Fig 1.
Our framework for task decomposition accounts for the computational cost of planning towards subgoals—task decompositions should jointly optimize task performance and the computational cost of search.
We formalize this in three nested layers of optimization: Action-Level Planning solves for a plan to accomplish a subgoal, which has a computational cost. Subgoal-Level Planning constructs subgoal sequences that maximize reward and minimize computational cost. Task Decomposition selects subgoals based on their value in Subgoal-Level Planning. This figure was adapted from a figure published in [12] (License: CC BY 4.0).
Fig 2.
Choosing task decompositions that make planning more efficient.
(a-c) State-specific search costs of the algorithms. The depicted task requires navigating on a grid from the start state (green) to the goal state (orange) with the fewest steps. Each column corresponds to a different algorithm and demonstrates two scenarios—Top: Search cost without subgoals, Bottom: Search cost when using the path midpoint as a subgoal (blue). We define the search cost as the number of iterations required for the search algorithm to find a solution. Larger states were considered more often during the search algorithm, resulting in greater search costs. (d) Plot of search cost for Iterative-Deepening Depth-First Search (IDDFS), Breadth-First Search (BFS), and a Random Walk (RW) in the task depicted in (a-c), with and without subgoals. Use of subgoals results in decreased search cost for BFS and IDDFS, but not for RW.
Table 1.
Descriptions of Normative Algorithms and Heuristics.
Fig 3.
Comparing predictions of the (a) RRTD-IDDFS, (b) Betweenness Centrality, (c) Solway et al. (2014) [8], and (d) Tomov et al. (2020) [7] models.
State color and size is proportional to model prediction when using the state as a subgoal. (Top) The 10-node, regular graph from Solway et al. (2014) [8]. (Middle, Bottom) Two eight-node graphs selected from the 11,117 included in our analysis.
Fig 4.
Correlation matrix comparing model predictions.
For each graph, correlations between two models are computed on the per-state subgoal values, then averaged across the 11,117 simple, connected, undirected, eight-node graphs. We discard correlations when either of the two models predicts a uniform distribution over subgoals because the correlation is not defined in those instances. References: Solway et al. (2014) [8], Tomov et al. (2020) [7].
Fig 5.
Screenshots from the experimental interface.
The depicted graph is the same as the top left graph of Fig 10. (a) An example navigation trial. The current state has a green background and the goal state has a yellow background. Only the edges connected to the current state are shown. (b) The interface used to show all graph edges between navigation trials. There is no indication of past or future trials on this screen. State icons are only shown when the cursor is placed on them. (c) An example implicit subgoal probe. (d) The final post-task assessment with the teleportation question. All icons were designed by OpenMoji and are reproduced here with permission.
Fig 6.
Participants that choose a subgoal instead of the goal more often in Explicit Probe trials have shorter average path length, relative to the optimal path length (r = −0.29, p < .001).
Fig 7.
Visualization of behavior and model predictions on four eight-node graphs selected from the 30 graphs used for the experiment.
(a) State color and size is proportional to subgoal choice, summed across participants and probe types. Each participant responded to a total of 21 subgoal probes and the number of participants per graph, from the top graph to the bottom graph, was 28, 26, 25, and 26. Visualized models are (b) RRTD-IDDFS, (c) RRTD-BFS, (d) RRTD-RW, (e) Betweenness Centrality, (f) Solway et al. (2014) [8], and (g) Tomov et al. (2020) [7]. For model predictions, state color and size is proportional to model prediction when using the state as a subgoal.
Fig 8.
Comparison of statistical analysis using mixed-effects multinomial regression to predict subgoal choice behavior for each subgoal probe.
Log likelihood (LL) is relative to the minimum model LL for each probe. Larger values indicate better predictivity. References: Solway et al. (2014) [8], Tomov et al. (2020) [7].
Table 2.
Estimated coefficients with standard errors from hierarchical multinomial regression predicting subgoal choice.
Likelihood-ratio test statistics compare regression models to the null hypothesis of sampling subgoals uniformly at random.
Fig 9.
Comparison of two-stage choice models to predict participant choice behavior among optimal paths.
Log likelihood (LL) is relative to the minimum model LL. Larger values indicate better predictivity. References: Solway et al. (2014) [8], Tomov et al. (2020) [7].
Fig 10.
The 30 undirected, eight-node graphs that were used in the experiment.