# Coercivity-aware memory optimizer

- ID: 811
- Canonical URL: https://synthcore.org/idea/811/coercivity-aware-memory-optimizer
- API JSON: https://synthcore.org/api/idea/811.json
- API Markdown: https://synthcore.org/api/idea/811.md
- Verification status: unverified
- Source: [arXiv:2607.12482](https://arxiv.org/abs/2607.12482)
- Category: dynamics
- Solves: stability, speedup, generalization
- ML areas: optimizer, training-dynamics
- Math tags: harmonic-analysis, functional-analysis, dynamical-systems
- Ratings: usefulness 6/10; difficulty 4/10; novelty 5/10

## Idea description

Add a positive completely monotone memory branch to an optimizer or recurrent state update, but retain an explicitly calibrated instantaneous gradient or input branch. Estimate the memory branch's finite-horizon coercivity and prevent the system from entering regimes where memory suppresses high-frequency corrections and causes slow or unstable training.

## Mathematical statement

The paper represents a completely monotone kernel as $k(t)=\int_{(0,\infty)}e^{-st}\,\mu(ds)$ for a nonnegative measure $\mu$. Its memory coercivity symbol is $m(\omega)=\operatorname{Re}\widehat{k}(\omega)=\int_{(0,\infty)}\frac{s}{s^{2}+\omega^{2}}\,\mu(ds)$, so $m(\omega)\to0$ as $|\omega|\to\infty$. For a time signal $x$, Parseval's identity gives the memory dissipation $D_k[x]=\int m(\omega)|\widehat{x}(\omega)|^{2}d\omega$, while the instantaneous energy is $E[x]=\int|\widehat{x}(\omega)|^{2}d\omega$; hence the exact spectral gap is $E[x]-D_k[x]=\int(1-m(\omega))|\widehat{x}(\omega)|^{2}d\omega$ in the paper's normalized units. On a finite horizon $[0,\mathfrak{T}]$, define $\Lambda_k(\mathfrak{T})=\inf_{x\neq0}D_k[x]/E[x]$ over nonzero signals supported on that interval. We use $\Lambda_k$ as a measured lower-bound proxy for memory responsiveness and enforce an instantaneous coefficient whenever $\Lambda_k$ is small; this compensates for the high-frequency no-go result rather than assuming that memory is uniformly coercive.

## Key formulas

- $$k(t)=\int_{(0,\infty)}e^{-st}\,\mu(ds),\qquad \mu\geq0,$$
- $$m(\omega)=\operatorname{Re}\widehat{k}(\omega)=\int_{(0,\infty)}\frac{s}{s^{2}+\omega^{2}}\,\mu(ds),\qquad m(\omega)\xrightarrow{|\omega|\to\infty}0,$$
- $$D_k[x]=\int m(\omega)|\widehat{x}(\omega)|^{2}\,d\omega,\qquad E[x]-D_k[x]=\int(1-m(\omega))|\widehat{x}(\omega)|^{2}\,d\omega,$$
- $$\Lambda_k(\mathfrak{T})=\inf_{x\in L^{2}(0,\mathfrak{T}),\,x\neq0}\frac{\int_{0}^{\mathfrak{T}}x(t)(k*x)(t)\,dt}{\int_{0}^{\mathfrak{T}}|x(t)|^{2}\,dt},\qquad (k*x)(t)=\int_{0}^{t}k(t-s)x(s)\,ds.$$

## Implementation notes

Integrate this into the optimizer's parameter-update line, or into the state update of a recurrent or SSM block. For the optimizer, let $g_t$ be the minibatch gradient and maintain $J$ exponential memory states with decay rates $s_j>0$: initialize $z_j=0$; at step $t$, update $z_j\leftarrow e^{-s_j\Delta t}z_j+(1-e^{-s_j\Delta t})g_t$; form the memory update $h_t=\sum_jw_jz_j$ with $w_j\geq0$; then use $\Delta\theta_t=-\eta[\alpha_tg_t+(1-\alpha_t)h_t]/(\sqrt{v_t}+\epsilon)$, where $v_t$ can be Adam's usual second moment. The resulting kernel is $k(t)=\sum_jw_je^{-s_jt}$. Every $K$ steps, collect the last $T$ gradients in a temporal buffer and estimate the finite-horizon coercivity. Apply the convolution recurrence to several random temporal probe vectors $q$: compute $C_kq$ with the same exponential states and evaluate $r(q)=\langle q,C_kq\rangle/\|q\|^{2}$; use the minimum of 3--5 probe values as $\widehat\Lambda$. Set $\alpha_t=\operatorname{clip}(\alpha_0+c(\lambda_*-\widehat\Lambda),\alpha_{\min},1)$, increasing the instantaneous gradient weight when memory coercivity falls below target $\lambda_*$. The paper supplies positivity of the mixture weights, the symbol decay $m(\omega)\to0$, and the spectral-gap identity; the finite-window estimate and controller constants are empirical. First test on CIFAR-10 using a small ResNet-18 or six-layer MLP, comparing AdamW, pure exponential-gradient memory, and the adaptive mixture at equal update FLOPs. Measure loss decrease per step, final accuracy, gradient-spike recovery, and sensitivity to $T$ and the decay rates. The method succeeds if it recovers faster after abrupt gradient changes and matches or improves AdamW accuracy without the divergence or sluggishness of the pure-memory baseline.

## Disclaimer

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