On Potential-Based and Direct Movements in Information Spaces

Abstract

Learning can be portrayed as a movement in a state space. Potential-based explanations of such movements hold that learners use a gradient-descent process to minimize a potential function. Direct explanations hold that learning is specific to information for learning. This study contrasts specific hypotheses derived from these general approaches. Participants estimated the length of arrow shafts of Müller-Lyer displays. Experiment 1 shows that learning in this paradigm can indeed be portrayed as a movement in a state space. In Experiment 2 nonveridical feedback was used in such a way that the contrasted hypotheses predicted learning in opposite directions. No learning was observed. In Experiment 3, movements in the state space were observed with practice conditions in which one of the hypotheses specified a movement in the state space whereas the other did not specify any movement. A tentative explanation for these findings is that both hypotheses are partly correct. However, more than to the empirical findings for this particular task, the authors wish to draw attention to the distinction between potential-based and direct processes—a distinction that they consider of general importance for the understanding of learning.

Notes

¹In mathematical terms the relation between a potential function and its gradient can be expressed as follows: If a state space is defined with the coordinate variables x ₁, …, x_n , and the potential function is given by f(x ₁, …, x_n ), then the gradient, typically denoted by ∇ f, is given by ∇ f = − (). In other words, the gradient can be computed from the partial derivatives of the potential function with respect to the state variables.

²We have previously referred to this approach as gradient-based approach (Jacobs et al., 2009, p. 268; cf. Michaels et al., 2008).

³It might be interesting to mention the following: For ecologically representative tasks the assumption that information for learning exists should be expected to be feasible only if one assumes this information (a) to be extended over space and time and (b) to be apparently complex. Apart from this general expectation, in this study we prefer to make as few task-independent assumptions as possible about the nature of the I_i s and g_i s in the system of equations. For speculative and critical remarks concerning more general equations we refer the reader to Jacobs et al. (2009, p. 284). Also note that in some cases direct learning is better described with difference rather than differential equations (e.g., Equations 7, 9, and 11 in Jacobs et al., 2009).

⁴The potential function and vector field illustrated in Figure 3 were computed as means over all trials used in the practice phase of Groups 2 and 3 of Experiment 1. To effect trial-to-trial steps in the variable space, however, one should assume that observers use means detected over fewer trials, or perhaps even error values detected on a single trial. We refer the reader to Figures 7 to 9 of Jacobs et al. (2009) and the text related to these figures for a speculative attempt to reveal the number of trials that are used to detect information for learning and information for calibration. The present study does not address this question.

⁵In a preliminary experiment reported in a conference abstract (Jacobs, Ibanez, & Travieso, 2010), we found that potential-only practice similar to the potential-only practice used in the present experiment led to less movement in a variable space than practice with veridical feedback. This led us to conclude that learning this task is at least partly direct. A comparison of the results of Experiments 1 and 3 indicates that veridical feedback might have led to more change in variable use also in the present study. Even so, whereas the change in variable use after potential-only practice reached statistical significance in the current experiment, it did not reach significance in the experiment reported in the conference abstract.