Generalization in Distant Regions of a Rule-Described Category Space: a Mixed Exemplar and Logical-Rule-Based Account

Abstract

An important question in the cognitive-psychology of category learning concerns the manner in which observers generalize their trained category knowledge at time of transfer. In recent work, Conaway and Kurtz (Conaway and Kurtz, Psychonomic Bulletin & Review 24:1312–1323, 2017) reported results from a novel paradigm in which participants learned rule-described categories defined over two dimensions and then classified test items in distant transfer regions of the category space. The paradigm yielded results that challenged the predictions from both exemplar-based and logical-rule-based models of categorization but that the authors suggested were as predicted by a divergent auto-encoder (DIVA) model (Kurtz, Psychonomic Bulletin & Review 14:560–576, 2007, Kurtz, Psychology of learning and motivation, Academic Press, New York, 2015). In this article, we pursue these challenges by conducting replications and extensions of the original experiment and fitting a variety of computational models to the resulting data. We find that even an extended version of the exemplar model that makes allowance for learning-during-test (LDT) processes fails to account for the results. In addition, models that presume a mixture of salient logical rules also fail to account for the findings. However, as a proof of concept, we illustrate that a model that assumes a mixture of strategies across subjects—some relying on exemplar-based memories with LDT, and others on salient logical rules—provides an outstanding account of the data. By comparison, DIVA performs considerably worse than does this LDT-exemplar-rule mixture account. These results converge with past ones reported in the literature that point to multiple forms of category representation as well as to the role of LDT processes in influencing how observers generalize their category knowledge.

Appendix

Appendix A. Illustration of the learning-during-test summed-similarity computation

Here we illustrate the LDT-exemplar model’s computation of the overall summed similarity to category B of an item presented on trial 5 of the test phase. The example assumes that the subject classified the test items from test trials 2 and 4 into category B, while classifying the test items from test trials 1 and 3 into category A. The short-term memory strength of a test item presented n trials back from the current test item on trial 5 is given by αⁿ. The relative long-term memory strength for all exemplars presented during the training phase is denoted ltmstr.

Trial number	1	2	3	4	5
Memory strength	${\alpha }^{4}$	${\alpha }^{3}$	${\alpha }^{2}$	$\alpha$	-
Category response	A	B	A	B	-

Thus, in this example, the overall summed similarity of the trial-5 test item [i(5)] to all category B exemplars, S_i(5),B, would be given by.

$$S_{i\left(5\right),\mathrm B}=ltmstr\times{\sum}_{\mathrm j\in B}s_{i\left(5\right),j}+{\alpha^3\times s}_{i\left(5\right),i\left(2\right)}+\alpha{\times s}_{i(5),i(4)},$$

where ${s}_{i\left(5\right),j}$ denotes the similarity of test item i(5) to each training-phase exemplar j from category B; and ${s}_{i\left(5\right),i\left(k\right)}$ denotes the similarity of test item i(5) to the test item presented on trial k of the test phase.

Appendix B. Fits of mixed-rule and LDT-exemplar models to the data from Conaway and Kurtz’s (2017) Experiment 1

In their Experiment 1, Conaway and Kurtz (2017) used the same training phase as in their Experiment 2A. However, at time of test, subjects were presented only with items from the lower-right portion of the Fig. 1 stimulus space. Specifically, they created a 7 × 7 grid of 49 stimuli occupying the lower-right region. This region included the six training items from the training phase as well as all items from the Quadrant-X transfer region. A total of 30 subjects were tested. The relative proportion of subjects who made each number of reduced-category (B) response in Quadrant X is listed in the “Observed” row of the Quadrant-X panel of our Table

Table 5 Predicted and observed relative-frequency distributions for the quadrant-X transfer stimuli and the training stimuli during the test phase of Conaway and Kurtz’s (2017) Experiment 1

5. Conaway and Kurtz did not report a frequency-distribution of correct responses for the training stimuli during the test phase; however, based on results reported in their article and in their Table 1, a reasonable estimate is that their subjects achieved mean accuracy roughly 0.93 on the training stimuli during the test phase. As an approximation, we assume that 13 of the 30 subjects (p = 0.433) classified 5 of the training stimuli correctly and that 17 of the 30 subjects (p = 0.567) classified all 6 of the training stimuli correctly. These estimated values are listed in the “Observed” row of the training-stimuli panel of our Table 5.

We fitted the mixed-rule and LDT-exemplar models to these data using the same procedures as described previously. The models are the same as described previously, although the sequence of stimuli experienced during the test phase (and that serves as input to the models) is different than in our applications to Experiment 2A. The maximum-likelihood predictions from the models for both the Quadrant-X and training-stimuli distributions are reported in Table 5. The best-fitting parameters from the models and their maximum-likelihood summary fits are reported in the Table 5 note. (These parameter settings are merely illustrative as a wide variety of parameter settings achieve roughly the same overall quantitative fit to the data.) As can be seen from inspection of Table 5, both models provide reasonably good accounts of the observed data. Because this version of the mixed-rule model uses fewer free parameters than does this version of the LDT-exemplar model, the fit that it achieves is more parsimonious than the one from the LDT-exemplar model. The main take-home message, however, is that both models do a reasonable job of accounting for the observed data.

Appendix C. Further details regarding the DIVA model-fitting procedure and results

Following Conaway and Kurtz (2017, p. 1314), the training exemplars (A and B in Fig. 8) for DIVA were represented within a ± 1 space, and we used linear interpolation and extrapolation to set the dimension values for the transfer stimuli in the generalization quadrants. Also following Conaway and Kurtz (2017, p. 1314), for any given simulation DIVA was initialized with random weights in the range (− R, + R) and trained with a random presentation sequence of the training stimuli that satisfied the constraints of the experimental design (12 blocks of each of the 6 training stimuli, with each of the B training exemplars presented twice). Following training, the weights were held fixed, and the trained network was used to classify each of the items presented during the transfer phase. Specifically, following Kurtz (2007, p. 565), for any given item i, one computes the sum-squared-error between the output activations and the teaching signals (input-pattern values) of each set of category output nodes, SSE_A(i) and SSE_B(i); the predicted probability that item i is classified in category A is then given by

$$\mathrm{P}\left(\mathrm{A}|i\right)=\left[1/{\mathrm{SSE}}_{\mathrm{A}}\left(i\right)\right]/\left[1/{\mathrm{SSE}}_{\mathrm{A}}\left(i\right)+1/{\mathrm{SSE}}_{\mathrm{B}}\left(i\right)\right].$$

(C1)

The predicted response-frequency distributions were then built using the same procedures as already described for the exemplar and rule-based models.

We fitted DIVA using a three-phase procedure. In the first phase, we used a coarse-grained grid search across the free parameters in the model, with the grid specified in Table

Table 6 Grid of parameters used in the initial coarse-grained grid search for fitting DIVA to the Experiment-1 data sets

6. In the second phase, we conducted a more fine-grained grid search with the finer grid centered around the best-fitting parameters from the coarse-grained search. Third, using the best-fitting parameters from the fine-grained grid search as starting values, we used the Hooke and Jeeves parameter-search algorithm to search for the final set of best-fitting parameters. In addition, we used 50 other random starting configurations of the parameters in these Hooke and Jeeve searches. The best-fitting parameters from the model for the three data sets are reported in Table 6.

Some comments are in order regarding the relation between our DIVA-fitting results and the presentation from Conaway and Kurtz (2017). First, in their Fig. 4, Conaway and Kurtz present a schematic, illustrative DIVA network that predicts that subjects will make predominantly reduced-category (B) responses in quadrant X. The authors acknowledge that the values depicted in the network are purely illustrative. However, the activation value for the illustrative hidden node is set at − 1, which is impossible for the logistic activation function (the activation value will be bounded between 0 and 1); therefore, it seems that an actual DIVA network would not be able to achieve this illustration. Second, in their Fig. 3, Conaway and Kurtz (2017) report results from simulations of DIVA showing that there is a small space of parameter values that allow DIVA to predict at least six (of nine possible) reduced category responses in quadrant X. It does not follow from this simulation, however, that there are parameter settings in DIVA that allow it to simultaneously predict high accuracy in the training region and the patterns of bimodal response-frequency distributions exhibited in the four generalization quadrants.