Introduction

A longstanding issue across both science and instruction has been to understand how various aspects of an educational curriculum or training program influence learning acquisition and generalization. One such aspect, which has received a great deal of research attention, is the variability of examples experienced during training (Raviv et al., 2022). The influence of training variation has been studied in numerous domains, including category learning (Cohen et al., 2001; Posner & Keele, 1968), visuomotor learning (Berniker et al., 2014 ; Schmidt, 1975), language learning (Perry et al., 2010), and education (Braithwaite & Goldstone, 2015; Guo et al., 2014). The pattern of results is complex, with numerous studies finding both beneficial (Braun et al., 2009; Catalano & Kleiner, 1984; Roller et al., 2001), as well as null or negative effects (Brekelmans et al., 2022 ; Hu & Nosofsky, 2024; Van Rossum, 1990). The present study seeks to contribute to the large body of existing research by examining the influence of variability in visuomotor function learning - a domain in which it has been relatively under-studied.

Function Learning and Extrapolation

The study of human function learning investigates how people learn relationships between continuous input and output values. Function learning is studied both in tasks where individuals are exposed to a sequence of input/output pairs (DeLosh et al., 1997; McDaniel et al., 2013), or situations where observers are presented with a an incomplete scatterplot or line graph and make predictions about regions of the plot that don’t contain data (Ciccione & Dehaene, 2021; Courrieu, 2012; Said & Fischer, 2021; Schulz et al., 2020).

Carroll (1963) conducted the earliest work on function learning. Input stimuli and output responses were both lines of varying length. The correct output response was related to the length of the input line by a linear, quadratic, or random function. Participants in the linear and quadratic performed above chance levels during extrapolation testing, with those in the linear condition performing the best overall. Carroll argued that these results were best explained by a ruled based model wherein learners form an abstract representation of the underlying function. Subsequent work by Brehmer (1974),testing a wider array of functional forms, provided further evidence for superior extrapolation in tasks with linear functions. Brehmer argued that individuals start out with an assumption of a linear function, but given sufficient error will progressively test alternative hypothesis with polynomials of greater degree. Koh & Meyer (1991) employed a visuomotor function learning task, wherein participants were trained on examples from an unknown function relating the length of an input line to the duration of a response (time between keystrokes). In this domain, participants performed best when the relation between line length and response duration was determined by a power, as opposed to linear function. Koh & Meyer developed the log-polynomial adaptive-regression model to account for their results.

The first significant challenge to the rule-based accounts of function learning was put forth by DeLosh et al. (1997) . In their task, participants learned to associate stimulus magnitudes with response magnitudes that were related via either linear, exponential, or quadratic function. Participants approached ceiling performance by the end of training in each function condition, and were able to correctly respond in interpolation testing trials. All three conditions demonstrated some capacity for extrapolation, however participants in the linear condition tended to underestimate the true function, while exponential and quadratic participants reliably overestimated the true function on extrapolation trials. Extrapolation and interpolation performance are depicted in Figure 1.

The authors evaluated both of the rule-based models introduced in earlier research (with some modifications enabling trial-by-trial learning). The polynomial hypothesis testing model (Brehmer, 1974; Carroll, 1963) tended to mimic the true function closely in extrapolation, and thus offered a poor account of the human data. The log-polynomial adaptive regression model (Koh & Meyer, 1991) was able to mimic some of the systematic deviations produced by human subjects, but also predicted overestimation in cases where underestimation occurred.

The authors also introduced two new function-learning models. The Associative Learning Model (ALM) and the extrapolation-association model (EXAM). ALM is a two layer connectionist model adapted from the ALCOVE model in the category learning literature (Kruschke, 1992). ALM belongs to the general class of radial-basis function neural networks, and can be considered a similarity-based model in the sense that the nodes in the input layer of the network are activated as a function of distance. The EXAM model retains the same similarity based activation and associative learning mechanisms as ALM, while being augmented with a linear rule response mechanism. When presented with novel stimuli, EXAM will retrieve the most similar input-output examples encountered during training, and from those examples compute a local slope. ALM was able to provide a good account of participant training and interpolation data in all three function conditions, however it was unable to extrapolate. EXAM, on the other hand, was able to reproduce both the extrapolation underestimation, as well as the quadratic and exponential overestimation patterns exhibited by the human participants. Subsequent research identified some limitations in EXAM’s ability to account for cases where human participants learn and extrapolate sinusoidal function Bott & Heit (2004) or to scenarios where different functions apply to different regions of the input space Kalish et al. (2004), though EXAM has been shown to provide a good account of human learning and extrapolation in tasks with bi-linear, V shaped input spaces Mcdaniel et al. (2009).

Variability and Function Learning

The influence of variability on function learning tasks has received relatively little attention. The study by DeLosh et al. (1997) (described in detail above) did include a variability manipulation (referred to as density in their paper), wherein participants were trained with either either 8, 20, or 50 unique input-output pairs, with the total number of training trials held constant. They found a minimal influence of variability on training performance, and no difference between groups in interpolation or extrapolation, with all three variability conditions displaying accurate interpolation, and linearly biased extrapolation that was well accounted for by the EXAM model.

In the domain of visuomotor learning, van Dam & Ernst (2015) employed a task which required participants to learn a linear function between the spikiness of shape stimuli and the correct horizontal position to make a rapid pointing response. The shapes ranged from very spiky to completely circular at the extreme ends of the space. Participants trained with intermediate shapes from a lower variation (2 shapes) or higher variation (5 shapes) condition, with the 2 items of the lower varied condition matching the items used on the extreme ends of the higher variation training space. Learning was significantly slower in the higher variation group. However, the two conditions did not differ when tested with novel shapes, with both groups producing extrapolation responses of comparable magnitudes to the most similar training item, rather than in accordance with the true linear function. The authors accounted for both learning and extrapolation performance with a Bayesian learning model. Similar to ALM, the bayesian model assumes that generalization occurs as a Gaussian function of the distance between stimuli. However unlike ALM, the bayesian learning model utilizes more elaborate probabilistic stimulus representations, with a separate Kalman Filter for each shape stimulus.

Figure 1: Generalization reproduced patterns from DeLosh et al. (1997) Figure 3. Stimulii that fall within the dashed lines are interpolations of the training examples.

Overview Of Present Study

The present study investigates the influence of training variability on learning, generalization, and extrapolation in a uni-dimensional visuomotor function learning task. To the best of our knowledge, this research is the first to employ the classic constant vs. varied training manipulation, commonly used in the literature on the benefits of variability, in the context of a uni-dimensional function learning task. Across three experiments, we compare constant and varied training conditions in terms of learning performance, extrapolation accuracy, and the ability to reliably discriminate between stimuli.

To account for the empirical results, we will apply a series of computational models, including the Associative Learning Model (ALM) and the Extrapolation-Association Model (EXAM). Notably, this study is the first to employ approximate Bayesian computation (ABC) to fit these models to individual subject data, enabling us to thoroughly investigate the full range of posterior predictions of each model, and to examine the the ability of these influential models of function learning to account for both the group level and individual level data.

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