Correction-aware tree optimizer MVP

Mechanism verification

The toy problem is a four-node scalar tree 0 -> {1,2}, 1 -> 3. For each edge, delta_s_i = rho * (u_i-u_parent), and the parent correction is the negative sum of its children's dual increments.

| Prediction | Predicted | Observed | |---|---:|---:| | Correction conservation error for rho = 0.01, 0.1, 0.4, 0.9 | 0 | [0, 0, 0, 0] | | Correction norm ratios for rho = 0.05, 0.10, 0.20, 0.40 | [1,2,4,8] | [1, 2.0000000000000004, 4.000000000000001, 8.000000000000002] | | Stale-dual error versus rho | linear, zero intercept | slope 0.9219544457, intercept 2.8e-17 |

Thus the core bookkeeping mechanism manifests numerically: the redistributed correction exactly cancels the parent aggregation error, and its magnitude scales linearly with the dual step size.

Mini-experiment

Fixed scalar quadratic local objectives, 160 rounds, rho=0.2, eta=0.16, penalty 0.8:

| Method | Final objective | Final consensus residual | Toy communicated bytes | |---|---:|---:|---:| | Corrected tree | 6.0937499891 | 3.23e-9 | 7680 | | Uncorrected tree | 6.0937499974 | 7.65e-10 | 7680 | | FedAvg | 6.0937500000 | 0 | 10240 |

The corrected and uncorrected methods converge to the same objective in this convex toy, so there is no demonstrated final-loss improvement. At round 40, corrected loss was 6.10348 versus 6.09493 for the uncorrected tree at rho=0.2; therefore correction is not uniformly better in this particular synchronous Jacobi discretization. The tree uses fewer toy bytes than FedAvg because it communicates along three edges rather than all worker-root links, but this is not a full systems benchmark.

Reproduce

/home/maxwelhelp/main/bin/python3 tree_optimizer_experiment.py

Outputs are written to results.json.

Limitations

This is a scalar quadratic proxy, not logistic regression or an MLP; it does not model network latency, asynchronous execution, minibatch noise, GPU placement, or realistic parameter-block bandwidth. The correction identity is tested exactly, while optimizer quality is only a small sanity experiment.