A rational model of function learning

Abstract

Theories of how people learn relationships between continuous variables have tended to focus on two possibilities: one, that people are estimating explicit functions, or two that they are performing associative learning supported by similarity. We provide a rational analysis of function learning, drawing on work on regression in machine learning and statistics. Using the equivalence of Bayesian linear regression and Gaussian processes, which provide a probabilistic basis for similarity-based function learning, we show that learning explicit rules and using similarity can be seen as two views of one solution to this problem. We use this insight to define a rational model of human function learning that combines the strengths of both approaches and accounts for a wide variety of experimental results. Keywords A rational model of function learningEvery time we get into a rental car, we have to learn how hard to press the gas pedal for a given amount of acceleration. Solving this problem-which is an important part of driving safely-requires learning a relationship between two continuous variables. Over the past 50 years, several studies of function learning have shed light on how people come to understand continuous relationships (Carroll 1963;Brehmer 1971;1974;Koh and Meyer 1991;Busemeyer et al. 1997;DeLosh et al. 1997;Kalish et al. 2004;McDaniel and Busemeyer 2005). It has become clear that people can learn and recall a wide variety of relationships, but demonstrate certain systematic biases that tell us about the mental representations and implicit assumptions that humans employ when solving function learning problems. For example, people tend to expect that relationships will be linear when extrapolating to novel examples , and find it more difficult to learn relationships that change direction than those that do not (Brehmer 1974;Byun 1995).Several models have been developed to understand the cognitive mechanisms behind function learning. These models tend to fall into two different theoretical camps. The first includes rule-based theories (e.g., Carroll, 1963, Brehmer, 1974, Koh and Meyer, 1991, which suggest that people learn an explicit function from a given family, such as polynomials (Carroll 1963;McDaniel and Busemeyer 2005) or power-law functions (Koh and Meyer 1991). This approach attributes rich representations to human learners, but has traditionally given limited treatment to how such representations could be acquired. A second approach includes similarity-based theories (e.g., DeLosh et al., 1997, which focus on the idea that 1194 Psychon Bull Rev (2015) 22:1193-1215 people learn by forming associations: if x is used to predict y, observations with similar x values should also have similar y values. This approach can be straightforwardly implemented in a connectionist architecture and thus gives an account of the underlying learning mechanisms, but faces challenges in explaining how people generalize so broadly beyond their experience. Most recently, hybrids of these two approaches have ...

Putting bandits into context: How function learning supports decision making.

“…Beyond that, it demonstrates a similar ordering of error rates to those of human learners across different tasks (McDaniel, Dimperio, Griego, & Busemeyer, 2009). Recently, Lucas et al (2015) proposed Gaussian process regression as a rational approach towards human function learning. Gaussian process regression is a Bayesian non-parametric model which unifies both rule-based and similarity-based theories of function learning.…”

“…From a psychological perspective, a GP model can in this way also be thought of as encoding "rules" mapping inputs to outputs. A GP can thus be simultaneously expressed as a similarity-based or rule-based model, thereby unifying the two dominant classes of function learning theories in cognitive science (for more details, see Lucas et al, 2015).…”

Communicating compositional patterns

“…We model human pattern description using a Bayesian inference over functions with a GP prior, an approach that has been successfully applied to a range of experimental and observational data (Griffiths, Lucas, Williams, & Kalish, 2009;Lucas, Griffiths, Williams, & Kalish, 2015;Schulz et al, 2019;Wu, Schulz, Speekenbrink, Nelson, & Meder, 2018). Given an observed pattern D = {x n , y n } N n=1 , where y n ∼ N (f (x n ), σ 2 ) is a draw from the latent function, the posterior predictive distribution for a new input x * is also normally distributed, where…”

Generalization as diffusion: human function learning on graphs

“…Most of function learning research has focused on how people learn a relationship between two continuous variables (Mcdaniel & Busemeyer, 2005;Lucas, Griffiths, Williams, & Kalish, 2015;. How much hot sauce should I add to enhance my meal?…”