Mark A. Gluck Rutgers University
Thomas J. Palmeri, Stephen C. McKinley
Indiana University
Paul Glauthier
Rutgers University
Robert M. Nosofsky
In recent years, there has been an avalanche of newly proposed models of category learning and representation. As such models grow increasingly more sophisticated, there is a need to develop increasingly more rigorous testing grounds to choose among them. Most previous attempts to test among alternative models have focused on the end-products of categorization by observing patterns of transfer data following an initial learning phase. In the spirit of developing more rigorous tests, there has been a renewed interest in understanding details of the category learning process (e.g., Estes, 1986; Estes, Campbell, Hatsopoulos, & Hurwitz, 1989; Nosofsky, Kruschke, & McKinley, 1992). Beyond simply predicting transfer data following the completion of category learning, the question is how well can alternative models predict patterns of classification during the entire learning sequence.
The purpose of our study was to collect a rich set of classification learning data that would provide a useful testing ground for the numerous models that have been proposed. An infinite variety of learning paradigms are available, but we hoped to collect some learning data that researchers might regard as fundamental. Although the ultimate goal is the development of a model that can account for all forms of classification phenomena, it seems worthwhile to focus initial efforts on primary and basic forms of classification learning data.
A classic study of category learning is the one reported by Shepard, Hovland, and Jenkins (1961), who studied the difficulty of learning six fundamental types of categorization problems. Their results proved to be highly diagnostic for ruling out various models of classification learning based solely on elementary principles of stimulus generalization and cue conditioning. As will be seen, their data continue to challenge current models of classification learning.
In the first part of our article we review the design and results of Shepard et al.'s (1961) elegant and influential study. We then report a partial replication and extension of their study. By using their basic paradigm, while collecting a more extensive data set, we provide a fundamental testing ground for formal models of classification learning. Finally, we begin the testing process by quantitatively fitting three formal models to the observed learning data: Anderson's (1991) rational model, Kruschke's (1992) ALCOVE model, and an extended version of Gluck and Bower's (1988b) configural-cue model.
The Shepard et al. (1961) tasks are examples of "rule-based" category-learning problems. There exist simple deterministic logical rules that allow one to classify all exemplars with perfect accuracy. By contrast, the computational models that we test in this article were developed primarily to account for the learning of fuzzy, ill-defined category structures. Nevertheless, we argue that it is important to understand how these models fare with data from simpler rule-based classification tasks. To the extent that these models are adequate for learning both rule-based and ill-defined structures, we may presume that a single unified process accounts for both types of learning. To the extent that the models apply to only one class of tasks, we might infer the existence of multiple learning strategies. This latter possibility would then give rise to further questions regarding how and why alternative learning strategies take precedence. Indeed, it is probably for the reasons just stated that current researchers have repeatedly reached back to the rule-based tasks of Shepard et al. (1961) as a canonical data base for evaluating models of human classification learning (Anderson, 1991; Estes, in press; Gluck & Bower, 1988b; Kruschke, 1992; Nosofsky, 1984).
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Insert Figure 1 about here
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The simplest category structure is the Type I problem.
Here, information about only one dimension (shape in the example
in Figure 1) is necessary to solve the problem. For Type II,
exactly two dimensions are relevant. In the Figure-1 example,
black squares and white triangles are assigned to Category 1,
whereas white squares and black triangles are assigned to
Category 2. Information about size is irrelevant to solving the
problem. Note that Type II is the exclusive-or problem along its
two relevant dimensions. The Type VI problem is the most complex
category structure, with all three dimensions being equally
relevant to solving the problem. Stating a logical rule for Type
VI in terms of values on each of the component dimensions amounts
to enumerating the stimuli in each of the categories.1 Finally,
Types III, IV, and V are intermediate in structural complexity
between Type II and Type VI. All three dimensions are relevant,
but to differing extents. These three types can be described as
"single dimension plus exception" structures. For example, for
Type V, squares are assigned to Category 1 and triangles to
Category 2, except the small white square is switched with the
small white triangle. We discuss the subtle structural
differences among the Type III, IV, and V problems later in our
article.
Shepard et al. (1961) measured the difficulty of learning these problem types in terms of the number of errors that subjects made until they reached a criterion. They found that Type I was the easiest classification to learn, followed by Type II, followed by Types III, IV, and V, which were about equal in difficulty, and finally Type VI. (This ordering of difficulty pertains to subjects' initial encounters with each problem type, not to some more intricate transfer effects that were also observed in their studies.)
Of particular interest was that the Type II problem was learned with fewer errors than were Types III, IV, and V, whereas a variety of models based on stimulus generalization and cue conditioning predicted the opposite ordering of difficulty. Recall that in the Type II problem only two dimensions are relevant, whereas in Types III-V all three dimensions are relevant. Shepard et al. (1961) suggested that for the Type II problem, subjects may have learned to focus attention on the two relevant dimensions, whereas for Types III-V, subjects had to spread attention across all three dimensions. Such a process might account for the observed ordering of difficulty. Support for this interpretation was provided in their original study and in additional theoretical analyses of Shepard et al.'s (1961) data reported by Nosofsky (1984) and Kruschke (1992). This theme of the role of selective attention in classification learning serves to highlight some of the model comparisons that we report later in our article.
Each of the 120 subjects was tested on 2 classification problems, for a total of 40 subjects per problem. All pairs of problems were tested equally often, and the order of problems within each pair was balanced across subjects. Subjects were given explicit instructions that the relevant rule and dimensions for the second problem were chosen independently of those that were relevant in the first problem.
The procedure for the learning of each problem was similar to the one used by Shepard et al. (1961). In the first and second block of eight trials, each stimulus appeared once in a random order. In each subsequent block of 16 trials, each stimulus appeared twice in a random order. (This procedure directly follows the one used by Shepard et al.) On each trial, a stimulus appeared on the screen, the subject classified it into Category 1 or 2, and feedback was provided. Learning continued until a subject reached a criterion of 4 consecutive sub-blocks of 8 trials with no errors, or for a maximum of 400 trials (25 blocks of 16 trials).
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As can be seen, the main results closely replicate the ones
reported previously by Shepard et al. (1961). The fewest errors
occurred for the Type I problem, followed by Type II, followed by
Types III, IV, and V, which were about equal in difficulty, and
finally Type VI. Pairwise t-tests showed that Type I was learned
with significantly fewer errors than was Type II, t(78) =3D 4.353,
p < .01; that Type II was learned with significantly fewer errors
than were Types III, IV, and V, average t(78) =3D 3.515, p < .01;
and that Types III, IV, and V were learned with significantly
fewer errors than was Type VI, average t(78) =3D 3.227, p<.01.
There were no significant differences among Types III, IV, and V.
Figure 3 shows the data separately for the first and second problems learned. As can be seen, there was an overall practice effect, with fewer average errors on the second problem than on the first problem. However, the overall ordering of difficulty for Types I-VI was the same for the first and second problems learned, and statistical tests yielded the same results as for the pooled data.2
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The average number of trials to criterion for Problem Types
I-VI was 44.0, 85.4, 121.6, 127.0, 133.8, and 189.2,
respectively. These results mirror the ones for average number
of errors, with Type I learned more quickly than Type II; Type II
learned more quickly than Types III, IV, and V; and Types III, IV
and V learned more quickly than Type VI. Across all 240 subject-problem combinations (120 subjects x 2 problems each), there were
only 6 cases of a subject failing to reach the learning
criterion: 4 in Type VI, 1 in Type III, and 1 in Type V.
In Problem Types I, II, and VI, all individual items have the same logical status, in the sense that their roles within the category structure are all logically the same. However, in Types III, IV, and V, all items do not have the same structural roles, and some may be easier to learn than others.
In Type IV, for example, Stimuli 1 and 8 can be characterized as a central members of their categories, whereas the remaining stimuli are peripheral members -- see Figure 1. (Recall that the Type IV structure can be described by a single-dimension-plus-exception rule. In the Type IV problem, a central member is one that always participates in the single-dimension rule and is never considered an exception, whereas peripheral members will sometimes serve as exceptions, depending on which dimension is used for the rule.) Likewise, in the Type III problem, Stimuli 1, 2, 7 and 8 can be described as central members and Stimuli 3, 4, 5 and 6 as peripheral members. And in the Type V problem, there are three distinct types of items: Stimuli 1 and 5 are central, Stimuli 2, 3, 6, and 7 are peripheral, and Stimuli 1 and 8 are exceptions. (The exceptions are those stimuli that violate the only single-dimension rule that is available for the Type V problem.)
We recorded detailed learning curves for each of these item types in Problems III, IV, and V, and found that for all problems, the central members were learned with fewer errors than were the peripheral members, whereas the exceptions had the most errors of all. Because the formal models that we test subsequently in our article all successfully predict this qualitative pattern of results, and the quantitative fits of the models to these data turned out not to be diagnostic, we do not consider the individual item-type data further.3
Because ALCOVE and the rational model have been discussed in detail in several previous articles (e.g., Anderson, 1990, 1991; Kruschke, 1992; Nosofsky & Kruschke, 1992), we only briefly summarize them here. A more extended discussion is provided of the elaborated configural-cue model.
A common assumption made when fitting all three models is that the stimuli are composed of three binary-valued dimensions, as illustrated by the structures in Figure 1. In other words, we assume that the psychological dimensions that compose the objects correspond directly to the physical dimensions. Furthermore, because assignment of physical dimensions to the logical structures that define each category was randomized in our experiment, the intrinsic salience of each logical dimension is assumed to be equal. Although some logical dimensions are more diagnostic than others for determining category membership, it is the models' job to learn these diagnosticities.
An important component assumption of ALCOVE is that selective attention processes can modify similarities among objects in the multidimensional space (Nosofsky, 1986). Selective attention is represented by a set of weights that "stretch" and "shrink" the space along its component dimensions. In general, ALCOVE learns to attend selectively to those dimensions that are relevant to solving a classification problem and to ignore those dimensions that are irrelevant.
Formally, ALCOVE is a three-layered, feed-forward connectionist network as illustrated in Figure 4. The input nodes code the values on the psychological dimensions that compose the stimuli. The hidden nodes represent locations in the multidimensional space in which the exemplars are embedded. Each hidden node corresponds to a unique training exemplar. The output nodes code the degree to which each category is activated. The input nodes are gated to the hidden exemplar nodes by dimensional attention-weights, and the hidden exemplar nodes are connected to the output nodes by a set of association weights.
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Insert Figure 4 about here
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When object i is presented, each exemplar node j is
activated by the function
aij =3D exp[- =FE =FE =E0m=FExim-xjm=FE], (=1)
where xim and xjm are the values of exemplars i and j on dimension m (constrained to be either 0 or 1 for the present binary-valued stimuli illustrated in Figure 1); =FE is an overall sensitivity parameter that scales distances in the psychological space; and =E0m is the (learned) attention weight on dimension m. This activation function assumes a city-block metric for computing distance in psychological space, and an exponential decay function for transforming distance into similarity (Shepard, 1987). Thus, the more similar an exemplar is to an input item, the greater is the activation of the hidden node that represents the exemplar.
The output of Category-node A is given by
OA =3D =FEaijwjA, (2)
where wjA is the (learned) association weight between exemplar node j and output node A, and the sum is over all exemplar nodes. The probability that item i is classified in Category A is given by
P(A=FEi) =3D (OA+b)/(OA+OB+2b), (3)
where b is a background-noise constant (Estes, in press; Nosofsky & Kruschke, 1992; Nosofsky, Kruschke, & McKinley, 1992).
ALCOVE learns the attention weights, =E0m, and association weights, wjA, on a trial-by-trial basis by means of backpropagation (Rumelhart, et al., 1986). The precise learning rules are derived and presented by Kruschke (1992).
In the present application of ALCOVE, there are four free parameters: the overall sensitivity parameter (=FE) for scaling distances in the space (Equation 1); the background-noise constant (b) for transforming category outputs into response probabilities (Equation 3); and learning rates, =FEw and =FE=E0, for updating the exemplar-category association weights and dimensional attention weights, respectively (see Kruschke, 1992).
In a previous theoretical analysis, Nosofsky (1984) showed that the exemplar-based context model correctly predicted the order of difficulty for the six types of problems in Shepard et al.'s study, as long as it was assumed that subjects came to optimize the distribution of attention weights over the dimensions that compose the exemplars. The ALCOVE model, which extends the context model in a connectionist framework, provides a mechanism by which these optimal weights can be learned, and Kruschke (1992) verified that ALCOVE indeed predicts the correct ordering. The model learns to attend to the single relevant dimension that defines the Type I problem, to split attention between the two relevant dimensions that define the Type II problem, and to distribute attention optimally among the three dimensions that are relevant for solving the Types III-VI problems. The present research represents the first attempt, however, to test ALCOVE's ability to quantitatively predict the classification learning data in Shepard et al.'s paradigm.
There is a mechanism in the rational model for computing the probability that membership in each cluster signals a given category label. Thus, when an item is presented, the probability of its category label can be computed by summing the probability that it belongs to each cluster, weighted by the probability that each cluster signals a given category label.
In the present situation, the baseline version of the rational model has three free parameters: a coupling parameter (c) that influences the prior probability for items to join pre-existing clusters or form new ones; a dimensional salience parameter (sD) for computing similarities between items and the central tendencies of the clusters; and a category-label salience parameter (sL). In Anderson's framework, the category labels assigned to stimuli are treated in the same manner as values on the other stimulus dimensions. However, because the category-label dimension has a special psychological status, it is reasonable to use a separate free parameter (sL) to represent that the salience of the category label may be distinct from the saliences of the other stimulus dimensions. In addition to influencing the similarity between items and clusters, the category-label salience parameter influences the estimated probabilities that individual clusters signal alternative category labels (for details, see Anderson, 1990, pp. 103, 105, 143).
To give the rational model additional flexibility for fitting data, we also provide it with a response-mapping parameter (r) for transforming estimated category-label probabilities to observed response probabilities. Let pA denote the rational model's estimated probability that item i signals category-label A. Then the actual probability that the subject makes response A is given by
PA =3D pAr/(pAr+pBr). (4)
When r=3D1, observed response probabilities match those that are estimated by the rational model, whereas when r>1, observed response probabilities will be more extreme (closer to 0 or 1) than the estimated probabilities. (See Maddox & Ashby [1993] for a previously successful use of such a response-mapping parameter in tests involving Nosofsky's [1986] generalized context model.)
It is critical to understand that providing the rational model with this response-mapping parameter generalizes Anderson's (1990, 1991) presentation of the model. If r=3D1, then the present model reduces to the version presented by Anderson (1990). Thus, this generalized model must fit the present classification learning data at least as well as the version without the response-mapping parameter. It is important to acknowledge that Anderson (1990, 1991) assumed only a monotonic relation between the estimated probabilities of the rational model and observed response probabilities. Nevertheless, it is critical to explore the ability of the model to yield good quantitative fits to data as well. Use of the present response-mapping parameter is a reasonable way to begin such an exploration.
In a previous theoretical analysis, Anderson (1991) showed that the rational model yielded fairly good qualitative predictions of the order of difficulty of learning the six problem types. Roughly, the model learns to classify items more efficiently when exemplars with the same category label are grouped into the same internal clusters, and exemplars with different category labels are grouped into distinct clusters. The Type I problem is learned most efficiently because all members of Category A are grouped into one cluster, and all members of Category B are grouped into a second cluster. The Type II problem is also learned efficiently, because only two distinct clusters are formed for the members of each category (e.g., pairs 1-2 and 7-8 form clusters for Category A -- see Figure 1). Types III, IV, and V are learned less efficiently, because, depending on the precise sequence of learning exemplars, there is often a "singleton" cluster that is formed which consists of only one exemplar, and this last exemplar takes a long time to learn. Type VI is learned least efficiently because the categories break up entirely into singleton clusters.
Although promising in the respects noted above, the qualitative predictions of the rational model presented by Anderson (1991, Figure 13) also differed in certain ways from the data observed in the present study. For example, in contrast to the present data, the advantage for Type II over Types III, IV, and V didn't emerge until around Block 5. Also, during the later learning blocks, performance on Type IV actually became worse than performance on Type VI. Finally, the overall level of performance on all of the problem types was considerably worse than observed in the present study. It remains to be seen whether or not a more exhaustive search of the parameter space will locate parameters that allow the rational model to provide adequate quantitative fits to the classification learning data in Shepard et al.'s (1961) paradigm.
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The activations on all input nodes are multiplied by the
connection weights currently existing in the network, which are
then summed to form outputs. Specifically, the output received
by Category-node A is given by
OA(t) =3D =FE ai(t)wiA(t), (5)
where wiA(t) is the weight on the connection that links input node i to output node A on trial t. An analogous expression is used to compute the output received by category node B. The probability of a Category A response is given by the same decision rule as used in ALCOVE (Equation 3).
All weights in the network are initialized to zero. Learning of weights takes place by using the delta or least-mean-squares (LMS) rule (Widrow & Hoff, 1960). When Category A feedback is presented, output node A receives a teaching signal of zA(t)=3D1 and output node B receives a teaching signal of zB(t)=3D0. (When Category B feedback is presented, the reverse teaching signals are provided.) The error at output node A is given by
=EBA(t) =3D zA(t)-OA(t), (6)
and likewise for the error at output node B. All weights in the network are then updated by using the rule
wiA(t+1) =3D wiA(t) + =FE=EBA(t)ai(t), (7)
where wiA(t) is the connection weight from input node i to output node A on trial t, and =FE is the learning rate parameter. In our present tests of the configural-cue model, separate learning-rate parameters are allowed for the connections from single nodes, double nodes, and triple nodes to the output nodes. The standard configural-cue model thus has four free parameters: the background noise constant (b) in Equation 3, and learning rates =FE1, =FE2, and =FE3 for single, double, and triple nodes, respectively.4
Gluck and Bower (1988b) used the configural-cue model to simulate learning of the six Shepard et al. (1961) tasks. A single learning rate was used for all weights. Regardless of the learning rate, the model predicted a consistent ordering of difficulty. Most important, the overall ordering late in learning was found to be I < III < II < IV < V < VI. With the exception of misordering Types II and III, therefore, the configural-cue model was fairly successful at accounting for the relative difficulty of the six problems. As explained by Gluck and Bower (1988b, pp. 187-189), the successes of the model derive from a combination of its configural representation of the stimuli, and the interactive, error-driven nature of the LMS learning rule, which learns associations for frequent and consistent cues more quickly than for infrequent and inconsistent cues.
Why, however, does the model fail to predict that people learn the Type II problem more quickly than Type III? Consider the structure of the Type II and Type III problems shown in Figure 1. There are no single-feature cues that are consistently associated with either category for either type of problem. For the Type II problem, there are only two double-feature cues that are consistently associated with Category A (black-square and white-triangle); whereas for the Type III problem, there are three double-feature cues that are consistently associated with Category A (small-black, large-square, and black-square). (An analogous set of cues with the same structure is associated with Category B.) Because more cue-configurations exist for classifying items in the Type III structure than in the Type II structure, the configural-cue model tends to learn the Type III structure more quickly.
Why do human subjects tend to learn the Type II structure more quickly? A likely reason is that in the Type II structure, both double-feature cues are defined over the same dimensions (i.e., color and shape for the example in Figure 1). By contrast, in the Type III structure, each of the relevant double-feature cues is defined over a different pair of dimensions (i.e., color-shape, size-shape, and size-color). Presumably, human subjects learn to attend selectively to dimensions when solving the classification problems, and this process facilitates the learning of the Type II problem. We hypothesized, therefore, that to improve the configural-cue model's ability to predict classification learning, mechanisms need to be added that will allow the model to learn to selectively attend to dimensions.
There are two key ideas that underlie the DALR model. First, whereas in the standard configural-cue model, the learning rates on each of the connections remain constant during training, in the extended model there is a meta-learning process that dynamically modifies each learning rate. Incorporating a dynamically modifiable learning rate provides a mechanism for adjusting the salience of individual cues. Specifically, the model learns to assign higher learning rates to those connections in the network that are highly diagnostic for solving a given problem. From a psychological perspective, this enhanced learning rule is analogous to adding an attentional component to the configural-cue model. Attention is conceptualized here as an enhanced tendency to change the strength of a given cue's associations. (See Frey & Sears [1978], Mackintosh, [1975], and Pearce & Hall [1980] for similar psychological ideas about the learning of cue-specific saliences and associabilities.)
A second key idea behind the DALR model is that it is provided with a "knowledge" of the underlying dimensional structure of the stimuli. In the standard configural-cue model, information about the dimensional structure of the stimuli is basically discarded from the input representation. For example, the model does not represent the relationship between black and white as being any different from the relationship between black and large. Each dimension value is represented by a unique input node that is either activated or not activated depending on its presence or absence in the input pattern.
It follows that even with the meta-learning process described above, there is still no form of dimensionalized attention learning in the standard configural-cue model, because separate learning rates can arise for each distinct configuration of dimension values. For example, separate learning rates can occur for the input nodes black-square and white-triangle, which are defined over the same dimensions.
Thus, we introduce a structural modification to the model by linking the learning rates for given dimensions and configurations of dimensions. For example, connections from the nodes for "large" and "small" share a single learning rate, allowing attention (in the form of a high learning rate) to be focused on the entire dimension of "size" rather than on specific dimension values. When the cue "small" is relevant to the classification task, the network increases the linked learning rate for size, thus assigning more attention to the cue "large" as well. Likewise, all connections from double-nodes defined over the dimensions of size and shape (large-square, large-triangle, small-square, small-triangle) share a single learning rate, and so forth.
We describe the formal implementation of this extended DALR model in the appendix. We use five free parameters to fit the DALR model to the classification learning data: an initial learning rate (=FE0) that applies to all connections in the network; a background noise constant (b) that is used in the decision rule (Equation 3); and three separate meta-learning parameters for adapting the learning rates on connections from single, double and triple nodes (=FE1, =FE2, and =FE3, respectively). These meta-learning parameters are used to dynamically adjust the learning rates on each of the (dimensionalized) connections in the network.
To fit each of the models, 100 random stimulus sequences were generated. The characteristics of each random sequence matched the constraints in our experimental design. For any given set of parameters, a model was used to generate predictions based on each random sequence. These 100 sets of predicted values were then averaged, and the averaged values constituted the predictions that were fitted to the observed data. The ALCOVE, configural-cue, and DALR models were fitted to the data by using a hill-climbing parameter-search routine. Various different starting configurations of the parameter values were used in these searches in an attempt to guard against local minima. The same hill-climbing routine was also used to fit the rational model. However, the rational model may be especially prone to local minima, because even small changes in parameter values can sometimes lead to dramatically different predictions from that model. (Small parameter changes can lead the model to form different clusters, resulting in markedly different behavioral predictions.) In a special attempt to guard against local minima for the rational model, therefore, a "grid search" was also used, which consisted of checking a huge number of possible combinations of parameter values. Both the hill climbing and the grid-search methods led to the same best-fitting parameters.
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Among the four models, ALCOVE provides the best overall
quantitative account of the learning data. ALCOVE accounts for
95.5% of the variance in the learning data for the six problems
taken together. The remaining three models, which provide
roughly the same overall fits as one another, account for an
average of 86.3% of the variance. The advantage for ALCOVE is
fairly consistent across all the problem types, as indicated by
the individual-problem SSDs that are reported in Table 2.
Furthermore, the advantage holds regardless of whether one adopts
the SSD or WSSD criterion of fit (see Table 2).
Visual comparison of Figures 6 and 2 reveals that ALCOVE does quite well at capturing the pattern of learning data. It predicts a clear advantage for the Type I problem, followed by Type II, followed by Types III, IV, and V, which are roughly equivalent, and finally Type VI, which lags clearly behind.
By comparison, inspection of Figures 7-9 reveals obvious shortcomings in the predictions of the remaining models. As expected, the standard configural-cue model (Figure 7) predicts incorrectly that the Type III problem is learned more quickly than the Type II problem. Furthermore, the model predicts only a slight advantage for Type II relative to Types IV and V, in contrast to the observed data. There is also not enough of an advantage for the Type I problem relative to the remaining problem types.
The extended DALR configural-cue model (Figure 8) improves on the standard model, at least in regard to the qualitative trends noted above -- but there is not sufficient improvement to yield a satisfactory quantitative fit. By incorporating the dimensionalized adaptive learning rates, the extended model correctly predicts the rapid learning that occurred for the Type I problem. In addition, it now predicts slightly better performance on the Type II problem than on the Type III problem. But the quantitative difference in performance between Types II and III is far too small in magnitude relative to what is observed in the data. We also tested versions of the DALR model in which separate initial learning rates were allowed for connections from single, double, and triple nodes, but these elaborated models provided only modest improvements in quantitative fit.
An intriguing question is why the DALR model fails to predict enough of an advantage for the Type II problem relative to Type III, despite the fact that a dimensionalized selective-attention mechanism has been added to that model. In an attempt to answer this question, we tracked the connection weights that were learned by the best-fitting version of the DALR model. As expected, for the Type II problem, the model learned to assign very large weights to the two perfectly diagnostic doublet-cues for each category. For example, using the illustration from Figure 1, the model learned to assign large weights to the connections linking black-square and white-triangle to Category A. No other connections received very much weight. Also as expected, for the Type III problem, the model learned to assign large weights to the three perfectly diagnostic doublet-cues for each category (black-square, large-square, and small-black for Category A). Because these doublet-cues are defined over different pairs of dimensions, however, the weights assigned to them were not as large as those for the Type II problem. Thus, this aspect of the DALR model worked precisely as anticipated.
A surprising result, however, was that for the Type III problem, the DALR model also learned to assign moderately large weights to connections from some of the singlet cues. Consider again Figure 1, and note that for the Type III problem, the individual dimensions of shape and color, although not perfectly diagnostic, are partially diagnostic of category membership. Three of the four squares are assigned to Category A, and three of the four triangles are assigned to Category B. Likewise, three of the four black objects are assigned to Category A, and three of the four white objects are assigned to Category B. Thus, overall performance is benefited if, in addition to attending to the perfectly diagnostic doublet cues, the model learns to attend to the partially diagnostic singlet cues. Indeed, the nature of the adaptive-learning rate mechanism in the DALR model is to assign higher learning rates to whatever cues are diagnostic of category membership. In a nutshell, then, a plausible explanation for why the DALR model learns the Type III problem nearly as efficiently as the Type II problem is because it takes simultaneous advantage of both the doublet-cue associations and singlet-cue associations that are relevant to solving that problem. We provide further discussion of the implications of this learning process in our General Discussion.
Like the configural-cue model, the rational model (Figure 9) also predicts better performance on the Type III problem than on the Type II problem, at least with the present parameter settings. Another shortcoming of the rational model is that learning on the Type I problem occurs too slowly, and there is not much separation between the Type I learning curve and the remaining learning curves. As we noted earlier, Anderson (1991, Figure 13) reported predictions for the rational model in which different parameter settings were used, and where performance on the Type I and Type II problems was markedly better than on the remaining problems. Unfortunately, with these alternative parameter settings, the rational model vastly underpredicts the overall level of performance that was observed in our experiment. (It also mispredicts other qualitative features of the learning data that we discussed earlier.) We have been unable to find parameter settings for the rational model that yield performance levels close to those observed in our experiment while at the same time correctly ordering the difficulty of the problem types. It may be that, like the configural-cue model, a form of dimensionalized attention learning needs to be added to the rational model to improve its quantitative predictions. In addition, perhaps an alternative mechanism for converting internal estimated probabilities to observed response probabilities would yield improved quantitative fits.
The attention-learning mechanism in ALCOVE is critical in allowing that model to correctly predict the data. A restricted version of ALCOVE arises by holding fixed the attentional learning rate at =FE=E0=3D0. This restricted model yields a fit that is far worse than that of the full model (SSD=3D.202, RMSD=3D.046, %Variance=3D85.3). It predicts worse performance on the Type II problem than on Types III and IV, and performance on Type II that is nearly as poor as on Type V. Learning on the Type I problem also proceeds too slowly. This conclusion about the importance of attention learning in exemplar-based models of classification agrees with earlier ones reached by Nosofsky (1984) and Kruschke (1992).
Our initial quantitative tests focused on ALCOVE, the rational model, and the configural-cue model. These models are among the leading candidates in the field today, but there have been few attempts to develop quantitative comparisons to test among them. When learning data from all six problem types were analyzed simultaneously, ALCOVE provided the best overall fit, yielding learning curves that nicely matched the observed data. A clear shortcoming of the configural-cue model and the rational model is that they failed to predict the magnitude of the advantage observed for the Type I and Type II problems relative to Types III-V and VI.
Because only a subset of dimensions is relevant for solving the Type I and II problems, it seems likely that some form of dimensionalized, selective-attention process is involved, and this process is well captured by the attention-learning mechanism in ALCOVE. It seems likely that the quantitative predictions of the configural-cue model and the rational model could be improved if they too incorporated forms of dimensionalized attention learning.
We made an attempt to incorporate a dimensionalized attention-learning mechanism in the elaborated DALR configural-cue model. Although this mechanism led to improvements in the qualitative predictions of the model, it still had quantitative shortcomings. An analysis of the pattern of connection-weight learning in that model indicated that a likely reason for its quantitative shortcomings is that the model learned to attend simultaneously to multiple configurations of cues (e.g., black-square, large-square, small-black, square, black). In other words, it attempted to take simultaneous advantage of all and whatever forms of diagnostic evidence were available in the stimulus patterns. It may be that people are more naturally inclined to learn to attend selectively to a more limited set of cues. If this suggestion is correct, then further improvements in the configural-cue model could probably be achieved by designing certain capacity limits into the attention-learning mechanisms. We leave this idea as a project for future exploration and research.
Another issue that arises is that the present models were fitted to averaged learning data, and not to the data of individual subjects. Although averaged data may sometimes obscure patterns observed at the individual subject level, we have good reason to believe that, in the present study, the averaged data are at least fairly reflective of individual-subject behavior. Before generating our averaged learning curves, we inspected histograms of individual subject errors on each of the problem types. These histograms appeared to be symmetric, unimodal, and with no extreme outliers. We believe that these histograms of individual-subject errors could be fitted quite well by simply allowing variability in the parameter values (e.g., the learning rates) across individual subjects. However, we leave this ambitious goal of actually fitting distributions of individual-subject behaviors as another issue for future research.
The learning tasks used in this study involved rule-based categories. In another recent theoretical investigation involving rule-based categories, Choi, McDaniel, and Busemeyer (1993) compared alternative network models on their ability to predict patterns of classification learning data in some concept-identification tasks reported by Salatas and Bourne (1974). Choi et al. (1993) found that, as long as one incorporated certain prior biases into the structure of the network, ALCOVE again provided an excellent description of the pattern of results. Thus, the applicability of ALCOVE in this domain of learning rule-based categories appears to have some generality. Whether similar patterns of results will be observed in situations involving more ill-defined categories remains an open question. Conceivably, alternative classification strategies operate under different learning conditions, and the configural-cue model and the rational model may show advantages in other domains.
A central claim of Shepard et al. (1961) was that their results implicate some form of abstraction or selective attention to dimensions during the course of classification learning. This attention-learning process is in some sense limited in capacity. Focusing attention on fewer dimensions allows for better performance on classifications that are defined over those dimensions. This same basic message echoes loudly in the present replication and extension of their study. Models of stimulus generalization and cue conditioning in human learning that do not incorporate some form of limited-capacity, selective-attention mechanism for dimensionalized stimuli have difficulty accounting for the pattern of results on these fundamental classification tasks.
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Adaptive-learning rate (ALR) methods are meta-learning algorithms for adapting step-size parameters (i.e. learning rates) during a learning process, which in the present case is the LMS rule. Consider, for example, one of the weights in a connectionist network and how it changes over time. If the weight changes are all in the same direction -- e.g., all increases -- this signifies that the step-size parameter is too small. The weight could reach its asymptotic value more quickly if it took larger steps. On the other hand, if the weight changes are in opposite directions -- e.g., first up and then down -- this signifies that the step-size parameter is too large. The basic idea behind current ALR methods is to adjust the step size according to the correlation between successive weight changes, with the goal of obtaining zero correlation. (If the correlation is positive, this signifies that weight changes have been in the same direction, whereas if the correlation is negative, this signifies that weight changes have been in opposite directions.) The particular mechanism used to form the dimensionalized ALR configural-cue model tested in this article was proposed by Sutton (1992a) and is known as the incremental delta-bar-delta (IDBD) algorithm.
Specifically, with this IDBD algorithm, there is a different learning rate =FEiA for each connection from input node i to output node A, and similarly for connections to any other output nodes. These learning rates change according to a meta-learning process. The standard learning rule (Equation 7) becomes
wiA(t+1) =3D wiA(t) + =FEiA(t+1)=EBA(t)ai(t), (A1)
where the learning-rate parameters are now indexed by trials, t+1. (The =FEiA are indexed by t+1 rather than t to indicate that their update, by a process described below, occurs before the wiA update.) To ensure that the learning rates remain positive, they are expressed and stored in the form =FEiA(t)=3Dexp[=FEiA(t)]. The IDBD algorithm updates the =FEiA by
=FEiA(t+1) =3D =FEiA(t) + =FE=EBA(t)ai(t)hiA(t)/=FE=FEiA(t), (A2)
where =FE is a positive constant, the meta-learning rate, and hiA is a memory variable associated with each connection from input node i to output node A. The memory variable indicates whether the recent errors associated with a given connection weight have been positive or negative. To indicate this information, the memory variable is initialized at zero and updated by:
hiA(t+1) =3D hiA(t)[1-=FEiA(t+1)a(t)]+ +
=FEiA(t+1)=EBA(t)ai(t), (A3)
where [x]+ is x, if x>0, else 0. The first term in the above equation is a decay term -- the product =FEiA(t+1)a(t) is normally zero or a positive fraction and this causes a decay of hiA towards zero. The second term increments hiA by the previous error, assuming that cue i was present in the input pattern [ai(t)=3D1]. The memory, hiA, is thus a decaying trace of the cumulative sum of recent errors associated with the presence of cue i (Sutton, 1992a).
Note from Equation A2 that if the present error, =EBA(t), matches the sign of the cumulative sum of recent errors, hiA(t), the learning rate on the connection from input node i to output node A will be increased. If the signs mismatch, the learning rate will be decreased. This algorithm, therefore, provides one way of instantiating the key idea behind ALR methods, namely, that of adjusting learning-rate parameters according to the correlation between successive weight changes.
As we discussed previously in the text, the DALR model introduces a form of dimensionalized attention learning by linking the learning rates for given dimensions and configurations of dimensions. As noted earlier, in the present experimental design, the configural-cue model has 26 input nodes. By linking learning rates, however, we reduce the number of unique learning rates from 26 to 7 for each of the two output nodes. (Unique learning rates occur on connections from nodes defined over the dimensions of size, shape, color, size-shape, size-color, shape-color, and size-shape-color). Note that each connection retains its own unique history variable, hiA; thus, there are 7 distinct learning rates and 26 history variables for each of the two outputs.