Scaling can lead to compositional generalization
Authors: Florian Redhardt
Yassir Akram
Simon Schug
Venue: NeurIPS 2025 spotlight ()
Keywords: compositionality
compositional generalization
scale
systematicity
generalization
Primary area: deep_learning
TL;DR: Scaling neural networks leads to compositional generalization if the training distribution sufficiently covers the task space.

Abstract: Can neural networks systematically capture discrete, compositional task structure despite their continuous, distributed nature?
The impressive capabilities of large-scale neural networks suggest that the answer to this question is yes.
However, even for the most capable models, there are still frequent failure cases that raise doubts about their compositionality.
Here, we seek to understand what it takes for a standard neural network to generalize over tasks that share compositional structure.
We find that simply scaling data and model size leads to compositional generalization.
We show that this holds across different task encodings as long as the training distribution sufficiently covers the task space.
In line with this finding, we prove that standard multilayer perceptrons can approximate a general class of compositional task families to arbitrary precision using only a linear number of neurons with respect to the number of task modules.
Finally, we uncover that if networks successfully compositionally generalize, the constituents of a task can be linearly decoded from their hidden activations.
We show that this metric correlates with failures of text-to-image generation models to compose known concepts.

Scaling can lead to compositional generalization
Florian Redhardt'
Yassir Akram"
Simon Schug
ETH Zurich
ETH Zurich
Princeton University
Abstract
Can neural networks systematically capture discrete, compositional task structure despite their continuous, distributed nature? The impressive capabilities of large- scale neural networks suggest that the answer to this question is yes. However, even for the most capable models, there are still frequent failure cases that raise doubts about their compositionality. Here, we seek to understand what it takes for standard neural network to generalize over tasks that share compositional structure. We find that simply scaling data and model size leads to compositional generalization. We show that this holds across different task encodings as long as the training distribution sufficiently covers the task space. In line with this finding, we prove that standard multilayer perceptrons can approximate a general class of compositional task families to arbitrary precision using only a linear number of neurons with respect the number of task modules. Finally, we uncover that if networks successfully compositionally generalize, the constituents linearly decoded from their hidden activations. We show that this metric correlates with failures of text-to-image generation models to compose known concepts.
a task can be
Code available at http
com http:///waaaaaaacceccccocoooooat
Introduction
The ability to understand and produce novel combinations from familiar constituents is a key faculty of intelligence. It has been debated for decades whether neural networks are ever able truly achieve such compositional generalization []] Regardless of these theoretical considerations, scaling neural networks continues to result in increasingly capable models 244 Naturally, as models are scaled up. their capacity to memorize grows, and it is perhaps unsurprising that as result of training on ever larger datasets their ability to recall more information grows too [5]. However, the nature of compositionality is an exponential growth and ultimately any attempt to exhaustively capture this breadth by scaling the training data will be confronted with physical constraints.
Many works therefore advocate that neural network architectures should be explicitly endowed with compositional structure [c.g.. 6-- allow making infinite use of their finite means [10 IT]. Capturing the underlying compositional procedure of the data is more efficient pathway to generalize. In particular, the algorithmic complexity of this generalizing solution is much smaller than the complexity of the memorizing solution [12]. But does this mean that architectures need to explicitly factorize according to the data's underlying compositional mechanisms ? For instance, monolithic networks have been shown to discover modular subnetworks which may enable compositionality without specialized symbolic mechanisms
[3] Maybe simply scaling the data and size of neural networks is then enough to achieve compositionality. Here, we attempt to answer this question:
Do neural networks compositionally generalize at scale?
"qqual contribution
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39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Scaling can lead to compositional generalization. We consider compositional task families that compose K out of M modules into tasks, each of which is modeled as function. This gives rise to an exponential number of (MM*) tasks. We train standard feedforward networks on a subset of tasks and evaluate compositional generalization on held-out tasks. We find that scaling the size of the model and the data leads to compositional generalization.
Our main contributions are follows
We demonstrate that standard multilayer perceptrons compositionally generalize on a variety of tasks data and model size scaled across task encodings if the training distribution sufficiently covers the task space.
We prove that multilayer perceptrons can approximate general class of compositional task families to arbitrary precision using only linear number of neurons with respect to the number of task modules.
We show that task constituents can be (linearly) decoded from the hidden activations of models that compositionally generalize, and demonstrate that this metric correlates with failures of image generation models to compose known concepts.
Compositionality and compositional generalization
We begin by formalizing compositionality and compositional generalization with the goal of capturing a variety of compositional data types including visual scenes, abstract reasoning and behavior policies.
2.1 Compositional task family
Specifically, we will consider compositional task families that specify a generative procedure over tasks with shared compositional structure. similar vein to [14), our definition uses algorithmic complexity theory, in particular the notion of Kolmogorov complexity, see [15 for formal treatment. This definition is a modified version of the definition introduced by [66]
Definition 2.1 (Compositional task family). compositional task family is tuple T = (C, :: p(z), p p(x)), where:
The task constituent space is set Z C (z E 10, : 1 | < K < M) with corresponding task distribution p(z). ). K is the number of task components and M is the number of task modules.
A task is a function fx X' + y that labels data points x ~ p(æ).
The composition operator is mapping C: Z - (X Y) that takes input task constituetss z E Z and maps them to task, C(z) = fx for which the following conditions hold:
(i C(z) + C(2') for all + z' with z, z' Z, i.e. C is injective.
(ii) The length of the shortest program that implements C as function of K grows sub- exponentially in K.
In the discrete case, where 2 (2 (0, 13 1 | < K < M) } restricted to the set of binary, Khoo vectors, Definition 2. essentially states that compositional task family compactly captures exponentially many tasks. Condition (i) ensures that all task components functionally
For the sake of simplicity, we are stating the definition slightly informally, For the asymptotic behavior over K used in condition (ii) to be defined, we are technically considering family of compositional task families.

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enter the composition, while condition (ii) excludes the case where compositions are purely context- sensitive, ensuring that there shared structure between tasks. For more detailed discussion of this definition, please refer to [16]
The notion of task is used in general sense here and allows to capture different types of composi- tional data. For instance, a task could refer to a visual scene, where modules are the set of possible objects and the composition operator renders a selection of such objects into scene. Similarly, task could refer to a behavior policy, where modules consist of different reward functions, subset of which is combined by the composition operator to induce an optimal policy.
2.2 Task encoding
Before we can continue to define compositional generalization using Definition we must first specify how to present the model with information about its current task, captured by the task constituents z. practice, such a task description might not be the task constituents themselves, but rather some encoding thereof. For example, a task could be described through a natural language instruction or by presenting example data points (xi, fa (x:)). To model this aspect, we define the task encoder as the mapping
8 (2, 2'
that maps task constituents 2 E and a random seed r E to a task encoding. Throughout paper, we mostly focus settings where the task is unambiguously specified, i.e. where the task encoding yp is information-preset preeterrnnnn and therefore injective.
2.3 Compositional generalization
With Definition 2.1 at hand, we can now formalize compositional generalization for a model that learns to perform tasks from compositional task family, given task encoding yp. This definition is a slightly modified version of the definition presented [66]
Definition 2.2 (Compositional generalization). A model parameterized by 0 is said to compositionally generalize with respect to the compositional task family T = (Z, C, c(z),, p(z)) if there exists a discrete training distribution ~ (2) with finite support such that the number of points in the support grows sub-exponentially K and it holds that
0' arg min E (xnpeminss) [Em-p(m) 1100 =]]]
0
0" min Be<p(*) [Ex=p(a)) [100, x. =)]]
where 1(0, 2) ({{{((()))) go (x sp(2) a loss function that measures the discrepancy between model predictions and the outputs of a task (x() for given datum æ and task encoding (p(z).
Note, that not considering fixed-size datasets but sampling directly from the data distribu- tion which reflects the nowadays common single-epoch training regime of large-scale foundation models [e.g.. 3
2.4 Hyperteachers: A general class of compositional task families
For the purpose of this study, it will be useful to instantiate concrete but nevertheless general class of compositional task families according to Definition 2.1 Given that neural networks are flexible function approximators and thus able cover wide range of behaviors, we parameterize both the composition operator as well as the task functions using neural networks. The resulting system, a composable neural network that generates another neural network, be interpreted as a hypernetwork [17]. Indeed, hypernetoorss have previously been used to study compositional
generalization [8]
we consider linear hypernetwork that sums out of M possible weight matrices to paraneterize a single hidden layer task network. We define the task constituent space to be : I| & K & M) and the composition operator as
Following 18
C {= E {0.5,00 0.6,0.,,,,100000. 1.0]
C(x) = + Ω ReLU
Σ K:
z4 2x
(I)

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Figure 2: Scaling data and model size leads to compositional generalization. Top-left Scaling the number of training tasks by increasing the number of modules, task components or decreasing the fraction of tasks held-out from training leads to compositional generalization the hyperteacher task family, Top-right The number of training tasks required to achieve compositional generalization, here defined as R2 > 0.95, scales sub-exponentially Bottom Scaling model size by increasing the number of hidden neurons and the number of hidden layers leads compositional generalization the hyperteacher across different task encodings. Error bars denote SEM over three seeds.
the number of total tasks in the task family grows.
where we have M sets of neural network parameters (O,, m E R Jmml with input neurons and H hidden neurons, Ω E R x0 is a shared readout projection with O output neurons and we sample R from the uniform distribution p(æ:)= u[-1, 1J. Each module O,,, also has associated bias vector which we omit here for conciseness. The resulting task functions, Jz R + RO, are thus single hidden layer ReLU networks.
3 Scaling can lead to compositional generalization
In the following, we will investigate if and under what circumstances standard multilayer perceptrons compositionally generalize. We will show that simply scaling the number of tasks in the training distribution as well as scaling model size leads to compositional generalization. The results presented in this section focus on various parameterizations of the hyperteacher compositional task family presented in Section 2.4 We reproduce all findings of this section on the compositional preference task family introduced by [18]. This task family requires learning optimal policies in a grid world with compositional reward functions. We present the corresponding results in Appendix B.1 in and Table For additional experimental details, please refer to Appendix
Figures
g 9
While there exist various architectures specialized for compositionality. here we are interested in un- derstanding if a standard fully connected neural network, a multilayer perceptron, can compositionally generalize. Fully connected layers are common building blocks of virtually all standard architectures, including transformers or recurrent neural networks. Specifically, we consider a multilayer perceptron with ReI.U nonlinearities that accepts the concatenation of æ: and (0(z, r) as input.
x
ReLU MLP
y
(2,,r)
In order to measure a model's ability to compositionally generalize, we will hold out tasks from training and evaluate the model's performance these tasks.

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3.1 Scaling the number of compositional tasks leads to compositional generalization
To investigate the main question of how scale affects compositional generalization, we will vary both data and model size. The former can be accomplished along two different dimensions: We can vary the total number of tasks in the compositional task family by changing both the number of modules M and the number of components K, and we can vary the fraction of distinct tasks that are held-out from training- Since the number of possible tasks grows exponentially, ((M)), the compositional task families we consider can easily contain very large number of distinct tasks. The top-left of Figure shows that as we scale the number of tasks, compositional generalization improves. Notably, the required number of training tasks to achieve compositional generalization grows more slowly than the total number of tasks as shown on the top-right of Figure 2 This implies that compositional generalization with sub-ex aubbexxonential number of tasks, in Definition 2.2 is indeed achievable at scale. In Appendix [ we further demonstrate that this scaling relationship is even more favorable for transformers and similarly holds for the compositional preference task family as well as hyperteacher with a deep target network.
In addition to scaling the data, we would like to understand how scaling model size affects composi- tional generalization. Shown at the bottom of Figure [ we vary both the number of hidden layers and the size of the hidden layers for various possible task encodings (2zz r) (more details on the task encodings will follow Section 3.3). We find that, given sufficient data, increasing model size consistently improves compositional generalization. This is noteworthy, given that increasing model size in principle increases the capacity to memorize training tasks without capturing the underlying compositional structure required for compositional generalization. As we will argue in the following however, these results can be interpreted as evidence that deep neural networks tend to prefer solutions of low algorithmic complexity [19]
3.2 Complexity of generalizing solution dominates memorizing solution asymptotically
Memorizing all tasks of a compositional task family by definition requires exponential network capacity. Intuitively, a solution that captures the underlying compositional structure and thus gen- eralizes should be more efficient. A priori, it is however not clear whether such a solution exists for finite-sized, multilayer perceptron. As we have argued before, hyperteachers can be regarded as a general class of compositional task families. It is therefore instructive to consider whether a finite-sized multilayer perceptron can implement any hyperteacher without having memorize the exponential number of possible tasks. The following theorem answers this question in the affirmative.
Theorem 3.1. Let (Om E RTxH) be a sequence of uniformly bounded matrices. Then, for any M E N, > and on any compact input set, X x Z with 2 = {z || S 1), there exists ReLU multilayer perceptron that approximates hyperteacher to within E error in the || 1loo norm using
O
+ M) neurons.
The corresponding constructive proof is presented in Appendix A.2 along with an extension to hyperteachers with multiple layers. Theorem notably states that the number of neurons required for the generalizing solution scales linearly in the number of modules M. Consistent with our experimental findings, this means that as .M grows, the simplicity of the fully generalizing solution will increasingly dominate the naive memorizing solution.
3.3 Compositional generalization emerges across task encodings
We now turn to the question, to what extent the way in which the task is specified to the model matters for its ability to compositionally generalize. Specifically, one might suspect that certain ways of encoding task are better suited to leveraging compositional structure than others. For instance, 20 argue that language in particular encodes task structure in a way that is beneficial for learning compositional representations. To study this question in the context of our setup, we experiment with different task encodings ((z,r) illustrated in Figure
Generally, we find that all task encodings lead to compositional generalization, including nonlinear encodings, although some require more model capacity as shown at the bottom of Figure Table (also see Table 2 in the appendix for corresponding results on the compositional preference task
5

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Task encodings
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Figure 3: Task encodings. Illustration of the different task encodings (0(2, r) used in Table 1 The first three encodings are linear with respect the task constituents while the last three nonlinear.
Table I: Compositional generalization emerges across task encodings. Comparison of the decodability of the task constituents from the hidden activations of the third layer (Task decoder) and compositional generalization performance for various task encodings for a hyperteacher with M 16. K 3. We additionally show the linear decodability of the task constituents directly from the task encoding itself (Input decoder), which allows to distinguish linear from nonlinear task encodings. + SEM over three seeds.
Task encoding
Task decoder (R2)
Input decoder (R2)
Comp. gen. (R2)
Identity
0.95 0.012
1.00 + 0.000
1.00 + 0.000
Orthogonal
0.96 + 0.002
1.00 + 0.000
1.00 + 0.000
Language
0.99 + 0.000
1.00 + 0.000
1.00 + 0.000
Invertible NN
0.94 + 0.001
0.56 + 0.000
0.95 0.000
Interval shuffle
0.96 + 0.011
0.73 + 0.082
0.98 + 0.010
Few shot
0.90 + 0.004
-0.23 + 0.008
0.97 + 0.001
family). Specifically, we observe no benefit of directly using the identity task encoding (z, 2. over random but fixed orthogonal projection s0(2r)) Q= where Q MM XM is orthogonal matrix. In the same way, encoding each task through a language instruction poses no issues. The latter is consistent with the observation that the task constituents instructions. Interestingly, even nonlinear encodings such as specifying the task through examples (denoted fewshot), via invertible neural network or from the highly nonlinear interval shuffle function (see Algorithm in the appendix for a definition) lead to compositional generalization the model size is sufficiently large. One possible explanation for these findings is that regardless of the task encoding, the model internally infers the task constituents up to linear transformation after which Theorem 3.1 guarantees that generalizing solution that scales linearly the number of modules M exists.
be linearly decoded from such
To verify this hypothesis, train linear decoder to predict the task constituents based on the hidden activations of the model solving the task. We report the ability of this task decoder to predict the task constituents the held-out compositional generalization tasks in Table Indeed, we find that also for nonlinear task encodings the task constituents can be linearly decoded from the hidden activations providing evidence that the models internally linearize the task constituents to achieve compositional generalization. We will expand on this finding in Section
3.4 The support of the training distribution needs to sufficiently cover the task space
In principle, it is easy come up with degenerate training distributions that make compositional generalization impossible. For instance, if a module is consistently absent from all training tasks, the model has opportunity to learn this module and will generally fail to generalize to tasks that contain this module. In this sense, the support of the training distribution *rain (z) needs to sufficiently cover the full constituent space for compositional generalization to succeed. In this section we investigate how various conditions over the training support affect compositional generalization.
Prior work has studied how the training distribution affects compositional generalization [e.g., 18 21 22). Following 21 we refer to the condition of having non-zero support for each module in the training distribution compositional support For the class of hyperteacher task families,

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Compostioonl support
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Figure 4: The support of the training distribution needs to sufficiently cover the task space.: Left Illustration of the different types of task support for the special case of a compositional task family with two components. Turquoise tiles denote module combinations that are part of the training distribution, red tiles reserved for evaluation. Right Compositional generalization as a function of the different types of task support on the hyperteacher for M = 16 and K 3.
the additional condition of connected support needs to be satisfied, which states that no subset of modules should appear solely in isolation from the rest. [88] show that teacher-student setting where both the teacher and the student are hypernetworks, this condition is required to guarantee compositional generalization. We can extend this result to the more general case of any kind of student, including the ReLU multilayer perceptron. The proof follows immediately by constructing examples of multiple different hyperteachers that have an identical training distribution if the training support is not connected. Please refer to Appendix A. for more details.
Figure illustrates the different types of training support we consider here well as their effect on compositional generalization in the hyperteacher. For a more detailed description for how each training support is constructed, please refer to Appendix C.3 Our findings empirically confirm that violating compositional and connected support interferes with compositional generalization. Interestingly, having small set of popular modules who appear more frequently poses no issue for achieving compositional generalization. The converse of having a small set of unpopular modules that are rarely encountered however does lead to a noticeable drop. These findings are consistent with prior work that find that module imbalance hampers compositional generalization [3 24). However, our experiments further suggest that such failures are due to underrepresentation of certain modules and not simply due to asymmetry in module popularity. Intuitively, if a module is seen only constant number of times during training, can be memorized with constant capacity.
4 Task constituents are linearly decodable in models that compositionally generalize
Section 3.3 has revealed that cases where compositional generalization succeeds, the task con- stituents can be linearly decoded from the hidden activations. This prompts the question whether models that succeed to compositionally generalize typically form an internal linear representation of the task constituents. Generally speaking, we can show that for any model that compositionally generalizes to most tasks, the task constituents must be decodable.
Theorem 4.1 (Decodability under compositional generalization). For any 8 > 0, assume we have student g that predicts labels f=(x)) given a lask encoding s22zr) r) and inputs a. Then there exists decoder map & that decodes c2(2,r) with probability at least V8, if:
(i) (2.m))(((2,))) - -C(=)()))
(ii) For each * x', Px( C(z') < 228
7

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Figure 5: Compositional generalization correlates with linear decodability of task constituents. Top Relationship between linear decodability of the task constituents and compositional generalization across hyperteachers with M 8 modules and varying K. Bottom Relationship between linear decodability of the task constituents and compositional generalization across different task encodings for varying model sizes on the hyperteacher with M = 16, K = 3. Error bars denote SEM over three seeds. Top/Bottom We report the R² and corresponding p-value for an ordinary least square estimator in the facet titles.
We provide the proof in Appendix A.3 In practice, we observe an even stronger version of this statement, namely that the task constituents are linearly decodable from the hidden activations in multilayer perceptrons that successfully compositionally generalize.
4.1 Compositional generalization correlates with linear decodability of task constituents
To further illuminate the connection between the observed linear decodability of the task constituents and compositional generalization. we attempt to decode the task constituents in models that (partially) fail to fully compositionally generalize. Figure 5 shows a remarkably clear correlation between decodability and compositional generalization across different data scales (top) and model sizes and task encodings (bottom). Particularly interesting is the case where the unmodified task constituents provided to the model, i.c. where the task encoding is the identity. In this case, the task constituents of course trivially linearly decodable from the input. However, training the decoder on the hidden activations of deeper layers, this information is lost in networks that do not compositionally generalize. In line with previous research, this implies that having access disentangled task representation is by itself not sufficient to achieve compositional generalization 25 26
4.2 Task constituents can be linearly decoded when image composition succeeds
Finally, can we leverage the decodability of task constituents to gain insights into the successes and failures of image generation models at composing scenes from text prompts? Image generation models have come a long way, displaying impressive abilities in creating novel compositions of known concepts (24). Nevertheless, there are still systematic failure cases [27] 28 Can such failures be related to the ability to infer the task constituents from the model's hidden activations?
To answer this question, we construct a large number of compositional tasks that require composing several concepts. Two examples are shown the top-left of Figure 6 For an extensive list of all the tasks we consider, as well as more details on the task construction, see Appendix C.7 Using these tasks. evaluate the ability of several diffusion-based image generation models to systematically generate image compositions 22 30. We use a vision-language model to label whether given generation was successful and train a linear decoder decode the task constituents based on the
"Code for image composition experiments available at https://github. https///ggituuu..co///oorraaaaaa com//lorian-tolll.
compgen-vision
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Example tasks
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Average constituant decodabiity (F1) Average constituett decodablitt (F1) Average consiitunt decodability (F1) Average constituent diccodability (F
Figure 6: Task constituents can be linearly decoded when image composition succeeds. Top Example tasks to test compositionality of image generation models as well as an overview of the pipeline used to decode task constituents and judge model outputs. Bottom Relationship between average constituent decodability and image generation success rate across image composition task families for four different image generation models. We report the R² and corresponding p-value for an ordinary least square estimator in the facet titles.
model's 's hidden activations. The full pipeline is shown in the top-right of Figure 6 Please refer to Appendix C.7 for further details. The results of this analysis are shown at the bottom of Figure We observe clear correlation between the average task constituent decodability and the average generation success rate across models, with the relative task difficulty being similar across models, as shown in Figure 12 of the Appendix. This provides evidence that models which succeed at systematically composing known concepts into scenes form an internal representation of the task constituents.
5
Related work
The study of compositional generalization in neural networks has long and rich history, with early critiques highlighting the challenges of connectionist models to exhibit systematicity and compositionality [e.g.. 01 31:33 and numerous work that in response explored mechanisms for representing and processing structured information using distributed representations [c.g., [3 35). In recent years, theoretical progress has been made in showing that compositional generalization can provably be achieved with neural networks in specific settings [18 22 36 39]. This typically
requires constraining the model architecture and the data generating process. Consistent with our results, the statistics of the training data play a crucial rule in enabling compositional generalization 11 21 39..
We aim to complement this work by showing how scaling generic neural networks can lead to compositional generalization in the absence of stronger architectural constraints. This is motivated by the finding that scaling neural networks can break the curse of dimensionality [40], and consistently results in improvements in model performance 2 41] with new capabilities emerging models are
scaled up B 4 42 Compositional abilities in particular have therefore seemingly moved within grasp in practice 43)48). However, even at larger scales, models often display a compositional
generalization gap which does not close as scale increases 49 57], despite standard transformers showing compositionality in controlled settings 60).
Image generation models in particular have made impressive leaps their ability to create novel
image compositions 22 30 61 62). Compositional abilities have been shown to emerge in such
diffusion-based models on synthetic tasks an order determined by the underlying data processes and
with performance showing sudden emergence due to multiplicative dependencies (24 63). Consistent

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with our finding that the task constituents are linearly decodable in models that successfully composi- tionally generalize, [64] find that diffusion image generation models learn factorized representations on number of synthetic tasks. However, as the number of concepts that need to be composed grows, the performance of image generation models starts to deteriorate, showing the limits of their productivity to arbitrarily complex compositions [27 28)
Discussion
We have shown that simply scaling standard multilayer perceptrons can lead to compositional generalization challenging the position that stronger architectural priors necessary endow neural networks with compositionality [6|-]] That being said, architectural priors do matter when it comes data efficiency, highlighted by the improved scaling of transformers over multilayer perceptrons (see Appendix [}. Our findings also emphasize that the particular structure of the training data plays a critical role, demonstrating that not any type of scaling will lead compositional generalization.
Interestingly, we find that when the training distribution is appropriately chosen, wide range of task encodings support compositional generalization, including language but crucially also nonlinear task encodings such specifying each task through examples. Prior work posits that language composi- tionally structures neural representations and thereby aids rapid adaptation to novel compositional tasks (20) Our results indicate that in fact any cnoommtiion-reseeveee constituents suffices to achieve compositional generalization. This might help explain why animals that do not use sophisticated languages can compositionally generalize [e.g.. 65..
mapping of the underlying task
Surprisingly, regardless of whether task encodings are linear or nonlinear, we consistently find that the degree to which a model compositionally generalizes is related to the task constituents being linearly decodable from its hidden activations. In particular, we show that this metric correlates with the success rate of image generation models at composing novel scenes from known constituents. This finding suggests that, even though nonlinear representations would in principle be possible, successful compositional generalization in neural networks depends linear representations of compositional structure.
Limitations While Theoren 3.1 reveals the existence of solution that enables compositional generalization without requiring an exponential number of neurons, identifying the theoretical conditions under which one can show that this solution is guaranteed to be discovered by stochastic gradient descent is an open question. Prior results indicate that deep neural networks trained with stochastic gradient descent often display preference towards simple solutions [19] 66)-68). In the
context of compositional generalization, the complexity of the memorizing solution is by definition exponential in the number of modules which might help explain why empirically we observe the discovery of the generalizing solution which in contrast scales linearly in the number of modules.
We have focused on settings, where the task is fully specified through the task encoding. In practice, task specification might be ambiguous or incomplete, requiring models to handle uncertainty and infer the task at hand. The strong in-context learning abilities of transformers that might allow extracting localized, and even cross-modal task vectors could play an important role in this context [69
More generally, scaling the data in a way that precisely controls the coverage of the training distri- bution requires an understanding of the true generative process underlying the compositional data. For this reason, our scaling experiments rely synthetic data generation that gives us the required experimental control. On real-worle data, the underlying generative process mostly unknown and accordingly identifying training distributions that can support compositional generalization remains an open question.
Broader impacts This paper conducts foundational research aiming to illuminate under what cireumstances neural networks compositionally generalize. While we foresee no immediate negative societal impact, we hope that it may improve our understanding of this widely deployed technology.
We would like to thank Seijin Kobayashi, Angelika Steger, Jack Brady and Brenden Lake for vital discussions and fruitful feedback. Simon Schug was supported by Post- doc.Mobility grant
Acknowledgments
from the Swiss National Science Foundation.
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19

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Appendix
Table of Contents
Proofs
21
A.I Connected support
21
A.2 2 Proof of Theorem]3.
21
A.3 Proof of Theorem 4.
24
B
Additional results
25
B.1
Compositional generalization on the preference grid
25
B.2
Scaling data leads to compositional generalization in transformers
26
B.3
Scaling data leads to compositional generalization deep hyperteacher
27
B.4
Difliculty of image composition task family
28
C
Experimental details
28
C.I
Task families
28
C.2
Task encoding
28
C.3
Training task support
29
Task constituent decoding
................................
29
C.5
Architecture
................................
29
C66
Hyperparameters
30
C.7
Compositional lext-lo- generation
30
Additional details
33
D.I Compute resources
33
Software libraris
33
20

---

A Proofs
A.1 Connected support
We briefly state the definition connected support as introduced by 18 using our notation.
Definition A.1 (Connected support), Let 7 (C, p. z + p(z), p: + p(z)) be a compositional task family, and let + p ( be a distribution over Z with discrete support supp(p"um) C Z. We say that (ppp(""am)) is connected if the graph G , E) is connected, where V {z Z
puming (z) > 0} is the set of vertices and E = [(z, x²) E V x V 3i E f1. M) such that 2/3 the set of edges.
other words, two training task constituents are connected by an edge and only if they share at least one element at the same position. Section A.3 of [18] provides example that demonstrates how student hypernetwork can perfectly fit the training distribution of a different teacher hypernetwork if the task support is compositional but disconnected. This implies that there exist two distinct hyperteachers that have the same training distribution. In such cases, any student, including the multilayer perceptron considered here, is not guaranteed to generalize even when perfectly fitting the training tasks. To avoid those cases, connected support is generally required for the hyperteacher.
A.2 Proof of Theorem
In the following, we will show that a multilayer perceptron can approximate a hyperteacher using linear number of neurons in the number of task modules. We first state the result for the single layer hyperteacher we primarily consider in the main text, before extending the result to hyperteachers with multiple layers.
A.2.1 Single layer hyperteacher
Let us recall the definition of the hyperteacher in Equation T with M modules, input neurons, H hidden neurons and O output neurons,
(ar, P ReLU
Σ
+ m. imiti
where (Om E RRIHIMM J==1 are the modules, Ω HR is readout projection and [0, 1j are the task constituents. We are restating Theorem 3.1 here for convenience.
Theorem 3.1. Let (Om E IRHH) be a sequence of uniformly bounded matrices. Then, for any M E N, and on any compact input set, x Z with = {z || S 1}, there exists ReLU multilayer perceptron that approximates hyperteucher within error in the || Iloo norm using +M neurons.
Proof. We will prove the statement, by providing an explicit construction of ReLU multilayer per- ceptron that approximates hyperteacher. The construction relies on two key lemmas: in Lemma A.2 we show that the square function can be approximated in a ReLU multilayer perceptron, which we use in Lemma A.3 to show how the multiplication of two numbers can be approximated. Based on these building blocks, we then provide the construction of the full hyperteacher approximation.
Lemma A.2. For any > 0, there exists ReLU multilayer perceptron that approximates
{
[0.1]
+
square ==
2
to within e error in the ||- oo norm using (1/822) neurons.
Proof. Let first consider how any function, be approximated by piecewise linear function L matching of on I; = 1 for i E n1), where n is a fixed integer. We define
L f(0) + Σ 1 - f(x,) ReLU(x xx) f(x;) - f(x=-1) ReLU 21-1))
F++1
2, Σ4-1
21

---

where [(o))-((-1)) is set to O by convention. 20-
It can easily be verified that L is linear any interval [z,, 2,+1 for i {0, 1,..,,,,, n), and coincides with f on & (0, 1,....... We now want to bound ||S Llloo While more general results can be shown for E C²2 for f = square can derive result in the following simple way: for any x on any interval 1141)
f(z141) f(xi) (a xz) + f(x))
L(z)
Zi+1 2/
1i+1 ?
[L(x) - f(x)
(r Fii + 27 x=
Fi+1 I;
(x xz) (3++++11 x)
E
for 2 1/VE
An immediate corollary is that this result holds on any fixed bounded set. We now show how to multiply two numbers using a ReLU multilayer perceptron:
Lemma A.3. For any there exists ReLU multilayer perceptron perceeptrooo
that approximates
multiply := [0, 1]2
+
I
(x, v)
+
xy
to within error in the || oo norm using O(1/E²) neurons.
Proof. Using the polarization identity zy = (atv) (atv)²-(x-g)) (x- w)² we can approximate multiply y)
with ReLU multilayer perceptron follows:
(x. v) + (xr + x - u) + ((x + y) (x - y) + ay
Again, this result holds on any fixed bounded set.
We state construction for multilayer perceptron that approximates the preactivation of the hyperteacher, namely (x, 2) DM Σ' mm,iZm using a linear number of neurons the number of task modules.
Lemma A.4. Let (Om E RR be sequence of uniformly bounded matrices. Then, for any M E N. 0, and any compact input set, X Z with Z = 1|21|11 S 1}, there exists ReLU multilayer perceptron that approximates ().2) + Σ M
1 Σ D/=
2 i,, Zm, to within e error in the
|| bl norm using O ( + +M) neurons.
Proof. We construct our multilayer perceptron layer by layer, tracking the number of neurons required for each layer and the error it incurs in the approximation:
The first layer copies the input and computes (x, 2) + EMM 2m Om,th), )a,h for each (1..... ! and h E (1.....O) O}, canceling the ReLUs by adjusting the biases accordingly given that the input is bounded. This requires ((M) neurons.
Using Lemma A.3 with error 7. the next three layers multiply ΣM 2m Om.shh by x, for each i E (1.....}
1) and h E O1, using ((1/VE) neurons.
The next layer sums over i for each h E (1...., O}. again canceling the ReLU nonlinearity by adjusting the biases. This requires M neurons and incurs final error most s since each output neuron sums the error coming from neurons.
22

---

In total, this construction uses ((M) O ( ( 0(1)) =0 0 + + neurons.
Since ReLU is contracting function, the output enror after applying the ReLU activation is also Finally, we can straightforwardly apply the output projection 5 using a constant number of neurons, scaling the error by constant.
A.2.2 Multilayer hyperteacher
We now consider a multilayer hyperteacher, where a linear hypernetwork configures a multilayer perceptron with L layers, Hi neurons for layer I, Ho = / input neurons and Hr. O output neurons. In this case, we have ((z) == 2hh,, a sequence of hidden layers,
h ReLU Σ mm θ 1,m tu).
:W((#)
with input ho x, sequence of modules for each layer. (0r. E I HixHit projection Ω E R-H--1
I= and output m=l....
Theorem A.5. Let (Ot.m E (RH(HHH))) be a sequence of uniformly bounded matrices. Then, for any M E N e > O, fixed L and on any compact input set, X x Z with 2 = {z =||=||| S 11,, there exists a ReLU multilayer perceptron that approximates an L-layer hyperteacher to within & error in
the ||
bo norm using O (Je + +M) neurons.
Proof. First, observe that we can copy the task constituents z to each layer using M neurons per layer. Because of this, will assume that have access to the task constituents at each layer. The proof then follows a similar approach to the proof of the single layer case of Theorem 3.1 but we must now consider how the errоr propagates through each layer. For the sake of simplicity, we will ignore the readout projection (i.e. assume Q = I), since this part of the proof is identical.
We prove by induction the number of layers L that in the || Il norm with O (+e + +M) neurons. The base case for L 0 holds by definition, i.c. hg = x. Now, assume the result holds for an I--laeer hyperteacher. Let e > 0 and X, be compact sets with Z {z || I 1]. By the induction hypothesis, let hy be a multilayer perceptron that approximates h2 on the compact set X' x Z, up error in the || 2l norm.
approximate h2 to within error
By Lemma A.4 let g be multilayer perceptron that approximates (æ., 2) Em= 2m O L++1,m m B, x Z, where B, is the closed e-ball around i (X' x 2). Since the image of compact set formed by continuous map is compact and the closed epsilon ball around compact set in dimensiimniii space is compact, B. also a compact set.
Let us now show that (z, 2) + ReLU (9(h x)) approximates h X' Z up to error (e)). We define A = hi (x, 2) hy for L layers. Then, for L + 1 layers it holds that,
|| |A4t||oo = |ReLU(g(h1 + A,, x)) - het1l120
l1g A1, Wr(z)hr.llo
(since ReLU is contracting)
llg(t + A..Z) Wz(2)(h + AL) + (L(2) Az) A
llg(ti A1, WL(=)(h, + AL)l [| loo AEllo
= E =(=)=((E))
where || 1o000+00 is the operator norm induced by the || 1lo0 norm. The final error can be reduced to be most by adjusting the number of neurons by a constant factor.
Finally, let us bound the number of neurons required for the full construction. We used O(M) neurons
for copying to each layer, O (+e + M) neurons for the construction of hz, and O (+z + M)
neurons for the construction of g, which sums up to O ( + M) neurons.
23

---

A.3 Proof of Theorem 4.1
Proof. Let X, Z, R be independent random variables sampled according their respective distribu- tions. We define the event made of the set of pairs of task constituents 2 and seeds r such that the accuracy is higher than 1 - V5:
= 11110)/|2BB.00000000004444442241 vo 2 2 1
A =
(2,r) 2xN
= ((=)
vō
where the expectation is taken over X. By Markov's inequality, and the first assumption, this set can't be too small. Indeed,
[Lgg(pp R), X) # C(Z)
P Pl-A]
<
by Markov's inequality
v8
by the first assumption
We now show that for all (z,r), (
p(z,r)= = = ==
2 Indeed, let ( N. with & = p(z,t) = ((2(,r)).
We have, by definition of A,
Pla(2, x) = ((z) 2 1 v8 Fl,(2,X) C(2))(X) 1- V8
so by union-bound, it holds
- PIC())(X)= = C(=) 2 21-20 2V8
which is only possible if z' given the second assumption. now define
2' + Z
E H z for any (2,r) E (2)
((A) otherwise
& 0
is uniquely defined because of the previous property.
We have
P()(p(Z,R)) 2) > E V5
which proves our statement.
24

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Composttonal genuraliation (anC.)
tasks Danng
6.155
025
0.5
0.75
03
2055
0.90
preferences
preferences
24 preferennces
32 preferences
componentss
componentts
components
componentt
Compostional ganes generalizaion (acc)
Hidcon loyers
Identity
Onihngonal
lang
mentile VN
Innerved shufll
2100k2
2100k
1000
F.MM
Modell size
Model
size
Model sire
Figure 7: Scaling data and model size leads to compositional generalization. Top Scaling the number of training tasks by increasing the number modules, task components or decreasing the fraction of tasks held-out from training leads compositional generalization the compositional preference task family. Bottom Scaling model size by increasing the number of hidden neurons or the number of hidden layers leads compositional generalization the compositional preference task family across different task encodings. Error bars denote SEM over three seeds.
generalization (acc.)
Compositional
0.5
Random
Balanced
Non-compaatie con-coopoutionall
Disconnected
Popular modules
Unpopular modules
Task support
Figure 8: The support of the training distribution needs to sufficiently cover the task space.: Compositional generalization as function of the different types of task support on the compositional preference task family for M = 16 and K = 3.
B
Additional results
B.1
Compositional generalization on the preference grid
The findings in Section 3 and Section 4.1 focus on the hyperteacher task family. In the following, we show that all findings can be reproduced the compositional preference family introduced by (18).
In Figure 7 we show, that scaling data and model size leads compositional generalization on the compositional preference task family. This holds for various linear and nonlinear task encodings as further shown in Table [ Note, that we are not able to evaluate the fewshot task encoding of the main text since the resulting input dimension is prohibitively large. Figure X shows that non- compositional and disconnected task support as well as rarely encountering few modules interferes with compositional generalization. And finally, Figure@ddemonstrates that also on the compositional preference task family, compositional generalization and linear decodability of the task constituents from the hidden activations of the model are correlated.
25

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Table 2: Compositional generalization emerges across task encodings. Comparison of the decodability of the task constituents from the hidden activations (Task decoder) and compositional generalization performance for linear and nonlinear task encodings on the compositional preference task family with M = 16, K = 3. For the linear Identity, Orthogonal and Language task encoding. we report the decodability of task constituents from the first layer whereas for the nonlinear Invertible NN (with layers) and Interval shuffle encodings we report for the second layer. As opposed to linear task encodings, for nonlinear task encodings the decodability of task constituents is higher in the second layer compared to the first layer suggesting that the first layer is used to linearize the task constituents. We additionally show the linear decodability of the task constituents directly from the task encoding itself (Input decoder), which allows to distinguish linear from nonlinear task encodings. + SEM over three seeds.
Task encoding
Task decoder (R2)
Emhedding decoder (R2)
Comp. gen. (acc.)
Identity Orthogonal
0.95 + 0.012
1.00 + 0.000
0.92 + 0.004
0.96 + 0.008
1.00 0.000
0.92 = 0.006
Language Invertible NN
0.98 = 0.004
1.00 = 0.000
0.92 = 0.004
0.93 = 0.005
0.74 0.018
0.82 = 0.009
Interval shuffle
0.90 + 0.022
0.72 + 0.083
0.84 + 0.017
Compostonnal generalbation (aoc)
Iraction of tasiss hwid--out from tharr compunents (fed ger 2012
0.75
0.9
0.55
0.98
bank componannis
Lank compooments #fot0tip-ooon
compareents sticute ge0
Lank camponens sh-oon toapen
(f-280xp-0001
0.4-
Tesk decoder (PP)
Task deccder (R2)
Task decoder (P))
Task decoder (RP)
Task decoder (R1)
Compositoral generaltattin (acc.
Task encoding:
by
Orthogonal
Lang-ege
Mocel size
15-0000000
jart 201)
lest
A24
6.5-
H
0.5 Task decoder (R))
0.
0.s
0.5 Task
decoder (R*)
decoder (R))
Task decoder (R+)
Figure 9: Compositional generalization correlates with linear decodability of task constituents. Top Relationship between linear decodability of the task constituents and compositional generalization across instantiations of the compositional preference task family with M 8 preferences and varying K. Bottom Relationship between linear decodability of the task constituents and compositional generalization across different task encodings for varying model sizes on the compositional preference task family with M = 16, K = 3. Error bars denote SEM over three seeds. Top/Bottom We report the R2 and corresponding p-value for an ordinary least square estimator in the facet titles.
B.2 Scaling data leads to compositional generalization in transformers
In addition to using multilayer perceptrons as done in the main text, we investigate whether the widely used transformer architecture similarly achieves compositional generalization through scale. For this purpose, we train a decoder-only transformer that takes a sequence consisting of the task constituents followed by the task inputs as input and predicts the corresponding labels. Given that the transformer contains multilayer perceptrons in the feedforward blocks, we expect it to be similarly capable of compositional generalization at scale.
The top of Figure 10 confirms this, showing that scaling data similarly leads to compositional generalization in transformers. Compared to the multilayer perceptron, the transformer requires less
26

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Fraction tasks held-out from training
Compositionss generalization (RP)
modules
0.6
02
components
components
Task companents
components
comp. gen.
MLP
Transformer
.... 5% of total Lasks tasks
comooments
componentts
componentts
2000
1590
Training
0-
Tasks in task family
Tasks in task family
Tasks in task family
Figure 10: Scaling data leads to compositional generalization in transformers. Top Scaling the number of training tasks by increasing the number of modules, task components or decreasing the fraction of tasks held-out from training leads to compositional generalization in transformers on the hyperteacher task family. Bottom Transformers require less training tasks to achieve compositional generalization (R² 0.95) compared to multilayer perceptrons (MLP).
Fraction of tasks
0.125
held-out from training
0.98
0.99
Compositional seneralization
task modules
16 task modules
32 rack modules
0.5
4
3
components
components
companents
Figure 11: Scaling data leads to compositional generalization on a deep hyperteacher. Scaling the number of training tasks by increasing the number of modules, task components decreasing the fraction of tasks held-out from training leads to compositional generalization on the hyperteacher task family with deep task network with three hidden layers.
distinct training tasks achieve compositional generalization, can be observed at the bottom of Figure 10
B.3 Scaling data leads to compositional generalization in deep hyperteacher
In the main text, we focus on hyperteacher with a task network with single hidden layer. natural question is how increasing the difficulty of the hyperteacher by equipping the task network with multiple hidden layers affects our results. For this purpose, we reproduce the data scaling plot shown in Figure of Section 3 for hyperteacher with three hidden layers, each with 16 hidden neurons. Figure [T demonstrates that while this makes the task noticeably more difficult, it reproduces our finding that scaling data leads to compositional generalization.
27

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FUX dev
austigee imggl JAni sdbn JO0BE 180
3. medium
-03
medium
Model
Model
6.6
artinaat with coffectiees
Etreet there coateraats
Uhe with rosees
conettig obberts
enieman
anitnall adffetiess
Veatts woth
eniir Storr, stylec the arromia
coomiing oijeccs
dit bung
sooghe Shapes
con bagle
toreet, celtuur
purter
instrrmmnn Inneraation Sharees
with
Dhrett coiturrs
fhree, adpertinns sporre
woree pepertian crttoratts
Task family
Task family
Figure 12: Image composition task difficulties sorted by linear decodability of task constituents. Left Average image generation success for each image composition task family for different text-to- image generation models. Left Average linear decodability of constituents from hidden activations for each image composition task family for different text-to-image generation models.
B.4 Difficulty of image composition task family
To complement Figure of the main text, Figure 12 shows the task difficulty well the average linear decodability of the task constituents from the hidden activations for all the image composition task families and text-to-image generation models consider.
C Experimental details
C.1 Task families
We create the hyperteacher task family described in Section 2.4 with I 16 input neurons, = 16 hidden neurons and O 16 output neurons to create family of compositional regression tasks to be learned by a student. We sample teacher modules from a truncated normal distribution with zero
mean and standard deviation
v3 Each teacher module also has bias term that is sampled from T
the uniform distribution over the interval [0, 0.5). We sample the fixed readout matrix shared by all
tasks from a truncated normal distribution with zero mean and standard deviation V H These values have been picked as to ensure that the hyperteacher creates tasks of sufficient diversity and difficulty.
//3
For the definition of the compositional preference task family, please refer [8] Following the notation of the hyperteacher, we denote the number possible preferences as M and the maximum number of preferences combined into a task K. [18] uses M number of possible preferences throughout. We increase the difficulty of the tasks by increasing the number of preference features to M 16 in Figure bottom, Figure 3 and Figure and to M 32 in Figure 7 top.
C.2 Task encoding
employ the invertible neural network introduced by 772 with layers (3 the compositional preference task family) that consists of series of bijective transformations, to obtain nonlinear information- preseiving encoding the task constituents. In addition, we use the Interval Shift function defined in Algorithm I as another example of nonlinear but bijective task encoding.
28

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Algorithm Interval Shift
1: Number of intervals: N
2: function ((2,
3:
GeneratePPrrmuaaioon//
Generate permutation of [O,N-1] using seed Determine which interval belongs to > Fractional position within interval
4: i 2
5:
a+ N.
6:
j f (i)
> Apply permutation
7:
return ++a N
Preserve relative position within new interval
8:
end function
C.3 Training task support
We investigate the effect of various procedures to construct the training task support compositional generalization, which we describe in the following. For this purpose, consider the graph G = (V, E)
where V {= 2 p train (z) > 0} is the set of vertices and E {(z, V V 3i {1, (1,2.....
K such that 2, = 2 the set of edges.
Random: Samples a random subset of vertices until graph that both compositional and connected is found.
Balanced: Similar to Random, but ensures that each module appears with equal frequency in the training distribution using greedy search over vertices.
Non-compositional:: Holds out all tasks that contain one random but fixed module.
Disconnected: Divides modules into two disjoint subsets and only uses vertices that use modules from either subset but do not contain modules from both subsets.
Defines a set of P popular modules and only includes vertices that contain least one popular module. P is determined such that the fraction of held-out tasks can be satisfied as specified. If it cannot be exactly satisfied, one module that is not in the set of popular modules receives additional vertices. Additionally ensures that the resulting set is compositional and connected.
Popular modules:
Unpopular modules: Defines a set of U unpopular modules and includes all vertices that do not contain any unpopular module. For each unpopular module, one vertex that includes the unpopular module and otherwise only not unpopular modules is added. U is determined such that the fraction of held-out tasks can be satisfied specified. If it cannot be exactly satisfied, one unpopular module receives additional vertices. Additionally ensures that the resulting set is compositional and connected.
C.4 Task constituent decoding
On the hyperteacher and compositional preference task family, fit linear decoder using ridge regression to predict the task constituents given the hidden activations of a particular layer of the multilayer perceptron solving the task. For this purpose, train pairs of hidden activations and ground truth task constituents from the training distribution, ( and evaluate the performance of the decoder held-out tasks, reporting the coefficient of determination (R2² score). Throughout, we employ a regularizer of À 1.0 for the ridge regression.
C.5 Architecture
Unless specified otherwise, we use a multilayer perceptron with four hidden layers with 1024 hidden neurons each for the hyperteacher and with two hidden layers for the compositional preference task family.
The transformer in Section B causally masked and consists of 4 layers with 4 attention heads, model dimension of 256, feedforward dimension of 1024 and separate projection matrices for the task constituents, inputs and the output.
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C.6 Hyperparameters
Throughout our experiments, we use the AdamW optimizer 73 with batch size of 128. On the hyperteacher task family, we use a mean- mnen-ssqureed error loss, on the compositional preference task family, we use a cross-entropy loss. We performed an initial grid search over the learning rate and weight decay to find a common set of hyperparameters for all experiments on the hyperteacher task family and a common set of hyperparameters for all experiments on the compositional preference tasks respectively. We report the search grid in Table 3
Table 3: Hyperparameters for experiments on the hyperteacher task family and compositional preference task family. Lists of values denote parameters explored via grid search with a bold number indicating the value found to perform best and used throughout the experiments.
Parameter
Hyperteacher
Compositional preferences
learning_rate
[0.001, 0.003, 0.0001, 0.0003] (0.003, 0.001, 0.0003]
[0.001, 0.003, 0.0001, 0.0003] [0.003, 0.001, 0.001,0.0003)
weight_decay
C.7
Compositional text-to-image generation
Models We compare four open-weight text-to-image generation models: FLUX. 1-dev 29 FLUX.1
dev Non-Commercial License], SD XL base 1.0 [74] Open RAIL++-M License], SD 3.0 medium
75
Stability AI Community License] SD 3.5 medium 76 Stability AI Community License].
Experimental setup In the following, we describe the experimental setup illustrated in Figure 6 For each image composition task family listed below, we prompt each model on all possible image combinations generate an image using 40 inference steps. During this process, hidden activations to be used by the image constituent decoder (see below). We then prompt a VLM to judge whether the image constituents have been correctly combined into coherent image by asking to readout the image constituents given the image and the full set of possible constituents. We count image generation as successful if the image constituents generated by the VLM exhaustively match the ground truth constituents. Here, we report results using Gemini 2.0 Flash 77 as the judge.
collect their
Image constituent decoder We train standard logistic regression classifiers predict image constituents given the hidden activations of the sat-to-immae generation models and report their F1 scores on held-out image compositions. Not all layers of the respective models contain this information and we need to carefully choose layers depending on the model architecture. In particular, trivially be linearly decoded from early layers of the model that encode the
the image constituents text prompt, as well as any layers that are directly connected to the text encoding either via residual connections via cross-attention. For this reason, select layers that require the model explicitly models FLUX. 1-dev,
learn to pass information about image constituents to. In the transformer-bast SD 3.5 medium um and SD 3.0 medium, we probe the hidden activation of the penultimate transformer block's 's MLP. SD XL base 1.0 contains bottleneck block, so we probe the hidden activations at the final layer of the bottleneck. Since the models we are considering are diffusion models that run inference over several time steps, we must specify at what time during the inference process to collect the hidden activations. In our experiments, we run 30 inference steps and collect hidden activations at time steps 9, 15, 21, and 30, as well as the average hidden activation over all time steps. We concatenate all hidden activations for a given model and image and use them as the input to the linear decoder.
Image composition task families We create a variety of image composition task families that each consist of a combinatorial space of objects and attributes that need to be combined into coherent image. Specifically, each task has the following structure with a corresponding prompt template:
Prompt template specifying how to compose (componentl| and (component2). componentl: modulel, module2, - component2: modulel, module2,
We list all the tasks in the following:
30

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animals_withh_oloure A fcolour] /animal]
animal: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
colour:
red, blue, green, yellow, orange, purple, pink, brown, black, white
animals_with_stylle A (animal) illustrated in (style/ style
animal: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
style: watercolor, pixel art, oil painting, sketch, cartoon, origami, stained glass, pop art, charcoal, clay sculpture
A (animal) in the (location]
animal: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
location:
top left, top center, top right, middle left, center, middle right, bottom left, bottom center, bottom right
hiiimals_witt_ddssrppppepe A (descriptor) [animal]
animal: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
descriptor: furry, scaly, feathered, leathery, smooth, wrinkled, young, old, three-legged, spotted
aninals__with_adfeccttt
A image of (adjective) [animal)
adjective: happy, sad, angry, sleepy, curious, playful, scared, proud, surprised, bored
animal: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
fruits_with_werbss A [fruit] is [verb)
fruit: apple, banana, orange, grape, strawberry, watermelon, pineapple, mango, blueberry, peach verb: dancing, flying. bouncing, sleeping, swimming, rolling, climbing, stretching, spinning, hiding
cuuntigg_animlss. An image with exactly (number) [animal)
number: one, two, three, four, five, six, seven, eight, nine, ten
animal: lions, elephants, giraffes, crocodiles, bears, snakes, eagles, cows, zebras, tigers
umimals_with_verbee A (animal) is /verb)
animal: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
verb: eating, sleeping, running, jumping, flying, swimming, climbing, dancing, playing, hiding
counting-objects: An image with exactly /number) (object)
number: one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty
object: tomatoes, onions, oranges, wolves, bears, apples, bananas, carrots, cucumbers, strawber-
ries, lemons, cherries, grapes, peaches, pears, foxes, rabbits, cats, dogs, sheep
posted_contaiinaen la disimal_ineer)
animals: A cunimal_outerr containing fanimal middle) containing
animal outer: unterr lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal _animal_middle: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal inner: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
three_ anieaass three materials: /animal f_re)) made fire, (animal ice) made of ice, and [animal wood| made of wood
animal fire: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal ice: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal :ood: lion, elephant, giraffe, crocodile, bear, snake, cagle, cow, zebra, tiger
Saree_animals.o awwmallsttheeeeeeeeeeeeeeeeeeee verbs: [animal_singing singing, [animal_euting eating, and (ani-
mal_sleeping/ sleeping
-animal_singnng lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
- aanimalccainng: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
-animal sleeping: lion, elephant, giraffe, crocodile, bear, snake, cagle, cow, zebra, tiger
31

---

three_aninne wtwpeaaaimmmlstttreeeeeeerreeeee A sad fanimal_mmee _hppyy)_ happy [animal_sser and an angry danimal_angry
animal animal_happy: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
hhimal_sade lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal angry: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
fanimal] wearing /clothing/
animal: cat, dog, bear, lion, elephant, giraffe, monkey, zebra, tiger, panda
clothing: hat, pair of sunglasses, scarf, bowtie, jacket, crown, tie, cape, sweater, necklace
animals_withe stthhh_oothhs__aaeeee __d___ddddddddd (animal) wearing (clothing) eating (food)
animal: cat, dog, bear, lion, elephant, giraffe, monkey, zebra, tiger, panda
clothing: hat, pair of sunglasses, scarf, bowtie, jacket, crown, tie, cape, pair of pants, pair of boots
food: pizza, banana, ice cream, cake, hamburger, apple, watermelon, donut, sandwich, salad
animals_we httpss_wwthfooddd__a_aeeee food_eyes, and_aadddddsl clothes: A (animal) with (food) as eyes wearing (clothing)
animal:
cat, bear, lion, elephant, giraffe, monkey, zebra, panda, wolf, rabbit strawberries, oranges, burgers, watermelons, donuts, cookies, cupcakes, pizza, lemons, tomatoes
food:
clothing:
crown, cowboy hat, scarf, cape, hawaiian shirt, leather jacket, pair of pants, tuxedo,
raincoat, sunglasses
stacked_foods: (food_top) on top of [food_middle) on top of (Sood_botton)
food top:
burger, pizza, salad, Sushi, taco, donut, Ice cream, pancake, Spaghetti, sandwich
mid_middee: burger, pizza, salad, sushi, taco, donut, ice cream, pancake, spaghetti, sandwich
food bottom: burger, pizza, salad, sushi, taco, donut, ice cream, pancake, spaghetti, sandwich
stacked_animalss [animal_tpp) on top of (animal_middles on top of [animal boottom
animal top: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal_me miimallmiddle: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal bottom: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger /animall) chasing (animal2] chasing (animal3)
caimmals_casinng.co
animall: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal2: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal3: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
cestted.containment (containerl) containing a (container2) containing a (object)
containerl: transparent cube, glass jar, wooden box, metal safe, woven basket, leather bag, ceramic pot, copper kettle, crystal sphere, rubber ball
small treasure chest, porcelain teacup, silk pouch, stone bowl, paper envelope, cardboard tube, tin can, shell, gold locket, velvet case
container2:
object: diamond, living butterfly, ticking clock, miniature planet, flickering flame, drop of mercury, hologram, glowing ember, snowflake, single cell organism
impossible catesibbl__mttrrillss www.emittaaas.... (object) made entirely of [material) sitting on (surface)
chair, bicycle, bookshelf, piano, computer, refrigerator, watch, umbrella. camera, guitar material: liquid water, fire, smoke, mirrors, ice, tree bark, glass noodles, gelatin, paper, soap bubbles
object:
surface: clouds, ocean waves, melting ice, sand dunes, moss, broken glass, spiderwebs, lily pads, autumn leaves, foam
A ponimal_pixelll in pixel art, (animal_oie _ill oil painting, and a [aaaimal_cartoo] in cartoon style
- sanimal_pixel lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal oil: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
annmal cartoon: lion, elephant, giraffe, crocodile, bear, snake, cagle, cow, zebra, tiger
32

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A amnimal_frryy with fur. ranimal_scally with
scales, and a mnnimal_feahhreed] with witthfeeeeerr
animal filiml_furry: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal _scll:ssclyy.. lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animal feathered: lion, elephant, giraffe, crocodile, bear, snake, eagle, cow, zebra, tiger
animals_ An image with fone (animal))).2.
giraffe, lion, elephant, crocodile, bear, snake, eagle, cow, zebra, tiger, rhino, hippo, wolf, fox, deer, monkey, panda, koala, kangaroo, penguin
animals:
objects__thre__colouue
An image with fone [object)]=.23
red cube, green cube, blue cube, blue sphere, orange sphere, red sphere, green cylinder, purple cylinder, yellow cylinder, yellow cone, blue cone, green cone, purple pyramid, red pyramid, orange pyramid, orange cuboid, yellow cuboid, blue cuboid
objects:
fruits_veggies__ogg An image with fone (fruit_ (rtuit_veggie/)1.2..
meggi_s apple, banana, orange, grape, strawberry, watermelon, pineapple, mango, blue- berry, peach, lemon, cherry, carrot, broccoli, tomato, cucumber, potato, onion, pepper, lettuce
An image with [one (element))1.2..
mnimals_vegetalll shapes: lion, elephant, giraffe, tiger, bear, zebra, monkey, carrot, broccoli, potato, tomato, cucumber, onion, pepper, cube, sphere, cylinder, cone, pyramid, triangular prism
D
Additional details
D.1
Compute resources
We used Linux workstation with two Nvidia RTX 3090 GPUs with 24GB of memory each for development and conducted hyperparameter searches and experiments on an internal Slurm cluster using Nvidia RTX 4090 GPUs and Nvidia A A100 GPUs.
A single of hyperteacher experiment takes 4- 10 minutes RTX 4090 depending model size, a single run of a compositional preference experiment takes 50-100 minutes. In total, reproducing all experiments reported for the hyperteacher task family with around 1000 distinct runs takes about 7 GPU days and reproducing all corresponding experiments for the compositional preference task family takes about 70 GPU days.
Generating the images one of the image composition task families takes GPU hours on an A100 for the Flux 1 dev model and 4 GPU hours RTX 4090 for each of the SD models. Running
all 27 tasks for all models takes a total of 23 GPU days.
D.2 Software and libraries
For the results obtained in this paper we built on free and open- source software. We implemented
our experiments in Python using JAX [78 Apache License 2.0], Flax (79 Apache License 2.0],
the Deepmind Jax Ecosystem 80 Apache License 2.0], PyTorrc [BSD-style license 81), L.M
[82, Apache License 2.0] and Scikit-learn
[3 BSD 3-Clause License]. We utilized WandB [84
MIT license] to monitor the progress and results of experiments, and Plotly [85] MIT license] for
generating the plots. We use uy for Python project dependency management 86 MIT License].
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41


Reviews:
[{"review_id": "k84f2wfmFj", "reviewer": "Reviewer_bGxw", "summary": "This paper investigates the impact of data and model scale on compositional generalization by considering a teacher-student setup. Specifically, they consider compositional tasks and a hypernetwork teacher (that receives as input the specific compositional task identity). The student receives some function of the compositional task identity as well as the given input. The authors find that with increasing model size and data scale, the network improves in its generalization to novel combinations of task. They tie this to an argument that MLPs can efficiently represent such compositional functions and therefore prefer learning such a solution over memorizing solutions for each task combination. They then suggest that MLPs that can learn such compositional functions will have representations that enable decoding the task identity. Notably, they replicate this phenomenon in image generation models.", "questions": "- It is quite surprising to me that the linear decoder of the task identity doesn't work. Given that the hidden layer of the neural network is fairly large, I would imagine that even a random nonlinear embedding would allow for memorizing the different tasks with a linear decoder. Can you give me an intuition for why this might happen? Is it related to the fairly large regularization parameter associated with the decoder? Does the neural network actually end up collapsing the representation?\n- I think it would be important to scale back the generality of the claims (as noted in my comments above). An example title would be \"Scale leads to compositional generalization: a teacher-student perspective\" --- more broadly, just giving a sense of the specific setup in the title would be good. There are many ways to study compositional generalization and the impact of scale and it would be helpful to be specific about what your contribution here is.", "limitations": "Yes (though see comments about scope of the work above).", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\nThis paper is a neat demonstration of how large data and model size can enable compositional generalization. This is particularly intriguing as the authors consider a very similar scenario to the one investigated by Schug et al., who demonstrated that modularity can be sufficient for generalization --- thus demonstrating that modularity is not necessary with sufficient data. I also appreciated that the authors provided theoretical arguments for why this might be the case. While this study focuses on relatively simple networks, I think such a simplified scenario can be a strength in itself as it can provide a useful case study. Finally, I thought the paper was well written and easy to follow.\n\n**Weaknesses**\n\nTheorem 3.1 does not actually show that neural networks in practice learn a compositionally generalizing solution; rather it demonstrates that this is an efficiently learnable solution. This is an important gap between the theory and the practical scenario. That said, theoretically understanding deep neural networks trained with backpropagation is often quite complicated and so in my mind, this is an acceptable gap.  However, it would be nice to provide some justification for why Theorem 3.1 might be useful for understanding networks trained in practice, e.g. by drawing on the literature showing that even overparameterized neural networks often end up learning a solution with a sparse hidden layer (indicating that they might be more likely to learn a solution that can be parameterized with few hidden neurons) [e.g 1-3].\n\nNotably, all of the results related to scale are based on simple MLPs and relatively simple teacher networks. While I think this is an interesting scenario to investigate (as noted above), I do think the authors should more strongly qualify their findings based on this limitation, in particular by making the title more specific to the scenario they are investigating.\n\n**References**\n1. https://proceedings.mlr.press/v125/chizat20a.html\n2. https://proceedings.mlr.press/v99/savarese19a.html\n3. https://proceedings.neurips.cc/paper_files/paper/2022/hash/7eeb9af3eb1f48e29c05e8dd3342b286-Abstract-Conference.html", "was_revised": true, "final_justification": "I continue to think that this is a valuable paper that sheds light on how sufficient training data can give rise to compositional generalization and I would like to see it accepted to Neurips.", "rebuttal": "Thank you for your positive evaluation of our work and having taken the time to provide informative and actionable feedback which has led us to revise several parts of the manuscript as well as triggered two new sets of experiments. We address the points you raise individually below and are looking forward to engaging in further discussion with you.\n\n> Theorem 3.1 does not actually show that neural networks in practice learn a compositionally generalizing solution; rather it demonstrates that this is an efficiently learnable solution. This is an important gap between the theory and the practical scenario. That said, theoretically understanding deep neural networks trained with backpropagation is often quite complicated and so in my mind, this is an acceptable gap. However, it would be nice to provide some justification for why Theorem 3.1 might be useful for understanding networks trained in practice, e.g. by drawing on the literature showing that even overparameterized neural networks often end up learning a solution with a sparse hidden layer (indicating that they might be more likely to learn a solution that can be parameterized with few hidden neurons) [e.g 1-3].\n\nTheorem 3.1 demonstrates the existence of a compositionally generalizing solution in standard neural networks. This is noteworthy since such compositional task families consist of exponentially many tasks and it is a priori not clear if neural networks would require an exponential number of neurons to learn all tasks. But as you correctly point out, proving that stochastic gradient descent will actually discover this solution is difficult in a general setting and hence this result is complemented with our empirical findings that show circumstances under which neural networks can learn a compositionally generalizing solution in practice. \n\nFollowing your suggestion, we added a paragraph to the limitations section to more openly discuss this point:\n\nWhile Theorem 3.1 reveals the existence of a solution that enables compositional generalization without requiring an exponential number of neurons, identifying the theoretical conditions under which one can show that this solution is guaranteed to be discovered by stochastic gradient descent is an open question. Prior results indicate that deep neural networks trained with stochastic gradient descent often display a preference towards simple solutions [1-4]. In the context of compositional generalization, the complexity of the memorizing solution is by definition exponential in the number of modules which might help explain why empirically we observe the discovery of the generalizing solution which in contrast scales linearly in the number of modules.\n\n[1] How do infinite width bounded norm networks look in function space? Savarese et al., PMLR 2019\n\n[2] Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss. Chizat & Bach, PMLR 2020\n\n[3] Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs. Boursier et al. NeurIPS 2022\n\n[4] Deep Learning is Not So Mysterious or Different. Wilson, arXiv 2025\n\n\n> Notably, all of the results related to scale are based on simple MLPs and relatively simple teacher networks. While I think this is an interesting scenario to investigate (as noted above), I do think the authors should more strongly qualify their findings based on this limitation [...]\n\nPlease allow us to clarify that as part of the original submission, we reproduce all results of Section 3 on the compositional preference task family, an optimal-policy imitation learning task over grid worlds with compositional reward functions (see L112-113 and Appendix B.1). We focus on the hyperteacher in the main text due to space constraints and to keep the exposition clear but we understand that it is easy to miss this aspect and we will make this reference more prominent in the manuscript.\n\nTriggered by your remark on the complexity of the teacher networks, **we conducted a new set of experiments and reproduced the data scaling plot on the top of Figure 1 for a deeper hyperteacher** with three hidden layers. This makes the task noticeably more difficult which is evidenced by the need for more data to achieve compositional generalization similar to the results on the compositional preference task family. Importantly, the qualitative finding that scaling data eventually leads to compositional generalization is reproduced in this setting. We regret that this time around we cannot show figures in the rebuttal: this plot looks qualitatively the same as the top of Figure 1 with all curves slightly shifted to the right due to the increased task difficulty.\n\nIn addition, we would like to emphasize that we see the simplicity of using MLPs as a strength of the work since arguably MLPs are basic building blocks of a variety of other neural network architectures. For instance, transformers explicitly use MLPs in their feedforward blocks. To convince our readers that this is not only a theoretical consideration, and in response to your remark (as well as reviewer kRyd), we conducted a new set of experiments and **reproduced the data scaling plot on the top of Figure 1 where we replace the MLP with a transformer**. We qualitatively reproduce our main result, i.e. as data is scaled, transformers achieve compositional generalization. Transformers are (maybe not unexpectedly) more sample efficient than MLPs at doing so and to further emphasize this point we create a plot where we compare the minimum number of training tasks required to achieve compositional generalization (OOD R² > 0.95) as the total number of possible tasks is increased (by increasing the number of modules M). In this plot, both the MLP and transformer clearly require a sub-exponential (in fact, sub-quadratic) number of training tasks compared to the total number of tasks with the transformer consistently requiring less than the MLP. Again, we regret that we cannot show the actual plots in the rebuttal but we have added these figures to the appendix of our updated version of the manuscript and refer to them in Section 3.1 of the main text. Please let us know if there are specific numbers for these results you would like to see in tabular form here.\n\n> It is quite surprising to me that the linear decoder of the task identity doesn't work. Given that the hidden layer of the neural network is fairly large, I would imagine that even a random nonlinear embedding would allow for memorizing the different tasks with a linear decoder. Can you give me an intuition for why this might happen? Is it related to the fairly large regularization parameter associated with the decoder? Does the neural network actually end up collapsing the representation?\n\nYour intuition is correct. The linear decoder does work when decoding the task modules and thereby the task identity on the in-distribution tasks (i.e. the R² scores on in-distribution tasks are > 0.99). However, after having trained the linear decoder to predict the constituent modules of the training tasks using the hidden representations of the neural network, we report whether it generalizes to predicting the constituent modules on the held-out, out-of-distribution tasks on which we measure compositional generalization. Thus, the instances you describe where the linear decoder fails are instances where it fails to generalize to held-out tasks indicating that in these instances the ground-truth modules are not linearly represented in the hidden representation of the neural network.\n\n> I think it would be important to scale back the generality of the claims (as noted in my comments above). An example title would be \"Scale leads to compositional generalization: a teacher-student perspective\" --- more broadly, just giving a sense of the specific setup in the title would be good. There are many ways to study compositional generalization and the impact of scale and it would be helpful to be specific about what your contribution here is.\n\nWe understand your concern that the current title might sound very general but we hope that our response above has also helped to clarify that your suggestion of adding the subtitle “a teacher-student perspective” might not accurately reflect that we run experiments outside the teacher-student setting, i.e. on the compositional preference task family for Section 3 as well as with image generation models in Section 4.2. \nWe would like to maintain a concise title that is easy to understand while avoiding to sound overly authoritative. In an attempt to accommodate both desiderata, we will change the title to “Scaling _can_ lead to compositional generalization\"."}, {"review_id": "eFImYWLkJE", "reviewer": "Reviewer_hkqv", "summary": "This paper investigates whether MLP neural networks can achieve compositional generalization through scaling alone, without compositionality enforced in the architecture. Using synthetic \"hyperteacher\" tasks where networks must compose modules to solve problems, the authors show that simply scaling data and model size leads to compositional generalization. The authors claim that the number of neurons needed for the generalizing solution scales linearly with task modules, making it asymptotically more efficient than memorization. They also find that compositional generalization emerges across different task encodings (identity, orthogonal, language, few-shot examples) and illustrate different shapes of the training data which are required to support compositional generalization. Finally, the authors find that models which successfully generalize can linearly decode task constituents from their hidden activations. The authors validate this on real image generation models, showing that diffusion models with better compositional abilities have more linearly decodable task representations.", "questions": "Q1: Are the runs for different model sizes FLOPs matched? (equal compute).\n\nQ2: do you have any flops to loss plots which demonstrate how saturated the training is?\n\nQ3: Figure 3 is very interesting! Is Compositional support and Connected support both R^2 ~ 1 ?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "S1: This paper addresses an important problem: Compositional Generalization. And the paper discusses this topic all the way from theory, to synthetic experiments to real experiments.\nS2: The paper is well formatted and easy to follow with well curated figures.\nS3: The results on compositional support are well illustrated.\nS4: The linear decodability result is often assumed but it's useful to see this tested both in synthetic and real image data.\n\nW1: **This is my main concern.** It is hard to determine whether, as the authors claim, “scale” alone leads to compositional generalization. Recently, several papers (albeit in different topics (e.g. https://arxiv.org/abs/2412.01003, https://arxiv.org/abs/2410.16531) has pointed out that it is often misleading that a scale-dependent effect is claimed when networks are trained with same number of steps, since wider networks are known in practice and in theory to learn faster. Would the authors clarify if all experiments are ran in equal flops settings? (This does not require more data for smaller models, but might need more epochs)\n\nW2: Another concern is that together with W1, it seems like the title is too general, (kind of the “attention is all you need” vibe), as it’s more the case that compositional support, data diversity, enough optimization *and* scale is needed for compositional generalization.\n\nW3: It would be much better if the experimental data is more explicitly described with examples.\n\nW4: The results are definitively useful and are well illustrated, but it is unclear to what degree these results are unexpected, surprising or actionable.", "was_revised": true, "final_justification": "This is a thorough paper ranging theoretical investigation to empirical validation. \n\nThe authors have also responded to my concerns adequately, I confirm my final score of 5.", "rebuttal": "Thank you for your positive and thoughtful review. We appreciate your detailed feedback that has helped us clarify important points in the manuscript and led us to conduct additional experiments, both of which we believe will help address your main concern. We are looking forward to engaging further with you throughout the discussion period.\n\n> W1: This is my main concern. It is hard to determine whether, as the authors claim, “scale” alone leads to compositional generalization. Recently, several papers (albeit in different topics (e.g. https://arxiv.org/abs/2412.01003, https://arxiv.org/abs/2410.16531) has pointed out that it is often misleading that a scale-dependent effect is claimed when networks are trained with same number of steps, since wider networks are known in practice and in theory to learn faster. Would the authors clarify if all experiments are ran in equal flops settings? (This does not require more data for smaller models, but might need more epochs)\n\n> Q1: Are the runs for different model sizes FLOPs matched? (equal compute).\n\nThank you for raising this important point. Indeed, when comparing the performance of models of varying size, it is important not to disadvantage smaller models by allocating them less compute (FLOPS). In our submission, we want to assess whether compositional generalization is at all possible at a given scale. This means we are giving smaller models as much compute as our budget allows, training them far beyond the point where the training loss has converged. As a result they are allocated more FLOPS than what is required for larger models to converge to the reported numbers.\nThis point was not sufficiently explained in our initial submission and we have now updated the main text in section 3.1 to add this information.\n\nTo further illustrate this point, **we performed two additional experiments**:\n\n- We ran training on a subset of the runs with smaller models for 100x longer after the training R² score reached 0.999 in an attempt to capture eventual “grokking”-like behavior but found no such phenomenon to occur, i.e. the compositional generalization R² remained comparable to training 100x less long.\n\n- We trained models of sizes ranging from 90K to 84M parameters on a fixed FLOPS budget of 5.16e16 on the hyperteacher with M=16 K=3 for five different fractions of tasks held-out from training (0.25, 0.75, 0.9, 0.95, 0.98). This experiment shows that the compositional generalization R² score increases with model size but eventually saturates at a given level, unless the fraction of tasks held-out from training is further increased. We will add a corresponding plot to the main text but for the sake of the rebuttal which does not allow showing figures this year, here are the results in a tabular format:\n\n\nTable 1: ISOFLOP analysis comparing models with different numbers of parameters (each entry shows the compositional generalization R² score on held-out tasks at the end of training with the same FLOPS budget of 5.16e16):\n\n| Fraction held-out              |90K    | 342K  | 1.3M   | 5.3M | 21M     | 84M    |\n|-----------------------------|-----------|-----------|-----------|---------|-----------|-----------|\n|                           0.98 |    -0.01 |     0.24 |     0.51 |   0.53 |     0.53 |     0.52 |\n|                           0.95 |     0.25 |     0.66 |     0.76 |   0.82 |     0.82 |     0.83 |\n|                           0.90 |     0.64 |     0.82 |     0.90 |   0.93 |     0.94 |     0.95 |\n|                           0.75 |     0.89 |     0.95 |     0.97 |   0.99 |     0.99 |     0.99 |\n|                           0.25 |     0.98 |     1.00 |     1.00 |   1.00 |     1.00 |     1.00 |\n\n> Q2: do you have any flops to loss plots which demonstrate how saturated the training is?\n\nThank you for this suggestion that will help to make the previous point more obvious in the manuscript. **We added FLOPS vs. loss plots** corresponding to the results of Figure 1 to the appendix. Again, since we cannot show figures in the response, here is a sample table that corresponds to the row above where 90% of the tasks are held-out from training.\n  \nTable 2: Training loss as more FLOPS are consumed comparing models with different numbers of parameters.\n\n| Training steps               |90K       | 342K    | 1.3M    |  5.3M | 21M     | 84M     |\n|-----------------------------|-----------|-----------|-----------|---------|-----------|-----------|\n|                   \t    25% |  3.1e-3 |  8.4e-4 |     5e-4 | 3.3e-4 | 3.8e-4 |     1e-3 |\n|                  \t    50% |  2.9e-3 |  7.5e-4 |  3.6e-4 | 2.8e-4 | 2.3e-4 |     8e-4 |\n|                   \t    75% |  2.8e-3 |   6.8e-4|     3e-4 | 2.4e-4 | 2.2e-4 |  6.5e-4 |\n|     \t     \t  100% |  2.6e-3 |  6.2e-4 |  2.8e-4 |    2e-4 | 2.0e-4 |  3.6e-4 |\n\n\n> W2: Another concern is that together with W1, it seems like the title is too general, (kind of the “attention is all you need” vibe), as it’s more the case that compositional support, data diversity, enough optimization and scale is needed for compositional generalization.\n\nThe “right” training data is indeed needed for scaling to lead to compositional generalization and we understand your point that the current title might be too absolute to accurately reflect this point. At the same time, we would like to maintain a concise title that is easy to understand. In an attempt to accommodate both desiderata, we will change the title to “Scaling _can_ lead to compositional generalization\" to give better justice to the fact that not any kind of data and model scaling yields such a result.\n\n> W3: It would be much better if the experimental data is more explicitly described with examples.\n\nThank you for your suggestion, we have extended section 2.4 and section B.2 respectively with a more explicit description of the experimental data we use. In addition, we added a conceptual figure that illustrates Definition 2.1 of the general class of compositional task families we consider.\n\n> W4: The results are definitively useful and are well illustrated, but it is unclear to what degree these results are unexpected, surprising or actionable.\n\nWe believe that these results are both surprising to the community and actionable. Please allow us to briefly restate our main arguments:\n\nIt is a decades-old question whether connectionist models are fundamentally able to capture compositional structure [1-4].\nDespite the success of large-scale models, we observe many failure cases that indicate deficits in capturing compositional structure making this a valid concern still today [5-9] (also see section 4.2 for systematic failures of image generation models). \nA number of recent papers argue that for compositional generalization to succeed, neural networks need to be endowed with architectural priors for modularity [e.g., 10-12]. \nOur results challenge this position, since even monolithic neural networks (MLPs) can achieve compositional generalization at sufficient scale. We believe it is actionable since it assigns a stronger importance to the structure of the training data over the model architecture than was previously thought, i.e. it is important how diverse the training data is (section 3.1) and how the training data is distributed (section 3.4) but shows that compositional generalization can ultimately be achieved via scaling and does not constitute a fundamental shortcoming of connectionist models.\n\nWe refined these points in the introduction, related work section and discussion of the paper.\n\n[1] Connectionism and cognitive architecture: A critical analysis. Fodor & Pylyshyn, Cognition 1988\n\n[2] Connectionism, Constituency and the Language of Thought. Paul Smolensky, Meaning in Mind 1991\n\n[3] Systematicity in connectionist language learning. Hadley, Mind & Language 1994\n\n[4] Connectionism and the problem of systematicity. Phillips 1995\n\n[5] Faith and Fate: Limits of Transformers on Compositionality. Dziri et al., NeurIPS 2023\n\n[6] The Reversal Curse: LLMs trained on “A is B” fail to learn “B is A”. Berglund et al., ICLR 2024\n\n[7] Not All LLM Reasoners Are Created Equal. Hosseini et al.,  NeurIPS 2024 Workshop MATH-AI\n\n[8] Do Large Language Models Have Compositional Ability? An Investigation into Limitations and Scalability. Xu et al., ICLR 2024 Workshop ME-FoMo\n\n [9] GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models. Mirzadeh et al., ICLR 2025\n\n[10] Discovering modular solutions that generalize compositionally. Schug et al., ICLR 2024\n\n[11] Compositional Generative Modeling: A Single Model is Not All You Need. Du & Kaelbling, ICML 2025\n\n[12] Breaking Neural Network Scaling Laws with Modularity. Boopathy et al., ICLR 2025\n\n\n> Q3: Figure 3 is very interesting! Is Compositional support and Connected support both R^2 ~ 1 ?\n\nYes! We ensure for both the “Random” and “Balanced” task distributions that their support is compositional and connected. We revised the caption to better reflect this point."}, {"review_id": "sn46yEda13", "reviewer": "Reviewer_kRyd", "summary": "the paper focuses on compositional generalization -- a whether a model can generalize correctly to new combinations of previously learned components.\n\nThe paper makes a very bold statement: just giving a model larger scale in terms of parameters and data is sufficient to learn compositional tasks -- without any explicit priors (say architectural priors) to learn compositionality.\n\nThe main contributions are:\n\n- Empirical results that compositional generalization emerges naturally as standard MLPs grow larger, provided the training data covers the compositional space of the task family.\n\n- A theoretical result showing that MLPs can approximate compositional task families efficiently, with neuron count growing linearly rather than exponentially with number of modules.\n\n- introduction of \"linear decodability,\" a metric that quantifies the extent to which compositional structure can be recovered linearly from internal activations", "questions": "1) I am not a mathematician but it seems to me that the theoretical contribution is rather informal. It appears less precise compared to say the analysis in Schug et al. where explicit theoretical conditions (connected and compositional support) and identifiability conditions were  formally proven. Do you agree? If so, can you clarify/refine the theoretical contribution?\n\n2) As I wrote earlier, a major concern is that the tasks are rather synthetic and the results only applicable to MLPs. Can you at least provide some insights -- why do you think that the same major result would be true say for transformers and for much more complex real-world tasks? \n\n3) I like what you propose about \"linear decodability\". However, it is unclear what linear decodability exactly captures about the structure learned by the model—particularly how it relates or contrasts with previously introduced measures such as \"coverage\" or \"synonymic invariance\" (see the earlier links I gave you)\n\n4) You suggest that scale along is enough for compositional generalization. However the paper that you cite (Schug 2024) shows that too much overparameterization can affect negatively compositional generalization. I cannot quite reconcile these two results/observations. Can you help?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Quality\n+ The experiments are carefully designed, showing that scaling standard MLPs indeed leads to compositional generalization under some conditions. \n+ A theoretical contribution showing that the number of neurons required scales only linearly with number of modules.\n- Rather artificial/synthetic compositional tasks. Not convincing that the main result of the paper would have more broad applicability.\n- The theoretical result is derived under a single hidden layer assumption. Very unclear if this would capture well what's happening with deeper or more complex architectures such as transformers.\n\nClarity\n+ The paper is well organized, and for the most part easy to follow. \n+ very good figures and other visualizations.\n- I would prefer to see more clear/formal definitions early on about key concepts such as \"compositional support\"\n- it seems that the paper assumes that all readers are familiar with Kolmogorov complexity\n\nSignificance\n+ The paper challenges a very prevalent assumption about the necessity of specialized architectures or explicit inductive biases for compositional generalization.\n- is it true however? I am not convinced because both the theoretical result and the empirical results are for very simplified settings. So it may be that the main \"message\" of this paper is challenged by others in the next 1-2 years. Of course this is not necessarily bad -- this is how research progresses.\n- the paper could contrast its insights to recent theoretical frameworks (such as the Coverage Principle or the Random Hierarchy Model -- see references below) that define conditions for compositional generalization. \n\nOriginality\n+ If the main result is true, the paper is definitely making a very original contribution: \"scale is all you need\" for compositional generalization\n- There are a couple of highly relevant papers that are not cited and discussed by the paper:\nhttps://arxiv.org/pdf/2307.02129 (the random hierarchy model)\nhttps://arxiv.org/pdf/2505.20278 (the coverage principle)", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your very positive review of our work. We are happy that you appreciated the design of our experiments and the significance of our findings. Your feedback has prompted us to extend Theorem 3.1 to the multilayer setting, run additional transformer experiments and clarify several parts in our manuscript. We are looking forward to hearing your thoughts on our detailed response below.\n\n> The theoretical result is derived under a single hidden layer assumption. [...] what's happening with deeper or more complex architectures such as transformers.\n\nTheorem 3.1 is indeed stated for a hyperteacher with a single hidden layer but just to avoid any confusion, we do not make a single hidden-layer assumption for the MLP.\n\nTriggered by your remark, **we were able to extend Theorem 3.1 to the L-layer hyperteacher setting** and have added it to our updated version of the manuscript. The main finding remains unchanged in the multilayer setting: The number of neurons required by the MLP to approximate the hyperteacher to arbitrary precision scales linearly in the number of modules M. To see this intuitively, consider the construction where we dedicate M neurons in the MLP at each layer to “pipe through” the task constituents and repeat the machinery to implement a single hidden layer hyperteacher we show in the submission L times. What becomes more complicated in the L>1 setting is keeping track of the constants needed to bound the approximation error as a function of L but this has no impact on the scaling in M.\n\nIn addition, we see using MLPs in Theorem 3.1 as a strength. It allows extending the result to more complex architectures which are built with MLPs. The transformer is a straightforward example since it explicitly uses MLPs in its feedforward blocks and thus transformers as well can approximate the hyperteacher to arbitrary precision using a number of neurons linear in M.\n\n> I would prefer to see more clear/formal definitions early on about key concepts such as \"compositional support\"\n\nThank you for this suggestion. We agree that the paper becomes more self-contained and easier to understand by restating these definitions from [1] and [2] early on. Accordingly, we have made changes to the manuscript and now introduce the concept of compositional and connected support more comprehensively.\n\n> 1. I am not a mathematician but it seems to me that the theoretical contribution is rather informal. It appears less precise compared to say the analysis in Schug et al. [...]\n\nPlease allow us to clarify this important point. We present two formal theorems (Theorem 3.1 and Theorem 4.1) which we prove in Appendix A. Please let us know if you have any concerns about technical flaws but these are fully formalized proofs rather than informal arguments. We briefly summarize Theorem 3.1 in the following and contrast it with the analysis in [1] that you mention.\n\nFor Theorem 3.1, we provide a constructive proof in Appendix A.2 that shows that there exists a standard MLP that can approximate a general class of compositional task families to arbitrary precision using only a linear number of neurons with respect to the number of task modules. This is noteworthy because there are potentially exponentially many different tasks in such a task family and it is a priori not clear whether as a result an MLP solving all tasks would require an exponential number of neurons.\n\nWe can compare this result to Theorem 1 in [2] which considers a more constrained setting, namely that the student, i.e. the model learning the compositional task family, has the same architecture as the teacher. That means in this case the student has the same compositional structure as the data built-in and the question of compositional generalization becomes one of identifiability. Now given this stronger and arguably less general assumption, [2] shows a strong result that guarantees compositional generalization using only a number of tasks linear in the number of modules when the training loss is zero.\n\nSo ultimately this demonstrates a typical tradeoff when doing theory - stronger assumptions can allow for stronger implications at the expense of generality. We would like to emphasize that this does not make either theoretical contribution more or less formal. In our submission, we pursued a more general question, i.e. can standard neural networks compositionally generalize, and we complement the theoretical existence proof with empirical evidence that MLPs can compositionally generalize in practice.\n\n\n> 2. As I wrote earlier, a major concern is that the tasks are rather synthetic and the results only applicable to MLPs. Can you at least provide some insights [...]\n \nYou raise two good questions:\n\na) Do the results extend beyond MLPs, e.g. to transformers?\n\nAs we argued in our response above, our theoretical result obtained with MLPs can be extended to transformers. We agree however that this point deserves more attention, so in response to your remark (as well as reviewer bGxw), we conducted a new set of experiments and **reproduced the data scaling plot on the top of Figure 1 where we replace the MLP with a transformer**. We qualitatively reproduce our main result, i.e. as data is scaled, transformers achieve compositional generalization. Transformers are (maybe not unexpectedly) even more sample efficient than MLPs at doing so and to further emphasize this point we create a plot where we compare the minimum number of training tasks required to achieve compositional generalization (OOD R² > 0.95) as the total number of possible tasks is increased (by increasing the number of modules M). On this plot, both the MLP and transformer clearly require a sub-exponential (in fact, sub-quadratic) number of training tasks compared to the total number of tasks with the transformer consistently requiring less tasks than the MLP. We regret that this time around, we cannot show plots in the rebuttal but we have added these plots to the appendix of our updated version of the manuscript and refer to them in Section 3.1. Please let us know if there are specific numbers for these results you would like to see in tabular form here.\n\nb) How can synthetic tasks help us learn something about compositional generalization in real-world tasks?\n\nIn our submission, we consider a number of different settings to reduce the likelihood that our results are specific to any one of them: the hyperteacher and the compositional preference task family in Section 3, the compositional image generation task in Section 4.2. In response to Reviewer bGxw we have in addition reproduced findings of Section 3.1 for a deeper hyperteacher.\n\nThat being said, an inherent limitation to investigating compositional generalization is that one needs precise control over the data distribution (a) to be able to specify what is considered a composition on which to measure generalization and (b) to systematically hold-out data from training to measure it. This is what makes it tricky to investigate compositional generalization beyond synthetic tasks. We have extended our discussion of this crucial point in the limitations section.\n\n> 3. I like what you propose about \"linear decodability\". [...]\n\nThank you for pointing us to these references. “Synonymic invariance” as used in [4] refers to the property that certain parts of the input in the data represent the same value, e.g. two inputs x1, x2 to a non-injective function that are mapped to the same element f(x1)=f(x2). Similarly, coverage in [3] refers to the set of inputs observed during training and all inputs that are functionally equivalent to them. Both are closely related to what is called “substitutivity” in linguistics where replacing a word with a synonym should not change the meaning of the whole [5]. In the settings we consider here, there are by construction no “synonyms”, i.e. each module that we consider for composition has a different meaning.\n\nWe are instead focused on assessing whether seeing a subset of compositions during training can lead to generalization to all possible compositions even though these held-out compositions are specifically not “synonymic” but result in an unobserved task. Linear decodability then captures whether a model, despite only being exposed to compositions of modules, internally represents the ground-truth modules that make up each task.\n\n> 4. You suggest [...] However the paper that you cite (Schug 2024) shows that too much overparameterization can affect negatively compositional generalization. [...]\n\nThis is a very important observation and indeed part of what makes our results interesting. The theoretical teacher-student analysis conducted in [2] suggests that overparameterization might hurt compositional generalization since there are counterexamples to their Theorem 1 in this setting. As a result, compositional generalization cannot be guaranteed in their setting if the student is overparameterized. However, this does not mean that the student cannot compositionally generalize, such a solution still exists. The question then becomes which solution the optimization algorithm finds. Already the empirical results of [2] show that in practice overparameterization does not always hurt (e.g. see Figure 2D in [2]). In our setting, we train a standard MLP (or transformer, see above) with SGD, instead of using gradient-based meta-learning to train a linear hypernetwork as in [2]. It turns out that in this arguably more common setting, overparameterization is beneficial to compositional generalization.\n\n[1] Compositional Generalization from First Principles. Wiedemer et al., NeurIPS 2023.\n\n[2] Discovering modular solutions that generalize compositionally. Schug et al., ICLR 2024\n\n[3] The Coverage Principle. Chang et al., arXiv 2025\n\n[4] The Random Hierarchy Model. Cagnetta et al., Phys. Rev. X 2024\n\n[5] Compositionality decomposed: how do neural networks generalise? Hupkes et al., JAIR 2020"}, {"review_id": "70jDPx0JjL", "reviewer": "Reviewer_239V", "summary": "The paper investigates whether standard neural networks can achieve compositional generalization through scaling data and model size. It presents theoretical results suggesting that MLPs can approximate compositional task families with linear scaling in the number of neurons relative to task modules. Empirically, it demonstrates that scaling data and model size improves compositional generalization across various task encodings, provided the training distribution sufficiently covers the task space. The paper also shows that task constituents can be linearly decoded from hidden activations in models that successfully generalize compositionally, correlating this decodability with the success of image generation models.", "questions": "1. The paper mentions that underrepresentation of certain modules leads to failures. Could the authors provide more detailed analysis on the types of failures observed and potential strategies to mitigate them beyond ensuring sufficient training data coverage?\n2. How do the findings translate to real-world applications where task specifications are not fully known or are ambiguous? \n3. I believe that these conclusions should be verified in more popular scenarios, such as in language models, to see if given certain capabilities, a small model cannot solve a task but scaling up can. I suggest referring to the experiments in \"Compositional task representations for large language models\" to provide a more in-depth analysis.", "limitations": "yes", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 2, "significance": 1, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. The paper offers a formal definition of compositional tasks.\n2. The authors conduct experiments across different task encodings and model sizes. \n\nWeaknesses:\n\n1. This paper lacks novelty. The problem it explores—whether scale can bring about compositional generalization—is not of significant importance to the industry. Moreover, we cannot gain any insights from it.\n2. The writing quality of this paper is very poor. The figures are poorly made, and the typesetting and writing are subpar, making it difficult to follow.\n3. The motivation of this paper is not clear enough. The abstract mentions that “there are still frequent failure cases that raise doubts,” but there is no more specific support in the introduction section provided. Moreover, it is not clear why this would lead to a focus on the issue of scale.\n4. The scale of the experiments in this paper is too small and lacks a clear experimental setting. The experiments are not extensive enough and are not convincing.\n5. The writing of the related work section is also unclear, making it difficult to understand the positioning of this work within the field.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review. We regret that you did not find our main research question, whether scale leads to compositional generalization, to be of significance. We hope our response helps to clear up potential misunderstandings and better contextualize the work.\n\n> This paper lacks novelty. The problem it explores—whether scale can bring about compositional generalization—is not of significant importance to the industry.\n\n> The motivation of this paper is not clear enough. The abstract mentions that “there are still frequent failure cases that raise doubts,” but there is no more specific support in the introduction section provided. Moreover, it is not clear why this would lead to a focus on the issue of scale.\n\n> The writing of the related work section is also unclear, making it difficult to understand the positioning of this work within the field.\n\nWe have revised the introduction and related work section to more clearly work out the motivation and significance of our research question. In addition, please allow us to copy our response to Reviewer hkqv to summarize the main arguments:\n\nIt is a decades-old question whether connectionist models are fundamentally able to capture compositional structure [1-4].\nDespite the success of large-scale models, we observe many failure cases that indicate deficits in capturing compositional structure making this a valid concern still today [5-9] (also see section 4.2 for systematic failures of image generation models). \nA number of recent papers argue that for compositional generalization to succeed, neural networks need to be endowed with architectural priors for modularity [e.g., 10-12]. \nOur results challenge this position, since even monolithic neural networks (MLPs) can achieve compositional generalization at sufficient scale. We believe it is actionable since it assigns a stronger importance to the structure of the training data over the model architecture than was previously thought, i.e. it is important how diverse the training data is (section 3.1) and how the training data is distributed (section 3.4) but shows that compositional generalization can ultimately be achieved via scaling and does not constitute a fundamental shortcoming of connectionist models.\n\nWe refined these points in the introduction, related work section and discussion of the paper.\n\n[1] Connectionism and cognitive architecture: A critical analysis. Fodor & Pylyshyn, Cognition 1988\n\n[2] Connectionism, Constituency and the Language of Thought. Paul Smolensky, Meaning in Mind 1991\n\n[3] Systematicity in connectionist language learning. Hadley, Mind & Language 1994\n\n[4] Connectionism and the problem of systematicity. Phillips 1995\n\n\n[5] Faith and Fate: Limits of Transformers on Compositionality. Dziri et al., NeurIPS 2023\n\n[6] The Reversal Curse: LLMs trained on “A is B” fail to learn “B is A”. Berglund et al., ICLR 2024\n\n[7] Not All LLM Reasoners Are Created Equal. Hosseini et al.,  NeurIPS 2024 Workshop MATH-AI\n\n[8] Do Large Language Models Have Compositional Ability? An Investigation into Limitations and Scalability. Xu et al., ICLR 2024 Workshop ME-FoMo\n\n [9] GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models. Mirzadeh et al., ICLR 2025\n\n\n[10] Discovering modular solutions that generalize compositionally. Schug et al., ICLR 2024\n\n[11] Compositional Generative Modeling: A Single Model is Not All You Need. Du & Kaelbling, ICML 2025\n\n[12] Breaking Neural Network Scaling Laws with Modularity. Boopathy et al., ICLR 2025\n\n\n> The paper mentions that underrepresentation of certain modules leads to failures. Could the authors provide more detailed analysis on the types of failures observed and potential strategies to mitigate them beyond ensuring sufficient training data coverage?\n\nWe assume that you are referring to the intervention on the training task support we term “unpopular modules” in Section 3.3 and describe in more detail in Appendix C.2, let us know if you had something else in mind: \n\nPlease allow us to clarify that this is an experimental intervention designed to elucidate why module imbalance can lead to failures in compositional generalization, as has also been observed in prior work [1,2]. Together with the “popular modules” training task support ablation, we find that such failures are due to underrepresentation of certain modules among the training tasks. Modules being disproportionately overrepresented is not an issue in this case. We can illuminate the type of failure more closely by noting that tasks with unpopular modules still achieve a low training loss/high training R² score, indicating that the network has memorized the specific task compositions present in the training data without having learned the underlying modules which would be required to generalize to held-out compositions that contain these modules. \n\n[1] Attribute-Centric Compositional Text-to-Image Generation. Cong et al., IJCV 2025\n\n[2] Compositional Abilities Emerge Multiplicatively: Exploring Diffusion Models on a Synthetic Task. Okawa et al., NeurIPS 2023\n\n> How do the findings translate to real-world applications where task specifications are not fully known or are ambiguous?\n\nWe discuss this point openly in the limitations section. In the case of ambiguous or unknown task specifications, it is no longer possible to cleanly measure compositional generalization without confounding it with failures in task inference. Since compositional generalization is the subject of our submission, we do not consider this case.\n\n> I believe that these conclusions should be verified in more popular scenarios, such as in language models, to see if given certain capabilities, a small model cannot solve a task but scaling up can. I suggest referring to the experiments in \"Compositional task representations for large language models\" to provide a more in-depth analysis. \n\nThe suggested paper proposes a method for learning and combining soft-prompts for pretrained language models to perform new tasks. It uses the term compositional generalization in the sense that soft-prompts are composed from a fixed number of codebook vectors. The model is then evaluated on whether it can perform a particular task such as Natural Language Inference using a combination of soft-prompts optimized for performing other tasks such as Summarization, Sentiment Analysis or Question Answering [1]. Unfortunately, we see no direct connection to our current submission since it is not clear to us how these tasks allow assessing compositional generalization.\n\n[1] Multitask Prompted Training Enables Zero-Shot Task Generalization. Sanh et al., ICLR 2022."}]

Author rebuttal:

Meta-review: The submission focuses on the compositionality of tasks in the context of neural networks. More specifically, if/how a neural network with a continuous nature can model a task that has an inherent discrete and compositional structure. The submission addresses this question from an experimental and partially theoretical angle and reports interesting findings, such as scaling the data and model size can lead to a computational generalization. 

All reviewers approve of the paper and recommend acceptance (except one reviewer). The AC also agrees that the paper has clear merits and should appear in NeurIPS. One review was deemed to be of insufficient quality was disregarded in the decision-making process. The review was reported for disciplinary action due to poor quality and failure to engage with the process despite multiple requests.
Decision: Accept (spotlight)