Lesson 17 established procedural memory as a system that improves with repetition, distinct from declarative memory. This lesson gives that improvement a precise mathematical shape — one of the most robust quantitative regularities in all of cognitive psychology.
Terms (standalone):
- The power law of practice: the empirical finding that performance time on a task decreases
with practice according to a power function of the number of trials:
T_N = T_1 × N^(−α), whereT_Nis the time to perform the task on trialN,T_1is the time on the very first trial, andα(alpha) is a task-specific learning-rate constant (typically between roughly 0.2 and 0.6 for real tasks). This holds astonishingly well across an enormous range of tasks — from cigar rolling to mental arithmetic to typing to playing Tetris — and across a huge range ofN, from a handful of trials to tens of thousands. - Diminishing returns: the direct consequence of the power-law shape — the absolute speedup
from one additional trial shrinks as
Ngrows. Going from trial 1 to trial 2 buys a large reduction in time; going from trial 1,000 to trial 1,001 buys almost nothing, even though the relative improvement per doubling of practice stays roughly constant (that constant-relative, shrinking-absolute pattern is the hallmark of any power law). - Log-log linearity: if you plot
log(T_N)againstlog(N), a power-law relationship produces a straight line — this is the classic diagnostic researchers use to confirm a power law is the right fit to practice data (as opposed to, say, an exponential curve, which would instead look straight on a plot with only the y-axis logged). - Why ACT-R predicts this, not just observes it (connecting to Lesson 10): ACT-R’s base-level activation formula for a chunk includes a term that sums decayed contributions from every past use of that chunk, and that summation — worked out mathematically — produces power-law-shaped learning curves as a direct consequence, not a separately bolted-on rule. The power law of practice isn’t just an empirical pattern ACT-R was built to match after the fact; it falls out of the same activation mechanism used for ordinary memory retrieval.
Part 1 — Numeric (above): compute a point on the curve
Using the simplified form T_N = T_1 / √N (i.e., α = 0.5), with T_1 = 12 seconds: what is
T_16, the time on the 16th trial?
Part 2 — Diminishing returns, made concrete
Using the same formula (T_1 = 12, α = 0.5), compute T_4, T_16, T_64, and T_256 — each is
16× the trial count of the previous. Look at the absolute drop in time between each consecutive
pair (T_4 → T_16, T_16 → T_64, T_64 → T_256). Does each 16×-more-practice step buy the same
absolute time savings, or does the savings shrink? Describe the pattern.
Part 3 — Reveal: what does NOT follow this curve indefinitely?
The power-law improvement in time cannot continue forever down toward zero — there’s a floor.
Name at least one concrete lower bound on how fast a human can perform a real skilled task (e.g.
typing, a sport, playing an instrument), and explain what kind of limit it is (motor/physical,
perceptual, or something else) — i.e., what stops the power law from predicting an impossible
result at very large N.
Part 4 — Connect to modern AI training
Neural network training (relevant to the ml track, and worth naming here since cognitive architectures and modern ML both show power-law learning curves) also famously shows power-law-like relationships between performance and training data / compute / model size — the so-called “neural scaling laws.” Given this lesson’s diminishing-returns logic, what does a power-law relationship between training compute and model performance imply about the cost of squeezing out each additional increment of performance, as a system gets more capable? Why might that matter for deciding when more training compute stops being worth it, versus other investments (better data, better architecture)?