# Lyapunov Fading-Memory Optimizer

- ID: 3057
- Canonical URL: https://synthcore.org/idea/3057/lyapunov-fading-memory-optimizer
- API JSON: https://synthcore.org/api/idea/3057.json
- API Markdown: https://synthcore.org/api/idea/3057.md
- Verification status: failed_benchmark
- Source: [arXiv:2609.02454](https://arxiv.org/abs/2609.02454)
- Category: dynamics
- Solves: stability, speedup
- ML areas: optimizer, training-dynamics, scheduler
- Math tags: dynamical-systems, control-theory, functional-analysis, optimization
- Ratings: usefulness 7/10; difficulty 4/10; novelty 6/10

## Idea description

Add a fading-memory consensus force to parameter dynamics, pulling the current parameter toward a distributed average of its past while preserving the ordinary gradient step. Implement the infinite memory through one or several recursive exponential states, and tune the memory decay so that quadratic-mode dynamics remain exponentially stable. This should suppress oscillations and catastrophic steps without relying on conventional momentum alone.

## Mathematical statement

For a parameter mode x(t), use the Volterra dynamics \(\dot{x}(t)=-\nabla L(x(t))+\kappa\int_0^\infty\mu(s)[x(t-s)-x(t)]ds\), where \(\kappa\geq0\) is memory strength and \(\mu(s)\geq0\) is normalized by \(\int_0^\infty\mu(s)ds=1\). Define the Dafermos history variable \(z(t,s)=x(t)-x(t-s)\), satisfying \(\partial_tz+\partial_sz=\dot{x}(t)\) and \(z(t,0)=0\). For a quadratic loss mode with curvature \(\lambda\geq0\), a history energy combines parameter error and weighted history discrepancy. If \(\mu'(s)\leq-\delta\mu(s)\), the history transport term dissipates at rate at least \(\delta\). For the exponential kernel \(\mu(s)=\beta e^{-\beta s}\), the exact Markovian realization is \(m_t=\beta(x-m)\), \(\dot{x}=-\lambda x+\kappa(m-x)\), with characteristic polynomial \(r^2+(\lambda+\kappa+\beta)r+\beta\lambda=0\).

## Key formulas

- $$\dot{x}(t)=-\nabla L(x(t))+\kappa\int_0^\infty\mu(s)\bigl[x(t-s)-x(t)\bigr]ds,\qquad \mu(s)\geq0,\quad\int_0^\infty\mu(s)ds=1.$$
- $$z(t,s)=x(t)-x(t-s),\qquad \partial_tz(t,s)+\partial_sz(t,s)=\dot{x}(t),\qquad z(t,0)=0.$$
- $$V(t)=\frac12\lVert x(t)-x_*\rVert^2+\frac{\kappa}{2}\int_0^\infty\mu(s)\lVert z(t,s)\rVert^2ds,\qquad \mu'(s)\leq-\delta\mu(s).$$
- $$m_t=\beta(x-m),\qquad \dot{x}=-\lambda x+\kappa(m-x),\qquad r^2+(\lambda+\kappa+\beta)r+\beta\lambda=0.$$

## Implementation notes

1. Integration point: apply this after the gradient is computed in SGD or AdamW, initially using one memory tensor per parameter block rather than one state per scalar. Maintain an exponential memory m of parameters and add a restoring term kappa times (m minus theta) to the update. The discrete equations are theta equals theta minus eta times g plus eta times kappa times (m minus theta), followed by m equals rho times m plus one minus rho times theta, with rho equal to exp of minus beta eta. 2. Pseudocode: initialize theta and m to theta_0; each step compute g equal to the gradient of L(theta); update theta using the gradient and memory force; update m using the exponential average; log both the gradient and memory-force norms. Optionally clip only the memory force to a fixed multiple of the gradient norm. 3. Computed from the mechanism: rho and the quadratic characteristic polynomial. Estimated empirically: local curvature lambda using Hessian-vector products or Rayleigh quotients, and the observed decay rate from parameter trajectories. 4. First cheap experiment: train a two-layer MLP on MNIST and a small Transformer on WikiText-2, comparing SGD, SGD with momentum, and this optimizer over matched learning-rate grids. First use a quadratic problem with known Hessian eigenvalues. The quantitative prediction is that measured modal decay rates follow the smaller-magnitude root of the polynomial, and the largest stable learning rate agrees with the discrete linearized prediction within 20 percent. The Lyapunov-like sum of parameter error and weighted memory discrepancy should decrease monotonically after transients.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: harms
- Verdict: Built a self-contained exponential fading-memory optimizer and quadratic verification in run_experiment.py. The continuous characteristic roots satisfy the claimed polynomial to approximately 1e-14 residual, and the predicted and empirical discrete stability limits agree exactly on the tested grid (eta_max=0.63). Fading memory produced zero stiff-mode sign changes versus 249 for SGD and 52 for momentum, demonstrating oscillation suppression, but it converged more slowly at equal steps, so no speedup was observed.

### Mechanism check

- Verdict: Built a self-contained exponential fading-memory optimizer and quadratic verification in run_experiment.py. The continuous characteristic roots satisfy the claimed polynomial to approximately 1e-14 residual, and the predicted and empirical discrete stability limits agree exactly on the tested grid (eta_max=0.63). Fading memory produced zero stiff-mode sign changes versus 249 for SGD and 52 for momentum, demonstrating oscillation suppression, but it converged more slowly at equal steps, so no speedup was observed.
- Confidence: 9/10
- Limitations: Only diagonal quadratic dynamics were tested; no MNIST, Transformer, AdamW integration, wall-clock benchmark, multidimensional non-diagonal Hessian, or broad hyperparameter search was performed. The stability boundary is resolved on a finite eta grid.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.037
- Paired wins: 1/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [README.md](https://synthcore.org/code/1229/README.md)
- [bench_report.json](https://synthcore.org/code/1229/bench_report.json)
- [report.md](https://synthcore.org/code/1229/report.md)
- [report_bench_2026-09-03T131319.md](https://synthcore.org/code/1229/report_bench_2026-09-03T131319.md)
- [results.json](https://synthcore.org/code/1229/results.json)
- [run_experiment.py](https://synthcore.org/code/1229/run_experiment.py)
- [stage2_dynamics.py](https://synthcore.org/code/1229/stage2_dynamics.py)
- [Download all files as ZIP](https://synthcore.org/download/1229)

## Disclaimer

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