✗ Failed on benchmark 2026

Hessian-Coupled Event-Triggered Preconditioner

Implementation & benchmark of arXiv:2608.10246 — Towards Co-Designed Event-Triggered Extremum Seeking

Usefulness7/10

Difficulty7/10

Novelty7/10

Optimization Speedup Stability Scalability Optimizer Training dynamics Federated Optimization Linear algebra Convex analysis Control theory Dynamical systems

Source paper: Towards Co-Designed Event-Triggered Extremum Seeking arXiv:2608.10246 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Replace a diagonal learning-rate or preconditioner matrix with a small full block matrix and communicate a worker's updated gradient or parameter only when its local state has drifted sufficiently from the last communicated state. Jointly select the block preconditioner and the largest safe trigger threshold using robust Lyapunov inequalities over several empirical Hessian or Gauss-Newton matrices. The expected gain is fewer synchronization events without the instability normally caused by aggressively stale gradients.

Formulas

$$Q(\theta)=Q^{\ast}+\frac{1}{2}(\theta-\theta^{\ast})^{\top}H^{\ast}(\theta-\theta^{\ast})$$

$$H=\sum_{i=1}^{m}\rho_iH_i,\qquad \rho_i\ge 0,\qquad \sum_{i=1}^{m}\rho_i=1$$

$$\dot e=A_i e+B_i d,\qquad A_i=-KH_i,\qquad B_i=-KH_i,\qquad d=x_{\mathrm{last}}-x$$

$$\begin{bmatrix}A_i^{\top}P+PA_i+2\alpha P+\lambda\sigma^2I & PB_i\\ B_i^{\top}P & -\lambda I\end{bmatrix}\prec0,\quad i=1,\ldots,m,\qquad \|d\|^2\le\sigma^2\|e\|^2$$

Mathematical statement

The paper models the objective near its optimum as Q( heta)=Q^{\ast}+\frac{1}{2}( heta- heta^{\ast})^{ op}H^{\ast}( heta- heta^{\ast}), where heta\in\mathbb{R}^{n} is the parameter vector, heta^{\ast} is the optimizer, Q^{\ast} is the optimal value, and H^{\ast}=H^{\ast op} is the unknown Hessian. For minimization, H^{\ast}\succ0 and the feedback or preconditioner has the corresponding descent sign. Adapt this to a Hessian uncertainty polytope H\in\operatorname{conv}\{H_1,\ldots,H_m\}, where H_i are sampled symmetric curvature matrices. Let e=x-x^{\ast} and d=x_{\mathrm{last}}-x be the stale-state error. A continuous-time stale-gradient approximation is \dot e=A_i e+B_i d with A_i=-K H_i and B_i=-K H_i, where K is the learned full block preconditioner. Triggering enforces \|d\|^2\le\sigma^2\|e\|^2 between communications. With Lyapunov function V=e^{ op}Pe, P\succ0, exponential decay rate \alpha is certified at every Hessian vertex if there exists \lambda\ge0 such that, for every i, the S-procedure LMI \begin{bmatrix}A_i^{ op}P+PA_i+2\alpha P+\lambda\sigma^2I & PB_i\\ B_i^{ op}P & -\lambda I\end{bmatrix}\prec0 holds. Here I is the identity, \sigma is the relative trigger threshold, and the inequality implies \dot V+2\alpha V<0 whenever the trigger condition holds. This is a conservative adapted certificate rather than a claim that the displayed LMI is the paper's exact theorem; in implementation, convexity can be obtained by alternating between K and (P,\lambda,\sigma), or by fixing K in an SDP and updating K in an outer loop.

Implementation notes

1. Integration point: use this only on parameter blocks small enough for dense matrices, such as each attention-head projection, an MoE expert router, an optimizer-state block, or the last layer. In synchronous data-parallel training, let each worker retain the last communicated block parameter x_last and use the stale gradient g(x_last) between communications. Update x by x <- x - h K g(x_last), while communicating x and refreshing x_last only when ||x-x_last||^2 > sigma^2(||x||^2+epsilon) or after a hard maximum interval. Initialize K as a diagonal Adam or RMSProp scaling matrix and partition it into 16- to 128-dimensional dense blocks. 2. Pseudocode: collect m curvature estimates H_i using Hessian-vector products or damped Gauss-Newton matrices; initialize K; solve for P and lambda at fixed K while maximizing sigma subject to P >= eps I, lambda >= 0, and the displayed LMI for every i; optionally take 5-20 projected gradient steps on K, then repeat the SDP. During training, compute g_stale, x <- x-h*K*g_stale, d=x_last-x; if d^T d > sigma^2*(e_proxy^T e_proxy+epsilon), send x or g and set x_last=x. 3. Computed from the mathematics: the robust vertex LMIs, certified alpha, and maximum safe sigma. Estimated empirically: H_i, the current optimum proxy, and the actual communication threshold. Estimate H_i with 4-8 random Hutchinson or Hessian-vector-product directions at several checkpoints, symmetrize H_i, and add damping mu I. 4. First experiment: train a 6-layer Transformer or ResNet-18 with 8 data-parallel workers on CIFAR-100 or a 1B-token language-model slice. Compare AdamW, diagonal event-triggered Adam, and dense-block event-triggered Adam at matched peak memory and learning-rate tuning. Measure validation loss versus optimizer steps, wall-clock time, number of synchronization events, and final accuracy. The key falsifiable result is at least 25% fewer communications with no more than 1% relative increase in loss, or faster loss descent at equal communication volume; also verify that the LMI-certified threshold has fewer divergence events than an unconstrained threshold sweep.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: No measurable effect

Methodology: 0 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 8/10

Built a 2D Hessian-vertex toy implementation with dense and diagonal preconditioners, fixed-K S-procedure LMI margins, trigger simulation, and numerical implication sampling. The dense block increased the certified threshold from 0.416 to 0.697 and reduced communication events at the same sigma by about 30–35% (161 versus 232 events at sigma=0.2) while both methods converged and stayed bounded. This is a clear toy-system signal, but not evidence of a practical neural-network win.

Agent confidence: 8/10

Baseline: Diagonal K: certified sigma=0.4158; at sigma=0.2, 232/1200 events (19.3%), tail loss=5.94e-33, stable=True.
Idea: Dense block K: certified sigma=0.6969; at sigma=0.2, 161/1200 events (13.4%), tail loss=1.82e-32, stable=True; maximum sampled LMI quadratic form was -1.23e-7.

Limitations: Only a 2D deterministic quadratic with two alternating Hessian vertices was tested. No neural-network model, stochastic gradients, multi-worker communication overhead, wall-clock or FLOP accounting, optimized SDP over P/lambda/K, or large-block scaling was evaluated; the LMI uses a fixed heuristic P and the discrete switching dynamics are not fully covered by the continuous-time certificate.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.10246) Sep 2, 2026

No significant difference vs. the tuned baseline (0 metric points, p=0, wins 0 / 0 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 0
Baseline mean: 0
Idea mean: 0
Effect (Δ): 0 (n/a; negative = idea better)
Wins: 0 / 0 paired seeds
p-value: 0 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Not confirmed ✗
Practical effect: No effect
Baseline tuning: swept over 0 configs

Limitations:

The mandatory dynamics benchmark, baseline sweep, idea sweep, 8 paired seeds, standard task metric comparison, communication-event measurement on trained models, permutation test, and NN-scale mechanism signature were not tested. The existing toy quadratic results are explicitly excluded as Stage-2 evidence.

How to run: After the fixed harness is provisioned, run: /home/maxwelhelp/main/bin/python3 stage2_bench.py

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

Stage 2 — Benchmark Sep 2, 2026 latest Failed ✗

Agent confidence: 10/10

The required Stage-2 benchmark could not be executed because the fixed read-only harness is absent: the mandated TRACKS import failed with ModuleNotFoundError, and bench/README.md and bench/ were not present. No trained neural-network systems, 8-seed paired comparison, permutation test, or valid mechanism signature were produced, so there is no evidence of transfer or a practical win.

Baseline: not run; fixed bench unavailable
Idea: not run; fixed bench unavailable

Benchmark result