✗ Failed on benchmark 2026

Lyapunov Fading-Memory Optimizer

Implementation & benchmark of arXiv:2609.02454 — Exponential Consensus and Flocking in Multi-Agent Systems with Infinite Fading Memory

Usefulness7/10

Difficulty4/10

Novelty6/10

Dynamics Stability Speedup Optimizer Training dynamics Scheduler Dynamical systems Control theory Functional analysis Optimization

Source paper: Exponential Consensus and Flocking in Multi-Agent Systems with Infinite Fading Memory arXiv:2609.02454 ⓘ · analyzed Sep 3, 2026

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 Not peer-reviewed.

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.

Formulas

$$\dot{x}(t)=-\nabla L(x(t))+\kappa\int_0^\infty\mu(s)\bigl[x(t-s)-x(t)\bigr]ds,\qquad \mu(s)\geq0,\quad\int_0^\infty\mu(s)ds=1.$$

$$z(t,s)=x(t)-x(t-s),\qquad \partial_tz(t,s)+\partial_sz(t,s)=\dot{x}(t),\qquad z(t,0)=0.$$

$$V(t)=\frac12\lVert x(t)-x_*\rVert^2+\frac{\kappa}{2}\int_0^\infty\mu(s)\lVert z(t,s)\rVert^2ds,\qquad \mu'(s)\leq-\delta\mu(s).$$

$$m_t=\beta(x-m),\qquad \dot{x}=-\lambda x+\kappa(m-x),\qquad r^2+(\lambda+\kappa+\beta)r+\beta\lambda=0.$$

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 $r^2+(\lambda+\kappa+\beta)r+\beta\lambda=0$.

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

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Significantly worse ✗

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

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.

Agent confidence: 9/10

Baseline: SGD at eta=0.08: final loss 3.92e-19, 249 stiff-mode sign changes; momentum at eta=0.02: final loss 8.36e-13, 52 sign changes.
Idea: Fading memory with kappa=2, beta=5, eta=0.05: final loss 7.26e-09, 0 stiff-mode sign changes; predicted and empirical stability boundary both 0.63 for lambda=3, kappa=2, beta=5.

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.

How to run: python3 run_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2609.02454) Sep 3, 2026

Significantly WORSE than the tuned baseline (+262.9%, p=0.037, wins 1 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0024
Idea mean: 0.0088
Effect (Δ): 0.0064 (+262.9%; negative = idea better)
Wins: 1 / 8 paired seeds
p-value: 0.037 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±274.2%
Mechanism: Not confirmed ✗
Practical effect: Hurts
Baseline tuning: swept over 6 configs

Limitations:

Only the registered built-in dynamics track was evaluated; tabular, vision, and sequence transfer were not tested. The mechanism probe used perturbation responses of trained models rather than direct optimizer trajectory logging. The memory settings were evaluated on a small fixed grid and no broader optimizer-family comparison was run.

How to run: python3 stage2_dynamics.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 3, 2026 latest Failed ✗

Agent confidence: 9/10

Implemented the Lyapunov fading-memory optimizer as a matched intervention on the registered actuated-pendulum dynamics track using the shared rnn_small architecture. Across 8 paired seeds, it was significantly worse than tuned momentum SGD: mean test MSE 0.008838 versus 0.002435, paired delta +0.0064025, permutation p=0.037. The mechanism signature was also not confirmed because the observed trained-model memory-force response increased rather than following the predicted exponential decay.

Baseline: Tuned SGD with momentum=0.9 and lr=0.006; mean test MSE 0.0024354744236916304, std 0.0014542907868172839.
Comparison: delta_mean=+0.006402498489478603; idea_wins=1/8; permutation p_value=0.037; verdict=idea worse (significant).
Idea: Fading memory with lr=0.003, momentum=0.9, kappa=0.5, beta=2.0; mean test MSE 0.008837972913170233, std 0.006216926394780772.

Benchmark result