Chunking as the result of an efficiency computation trade-off

Abstract

How to move efficiently is an optimal control problem, whose computational complexity grows exponentially with the horizon of the planned trajectory. Breaking a compound movement into a series of chunks, each planned over a shorter horizon can thus reduce the overall computational complexity and associated costs while limiting the achievable efficiency. This trade-off suggests a cost-effective learning strategy: to learn new movements we should start with many short chunks (to limit the cost of computation). As practice reduces the impediments to more complex computation, the chunking structure should evolve to allow progressively more efficient movements (to maximize efficiency). Here we show that monkeys learning a reaching sequence over an extended period of time adopt this strategy by performing movements that can be described as locally optimal trajectories. Chunking can thus be understood as a cost-effective strategy for producing and learning efficient movements.

Figure 1. Movements become more regular with learning.

(a) Reaching task. Monkeys move a cursor through 5 out-and-back reaches (10 elemental movements) between central and peripheral targets. White-filled circular cues indicate which target to capture. Each successful element is rewarded. (b) Hand trajectories: left, position; right, speed. Each trial is stretched to a duration of 5 s. Grey traces indicate single trials and bold coloured traces indicate mean traces. The coloured envelopes around the mean trace indicate one s.d. on either side of the mean.

Figure 2. Modelling chunks as locally optimal-control trajectories.

(a) Illustration of canonical HALT and VIA models: green, VIA points; red, HALT points. (b) Computing the trade-off between efficiency and the complexity of being efficient. Each grey dot represents one potential chunk structure plotted against its maximally achievable efficiency under the model, and the corresponding computational complexity. The red curve is the convex hull of these points, and represents the Pareto frontier of the efficiency–computation trade-off curve. (c) Kinematics (black) and minimum-jerk model (blue). Left: trajectories become more looped as monkeys optimize over longer horizons. Middle: speed traces. Initially, trajectory optimization appears to happen over several chunks. Later in learning, a smaller number of chunks reveal increasingly efficient movements. Right: the squared jerk of the kinematic data and the model suggest that the behaviour approaches the efficiency of the minimum jerk model after learning. (d) Goodness of fit (Pearson's correlation coefficient) between the speed profiles of the minimum-jerk model and the kinematic data (mean±2 s.e.m.'s) across days of learning.

Figure 3. Efficiency and costs are traded off.

(a) Over the course of practice, movement efficiency (negative normalized squared jerk; mean±2 s.e.m.'s) increases with the number of days. (b) The number of chunks (mean±2 s.e.m.'s) estimated by the model decreases with number of days. (c) Computational complexity (median±2 s.e.m.'s) of chunk structures, defined as the de novo cost of computation, increases with increasing chunk length due to longer planning horizons. ***P<0.001.

Figure 4. The effect of chunking on cumulative computational resources.

(a) The black trace shows the Pareto frontier representing the maximum efficiency that can be achieved for a given complexity. Thus, the greyed-out region cannot be achieved. This represents a fundamental trade-off between efficiency and computational complexity. The coloured paths illustrate potential learning strategies, with increasing budgets from scenarios 1 through 5 for the cumulative outlay of computation. (b) For different rates of efficiency improvement as a function of trials (curves E1–E3), we can compute the cumulative computation of each learning strategy. (c) Each learning strategy is associated with a different cumulative outlay of computation over the course of learning. These outlays are ranked for the three efficiency improvement rate curves (E1–E3), with rank 1 indicating the least cumulative outlay of computation and outlays increasing from scenario 1 to 5.

Figure 5. Monkeys learn to be efficient cost-effectively.

They first increase their efficiency at low computational complexity and then select chunk structures with greater complexity to achieve even greater efficiencies. Coloured dots show mean efficiency against computational complexity for each day of learning, going from blue to red over the course of learning. Error bars show 2 s.e.m.'s across trials in each day. Grey dots show all possible minimum-jerk model trajectories plotted against their respective computational complexity. The red trace shows the best achievable frontier of the efficiency–computation trade-off.

Figure 6. Optimization of efficiency at fixed computational complexity.

(a) Method to identify degenerate sets of chunk structures with the same computational complexity. (b) Efficiency improves for fixed computational complexity with learning. Top panel: monkey E; bottom panel: monkey F. The squared jerk decreases as a function of trials for each degenerate set of chunks (the eight most commonly occurring sets are shown). The number of chunks for each degenerate set is indicated in the header of each plot. The red fit line is the fixed effect model that is identical for each set and the yellow fit line is the mixed effect model that is unique to each set.