Discovering Symbolic Cognitive Models from Human and Animal Behavior

We thank the reviewer for their careful review of our submission, their useful comments, and are glad they found it “exceptionally well-written and (surprisingly) easy to follow”, and are thrilled (and grateful) they “enjoyed reviewing the paper and learnt a lot”. Below, we address the main issues raised.

W1. “The authors only evaluate Gemini 1.5 Flash… the paper could be made stronger if more LLMs are tested.”

Our main intent with this work was to demonstrate that the use of LLMs with CogFunSearch was capable of producing state of the art results. An important consideration in methods that use LLMs for the mutation function in an evolutionary algorithm is the tradeoff between model quality (relating to expected performance improvement per sample) and model throughput (relating to the number of samples that can practically be drawn). We chose Gemini 1.5 Flash specifically because it was designed to provide a reasonable tradeoff between these concerns, rather than providing the absolute best quality samples. Although we did not explore the use of other LLMs systematically, we agree this would be a natural idea for pushing the performance of this approach even further.

W2. “For RNN results it's unclear (1) whether the authors trained the RNNs themselves, (2) what architecture they used, (3) what hyperparameters they tuned.”

We trained our own RNNs, using a GRU model (Chung et al., 2014) with a single hidden layer, and trained with early stopping. We performed a sweep over a set of possible hidden units ([1, 2, 4, 8, 16, 32, 64, 128]) and picked the value that gave the best performance. All the variants were trained with the Adam optimizer (Kingma & Ba, 2015) with a learning rate of 1e-3.

Thank you for raising this point, as we inadvertently missed including these details, and will be including them in the revised version of the paper.

W3. “recommend moving related work discussion to the main text.”

We agree, and will be following your suggestion to move the related work (or the most important parts of it) to the main paper.

C2. “connecting to the work that uses LLMs to propose hypotheses in other CS-related areas would be helpful to provide a comprehensive background to the general ICML audience”

Thank you for this suggestion and for the provided references. We agree it would be valuable to connect to this body of related work, and will be including it in our revised related work section.

C3 and C4

We will address these issues, thank you for pointing them out.

Q1. “Is Figure 2B necessarily a Bayesian program synthesis pipeline? It seems that the "prior" $\theta$ is not necessarily a distribution, but a point estimate. Is there any obstacle to generalize it to a distribution?”

We would like to clarify that $\theta$ is not a prior, but rather the parameters that are fit in the internal optimization process (Fig. 2). However, we do essentially maintain a "distribution" over programs, via our program database (Fig 2A). The prior distribution over these programs is governed by the score $\Omega$ of each program, which is proportional to the likelihood of it being sampled for evolution. When a new program is generated, this "prior" distribution is updated as a new program (with its score $\Omega$) is added to the database.

Q2. “L155 (right): Why is $c \in 0, \ldots ,n$ instead of $c \in 1, \ldots ,n$? Is it allowed to have a "null" choice?”

Thank you for pointing this out, it should be $c \in 1, \ldots ,n$, as the reviewer suspected. We will correct this in the revised version.

Q3. “If I understood correctly, CogFunSearch is a general framework and should work for general data science tasks that involve symbolic modeling. Are there anything specific things about cognitive modeling that make CogFunSearch particularly suitable for it?”

Our bilevel optimization arose from a common modeling choice in cognitive science, which is that programs capture across-subject patterns and parameters capture individual variations. For example, a cognitive modeling setup might assume or hypothesize that all subjects are executing the same RL algorithms, but each subject has a different learning rate and exploration parameter. In our case, CogFunSearch is set up to fit unique per-subject parameters in the inner loop, and programs are evolved based on the average across-subject score in the outer loop.

However, as the reviewer suggests, this can be applicable to other situations with similar setups. The core of our approach is using FunSearch to discover parameterized programs that can be fit to a dataset, which is applicable to other areas of data-driven scientific discovery.

We thank the reviewer for their careful review of our submission, their useful comments, and are glad they found our approach “novel” and yielding “new behavioral insights”.

“missing reference to [1].”

Thank you for pointing out this relevant work! We will cite it and include it in our discussion.

W1. “requiring programs to be differentiable may hinder the application to other settings.”

This is true. We considered it a reasonable limitation, given that the most popular models for these datasets are all differentiable. One could in principle use a gradient-free or hybrid optimizer for the inner loop (e.g. Acerbi & Ma, 2017), and this would expand the space of models that could be considered. However, this would affect the computational cost required for scoring the models. We will revise the manuscript to be clear about this limitation.

W2 & Q2a. “how do the relative complexity scores compare to a human expert judgement.”

Automatic complexity of programs is an open problem with multiple proposed solutions in the literature, which is why we use multiple methods. Halstead Difficulty is a classic and widely-used measure designed to capture the difficulty of constructing or parsing the program. Several studies do exist linking it to behavioral measures and subjective complexity (e.g. Curtis et al., 2006; De Silva et al., 2012; Gao et al., 2025), but to our knowledge it has not been thoroughly evaluated for Python. LLMs, having been trained on human data and optimized for human tasks, can also be considered a proxy for human judgments, though this approach also lacks thorough validation. For our programs, we find that these scores largely align (as shown in Figs 7, 12, 13).

Thoroughly validating these measures ourselves was beyond the scope of this work, but we do include example programs in the supplement so that readers can assess how they compare to their own intuitive notions of complexity.

W3. “CogFunSearch requires a significant amount of compute”

While this is true, it is not clear how to compare compute cost appropriately against alternative approaches. The baselines models considered represent the results of years of human effort, and were therefore clearly vastly more expensive to produce than the programs we generate automatically. We expect that advances in hardware, model efficiency, and alternative evolution techniques (our rejection sampling being a simple example of one) will result in reduced computational costs for applying approaches such as CogFunSearch.

W4. “symbolic models may contain errors or be overly complex, hindering readability.”

While discovered programs showed sufficient interpretability/readability to afford insights about learning strategies, it is true that there is room for improvement. Exploring tools for minimizing extraneous complexity or improving the readability of the discovered programs is a compelling direction for future work.

It is also worth mentioning that while the best-fitting models were more complex than human-discovered baselines, there were many discovered programs that were less complex but still outperformed the baselines.

Q1. “Are the set of parameters the same for all the programs? If so, how does one decide which parameters to use?”

All programs are provided with the same number of free parameters (10), but each program will use these differently (and some may even use fewer than 10).

As a concrete example: param[0] may be learning_rate for one program, and lapse_rate in another. Furthermore, each subject in a dataset can have its own value for each parameter, so learning_rate for rat 1 may be different than learning rate for rat 13, but the parameter will be used in the same way.

Q2b. “ optimal tradeoff between complexity and performance?”

This would depend on what the model will be used for. In some use cases, greater interpretability (possibly at the expense of accuracy) may be more desirable, and in others, the opposite may hold. One of the advantages of our approach is that it produces multiple programs along this efficiency frontier, enabling researchers to identify the program(s) that are useful to them. We expect a major use case for the tool will be to surface useful ideas about how to model the dataset, & that the best approach will be to examine multiple programs at different points along the complexity/performance tradeoff.

“How data-efficient is CogFunSearch compared to the RNN? For example, would CogFunSearch outperform the RNN if trained on only 10% of the subjects?”

While we have not run this experiment directly, our comparison on the human dataset, where RNN struggles with per-subject parameters, is an indication that CogFunSearch is able to find more generalizable solutions that are still accurate.

We thank the reviewer for their careful review of our submission and their useful feedback. We are glad the reviewer found the submission “well-structured, comprehensive, and methodologically rigorous”. Below we provide responses to some of the main concerns raised.

“Can researchers leverage these models to develop new methodologies?”

Computational cognitive models for tasks of this kind are widely used for interpreting neuroscience data (do the internal variables of the model have correlates in the brain?) as well as understanding the effects of causal experiments (does damage to particular brain regions affect behavior in ways that are similar to ablation of the models?) and naturally-occurring variability (do patients with particular psychiatric diagnoses differ systematically in their model parameters?)

A widely-held belief is that models which better capture behavior are more likely to be useful for these kinds of neuroscience applications. Exploring whether this is true for the models we have discovered here remains a direction for future work, but it is one we are optimistic about. This is especially true for the models discovered for the human bandit dataset, which fit data dramatically better than existing models and which contain structure qualitatively unlike the ones in those models.

“Do these models generalize to novel datasets, or do they merely excel in fitting existing data?”

In our evaluation we do cross-validation over held-out subjects. This can be considered as a form of generalization to new datasets, where the datasets come from new (held-out) animals performing the same task. Whether these models can generalize to entirely new tasks or different organisms is an interesting question for future work, but was not within the scope of our work here.

“Do the variable names in the high-performing models genuinely reflect their functional roles as implied by their nomenclature?”

Thank you for raising this question. We have manually inspected the top performing programs and found that, for the majority of cases, variable names are semantically meaningful with respect to their functional roles. We find that the meaningfulness of the variable names tends to correlate with the strength of the seed programs (i.e.starting from FullModel tends to result with more meaningful variable names versus starting from LowInfo).

“Lack of Comparison with Other AI Discovery Methods”

Our focus in this work was on whether LLM-based discovery methods could produce accurate, yet interpretable, models. We were particularly compelled to explore this due to the fact that LLMs are able to utilize the semantics of its inputs (such as variable names) to guide its generation. Some of the alternative approaches suggested would lack this semantic “understanding”. Nonetheless, we do agree it’s an interesting alternative to explore and will add a discussion of these ideas to our conclusions.

“Do you believe that similar methods could be successfully applied to other domains, such as physics, chemistry, or economic modeling?”

Yes! At the core of our approach is using funsearch to discover parameterized programs that can be fit to a dataset, and this can be applied to other areas of data-driven scientific discovery. One recent work in this vein is [1]. We will add a brief discussion of this extension in the final version of our manuscript.

“what characteristics of Symbolic Cognitive Models made this field particularly suitable for your approach”

It can be argued that this field has relied heavily on models fit to data of one type (behavior) to understand data of very different types (neural activity, psychiatry, etc). This field is in crisis: deep learning has made it possible to benchmark longstanding models and find out that they just do not fit very well. It's therefore both theory-poor but also well-equipped to make use of new theories should they appear. Increasingly, it is also data-rich. These considerations make it a good place for data-driven theory discovery to have outsized impact. Further, the semantic-grounding of LLMs (discussed above) and the human-interpretable nature of Python programs allow us to leverage, and go beyond, prior cognitive models.

[1] Grayeli et al., 2024, “Symbolic Regression with a Learned Concept Library”

We thank the reviewer for their careful review of our submission, their useful comments, & are glad they found our proposed methodology “particularly impressive”.

RNN trained on data from all subjects…rather than…separate RNNs…

To clarify: This concern applies only to the human bandit dataset. We find that per-subject RNNs highly overfit to the training data & evaluated poorly on validation data, likely due to the small number of sessions & trials per subject. We have revised the manuscript to include fits of RNNs to individual subjects’ data, with a cross-validation setup comparable to CogFunSearch. We sweep network hidden sizes, used early stopping to mitigate overfitting, & find that the avg normalized score is 40.27$\pm$0.50, which is considerably worse than the cognitive model baseline (55.55$\pm$0.64), the best discovered program (60.93 $\pm$0.63), and the multisubject RNN (61.83$\pm$0.81). For the rat dataset, RNNs were trained separately for each subject. For the fly dataset, both RNNs and CogFunSearch programs were fit to the entire dataset, since each fly performed only one session, so it was not possible to compute held-out per-animal scores.

does building computational process models contribute to…scientific understanding…?

We thank the reviewer for raising this important philosophical question, which we agree is highlighted by our work. Computational models developed by humans reflect human-developed theories about the mechanisms at play, and are traditionally seen as tools for exploring the implications of these theories. These models can be used to make quantitative predictions about behavior in different situations as well as different data modalities like neural recordings. They are also commonly used to understand differences in behavior, both experimentally-induced or naturally-occuring.

We believe that automatically-discovered models can play a similar set of roles, so long as the theories that they express are interpretable to scientists. They play an additional role in surfacing new ideas that might be different from those that would occur to researchers, as we discuss in section 4, as well as in accelerating their discovery.

How valuable is it…to dedicate years to mastering a skill that LLMs can now acquire with ease?

We agree that as technology evolves, the set of skills that will be valuable to learn will also evolve. We see methods like ours as adding to the toolkit available, but not necessarily as reducing the skillset required of our students. Many of the skills now required in order to build and use computational models are still required to interpret & use automatically-discovered models. We agree with reviewer fWgP that CogFunSearch very much does not mean “LLMs can substitute cognitive scientists”. Instead, we hope that by facilitating discovery of interpretable quantitative models we can accelerate scientific research & enable scientists to tackle increasingly ambitious challenges.

"Can a two-step task truly capture the richness of cognitive behaviors…?” and “the behaviors studied…are too low-dimensional…”

The bandit tasks we consider strike a nice balance of not being too simple (so there’s something to learn from them) & not being too complex (so we stand a chance of learning it). Indeed, cognitive scientists have struggled with these bandit tasks for decades, despite their apparent simplicity.

LLMs have already reached a level of intelligence sufficient to model the intelligent behaviors of other agents

It is worth clarifying that currently LLMs can’t directly model this data: we are using LLMs to create variants of Python programs which serve as the cognitive models within the framework of FunSearch’s evolutionary algorithm. It is also worth noting that these LLMs are leveraging the knowledge we have built over decades of study via the seed programs provided, & exemplified in the semantically-meaningful variable names used. Reviewer fWgp phrased our goal well: “the authors did not claim "LLMs can substitute cognitive scientists" but rather focused on the specific problem of symbolic cognitive modeling, & the experiments have shown convincing results that the discovered models are of high quality.”

“the findings suggest mainstream symbolic…models may be oversimplifying…in problematic ways.” … “If AI can so readily outperform…do these models…meaningfully advance our understanding…?”

We partly agree, since RNNs (believed to achieve ceiling predictability) easily outperform previous cognitive models. Our approach largely closes this gap and arguably addresses this shortcoming. We feel that the issue is not with constructing cognitive models themselves, but rather with how we were finding them. Our results demonstrate both that better models are possible, & a novel mechanism for searching for them.

“are models…exploiting shortcuts in datasets?”

All our models were validated on held-out subjects that were never seen during training.