# Floquet Loadability Monitor

- ID: 2654
- Canonical URL: https://synthcore.org/idea/2654/floquet-loadability-monitor
- API JSON: https://synthcore.org/api/idea/2654.json
- API Markdown: https://synthcore.org/api/idea/2654.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.21256](https://arxiv.org/abs/2608.21256)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: optimizer, training-dynamics, scheduler
- Math tags: dynamical-systems, bifurcations, linear-algebra, numerical-analysis
- Ratings: usefulness 8/10; difficulty 6/10; novelty 7/10

## Idea description

Treat a periodically forced optimizer as a discrete nonautonomous dynamical system and monitor its periodic parameter orbit rather than using only an average learning rate. Increase the forcing amplitude or base learning rate until the largest Floquet multiplier approaches +1, then reduce the schedule magnitude before the cyclic-fold instability.

## Mathematical statement

Let theta_k in R^d be neural-network parameters and let F_k(theta;p) be the optimizer update at phase k of a T-step periodic schedule, where p is the forcing amplitude. The period map is Phi_T(theta;p)=F_{T-1} composed with ... composed with F_0. A periodic orbit satisfies theta_star=Phi_T(theta_star;p). Its monodromy matrix is M=D_theta Phi_T(theta_star;p), and perturbations over complete periods satisfy delta theta_{q+1}=M delta theta_q. Linear stability requires the spectral radius rho(M)<1. A generic cyclic fold occurs when a simple Floquet multiplier mu reaches +1, equivalently det(I-M)=0, provided the parameter crossing is nondegenerate.

## Key formulas

- $$\theta_{k+1}=F_k(\theta_k;p),\qquad F_{k+T}=F_k$$
- $$\Phi_T(\theta;p)=F_{T-1}\circ\cdots\circ F_0(\theta;p),\qquad \theta^\star=\Phi_T(\theta^\star;p)$$
- $$M=D_\theta\Phi_T(\theta^\star;p)=\prod_{k=T-1}^{0}D_\theta F_k(\theta_k^\star;p),\qquad \rho(M)<1$$
- $$\det(I-M)=0\qquad\Longleftrightarrow\qquad \mu=+1$$

## Implementation notes

Use a small MLP, RNN, or state-space model with a deliberately periodic schedule. For example, set the learning rate to eta_k=eta_0 times [1+p sin(2 pi k/T)], or alternate easy and hard minibatches every T steps. The integration point is after each optimizer update: retain the trajectory over one complete period and estimate the Jacobian of the period map. Pseudocode: run T updates from theta_0; repeat periods until the normalized difference between theta_T and theta_0 is below epsilon; estimate the leading eigenvalue by power iteration; obtain each matrix-vector product with forward-mode JVPs through the T updates; sweep p or eta_0 by continuation; stop when the leading multiplier is within delta of +1. The exact mechanism supplies the fixed-point equation, monodromy product, and multiplier criterion. The orbit and Jacobian are estimated numerically with JVPs or truncated differentiation. The first experiment should use a two-layer MLP on MNIST, comparing constant and sinusoidally modulated learning rates across p and T. The quantitative prediction is a sharp divergence or loss-amplification boundary when the leading multiplier crosses +1, with the critical amplitude depending on T and agreeing with the Floquet estimate within about 20 percent. An average-learning-rate calculation should not predict this boundary.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: inconclusive
- Verdict: Built a reproducible Floquet monitor for periodic heavy-ball optimization on a 1-D quadratic, including exact period-map Jacobians, determinant-root continuation, multiplier sweeps, and trajectory monitoring. The mechanism manifested: the leading multiplier crossed +1 at p=0.866886, with rho approximately 1 and det(I-M) approximately 5.6e-10; p=0.87 was unstable with rho=1.01384 and rapid amplification. Below the boundary, p=0.7869 had rho=0.88255 and contracted, while above it p=0.9469 had rho=1.37136 and grew strongly.

### Mechanism check

- Verdict: Built a reproducible Floquet monitor for periodic heavy-ball optimization on a 1-D quadratic, including exact period-map Jacobians, determinant-root continuation, multiplier sweeps, and trajectory monitoring. The mechanism manifested: the leading multiplier crossed +1 at p=0.866886, with rho approximately 1 and det(I-M) approximately 5.6e-10; p=0.87 was unstable with rho=1.01384 and rapid amplification. Below the boundary, p=0.7869 had rho=0.88255 and contracted, while above it p=0.9469 had rho=1.37136 and grew strongly.
- Confidence: 8/10
- Limitations: This is an exactly linear two-state quadratic toy, not the proposed two-layer MNIST MLP. It does not test CUDA, wall-clock speed, minibatch noise, high-dimensional JVP estimation, or continuation on a learned neural-network orbit. The rho cap controls asymptotic linear stability but can still allow transient amplification because the monodromy matrix is nonnormal.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.25
- Paired wins: 0/0
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/1002/bench_report.json)
- [fastsearch.py](https://synthcore.org/code/1002/fastsearch.py)
- [floquet_mvp.py](https://synthcore.org/code/1002/floquet_mvp.py)
- [report.md](https://synthcore.org/code/1002/report.md)
- [report_bench_2026-09-01T180545.md](https://synthcore.org/code/1002/report_bench_2026-09-01T180545.md)
- [results.json](https://synthcore.org/code/1002/results.json)
- [search_plus.py](https://synthcore.org/code/1002/search_plus.py)
- [stage2_bench.py](https://synthcore.org/code/1002/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/1002)

## Disclaimer

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