The learning curve: implications of a quantitative analysis

Abstract

The negatively accelerated, gradually increasing learning curve is an artifact of group averaging in several commonly used basic learning paradigms (pigeon autoshaping, delay- and trace-eye-blink conditioning in the rabbit and rat, autoshaped hopper entry in the rat, plus maze performance in the rat, and water maze performance in the mouse). The learning curves for individual subjects show an abrupt, often step-like increase from the untrained level of responding to the level seen in the well trained subject. The rise is at least as abrupt as that commonly seen in psychometric functions in stimulus detection experiments. It may indicate that the appearance of conditioned behavior is mediated by an evidence-based decision process, as in stimulus detection experiments. If the appearance of conditioned behavior is taken instead to reflect the increase in an underlying associative strength, then a negligible portion of the function relating associative strength to amount of experience is behaviorally visible. Consequently, rate of learning cannot be estimated from the group-average curve; the best measure is latency to the onset of responding, determined for each subject individually.

Fig. 1.

Group-average (n = 20 pigeons) rate of key pecking (the conditioned response in the pigeon autoshaping paradigm) as a function of number of sessions (with 50 trials per session). Coordinate frame and jagged data line were traced from Gamzu and Williams (figure 2 in ref. , p. 227). We have superposed a Weibull function approximation (smooth curve), to show that this function, y = A {1 – 2^{^} – [(x/L)^{^}S]} can capture the kind of prolonged increase seen in these averages. A is the asymptote and L is the onset latency or location (the value of x at which y is half of its asymptotic value). Note the value (1.4) of the shape parameter S, which determines the shape and steepness of the function.

Fig. 2.

The cumulative number of pecks versus the number of trials for the nine birds in Condition CR_CS6_IT9.

Fig. 3.

Two examples of Pecks versus Trials plots. The dashed curve in each panel is the best-fitting Weibull function. (Upper) The subject did not respond at all for ≈40 trials; then, within the space of ≈10 trials, it transitioned to making between 5 and 15 pecks on each trial. These data are summarized fairly well by the best-fitting Weibull function. (Lower) The subject did not respond at all for the first 30 trials; then, it began to make between zero and three pecks per trial. This pattern of weak and highly intermittent responding persisted for 600 more trials. Although the plot is visually confusing, the Weibull function again captures the structure of the data. The asymptote is at 0.5 pecks per trial because the subject did not peck on substantially more than half the trials. The function rises with step-like abruptness, because after the first trial on which there was a peck (Trial 30), there was no further increase in the weak and intermittent pecking tendency. In fact, there was a modest decrease after Trial 200.

Fig. 7.

Determination of the dynamic interval, by using either the Weibull representation or the steps representation. This data set had the lowest value (0.49) for the S parameter of the Weibull function among the 105 data sets analyzed. Nonetheless, the initial rise is very rapid because the onset latency is so short (seven trials). The dynamic interval based on the Weibull representation is the number of trials between the first and ninth decile. The dynamic interval based on the Slopes representation is the number of trials between the first upward change point and the change point at which the postchange point slope is >80% of the asymptotic rate.

Fig. 4.

Best-fitting Weibull functions for the Pecks versus Trials plots of the nine subjects whose data were first shown in Fig. 2 (smooth curves). The heavy jagged line is the group-average pecks per trial.

Fig. 5.

In this illustration, the algorithm for finding change points is applied to the cumulative record as of Trial 27. (In practice, it is applied iteratively to each successive point in the cumulative record.) In this record, there were no pecks until Trial 20, where pecking began. The slanted dashed line is a straight line drawn between the origin and the cumulative record at end of Trial 27. The cumulative record deviates maximally from this straight line between Trials 19 and 20, so that is the putative change point. It divides the record up to Trial 27 into two portions: Trials 1–19 and Trials 20–27. If the change point is accepted as valid, then the algorithm begins over again, with the pecks on Trial 20 as the first datum.

Fig. 6.

Different representations of the course of the acquisition of key pecking by a pigeon in a Pavlovian appetitive conditioning experiment. Data are either the number of pecks on each trial (left axis) or the cumulative number of pecks (right axis). The first representation is by means of a Weibull function fit to the pecks per trial (dotted curve). The second is by means of the slopes of the straight lines connecting successive change points in the cumulative record (dashed sequence of steps). These slopes are the average pecks per trial between two change points. The change points, as found by the change-point algorithm, are superposed on the cumulative record (▪).

Fig. 8.

The rate of pecking between the first and second change points (determined by using t test and criterion = 2) as a fraction of the asymptotic rate of pecking, on a log scale, plotted as a function of the trial at the first change point.

Fig. 9.

A random sample of the diverse but commonly seen ups and downs in the level of conditioned responding after its first appearance. The change points that generated these plots were detected by using the change-detecting algorithm, with a t test and a decision criterion (logit) of 6, which corresponds to a P value of <0.000001. Significant, substantial, and long-lasting decreases in performance are often seen. In other words, conditioned performance is asymptotically unstable.

Fig. 10.

Cumulative records of conditioned blinks from Kehoe's data. (A) Records of the most abrupt and steadiest rabbits. (B) The more variable records from Kehoe's data.

Fig. 11.

Spatial learning. (A) Three examples of cumulative correct choices in a plus maze (solid curves, plotted against the left axes) as a function of elapsed trials, with significant change points superposed (small circles). The slopes between the change points (heavy dashed lines) are plotted against the right axes (probability of correct choice). The chance level of performance (0.33) is indicated by the thin dashed lines. The thin vertical line at Trial 20 indicates the end of the pretraining period, during which the bait was randomly relocated from trial to trial. Data were from D. Smith and S. Mizumori. (B). Three examples of cumulative efficiencies for mice in a water-maze paradigm. The efficiency is the straight line distance from the point of placement to the platform divided by the distance swum. For reference, each plot also has lines with slopes equal to the group mean on the first trial and the group mean + 2 SE.