✗ Failed on benchmark 2026

Variationally Relaxed Optimizer State

Implementation & benchmark of arXiv:2607.06723 — Optimization Geometrodynamics: Variational Reduction and Interaction Curvature

Usefulness7/10

Difficulty6/10

Novelty6/10

Optimization Stability Accuracy Sample efficiency Optimizer Training dynamics Optimization Convex analysis Linear algebra Dynamical systems

Source paper: Optimization Geometrodynamics: Variational Reduction and Interaction Curvature arXiv:2607.06723 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Replace an opaque adaptive-optimizer state update with a small controller variable obtained by minimizing a strongly convex energy jointly associated with the proposed parameter motion. The controller is allowed to relax toward the current gradient before the parameter update, while the visible update uses the reduced energy and its envelope gradient. This creates an optimizer whose hidden geometry is optimized rather than inherited from a fixed exponential-moving-average recurrence.

Formulas

$$(\pi_{!}E)(y)=\inf_{\pi(x)=y}E(x)$$

$$D\sigma_{\star}=-E_{\sigma\sigma}^{-1}E_{\sigma y},\qquad D^{2}(\pi_{!}E)=E_{yy}-E_{y\sigma}E_{\sigma\sigma}^{-1}E_{\sigma y}$$

$$\Delta\theta_t=y_t,\qquad (\sigma_t,y_t)\approx\arg\min_{\sigma,y}\;E_t(\sigma,y),\qquad \theta_{t+1}=\theta_t+y_t$$

$$\widehat{D^2F}=E_{yy}-E_{y\sigma}(E_{\sigma\sigma}+\lambda I)^{-1}E_{\sigma y},\qquad \lambda>0$$

Mathematical statement

For visible action y and hidden controller state sigma, define an energy E(sigma,y), and the reduced energy F(y)=(pi_!E)(y)=inf_sigma E(sigma,y). At a unique stationary minimizer sigma_star(y), let H=E_{sigma sigma}(sigma_star,y) be positive definite and let G=E_{sigma y}(sigma_star,y). The paper gives the controller sensitivity D sigma_star=-H^{-1}G and the reduced curvature D^2F=E_{yy}-E_{y sigma}H^{-1}E_{sigma y}. The first term is the direct curvature and the second is the Schur-complement correction caused by relaxed hidden state. In an optimizer, y is a candidate parameter displacement, sigma is a low-dimensional preconditioner or momentum controller, E is a scalar controller energy, and gradients of F are computed by the envelope theorem as D_yF=E_y(sigma_star(y),y), without differentiating through the exact inner minimizer. Positive definiteness of H is the required local stability assumption; a practical implementation enforces H\succeq\mu I with damping.

Implementation notes

Integrate this into AdamW or SGD as a two-level optimizer with a controller dimension k much smaller than the number of parameters. Let g_t be the minibatch gradient and let sigma_t contain k coefficients defining a structured preconditioner, for example P(sigma)=diag(exp(U sigma)) for a fixed random or learned feature matrix U, or a rank-k correction P(sigma)=I+U diag(sigma) U^T. Define the candidate visible displacement y=-alpha P(sigma)g_t and an inner energy E_t(sigma,y)=L_t(theta_t+y)+rho/2||sigma-sigma_{t-1}||^2+mu/2||y+alpha P(sigma)g_t||^2. Use one to three damped Newton or gradient steps on sigma and y, or eliminate y by the displayed parameterization and optimize sigma only. In pseudocode: compute g; initialize sigma=sigma_{t-1}; repeat q times: compute y=-alpha P(sigma)g, compute the gradient of E with respect to sigma, estimate H=E_{sigma sigma} by exact autograd or finite differences, set sigma <- sigma-(H+lambda I)^{-1} grad_sigma E; then recompute y and update theta <- theta+y with decoupled weight decay. The paper-derived computation is the implicit sensitivity and Schur matrix; empirical quantities are H, G, and the damping lambda. Estimate H^{-1}G with a k-dimensional Cholesky solve, not a parameter-sized inverse. First test on CIFAR-10 with a 2-layer MLP and small ResNet, comparing SGD, AdamW, and the relaxed optimizer at equal forward/backward FLOPs. Measure loss decrease per optimizer step, gradient-norm variance, controller condition number, and final accuracy. Success is faster loss descent or lower minibatch-gradient variance without divergence when alpha is increased beyond the AdamW stability limit.

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 bounded, damped variational controller optimizer and an analytic quadratic verification in experiment.py. The mechanism matched all three predictions: sensitivity, Schur curvature, and coupling/damping scaling errors were below 1.2e-12; the observed stability boundary was 0.909015 versus the predicted 0.909091. In the small anisotropic quadratic trial, relaxed optimization reached loss 0.2631 versus SGD 0.4077 after 100 steps and 2.2283 versus 4.4976 after 10 steps, but this is only a toy signal rather than evidence of a general ML win.

Agent confidence: 9/10

Baseline: SGD: loss at step 10 = 4.4976339343; step 100 = 0.4077188348; stable for 100 steps
Idea: Variational controller: loss at step 10 = 2.2283102375; step 100 = 0.2630654712; stable for 100 steps
Math check: Maximum sensitivity error = 1.18e-12; curvature and Schur correction errors <= 4.44e-16; damped-curvature error = 0; predicted stability boundary = 0.909091, observed = 0.909015

Limitations: No CIFAR-10, MLP, ResNet, AdamW, minibatch-gradient variance, FLOP-normalized comparison, or beyond-stability-limit neural-network test was run. The practical controller uses bounded tanh parameterization and first-order inner steps rather than exact Newton or implicit envelope gradients, so the toy improvement should not be generalized.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.06723) Sep 2, 2026

Significantly WORSE than the tuned baseline (n/a, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Tabular regression (Friedman)
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 0
Idea mean: 8.6763
Effect (Δ): 2.1872 (n/a; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Not confirmed ✗
Practical effect: Hurts
Baseline tuning: swept over 0 configs

Limitations:

Only the optimizer-matched tabular track was tested, with mlp_tiny, 800 training samples, 400 test samples, 20 epochs, and batch size 128. The implementation uses four parameter-block controller coefficients and a first-order linearized energy proxy rather than exact Newton/implicit envelope optimization; no CIFAR-10, sequence, dynamics, FLOP-normalized, or beyond-Adam stability experiment was run.

How to run: python3 bench_experiment.py

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

Stage 2 — Benchmark Aug 30, 2026 latest Failed ✗

Agent confidence: 10/10

Implemented the variational controller as a local optimizer intervention on the structurally matched tabular Friedman regression track using the shared mlp_tiny architecture. Across 8 paired seeds, the tuned Adam baseline achieved test MSE 6.4891 versus 8.6763 for the idea; the idea was significantly worse (delta +2.1872, permutation p=0.0081), and the trained-model mechanism signature was not confirmed because scale/gradient correlation was -0.129.

Baseline: Adam mlp_tiny, best lr=0.01: mean test MSE 6.4890658259, std 1.3862433020, 8 seeds
Idea: Variational controller, selected lr=0.003 rho=0.05: mean test MSE 8.6762773991, std 0.9955932280, 8 seeds
Mechanism signature: Observed mean scale-gradient correlation = -0.1285149642; confirmed=false
Paired delta: idea - baseline = +2.1872115731 MSE; 0/8 idea wins; permutation p=0.0081

Benchmark result