# Geometric Feedback Compute Scheduler

- ID: 871
- Canonical URL: https://synthcore.org/idea/871/geometric-feedback-compute-scheduler
- API JSON: https://synthcore.org/api/idea/871.json
- API Markdown: https://synthcore.org/api/idea/871.md
- Verification status: beats_baseline
- Source: [arXiv:2607.13648](https://arxiv.org/abs/2607.13648)
- Category: architecture
- Solves: speedup, scalability, sample-efficiency
- ML areas: moe-routing, inference-speedup
- Math tags: probability, stochastic-processes, optimization
- Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10

## Idea description

Treat unresolved inference items as active threats and allocate a fixed budget of C module evaluations per round. Each evaluation has an item-dependent probability of completing the item, while the scheduler observes only completion or failure after the round. Use fair allocation when completion probabilities are unknown or nearly homogeneous, then switch to a marginal-success greedy policy as feedback estimates become reliable.

## Mathematical statement

The paper models a feasible policy as an allocation rule satisfying $\sum_{i\in\mathcal{M}}[\pi_t(\mathcal{M})]_i=C$ for every nonempty active set $\mathcal{M}$ and every round $t$, where $C$ is the total number of effectors and $[\pi_t(\mathcal{M})]_i$ is the number assigned to active item $i$. The active sets form a random nested sequence $[n]=\mathcal{M}_1\supseteq\mathcal{M}_2\supseteq\cdots\supseteq\emptyset$. For item $i$, let $q_i\in[0,1)$ be the probability that one attempt fails and let $L_i$ be its number of attempts required for completion. The extracted paper states $\mathbb{E}[L_i]=1/(1-q_i)$ and uses $\Pr(L_i\le k)=1-q_i^k$, $\Pr(L_i>k)=q_i^k$, and $\Pr(L_i=1)=1-q_i$. In the neural adaptation, $i$ indexes an unresolved token, sample, branch, or latent trajectory; $a_i$ is its cumulative number of allocated module calls; and the probability that its next call completes it is estimated as $p_i=1-q_i$. A fair policy assigns $a_i$ as evenly as possible across active items. A geometric marginal-gain policy assigns the next slot to the item maximizing $q_i^{a_i}(1-q_i)$, the probability that this next attempt is the first successful completion attempt under the geometric model.

## Key formulas

- $$\Pi:=\left\{\pi:\sum_{i\in\mathcal{M}}\left[\pi_t(\mathcal{M})\right]_i=C,\ \forall\mathcal{M}\neq\emptyset,\ \forall t\in\mathbb{Z}_{\geq 1}\right\}$$
- $$[n]=\mathcal{M}_1\supseteq\mathcal{M}_2\supseteq\ldots\supseteq\emptyset$$
- $$\mathbb{E}[L_i]=\frac{1}{1-q_i},\qquad \Pr(L_i\le k)=1-q_i^k,\qquad \Pr(L_i>k)=q_i^k$$
- $$\Delta_i(a_i)=\Pr(\text{item }i\text{ completes on its next call})=q_i^{a_i}(1-q_i)$$

## Implementation notes

Integrate this at inference time as a batch-level scheduler around an expensive module: an MoE expert call, verifier, recurrent refinement block, diffusion denoising update, or test-time reasoning step. Maintain an active set of items that have not yet met a stopping criterion. The scheduler receives a fixed per-round budget C and cumulative call counts a[i]. Estimate each item’s failure probability q[i] from a difficulty head or online feedback. For a confidence-based system, define success as confidence exceeding a threshold or verifier reward exceeding a threshold; update q[i] from observed failures and successes with a Beta prior, using p[i]=(s[i]+alpha)/(s[i]+f[i]+alpha+beta) and q[i]=1-p[i]. At every round, initialize a_round[i]=0. If estimates are unavailable, assign slots in round-robin order over active items, implementing fair allocation up to integer rounding. Otherwise repeat C times: compute score[i]=q[i]**a[i]*(1-q[i]) for each active item, assign one slot to argmax(score), and increment its temporary and cumulative count. Execute selected calls in a batch, update feedback statistics, and remove completed items. The geometric equations are the model assumption; q estimates, stopping thresholds, and calibration are empirical. First test on a small MoE language model or verifier benchmark, comparing round-robin, static confidence allocation, and greedy scheduling at equal total calls. Measure accuracy, average calls, p95 latency, and unfinished items by a deadline. Success means lower compute or tail latency at matched accuracy, particularly on heterogeneous-difficulty batches.

## Verification

- Status: beats_baseline
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: helps
- Verdict: Built and numerically verified the geometric scheduler. The core predictions were confirmed: completion-probability error was at most 0.00119, mean-attempt scaling error at most 0.17%, and greedy allocation matched exhaustive optima exactly on all tested cases. However, feedback greedy did not produce a practical win over fair allocation in the toy deadline benchmark; the mechanism is validated, but the inference-speed benefit remains unproven.

### Mechanism check

- Verdict: Built and numerically verified the geometric scheduler. The core predictions were confirmed: completion-probability error was at most 0.00119, mean-attempt scaling error at most 0.17%, and greedy allocation matched exhaustive optima exactly on all tested cases. However, feedback greedy did not produce a practical win over fair allocation in the toy deadline benchmark; the mechanism is validated, but the inference-speed benefit remains unproven.
- Confidence: 9/10
- Limitations: Only a small synthetic geometric model was tested; no real MoE, verifier, neural inference module, wall-clock latency, GPU execution, calibration study, or larger batch/deadline sweep was evaluated. The practical benchmark used oracle geometric probabilities for comparison and a simple Beta posterior for feedback.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0312
- Paired wins: 6/8
- Benchmark verdict: idea better (significant)

## Artifacts

- [REPORT.md](https://synthcore.org/code/352/REPORT.md)
- [bench_report.json](https://synthcore.org/code/352/bench_report.json)
- [report.md](https://synthcore.org/code/352/report.md)
- [report_bench_2026-08-31T015902.md](https://synthcore.org/code/352/report_bench_2026-08-31T015902.md)
- [results.json](https://synthcore.org/code/352/results.json)
- [scheduler_experiment.py](https://synthcore.org/code/352/scheduler_experiment.py)
- [stage2_bench.py](https://synthcore.org/code/352/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/352)

## Disclaimer

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