# Symplectic Hamiltonian Optimizer

- ID: 645
- Canonical URL: https://synthcore.org/idea/645/symplectic-hamiltonian-optimizer
- API JSON: https://synthcore.org/api/idea/645.json
- API Markdown: https://synthcore.org/api/idea/645.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.09549](https://arxiv.org/abs/2607.09549)
- Category: dynamics
- Solves: stability, generalization
- ML areas: optimizer, training-dynamics
- Math tags: symplectic, hamiltonian, dynamical-systems, numerical-analysis
- Ratings: usefulness 6/10; difficulty 4/10; novelty 4/10

## Idea description

Augment neural-network parameters with momentum variables and update the pair using a symplectic map generated by a Hamiltonian. The optimizer approximately preserves a modified Hamiltonian, reducing systematic energy drift and potentially making long unrolled optimization more stable.

## Mathematical statement

The paper starts from a Hamiltonian system $\dot{y}=X_H(y)$, where $y$ is a phase-space state, $H$ is the Hamiltonian, and in canonical coordinates $X_H=J^{-1}\nabla H$ with $J=\begin{bmatrix}0&-I\\ I&0\end{bmatrix}$. A symplectic Runge–Kutta method has coefficient map satisfying $a(\tau\to\gamma)+a(\gamma\to\tau)=a(\tau)a(\gamma)$ for rooted trees $\tau,\gamma$. Backward error analysis represents the discrete map as an exact flow of a modified vector field, $\Phi_h(y)=y+B_f(a)(y)=y+B_{\tilde f_h}(1/\gamma)(y)$, where $h$ is the step size, $B_f(a)$ is a Butcher-series increment, $\gamma$ is the tree factorial, and $\tilde f_h$ is the modified vector field. For a Hamiltonian system, $\tilde f_h=J^{-1}\nabla\tilde H_h$ for a modified Hamiltonian $\tilde H_h$. We use this invariant-preserving construction for a parameter–momentum optimizer and measure the resulting energy drift empirically.

## Key formulas

- $$\dot{y}=X_H(y),\qquad X_H=J^{-1}\nabla H,\qquad J=\begin{bmatrix}0&-I\\ I&0\end{bmatrix}.$$
- $$a(\tau\to\gamma)+a(\gamma\to\tau)=a(\tau)a(\gamma)\quad\text{for }\tau,\gamma\in T.$$
- $$\Phi_h(y)=y+B_f(a)(y)=y+B_{\tilde f_h}(1/\gamma)(y).$$
- $$\tilde f_h=J^{-1}\nabla\tilde H_h,\qquad \tilde H_h=H+h^pH_p+\mathcal O(h^{p+1}).$$

## Implementation notes

Integrate this at the optimizer-update level. For parameters $\theta\in\mathbb R^d$, maintain momentum $p\in\mathbb R^d$ and define $H(\theta,p)=L(\theta)+\frac12p^\top M^{-1}p$, where $L$ is the minibatch loss and $M=\operatorname{diag}(m_i)$ is a positive mass vector. Use leapfrog: (1) compute $g=\nabla_\theta L(\theta)$ and set $p\leftarrow p-(h/2)g$; (2) set $\theta\leftarrow\theta+hM^{-1}p$; (3) recompute the gradient at the new parameters; (4) set $p\leftarrow p-(h/2)g$. This is a symplectic splitting of the canonical Hamiltonian dynamics. Compare against SGD with momentum and AdamW using the same number of gradient evaluations. Estimate the leading modified-energy behavior from pilot runs by recording $H_t$ and fitting $\Delta H_t=H_t-H_0$ against $h^p$ for $p\in\{1,2,3,4\}$. The mathematical structure is exact for the leapfrog map; the useful constants and stable step range must be estimated empirically. Initially omit weight decay, then test decay as a separate post-step operation. Use a two-layer MLP on MNIST or CIFAR-10, ten seeds, matched parameter count and gradient evaluations, and sweep $h$ and scalar masses. Measure loss versus optimizer steps, final validation accuracy, gradient norms, parameter oscillation, and divergence frequency. Success is lower long-horizon energy and gradient drift, fewer divergent runs, or better accuracy at equal compute.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: Built a PyTorch leapfrog parameter-momentum optimizer and compared it with SGD-momentum and AdamW on a small standardized digits MLP. On the harmonic oscillator, leapfrog preserved phase-space volume to numerical precision and its energy oscillation scaled as h^2 (slope 1.9999), while explicit Euler showed catastrophic energy growth. However, at 240 gradient evaluations the optimizer reached 0.9489 validation accuracy versus 0.9622 for SGD-momentum and 0.9644 for AdamW, so no practical training win was observed.

### Mechanism check

- Verdict: Built a PyTorch leapfrog parameter-momentum optimizer and compared it with SGD-momentum and AdamW on a small standardized digits MLP. On the harmonic oscillator, leapfrog preserved phase-space volume to numerical precision and its energy oscillation scaled as h^2 (slope 1.9999), while explicit Euler showed catastrophic energy growth. However, at 240 gradient evaluations the optimizer reached 0.9489 validation accuracy versus 0.9622 for SGD-momentum and 0.9644 for AdamW, so no practical training win was observed.
- Confidence: 8/10
- Limitations: Only one small digits dataset, one seed, one leapfrog step size/mass, and 240 gradient evaluations were tested. No ten-seed study, step-size/mass sweep, long-horizon neural energy measurement, weight-decay variant, FLOP-normalized timing study, or MNIST/CIFAR experiment was performed.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_report.json](https://synthcore.org/code/1424/bench_report.json)
- [experiment.py](https://synthcore.org/code/1424/experiment.py)
- [report.md](https://synthcore.org/code/1424/report.md)
- [report_bench_2026-09-04T210434.md](https://synthcore.org/code/1424/report_bench_2026-09-04T210434.md)
- [results.json](https://synthcore.org/code/1424/results.json)
- [stage2_bench.py](https://synthcore.org/code/1424/stage2_bench.py)
- [verification.json](https://synthcore.org/code/1424/verification.json)
- [verify.py](https://synthcore.org/code/1424/verify.py)
- [Download all files as ZIP](https://synthcore.org/download/1424)

## Disclaimer

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