SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this appendix, we explore whether similar results might have been obtained using classic neural network architectures, either by constraining the size of the network [2] or by fitting a larger network and considering only the top few principal components of its activity space. We consider networks using the popular Gated Recurrent Unit (GRU) architecture [9], because related work has reported good performance with this architecture on similar datasets [2, 16] and because having only a single type of recurrent unit makes them simpler to analyze.
First, we explore quality of fit for GRU networks of different sizes, ranging from one to fourteen hidden units, fit to each of the five datasets (table 1). We find that for the synthetic datasets, "tiny" networks with just one or two recurrent units do provide a quality-of-fit that is only slightly less than that of larger networks. This raises the possibility that these networks might be able to discover the true model structure for these synthetic datasets. For the rat datasets, we find that larger networks provide a meaningfully better quality-of-fit. This indicates that very small networks do not quantitatively fully capture the structure present in the rat datasets, though it does not rule out the possibility that they might nevertheless discover scientifically-useful model structures.
Next, we inspected the fit of the "tiny" two-unit GRU networks to our three synthetic datasets, examining plots of their update rules (Figure A4). We see that these typically do not have a 1:1 relationship with the true generative latent variables. The exception is the Q-Learning agent, for which two-unit GRUs do often discover a disentangled solution. Solutions found for the Actor-Critic and Bounded Accumulator datasets are fully entangled, with each unit's update dependent on the value of the other unit and on both input variables. We interpret this to mean that very small conventional networks can discover dynamics that recapture those of certain generative processes, but that they do not do so reliably.
Finally, we inspected the fit of larger ten-unit GRU networks to our three synthetic datasets (Figure A5). We summarize their dynamics by plotting update rules for the first two principal components. While some interpretable patterns are apparent, there is still not a 1:1 mapping between PCs and
the latent variables of the generative process. The dynamics are entangled, with each PC's update depending on all inputs and on the value of the other PC.
Taken together, we interpret these results to indicate that conventional neural networks like GRUs can be a viable route to cognitive model discovery in some circumstances, but also that they have important limitations. One limitation is that, while task training ensures that the dynamics they contain are sufficient to solve the task, nothing ensures that all aspects of these dynamics are necessary (they are free to retain null-space dynamics). Another limitation is that, while the number of latent variables can be constrained by limiting network size or by only considering the top few PCs, nothing ensures that the update rules for these variables are sparse, and nothing encourages them to be "axis aligned", mapping 1:1 onto the true generative dynamics. Note that the "tiny RNNs" approach of Ji-An et al., [2] introduced additional architectural changes to the networks, which may help address some of these limitations.
Hidden	Q-Learning	Actor-Critic	Bandit Rats	Accumulator	Clicks Rats
1	-0.72	-1.38	-2.16	-	-0.36
2	-	-	-	-0.008	-
3	0.002	0.01	0.34	0.001	0.08
4	0.003	0.02	0.26	0.003	0.07
5	0.002	0.02	0.39	0.004	0.08
6	0.002	0.02	0.38	0.004	0.09
7	0.002	0.02	0.35	0.004	0.08
8	0.002	0.02	0.46	0.004	0.10
9	0.002	0.02	0.44	0.004	0.11
10	0.002	0.02	0.38	0.004	0.05
11	0.002	0.02	0.35	0.004	0.10
12	0.002	0.02	0.51	0.004	0.10
13	0.001	0.02	0.53	0.004	0.10
14	0.001	0.02	0.50	0.004	0.13
Table 1: Model Comparison: GRUs of Different Sizes Difference in cross-validated normalized likelihood (percentage points) between GRU networks with different numbers of hidden units. Score for each synthetic dataset is the average of three networks fit with different random seeds. Score for the rat two-armed bandit dataset is the average of three networks for each of 20 rats. Score for the rat Poisson clicks dataset is the average of three networks for each of 19 rats. Scores are reported as the difference between the average score for that combination of dataset and network size and a reference size for each dataset (two units for Q-Learning, Actor-Critic, Bandit Rats, and Clicks Rats; one unit for Bounded Accumulator)
GRU Trained on Q-Learning Dataset
GRU Trained on Actor-Critic Dataset
Update Rule: Unit Two
GRU Trained on Bounded Accumulator Dataset
Update Rule: Unit Two
Figure A4: Update Rules for Two-Unit GRUs Fit to Synthetic Datasets. The update rules for the GRU fit to the Q-Learning dataset are strikingly similar to the true update rules of the Q-Learning agent (Figure 3). The update rules for GRUs fit to the Actor-Critic and Bounded Accumulator datasets are quite different from the true update rules of these agents.
GRU Trained on Q-Learning Dataset
Update Rule: PC One
GRU Trained on Actor-Critic Dataset
Update Rule: PC One
Update Rule: PC Two
GRU Trained on Bounded Accumulator Dataset
Update Rule: PC Two
Update Rule: PC One
Update Rule: PC Two
Figure A5: Update Rules for Top Two PCs of Large GRUs fit to Synthetic Datasets. For all three networks, the update rules for the top two PCs are quite different from the true update rules of the corresponding generative agents.