The conceptual basis of function learning and extrapolation: Comparison of rule-based and associative-based models

The purpose of this article is to provide a foundation for a more formal, systematic, and integrative approach to function learning that parallels the existing progress in category learning. First, we note limitations of existing formal theories. Next, we develop several potential formal models of function learning, which include expansion of classic rule-based approaches and associative-based models. We specify for the first time psychologically based learning mechanisms for the rule models. We then present new, rigorous tests of these competing models that take into account order of difficulty for learning different function forms and extrapolation performance. Critically, detailed learning performance was also used to conduct the model evaluations. The results favor a hybrid model that combines associative learning of trained input—prediction pairs with a rule-based output response for extrapolation (EXAM).     

Learning and extrapolating a periodic function

How people learn continuous functional relationships remains a poorly understood capacity. In this article, I argue that the mere presence of nonmonotonic extrapolation of periodic functions neither threatens existing theories of function learning nor distinguishes between them. However, I show that merely learning periodic functions is extremely difficult. It is only when stimuli are presented numerically, rather than as numberless quantities, that participants learn anything like a periodic function. In addition, I show that even then, people do not regularly extrapolate periodically. The lesson is that careful methodologies will be required to understand a psychological capacity that is as idiosyncratic as the learning of complex functions appears to be.     

What is learned in sequential learning? An associative model of reward magnitude serial-pattern learning

A computational model of sequence learning is described that is based on pairwise associations and generalization. Simulations by the model predicted that rats should learn a long monotonic pattern of food quantities better than a nonmonotonic pattern, as predicted by rule-learning theory, and that they should learn a short nonmonotonic pattern with highly discriminable elements better than 1 with less discriminable elements, as predicted by interitem association theory. In 2 other studies, the model also simulated behavioral "rule generalization," "extrapolation," and associative transfer data motivated by both rule-learning and associative perspectives. Although these simulations do not rule out the possibility that rats can use rule induction to learn serial patterns, they show that a simple associative model can account for the classical behavioral studies implicating rule learning in reward magnitude serial-pattern learning.     

Why people underestimate y when extrapolating in linear functions

E. L. DeLosh, J. R. Busemeyer, and M. A. McDaniel (1997) found that when learning a positive, linear relationship between a continuous predictor (x) and a continuous criterion (y), trainees tend to underestimate y on items that ask the trainee to extrapolate. In 3 experiments, the authors examined the phenomenon and found that the tendency to underestimate y is reliable only in the so-called lower extrapolation region--that is, new values of x that lie between zero and the edge of the training region. Existing models of function learning, such as the extrapolation-association model (DeLosh et al., 1997) and the population of linear experts model (M. L. Kalish, S. Lewandowsky, & J. Kruschke, 2004), cannot account for these results. The authors show that with minor changes, both models can predict the correct pattern of results.     

Parallels between remembering and predicting an object's location

Three experiments explore effects of rotational axis and velocity on observers' predictions regarding an object's future position, and begin to explore connections between this extrapolation task and representational momentum (RM). In general, observers accept as "correct" positions that are behind the correct extrapolated location, as shown in previous work (Cooper & Munger, 1993; Finke & Shyi, 1998). These prediction distortions are in the opposite direction of typical RM memory distortions, suggesting that these tasks are unrelated. However, new effects of axis for the extrapolation task are reported which parallel axis effects previously observed for RM and mental rotation tasks. Specifically, participants make larger negative distortion errors for axes of rotation that are not coincident with the viewer's coordinate system, as if these paths of rotation are harder to extrapolate. Effects of velocity and axis are examined for RM and extrapolation tasks, and the overall pattern of distortions supports a link. In particular, participants who misremember the location of an object as further along the implied path of motion also accept as "correct" positions relatively further along the extrapolated path of motion, suggesting that these apparently opposite types of errors are indeed related.     

Representation versus process in simplicity of serial pattern completion

This study deals with the question whether preferred extrapolations of serial patterns are determined by the extrapolation process or the representation of serial patterns. Three experiments are reported. The first deals with series of letters from the alphabet. Each series gave rise to extrapolations with varying frequencies, and we investigated how well these frequencies are predicted by the information loads, stemming from three models. The model of Klahr and Wallace (1970) is mainly concerned with the process of extrapolation. The model of Simon and Kotovsky (1963) describes both the process and the representation of extrapolated series. A third model is introduced here that exclusively deals with the representation of series. The outcome indirectly indicates that the preferred extrapolation is determined by the simplest representation of a series and not by the simplest process towards this representation. To strengthen the last conclusion two experiments are added dealing with bar graph stimuli. They are designed to experimentally disentangle representation and process complexity. In the second experiment the task was to judge the complexity of each series after having extrapolated the series. In the third experiment, subjects had to judge complexity of extrapolated series without the task to extrapolate the series. The predictions from the three models make plausible that the complexity judgments of second experiment reflect the process of extrapolation and the complexity judgments of the third experiment still reflect the representation of series. The latter results again support the assumption that the preferred extrapolation is determined by the simplest representation of a series.     

Learning a two-cue inference task when one of the cue-criterion relations is known

Learning two-cue probability learning (MPL) tasks when the functional relation between the first cue and the criterion was known to the subject was studied in a factorial experiment. Contrary to the predictions from a hypothesis sampling model assuming independent learning of each cue in MPL, the function form for the first cue was found to affect learning of the second cue, facilitating it when the functional relations were the same for both cues and inhibiting it when the functions were different. These results suggest that it is not possible to extrapolate findings from single-cue probability learning directly to MPL.     

Representation of the numerosities 1-9 by rhesus macaques (Macaca mulatta)

Three rhesus monkeys (Macaca mulatta) were trained to respond to exemplars of 1, 2, 3, and 4 in an ascending, descending, or a nonmonotonic numerical order (1-->2-->3-->4, 4-->3-->2--1, 3-->1-->4-->2). The monkeys were then tested on their ability to order pairs of the novel numerosities 5-9. In Experiment 1, all 3 monkeys ordered novel exemplars of the numerosities 1-4 in ascending or descending order. The attempt to train a nonmonotonic order (3-->1-->4-->2) failed. In Experiment 2A, the 2 monkeys who learned the ascending numerical rule ordered pairs of the novel numerosities 5-9 on unreinforced trials. The monkey who learned the descending numerical rule failed to extrapolate the descending rule to new numerosities. In Experiment 2B all 3 monkeys ordered novel exemplars of pairs of the numerosities 5-9. Accuracy and latency of responding revealed distance and magnitude effects analogous to previous findings with human participants (R. S. Moyer & T. K. Landaeur, 1967). Collectively these studies show that monkeys represent the numerosities 1-9 on at least an ordinal scale.     

The effects of positive versus negative feedback on information-integration category learning

A number of studies have shown that in category learning, providing feedback about errors allows faster learning than providing feedback about correct responses. However, these previous studies used explicit, rule-based tasks in which the category structures could be separated by a simple rule that was easily verbalized. Here, the results of the first experiment known to compare the efficacy of positive versus negative feedback during information-integration category learning are reported. Information-integration tasks require participants to integrate perceptual information from incommensurable dimensions, and evidence suggests that optimal responding recruits procedural learning. The results show that although nearly all of the full-feedback control participants demonstrated information-integration learning, participants receiving either positive-only or negative-only feedback generally used explicit, rule-based strategies. It thus appears that, unlike rule-based learning, consistent information-integration learning requires full feedback. The theoretical implications of these findings for current models of information-integration learning are discussed.     

Role of expertise and action in motion extrapolation from real road scenes

The tendency of participants to “remember” the stopping point of an event as being farther along in the direction of motion has been a topic of study for about 30 years. The purpose of the present experiment was to test the influence of two factors on motion extrapolation: (1) The involvement of participants in the action, and (2) their expertise in the domain of automobile driving. Participants viewed real driving scenes from the driver's point of view. They were divided into four groups depending upon their involvement in the action (more or less active) and their driving experience (inexperienced or experienced). In order to get half of the participants involved in the driving situation, they had to use a steering wheel to follow the contour of the road. The results showed that both of these factors increased motion extrapolation. The discussion deals with how the interrelationships between real road scenes, expertise, and action-related knowledge affect motion extrapolation.     

Applying an exemplar model to an implicit rule-learning task: Implicit learning of semantic structure

Studies of implicit learning often examine peoples’ sensitivity to sequential structure. Computational accounts have evolved to reflect this bias. An experiment conducted by Neil and Higham [Neil, G. J., & Higham, P. A.(2012). Implicit learning of conjunctive rule sets: An alternative to artificial grammars. Consciousness and Cognition, 21, 1393–1400] points to limitations in the sequential approach. In the experiment, participants studied words selected according to a conjunctive rule. At test, participants discriminated rule-consistent from rule-violating words but could not verbalize the rule. Although the data elude explanation by sequential models, an exemplar model of implicit learning can explain them. To make the case, we simulate the full pattern of results by incorporating vector representations for the words used in the experiment, derived from the large-scale semantic space models LSA and BEAGLE, into an exemplar model of memory, MINERVA 2. We show that basic memory processes in a classic model of memory capture implicit learning of non-sequential rules, provided that stimuli are appropriately represented.     

Persistent coordination patterns in a complex task after 10 years delay: Subtitle: How validate the old saying “Once you have learned how to ride a bicycle,you never forget!”

Motor learning studies have for a long time focused on performance variables (in terms of speed or accuracy) in assessing learning, transfer and retention of motor skills. We argue, however, that learning essentially resides in changes in coordination variables (in terms of qualitative organization of behavior) and that relevant tests for assessing the effectiveness of learning and retention should consider these variables. The aim of this experiment was to test the retention of a complex motor skill, after a long-term delay. Ten years ago, five participants were involved in an experiment during which they practiced for 39 sessions of ten 1-min trials on a ski-simulator. All participants volunteered for a retention test, ten years after, for one session of ten 1-min trials. Analyses focused on the oscillations of the platform of the simulator. Performance was assessed in terms of amplitude and frequency. Coordination was accounted for by an analysis of dynamical properties of the motion of the platform, and especially the nature of the damping function that was exploited for sustaining the limit cycle dynamics. Results showed a significant decrement in performance variables. In contrast, all participants adopted from the first trial onwards the coordination mode they learned 10 years ago. These results confirm the strong persistence of coordination modes, once acquired and stabilized in the behavioral repertoire. They also support the importance of coordination variables for a valid assessment of learning and retention.     

How drivers adapt to drive in driving simulator,and what is the impact of practice scenario on the research?

Adaptation is an important precondition for validity of experiments carried out using a driving simulator. Learning how to control a simulated vehicle imposes mental load on participants, which can potentially distract them from performing the main task and bias the results of experiments. Most researchers have a practice session before the main scenario to ensure participants have adapted. However, the practice scenarios vary greatly both in duration and form. Moreover, in almost all cases, it is difficult to verify that a participant is adapted at the end of the practice session. It was previously shown that using a power curve can mathematically model the learning pattern of subjects to steering and pedal control and can also help identify adapted and non-adapted subjects at the end of practice scenarios. The same concept is implemented in the current study to evaluate the appropriateness of such a methodology for a different practice scenario. The results showed that adaptation time and learning rate did not differ significantly between male and female participants. More importantly, statistical evidence verified that adaptation to a driving simulator is task-independent. The impact of the practice scenario on the main task was also analyzed, leading to the observation that during the experiment scenario participants tend to continue to improve the sub-skills they focused on during the practice scenario. Based on the results of these adaptation sessions, recommendations are provided to improve the quality of design for the practice scenario and to minimize its impact on the experiment scenario.     

On discontinuities in motor learning: a longitudinal study of complex skill acquisition on a ski-simulator

The qualitative behavioral reorganizations that occurred during the acquisition of a complex motor skill were examined. Five novice participants practiced for 39 sessions of ten 1-min trials on a modified version of the ski-simulator. Analyses focused on the motion of the apparatus platform, modeled as a self-sustained oscillator. At the beginning of the experiment, all participants adopted a behavior that could be modeled with a highly nonlinear stiffness function and a Rayleigh damping function. The behavior in the final part of the experiment was captured by a qualitatively different model, with a linear stiffness function and a van der Pol damping behavior. The transition from the initial to the final model was gradual and was marked in most cases by an abrupt increase of oscillation frequency. During the transition stage, the 2 damping behaviors seemed alternately exploited within each trial. The results are discussed in the framework of the dynamical systems approach to motor coordination and learning, considering motor skill acquisition as a phase transition.     

Stereoscopic perception with brief exposures

In this report we describe the results of an experiment in which we demonstrated that a powerful and compelling stereoscopic experience is elicited with very brief (< 1 msec) stimulus durations. The observers were highly successful in recognizing briefly flashed, stereoscopic, random-dot surfaces in the absence of monocular contours. The results are shown to be closely related to the range of depths for any stimulus form; however, the recognition thresholds were nonmonotonic as a function of disparity. Previous investigators have disagreed about the existence of a temporal threshold for stereopsis. We believe that prior findings suggesting that stereopsis cannot occur at short exposure durations are probably due to inadequate control of fixation disparity. Therefore, there is poor dichoptic image registration when a stereoscopic stimulus is presented. The present results also raise difficulties for any theory of stereopsis that requires eye movements.     

A test of foreign language acquisition: paired‐association learning

This experiment investigated college student difficulties with foreign language acquisition. We developed a word‐sound, paired‐associates learning test that discriminated 92% of the control participants and 95% of the experimental participants. The results are interpreted in terms of an associative model of foreign language learning disability. We discuss how to integrate the test into a larger diagnostic battery for predicting the facility with which a person will be able to learn a foreign language. Copyright © 2002 John Wiley & Sons, Ltd.     

Tracking the emergence of memories: A category-learning paradigm to explore schema-driven recognition

Previous research has shown that prior knowledge structures or schemas affect recognition memory. However, since the acquisition of schemas occurs over prolonged periods of time, few paradigms allow the direct manipulation of schema acquisition to study their effect on memory performance. Recently, a number of parallelisms in recognition memory between studies involving schemas and studies involving category learning have been identified. The current paper capitalizes on these findings and offers a novel experimental paradigm that allows manipulation of category learning between individuals to study the effects of schema acquisition on recognition. First, participants learn to categorize computer-generated items whose category-inclusion criteria differ between participants. Next, participants study items that belong to either the learned category, the non-learned category, both, or neither. Finally, participants receive a recognition test that includes old and new items, either from the learned, the non-learned, or neither category. Using variations on this paradigm, four experiments were conducted. The results from the first three studies suggest that learning a category increases hit rates for old category-consistent items and false alarm rates for new category-consistent lures. Absent the category learning, no such effects are evident, even when participants are exposed to the same learning trials as those who learned the categories. The results from the fourth experiment suggest that, at least for false alarm rates, the effects of category learning are not solely attributable to frequency of occurrence of category-consistent items during learning. Implications for recognition memory as well as advantages of the proposed paradigm are discussed.     

Relational learning with and without awareness: Transitive inference using nonverbal stimuli in humans

Learning complex relationships among items and representing them flexibly have been shown to be highly similar in function and structure to conscious forms of learning. However, it is unclear whether conscious learning is essential for the exhibition of flexibility in learning. Successful performance on the transitive inference task requires representational flexibility. Participants learned four overlapping premise pairs (A > B, B > C, C > D, D > E) that could be encoded separately or as a sequential hierarchy (A > B > C > D > E). Some participants (informed) were told prior to training that the task required an inference made from premise pairs. Other participants (uninformed) were told simply that they were to learn a series of pairs by trial and error. Testing consisted of unreinforced trials that included the non-adjacent pair, B versus D, to assess capacity for transitive inference. Not surprisingly, those in the informed condition outperformed those in the uninformed condition. After completion of training and testing, uninformed participants were given a postexperimental questionnaire to assess awareness of the task structure. In contrast with expectations, successful performance on the transitive inference task for uninformed participants does not depend on or correlate with postexperimental awareness. The present results suggest that relational learning tasks do not necessarily require conscious processes.     

How useful are corpus-based methods for extrapolating psycholinguistic variables?

Subjective ratings for age of acquisition, concreteness, affective valence, and many other variables are an important element of psycholinguistic research. However, even for well-studied languages, ratings usually cover just a small part of the vocabulary. A possible solution involves using corpora to build a semantic similarity space and to apply machine learning techniques to extrapolate existing ratings to previously unrated words. We conduct a systematic comparison of two extrapolation techniques: k-nearest neighbours, and random forest, in combination with semantic spaces built using latent semantic analysis, topic model, a hyperspace analogue to language (HAL)-like model, and a skip-gram model. A variant of the k-nearest neighbours method used with skip-gram word vectors gives the most accurate predictions but the random forest method has an advantage of being able to easily incorporate additional predictors. We evaluate the usefulness of the methods by exploring how much of the human performance in a lexical decision task can be explained by extrapolated ratings for age of acquisition and how precisely we can assign words to discrete categories based on extrapolated ratings. We find that at least some of the extrapolation methods may introduce artefacts to the data and produce results that could lead to different conclusions that would be reached based on the human ratings. From a practical point of view, the usefulness of ratings extrapolated with the described methods may be limited.     

How the Learning Path and the Very Structure of a Multifloored Environment Influence Human Spatial Memory

Few studies have explored how humans memorize landmarks in complex multifloored buildings. They have observed that participants memorize an environment either by floors or by vertical columns, influenced by the learning path. However, the influence of the building’s actual structure is not yet known. In order to investigate this influence, we conducted an experiment using an object-in-place protocol in a cylindrical building to contrast with previous experiments which used rectilinear environments. Two groups of 15 participants were taken on a tour with a first person perspective through a virtual cylindrical three-floored building. They followed either a route discovering floors one at a time, or a route discovering columns (by simulated lifts across floors). They then underwent a series of trials, in which they viewed a camera movement reproducing either a segment of the learning path (familiar trials), or performing a shortcut relative to the learning trajectory (novel trials). We observed that regardless of the learning path, participants better memorized the building by floors, and only participants who had discovered the building by columns also memorized it by columns. This expands on previous results obtained in a rectilinear building, where the learning path favoured the memory of its horizontal and vertical layout. Taken together, these results suggest that both learning mode and an environment’s structure influence the spatial memory of complex multifloored buildings.