operates over
learned representations of abstract task-relevant variables, without direct
access to the object embeddings, and 3) an external memory system responsible
for binding and associating representations between these two pathways. The
ESBN processes perceptual observations sequentially. For each observation, a
pair of embeddings is added to memory, one from the perceptual pathway
(referred to as a key), and one from the control pathway (referred to as a
value)111Note that the use of the terms ‘key’ and ‘value’ here is reversed
relative to the original paper [33] in order to be more consistent with their
usage in describing the CoRelNet and Abstractor architectures.. To read from
this memory, the object embedding for the current observation (referred to as
a query) is compared to all of the keys in memory via an inner product,
yielding a set of scores (one for each key) that govern the retrieval of the
associated values in the abstract control pathway.

Importantly, in this retrieval operation, the control pathway does not gain
access to the content of the representations in the perceptual pathway.
Instead, the interaction is mediated only by the comparison of perceptual
representations with each other. The ESBN thus implements the relational
bottleneck as an architectural prior, separating the learning and use of
abstract representations by the controller from the embeddings of perceptual
information. Thanks to this design feature, the ESBN is capable of rapidly
learning relational patterns (such as the identity rules displayed in Figure
1), and generalizing them to out-of-distribution inputs (e.g., to previously
unseen shapes) [33]. Critically, it can be shown to use precisely the same
representation for a given role, irrespective of filler, thus exhibiting a
critical feature of abstract, symbolic processing [24]. In this sense, the
representations in the model’s control pathway can be viewed as a form of
learned ‘symbols’.

It is instructive to compare this model with similar approaches that do not
implement the relational bottleneck. The ESBN is part of a broader family of
neural network architectures that use content-addressable external memory – a
separate store of information with which a neural network can interact via
learnable read and write operations [39, 40, 41]. Notably, these read and
write operations typically rely on a similarity computation (based on inner
products). These have often been cast as simplified implementations of the
brain’s episodic memory system [42, 43]. Standard external memory
architectures do not typically isolate the control and perceptual pathways.
Instead, perceptual inputs are passed directly to a central controller, which
is then responsible for writing to and reading from a single, monolithic
memory. Though it is possible for a role-filler structure to emerge in these
systems given a sufficiently large amount of training data [44], they take
much longer to learn relational tasks (requiring approximately an order of
magnitude more training data), and do not display the same degree of
generalization [33]. Thus, although external memory plays an important role in
the ESBN architecture, the presence of external memory alone is insufficient
to implement a relational bottleneck. Rather, it is the isolation of the
perceptual and abstract processing components from one another that implements
the relational bottleneck. Furthermore, as we illustrate in the following
sections, it is possible to achieve this isolation without the use of external
memory.

### Relation matrices

An alternative approach to implementing the relational bottleneck is
illustrated by the Compositional Relation Network (CoRelNet) (Figure 2(b))
[34]. In that approach, a set of perceptual observations are first processed
by an encoder, yielding a sequence of object embeddings. A relation matrix is
then computed over all pairs of objects, in which each entry consists of the
inner product between a pair of object embeddings. Finally, this relation
matrix is passed to a downstream decoder network (the architecture of this
network can vary, e.g., using a multilayer perceptron or transformer). This
decoder is subject to a relational bottleneck, in that it only has access to
the relation matrix, and does not have direct access to the object embeddings.
As with the ESBN, this relational bottleneck enables CoRelNet to rapidly learn
and systematically generalize relational patterns.

CoRelNet can be viewed as a feedforward, parallelized implementation of the
sequential process (of encoding and similarity-based retrieval from external
memory) carried out by the ESBN. This results in performance benefits, as
CoRelNet does not suffer from the vanishing gradient problem that is a
challenge for recurrent neural networks used to implement such sequential
processing [45]. It also makes the key relational inductive bias underlying
the ESBN more explicit. The ESBN’s memory retrieval procedure, in which the
current observation is compared to the entries in memory, can be interpreted
as computing a single row of the relation matrix. In both architectures,
downstream processing is constrained so as to depend only on this relation
matrix, though the details of this dependency differ.

Here too, a useful contrast can be made with related architectures that do not
incorporate a relational bottleneck. In particular, architectures such as the
Relation Net [46] (see [47] for related approaches) explicitly perform a
comparison between each pair of inputs, leading to improved performance in
relational tasks. However, whereas CoRelNet represents pairwise relations
using inner products, the Relation Net utilizes generic neural network
components (e.g., multilayer perceptrons) that are learned in a task-dependent
manner. While this is in principle more flexible, it does not constrain the
network to learn representations that only capture relational information. As
a consequence, this architecture is susceptible to learning shortcuts
consistent with the training data (i.e., overfitting to perceptual details),
compromising its ability to reliably generalize learned relations to out-of-
distribution inputs [48, 33, 49]. This is in contrast to the inner product
operation employed by the ESBN and CoRelNet, which is inherently relational,
and therefore guarantees that downstream processing is based only on
relations.

### Relational attention

The recently proposed Abstractor architecture (Figure 2(c)) [35] illustrates
how the relational bottleneck can be implemented within the broader framework
of attention-based architectures (including the Transformer [50]). The
Abstractor is built on a novel attention operation termed relational cross-
attention. In this operation, a set of object embeddings (which may be
produced by an encoder given perceptual observations) is converted to form
keys and queries, using separate linear projections. A relation matrix is then
computed, in which each entry corresponds to the inner product between a query
and key. The relation matrix is used to attend over a set of learned values,
which reference objects but are independent of their attributes.

Relational cross-attention can be contrasted with the standard forms of
attention employed in Transformers: self-attention and cross-attention. In
self-attention, the same set of object embeddings are used to generate keys,
queries, and values. In cross-attention, object embeddings are used to
generate keys and values, and queries are generated by a separate decoder
network. In both cases, the values over which attention is performed are based
directly on the object embeddings, and the information contained in these
embeddings is therefore passed on for downstream processing (thus contravening
the relational bottleneck). By contrast, in relational cross-attention, keys
and queries are generated from object embeddings, but a separate set of
learned vectors are used as values. As in the ESBN, these values can be viewed
as learned ‘symbols’, in the sense that they are isolated from the perceptual
content of the objects with which they are associated.

This implementation of the relational bottleneck yields the same benefits
observed in others: the Abstractor learns relational patterns faster than the
Transformer, and displays better out-of-distribution generalization of those
patterns [35]222Although, as noted in the introduction, there is evidence that
the standard Transformer architecture can learn to perform relational tasks
(e.g., in the case of large language models [18]), this requires considerable
amounts of data. Experiments comparing the standard Transformer architecture
with the various implementations of the relational bottleneck highlighted
above suggest that the latter may be substantially more data efficient, though
this remains to be demonstrated at scale, and for the full range of tasks over
which Transformers have been applied.. The Abstractor also has a few
advantages relative to existing implementations of the ESBN and CoRelNet.
Because the relation matrix is computed using separate key and query
projections, the Abstractor is capable of representing asymmetric relations
(e.g., can capture the difference in meanings between ‘A is greater than B’
and ‘B is greater than A’). In addition, multi-headed relational cross-
attention enables the Abstractor to model multi-dimensional relations. As
proposed, ESBN and CoRelNet are limited to relations along a single feature
dimension only. Finally, similar to Transformers, the Abstractor is a
generative architecture, whereas the ESBN and CoRelNet are purely
discriminative333It should be noted that these are not fundamental limitations
of the ESBN and CoRelNet architectures. For instance, both of these
architectures can be modified so as to employ separate key and query
embeddings, enabling asymmetric relations to be modeled [34]. Furthermore, an
alternative implementation of the ESBN has been proposed that can perform
generative tasks [51].. This enables the Abstractor to perform a broader range
of tasks, including the sequence-to-sequence tasks that are common in natural
language processing.

As the examples we have considered illustrate, the relational bottleneck can
be implemented in a diverse range of architectures, each with their own
strengths and weaknesses. In each case, the inclusion of a relational
bottleneck enables rapid learning of relations without the need for pre-
specified relational primitives. In the remainder of the review, we discuss
the implications of this approach for models of cognition, and consider how
the relational bottleneck may relate to the architecture of the human brain.

#### Box 2: Neuro-symbolic modeling approaches

Many approaches have been proposed for hybrid systems that combine aspects of
both neural and symbolic computing. Early work in this area focused on
incorporating a capacity for variable-binding – a key property of symbolic
systems – into connectionist systems. Notable examples of this approach
include binding-by-synchrony [37], tensor product variable-binding [36], and
BoltzCONS [52]. A number of vector symbolic architectures have since been
proposed that build on the tensor product operation, but enable more elaborate
symbolic structures to be embedded in a vector space of fixed dimensionality
[53, 54, 55, 56]. These approaches have all generally relied on the use of
pre-specified symbolic primitives.

More recently, hybrid systems have been developed that combine deep learning
with symbolic programs. In this approach, deep learning components are
typically employed to translate raw perceptual inputs, such as images or
natural language, into symbolic representations, which can then be processed
by traditional symbolic algorithms [57, 58, 59, 60, 61]. This approach is
complemented by recent neuro-symbolic approaches to probabilistic program
induction, in which symbolic primitives are pre-specified (following earlier
symbolic-connectionist modeling efforts), and then deep learning is used to
assemble these primitives into programs [28].

An alternative approach (which might also be viewed as neuro-symbolic in some
sense) involves the integration of key features of symbolic computing within
the framework of end-to-end trainable neural systems. Examples of this
approach include neural production systems [62], graph neural networks [47],
discrete-valued neural networks [63], and efforts to incorporate tensor
product representations into end-to-end systems [64, 65]. The relational
bottleneck falls into this broad category, as it incorporates key elements of
symbolic computing – variable-binding and relational representations – into
fully differentiable neural systems that can be trained end-to-end without the
need for pre-specified symbolic primitives. Relative to these other
approaches, the primary innovation of the relational bottleneck framework is
the emphasis on architectural components that promote the development of
genuinely relational representations.

## The relational bottleneck in the mind and brain

### Modeling the development of counting: a case study in learning
abstractions

A core requirement for cognitive models of abstract concept acquisition is to
account for the timecourse of acquisition during human development. A useful
case study can be found in the early childhood process of learning to count
[66, 67, 68]. Children typically learn to recite the count sequence (i.e.
‘one, two, three,…’ etc.) relatively early, but their ability to use this
knowledge to count objects then proceeds in distinct stages. Each stage is
characterized by the ability to reliably count sets up to a certain size
(i.e., first acquiring the ability to reliably count only single objects, then
to count two objects, and so on). Around the time that children learn to count
sets of five, an inductive transition occurs, in which children rapidly learn
to counts sets of increasing size. It has been proposed that this transition
corresponds to the acquisition of the ‘cardinality principle’ – the
understanding that the last word used when counting corresponds to the number
of items in a set [68].

A recent study investigated the development of counting in deep neural network
architectures [69]. These included the ESBN, the Transformer, and long short-
term memory (LSTM) [70] (a type of recurrent neural network). Each
architecture displayed a distinct developmental timecourse. The Transformer
displayed a roughly linear timecourse, taking approximately the same amount of
time to master each number. The LSTM displayed an exponentially increasing
timecourse, taking more time to learn each new number. Only the ESBN displayed
a human-like inductive transition, gradually learning to count each number
from one to four, and then rapidly acquiring the ability to count higher after
learning to count to five. This was due to the ability of the ESBN to learn a
procedure over the representations in its control pathway that was abstracted
away from the specific numbers in the count sequence (represented in the
model’s perceptual pathway), allowing it to rapidly and systematically
generalize between numbers. This case study illustrates how the relational
bottleneck can emulate a human-like developmental trajectory for learning
abstract concepts.

### Cognitive models of analogical reasoning

The relational bottleneck also has some important parallels with cognitive
models of analogical reasoning. In particular, both approaches afford an
especially important role to patterns of similarity. In traditional symbolic
models, this typically takes the form of literal identicality between symbols
[71]. However, more recent models employ a graded measure of similarity that
can be easily applied to distributed representations, such as those derived
from deep learning systems (e.g., word or image embeddings) [72, 73, 74]. In
those models, a similarity matrix is computed, which is then used to identify
a mapping from the elements in one situation to the elements in another
situation. This use of similarity matrices has a close connection to the
relation matrices (both of which are based on inner products) employed
explicitly in architectures such as CoRelNet and the Abstractor, and
implicitly in the retrieval operation of the ESBN. This suggests the
intriguing possibility that these architectures, aided by the inductive bias
of the relational bottleneck, may learn to implement a procedure similar to
the mapping algorithm proposed by cognitive models of analogy.

### Capacity limits and the curse of compositionality

The relational bottleneck principle may also help to explain the limited
capacity of some cognitive processes (e.g., working memory) [75]. Recent work
has demonstrated that human-like capacity limits naturally emerge in an
architecture that implements the relational bottleneck [76]. In that
architecture, two separate representational pools (each representing distinct
feature spaces, e.g., color and location) interact via a dynamic variable-
binding mechanism (in that case, implemented using rapid Hebbian learning).
This architecture is conceptually similar to the ESBN, but is subject to
classical efficient coding constraints—that is, limits not only on the amount
of available data, but also time and memory available for optimizing a loss
function. This mechanism, which is intimately related to classic neural
network models of rapid learning and memory retrieval [77], enables the model
to flexibly construct compositional representations (e.g., representing a
visual scene by binding together spatial locations and visual features).
However, this flexibility comes at the cost of relying on compositional
representations that, by definition, are shared across many different,
potentially competing processes (an instance of the general relationship
between shared representations and cognitive capacity [78]). The model
quantitatively captures capacity limits observed in three distinct cognitive
domains: working memory [75], subitizing (the ability to rapidly identify the
number of items in a display) [79], and absolute judgment (the ability to
correctly label specific feature values such as pitch or loudness) [80].

### Brain mechanisms supporting the relational bottleneck

We close by considering how the relational bottleneck might relate to the
architecture of the human brain. A central element of this framework is the
presence of segregated systems for representing abstract vs. perceptual
information (i.e., abstract values vs. perceptual keys/queries in the ESBN or
Abstractor). A large body of findings from cognitive neuroscience suggests the
presence of distinct neocortical systems for representing abstract structure
(e.g., of space or events) vs. concrete entities (e.g., people or places),
located in the parietal and temporal cortices respectively [81, 82, 83, 84,
85]. This factorization has also been explored in a number of recent
computational models [86, 87, 88].

However, this segregation raises the question of how representations in these
distinct neocortical systems are flexibly bound together. Though many
proposals have been made for how the brain might solve this variable-binding
problems (see Box 2), one intriguing possibility involves the episodic memory
system [42]. A common view holds that episodic memory is supported by rapid
synaptic plasticity in the hippocampus, which complements slower statistical
learning in the neocortex [43, 89]. According to this view, episodes are
encoded in the hippocampus by the rapid binding of features that co-occur
within an episode, while the features themselves are represented in
neocortical systems. This same mechanism could in principle support an
architecture similar to the ESBN, by enabling rapid binding of abstract and
perceptual neocortical representations. This is in fact very similar to models
of cognitive map learning, according to which distinct representations of
structural vs. sensory information, corresponding to the medial vs. lateral
entorhinal cortices (often viewed as extensions of the parietal and temporal
neocortical systems referenced above), are bound together by rapidly formed
conjunctive representations in the hippocampus [90].

That said, the extent to which variable-binding relies on the hippocampus
remains an open question. Some lesion evidence suggests that hippocampal
damage does not lead to impairments of abstract reasoning [91] (see Box 3 for
further discussion). Other alternatives are that variable-binding may be
supported by other structures capable of rapid synaptic plasticity (e.g., the
cerebellum, which has been increasingly implicated in higher cognitive
functions [92, 93, 94]), or by other structures (such as the prefrontal
cortex) that use other mechanisms for binding (such as selective attention
[95] or working memory gating [96]). The latter possibilities are consistent
with findings that prefrontal damage often leads to severe deficits in
abstract reasoning tasks [97, 98], and prefrontal activity is frequently
implicated in neuroimaging studies of abstract reasoning [99, 100]. However,
this may also reflect the role of prefrontal cortex in representing abstract
structure (along with the parietal system described above), rather than the
binding of that structural information to concrete content. Of course, it is
also possible that variable-binding is supported by a collection of distinct
mechanisms, rather than a single mechanism alone. These are all important
questions for future work that we hope will be usefully guided by the
formalisms and computational models reviewed here.

#### Box 3: Episodic memory and the relational bottleneck

The proposal that episodic memory (EM) plays a crucial role in abstract
reasoning may seem to be at odds with conventional wisdom for several reasons.
First, the capacity for abstraction may be assumed to fall more naturally
within the province of semantic memory, which is generally assumed to encode
the abstract (e.g., statistical) structure of relationships among concepts
[43, 89]. The proposal considered here is not that EM represents such
structure, but rather that it is used to apply structural information (e.g.,
roles) to specific instances (e.g., fillers) by serving as a binding
mechanism.

Another concern might be that reasoning processes are generally associated
with working memory (WM) function [101, 95] rather than EM. However, a growing
body of recent findings have suggested the potential involvement of EM in
tasks that are traditionally associated with WM [102, 103, 104]. Furthermore,
the functional properties of EM are well suited to perform the variable-
binding operation that plays a critical role in the relational bottleneck
framework. In traditional accounts of EM, rapid hippocampal plasticity serves
to bind together the features of an episode, but this same mechanism is in
principle capable of binding together abstract and perceptual representations
(such as the key and value representations in the ESBN) in the service of an
ongoing reasoning process.

As noted in the text, there is some evidence that could be taken as evidence
against this account: lesion studies suggesting that hippocampal damage, which
leads to severe EM deficits, does not lead to comparably severe deficits in
abstract reasoning [91], whereas reasoning deficits often arise from damage to
prefrontal cortex [97, 98]. It is of course possible that both hippocampal and
prefrontal mechanisms contribute to variable-binding in the healthy brain, but
that prefrontal mechanisms alone can support variable-binding in the event of
hippocampal damage. However, an alternative possibility is that EM-like
processes – i.e., the rapid encoding of arbitrary but durable associations
subject to similarity-based retrieval – may be subserved by other brain
regions not traditionally associated with EM, such as the prefrontal cortex,
cerebellum, or other structures. From this perspective, the relational
bottleneck framework points to a number of intriguing directions for future
research concerning the nature of EM and its relationship to the capacity for
abstraction.

## Concluding remarks and future directions

The human mind has a remarkable ability to acquire abstract relational
concepts from relatively limited and concrete experience. Here, we have
proposed the relational bottleneck as a functional principle that may explain
how the human brain accomplishes such data-efficient abstraction, and
highlighted recently proposed computational models that implement this
principle. We have also considered how the principle relates to a range of
cognitive phenomena, and how it might be implemented by the mechanisms of the
human brain.

It should be noted that the framework reviewed here is not necessarily at odds
with the existence of certain forms of domain-specific innate knowledge. In
particular, a range of evidence from developmental psychology has suggested
that humans possess certain ‘core knowledge’ systems, such as an innate
capacity to represent objects [105, 106, 107]. These findings have motivated
the development of neuro-symbolic models endowed with these innate capacities
[108], although it is also possible that these findings may ultimately be
accounted for by the inclusion of additional inductive biases into
connectionist systems, such as mechanisms for object-centric visual processing
[109, 110, 111, 112] (which have also been combined with the relational
bottleneck [113]). Critically, however, it is important to emphasize that the
relational bottleneck is, in principle, orthogonal to questions about these
domain-specific capacities, and is focused instead on explaining the induction
of abstract, domain-general concepts and relations.

There are a number of important avenues for further developing the relational
bottleneck framework (see Outstanding Questions). Further work is needed to
integrate the relational bottleneck with a broader range of cognitive
processes relevant to abstraction, including attentional processes [114] and
semantic cognition [115]. Additionally, much work has suggested that human
reasoning is not purely relational, but instead depends on a mixture of
concrete and abstract influences [116, 117, 118, 119]. This suggests the
potential value of a more graded formulation that controls the amount of non-
relational information allowed to pass through the bottleneck. Finally, the
human capacity for abstraction surely depends not only on architectural biases
such as those that we have discussed here, but also on the rich educational
and cultural fabric that allows us to build on the abstractions developed by
others [120]. In future work, it will be important to explore the interaction
between education, culture and relational inductive biases.

## Outstanding Questions

  * •

Human reasoners often display so-called ‘content effects’, in which abstract
reasoning processes are influenced by the specific content under consideration
(and therefore are not purely abstract or relational). Can a more graded
version of the relational bottleneck capture these effects, while preserving a
capacity for relational abstraction?

  * •

How can other cognitive processes (perception, attention, memory, etc.) be
integrated with the relational bottleneck?

  * •

How is the relational bottleneck implemented in the brain? To what extent does
this rely on mechanisms responsible for episodic memory, attentional
mechanisms, and/or other mechanisms that remain to be identified? What role do
the hippocampus, prefrontal cortex, and/or other structures play in these
computations?

  * •

How do architectural biases toward relational processing interact with
cultural sources of abstraction (e.g., formal education)?

## Glossary

#### Connectionism

A modeling framework in cognitive science that emphasizes the emergence of
complex cognitive phenomena from the interaction of simple, neuron-like
elements organized into networks, in which connections are formed through
learning.

#### Empiricism

An epistemological view according to which knowledge is ultimately derived
from experience. Often contrasted with nativism.

#### Episodic memory

A form of memory in which arbitrary, but durable, associations can be rapidly
formed. Often thought to be implemented by hippocampal mechanisms for rapid
synaptic plasticity and similarity-based retrieval.

#### External memory

In the context of neural networks, an approach that combines these with
separate external stores of information, typically with learnable mechanisms
for writing to and reading from these stores, and in which retrieval is
usually similarity-based (i.e, ‘content-addressable’). Often used to implement
a form of episodic memory.

#### Inductive bias

An assumption made by a machine learning model about the distribution of the
data. In deep learning models, this often takes the form of architectural
features that bias learning toward certain (typically desirable) outcomes.
Genetically pre-configured aspects of brain structure can be viewed as a form
of inductive bias.

#### Out-of-distribution generalization

In machine learning, generalization to a distribution that differs from the
distribution observed during training.

#### Nativism

The view that certain concepts and mental capacities are innate rather than
learned from experience. Often contrasted with empiricism.

## Declaration of interests

The authors declare no competing interests.

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