Turnpike-Calibrated Short-Window Training
Implementation & benchmark of arXiv:2609.02071 — Turnpike properties in nonlinear system identification
Source paper: Turnpike properties in nonlinear system identification arXiv:2609.02071 ⓘ · analyzed Sep 3, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Train a recurrent or neural state-space model on fixed-initial-state subsequences, but select the training horizon and burn-in from an empirically estimated turnpike bound instead of choosing them arbitrarily. If the cumulative discrepancy between fixed-initial-state and free-initial-state optima is bounded, the average discrepancy decreases as 1/N, allowing shorter windows while preserving the long-horizon optimum.
Formulas
Mathematical statement
The model is xₖ₊₁ = fθ(xₖ,uₖ) and yₖ = hθ(xₖ,uₖ). For a horizon N, the fixed-initial-state objective is J_N(θ,x₀) = Σₖ₌₀ᴺ⁻¹ ℓₖ(yₖ(x₀,θ,U)), while the free-initial-state objective is J_N^free(θ) = inf over x₀ of J_N(θ,x₀). Let yₖ^fix,N be an optimum with a prescribed initial state and yₖ^free,N be the closest free-initial-state optimum. The cumulative turnpike property is Σₖ₌₀ᴺ⁻¹ d(yₖ^fix,N,yₖ^free,N) ≤ C, where C does not depend on N. Therefore the mean discrepancy is at most C/N. Estimate C and the transient length from paired fixed/free rollouts, then choose N so that the predicted mean gap is below a target ε.
Implementation notes
Integrate this into truncated backpropagation through time for a GRU, RNN, or neural state-space model. For each minibatch sequence, use the current fixed initial state supplied by an encoder, a zero state, or a cached state. Optimize the normal rollout loss over a candidate horizon N. Every K optimizer steps, create an auxiliary free-initial-state problem by treating x₀ as a differentiable variable and taking 5 to 20 gradient steps on the same sequence loss. Roll out both trajectories with identical inputs, compute C_hat_N = Σₖ ||yₖ^fix − yₖ^free||₂, and update an exponential moving estimate of C_hat. Select the shortest horizon satisfying C_hat/N ≤ ε, or discard the first estimated burn-in steps when computing the loss. The paper provides the cumulative-turnpike mechanism; C_hat, the free initial state, and burn-in are estimated empirically. First experiment: train a GRU and neural state-space model on Duffing-oscillator or Lorenz-63 sequences, comparing full BPTT, fixed windows, and the adaptive rule. Sweep N and plot the average fixed/free gap. The predicted signature is a log-log slope near −1 for average gap versus N, with the selected N achieving gap below ε within roughly 20 percent. If C_hat grows proportionally with N, the assumed turnpike regime is absent.
Verification
Stage 1 · Toy mechanism gate: Failed ✗
Stage 2 · Mechanism transferred to benchmark: Not tested
Stage 2 · Practical benchmark result: Not run
Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works
Stage 1 — Mechanism check agent confidence 9/10
Built a reproducible scalar state-space verification, fixed/free initial-state calibration, horizon sweep, and training comparison in run_experiment.py. The analytic toy system clearly shows bounded cumulative discrepancy and mean gap scaling near 1/N (log-log slope -0.959), while the learned-model sweep is approximately decreasing (slope -0.767). However, strict calibration with epsilon=.05 selected the full horizon 40 and gave essentially the same MSE as full training (0.00710 vs 0.00701); relaxed calibration selected 27 but substantially worsened MSE to 0.05865, so no practical speedup with preserved performance was observed.
- Agent confidence
- 9/10
- Baseline
- Full horizon 40: MSE 0.007013; fixed horizon 8: MSE 0.024971; empirical gap sweep slope -0.767.
- Idea
- Calibrated epsilon=.05: selected horizon 40, MSE 0.007104, mean fixed/free gap 0.1233. Relaxed epsilon=.20: selected horizon 27, MSE 0.05865, mean gap 0.0972.
Limitations: Only a scalar linear state-space model and one fixed random seed were tested; no GRU, Duffing, Lorenz-63, minibatch dataset, wall-clock/FLOP measurement, multi-sequence calibration, or statistical repeats were included. The free-initial-state optimization is a small approximate inner solve rather than an exact global optimum.
How to run: python3 run_experiment.py
Verdict computed by deterministic test code from paired-seed statistics — not by the language model.
Artifacts
- report.md 1.6 KB View
- results.json 7.1 KB View Raw
- run_experiment.py 3.7 KB View Raw