# Topological Reachable-Set Coverage Scheduler

- ID: 2082
- Canonical URL: https://synthcore.org/idea/2082/topological-reachable-set-coverage-scheduler
- API JSON: https://synthcore.org/api/idea/2082.json
- API Markdown: https://synthcore.org/api/idea/2082.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.10332](https://arxiv.org/abs/2608.10332)
- Category: training
- Solves: stability, sample-efficiency, generalization
- ML areas: optimizer, training-dynamics, rl, world-model
- Math tags: control-theory, dynamical-systems, geometry, optimization
- Ratings: usefulness 7/10; difficulty 6/10; novelty 7/10

## Idea description

Use the reachable-safe-set viewpoint to make training data generation adaptive: maintain an approximation of the states reached by the current neural policy, identify boundary regions with weak barrier margin, and sample there until the set is sufficiently covered. This replaces random rollout expansion with a measurable coverage condition that can support finite-sample safety claims.

## Mathematical statement

Let R_theta denote the set of states reachable under policy pi_theta from the initial-state set over the training horizon. Let r_theta(x) be the CBF residual and suppose it is L_r-Lipschitz on R_theta. A finite sample set S is a delta-net of R_theta if every x in R_theta has a sample x_i in S with distance at most delta. If all samples have residual at least epsilon, then every reachable state has residual at least epsilon - L_r delta. Define the empirical covering radius as delta_hat = max over validation states x of the distance from x to its nearest training sample. Training or certification stops only when delta_hat is below epsilon divided by the estimated L_r.

## Key formulas

- R_theta = {x(t; x_0, pi_theta) : x_0 in X_0 and 0 <= t <= T},    C = {x : h(x) >= 0}
- delta(S, R_theta) = sup over x in R_theta of min over x_i in S of ||x - x_i||
- min over x_i in S of r_theta(x_i) >= epsilon and r_theta is L_r-Lipschitz implies inf over x in R_theta of r_theta(x) >= epsilon - L_r delta(S, R_theta)
- delta(S, R_theta) < epsilon / L_r implies r_theta(x) > 0 for every x in R_theta

## Implementation notes

1. Integration point: attach the scheduler to offline policy training or differentiable model-predictive rollouts. Store state-action trajectories, barrier residuals, and a k-d tree over training states. 2. Pseudocode: initialize a seed set; train the policy for several epochs; generate a large validation cloud of reachable states; for each validation state compute nearest-training-sample distance and barrier residual; select states with high distance, low residual, or both; add them to the training set; retrain; stop only when delta_hat is below epsilon divided by estimated L_r. A practical priority score is q(x) = distance_to_nearest_sample + gamma times max(0, epsilon - r_theta(x)). 3. The theory supplies the margin relation and threshold epsilon/L_r. Estimate the reachable set, coverage radius, and L_r empirically, and report conservative upper bounds. For high-dimensional states, use a latent metric only when a decoder reconstruction-error bound is available. 4. First cheap experiment: use a two-dimensional pendulum or planar robot with a known safe region. Compare random sampling, uniform initial-state sampling, and adaptive sampling at equal simulator-call budgets. Measure maximum dense-grid violation, estimated coverage radius, minimum sampled residual, and training-set size. The predicted signature is monotonic reduction in violation as delta_hat decreases, with a transition near delta_hat = epsilon / L_r. Adaptive sampling should reach zero violation with fewer trajectories than random sampling.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: inconclusive
- Verdict: Built a readable 2D reachable-set coverage scheduler with random and adaptive/farthest sampling, exact KD-tree coverage measurement, and reproducible sweeps. The Lipschitz safety bound held in 80/80 checks, residual-gap scaling was linear in L with constant gap/L=0.141421, and adaptive sampling reduced median coverage radius and reached certification substantially sooner for feasible thresholds. This validates the mechanism in the controlled toy setting, but not yet in a learned neural-policy or physical rollout setting.

### Mechanism check

- Verdict: Built a readable 2D reachable-set coverage scheduler with random and adaptive/farthest sampling, exact KD-tree coverage measurement, and reproducible sweeps. The Lipschitz safety bound held in 80/80 checks, residual-gap scaling was linear in L with constant gap/L=0.141421, and adaptive sampling reduced median coverage radius and reached certification substantially sooner for feasible thresholds. This validates the mechanism in the controlled toy setting, but not yet in a learned neural-policy or physical rollout setting.
- Confidence: 8/10
- Limitations: The experiment uses a known static 2D unit-square reachable set and an analytically Lipschitz residual on a finite grid. It does not test a learned policy, simulator-generated trajectories, estimated reachable-set error, noisy residuals, high-dimensional metrics, latent reconstruction bounds, or confidence intervals; the low-margin score was inactive because all residuals exceeded epsilon.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.51515
- Paired wins: 4/8
- Benchmark verdict: no significant win

## Artifacts

- [REPORT.md](https://synthcore.org/code/836/REPORT.md)
- [bench_report.json](https://synthcore.org/code/836/bench_report.json)
- [coverage_scheduler_experiment.py](https://synthcore.org/code/836/coverage_scheduler_experiment.py)
- [report.md](https://synthcore.org/code/836/report.md)
- [report_bench_2026-09-01T115634.md](https://synthcore.org/code/836/report_bench_2026-09-01T115634.md)
- [results.json](https://synthcore.org/code/836/results.json)
- [stage2_bench.py](https://synthcore.org/code/836/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/836)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.