$Department of Kinesiology, Penn State University, University Park, PA16801, ^#Taipei Physical Education College, Taipei, Taiwan

Presented at: International Conference on Complex Systems - October 25-30, 1998, Nashua, NH

A common feature of all complex system is that in one way or another they show a form of adaptation that we can call learning in a broad sense of the word. In the following we present a number of historical examples of human motor learning that suggests a power law behavior of performance as a function of practice time. There have been theoretical explanation attempt based on "chunking" of information that is learned algorithmically [Newell & Rosenbloom, 1981]. We want to re-examine the generic form of learning curves in the framework of non-linear complex systems where we would expect both exponential [Shaw & Alley, 1985] as well as power-law regularities in learning curves, depending on the dynamical state the system is in.

Already in the beginning of this century it was observed that there are strong regularities in the general shape of learning curves in different areas of human motor learning [Thurstone, 1919]. Without much theoretical foundation a number of different functions have been proposed to fit the data [Thorndike, 1927]. It turns out, however, that all these functions fall into two categories [Newell, et all. 1998]:

where A is a constant corresponding to the asymptotic performance parameter, B indicates the difference to the initial performance, g indicates the magnitude of the learning rate (generally g is negative) and therefore its inverse defines an intrinsic time-scale of the system. The constant n₀ allows taking into account for a shift in the start time (in the following we assume n₀ = 0). If we plot the logarithm of the performance function we obtain a linear graph with slope g :

2. Powerlaw functions: The learning rate is decreasing and there is no single time scale ("fractal scaling", self similarity) [Schroeder, 1991]. The general form of the performance function p(n) is given by:

The parameters have analogous meanings as in the exponential case 1. The main difference is that a does not play the role of a time-scale as g did but is an example of a "scaling exponent". For certain classes of complex systems scaling exponents play a central role and are "universal" in the sense that their value does not depend on system details. The exponent a can be estimated from the slope of the performance function (minus its asymptotic value) in a double logarithmic representation:

In his classic experiment Snoddy asked his subjects to trace a circuit of a 12-edge, star shaped path with one fourth inch width [Snoddy, 1926]. The direct vision of the tracing instrument and the hands were obstructed by a screen so that only the indirect mirror image of the tracing device and hands was available to the participants. The instruction to the participants was to "move around as fast as possible and avoid making contact". Each trial consisted of completing one circuit, and the ratio of 1000 over the sum of tracing time (T) and number of contact made (E) within each trail [1000/(T+E)] was used as the trial score and performance parameter. Figure 1 shows the experimental data scanned from the original figure.. They are from 100 participants practicing 20 trials per day, 4 days' performance. The performance parameter is plotted as a function of "practice time" which is given by the number of trials a subject has performed that task. In figure 1 we have plotted the results in two different representations:

Figure. 1. Two different representation of Snoddy's mirror tracing experiments [Snoddy, 1926]. Each data point represents the median performance of 100 participants practicing 1 circuit of mirror tracing task. The entire practice consisted of groups of 20 circuits, the circuits being separated by 1 minute interval and the groups by 24 hours interval. The first 10 circuits of the first group were not shown on the figure. In (a) we fit an exponential to each group and to the whole data set, in (b) we fit a power law function.

In the following example we approximate the power law P₇(n) = n^-0.7 (red line in the log-log plot of Figure 2) with the sum of two exponential functions with only real exponents g (see eq.(1)). The two amplitudes and exponents in Figure 2 are B₁= 0.3961, g ₁=-1.36 (blue) and B₂= 0.6038, g ₂=-0.161 (yellow)). We can see that the sum of the two exponentials (green) approximates the power law fairly well over a certain range of practice times.

Figure.2. Approxiation of a power law function P₇(n) = n^-0.7 (red) as the sum of two exponential functions E(n) = E₁(n) + E₂(n), where E₁(n) = 0.3961 e^{-1.36 n}, E₂(n) = 0.6038 e^{-0.161 n}. We observe a reasonably good approximation over seven trials.

As an example, let us consider a learning curve that measures errors E_n of consecutive trials and is characterized by a power law with exponent: E_n = E n^g. Here n represents trial number (time) and g the rates at which the errors are assumed to decrease with trial number. For each trial number we can approximate the power law by an exponential function by estimating the learning rate R(n) at that specific trial number n. Under the assumption that we are far away enough from the target so that we have a continuous improvement of the performance (i.e. E_n > E_n+1 , E_n - E_n+1 small compared to E_n) this can be done by dividing the difference in consecutive errors (E_n - E_n+1, performance increase due to learning) by the value of the error E_n at that trial.

If we do this for consecutive trials then we observe a systematic decrease not only of the error but also of the rate at which the error changes, the learning rate R(n). In the situation of the power law learning curve (E_n = E n^g) as shown in Figure 3 the learning rate R(n) decreases as:

4. Power Laws From Concatenated Exponentials.

In this example we show that one can approximate a power law by concatenated exponentials. This is a feature that will arise when different learning rates dominate in different phases of learning. Since we have a given (local) learning rate R(n) at any point of the learning curve, we can approximate a power law by a sequence of processes with fixed but decreasing rates. For instance we can interpolate an exponential function h_mn(t) between values of the learning curve E_m and E_n for times t between m and n. In Figure 4 we illustrate that process with the example of the power-law E_n = E ng from above with g=-0.7 (green) and exponential interpolation between consecutive trials, i.e. n-m = 1 (red) and n-m=2 (blue) for trials n,m < 10.

Figure.4: Log-Log plot of power law E_n = E n^g with g=-0.7 (red) and exponential interpolation between consecutive trials (red) and next to consecutive trials (blue).

. Although the class of exponential functions with real exponents does not constitute a basis set in function space (as opposed to, for example, trigonometric functions as Fourier components) they nevertheless can approximate power-law functions over a finite range with limited precision. To illustrate this case let us assume for the moment that we have N participants with exponential learning curves with fixed rates g_i > 0 centered at a mean rate g and with a Gaussian distribution of width Dg.

This means that for each participant "i" we have a learning curve: E_i,n = E_i e^g(i)n. If we average across participants then we observe an approximate averaged learning curve E_n = E e^gn. For large values of Dg, however, the observed averaged data can have a better fit with a power law than with an exponential. In Figure 5 we illustrate such an example. In summary, this simulation shows that averaging learning data across individuals that exhibit an exponential function with different exponents can lead to a power law for the collective function of change. To address the hypothesis of exponential functions being more likely to emerge in simple tasks, we have reanalyzed data from a study that was set-up to examine the time evolutionary features of learning a single biomechanical freedom timing task [Newell, Liu, & Mayer-Kress, 1997, Liu, Mayer-Kress, & Newell, 1998].

Figure 5: Log-Log plot of distribution of simulated exponential learning curves Ei(n) (dots) with average rate g = 0.7 and variance s ² = 0.25. The average E(n) was calculated over N = 1000 simulated participants with 10 trials each.