✗ Failed on benchmark 2026

Removable-Pole Negative-Shifted Optimizer

Implementation & benchmark of arXiv:2607.22474 — Beyond Negative-Ridge Endpoints: Mixed-Sign Spectral Regularization via Negative-Shifted Gradient Descent

Usefulness7/10

Difficulty5/10

Novelty7/10

Optimization Accuracy Sample efficiency Speedup Optimizer Fine tuning Training dynamics Spectral theory Dynamical systems Optimization Linear algebra

Source paper: Beyond Negative-Ridge Endpoints: Mixed-Sign Spectral Regularization via Negative-Shifted Gradient Descent arXiv:2607.22474 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Replace ordinary gradient descent in a chosen approximately linear parameter block with gradient descent plus a controlled negative quadratic penalty, and stop before the unstable directions explode. The finite-time spectral filter can amplify well-supported directions while retaining shrinkage or limited exposure on weak directions, which is unavailable to a stable negative-ridge endpoint.

Formulas

$$\dot{\beta}_{t}=n^{-1}X^{\top}(y-X\beta_{t})+\nu\beta_{t}=b-(\widehat{\Sigma}-\nu I)\beta_{t},\qquad\beta_{0}=0.$$

$$g_{\nu,t}(s)=\frac{1-e^{-(s-\nu)t}}{s-\nu},\qquad g_{\nu,t}(\nu)=t,\qquad f_{\nu,t}(s)=s\,g_{\nu,t}(s),\qquad f_{\nu,t}(\nu)=\nu t.$$

$$\beta_{k+1}=\beta_k-\eta\left(\nabla_{\beta}L(\beta_k)-\nu\beta_k\right)=(1+\eta\nu)\beta_k-\eta\nabla_{\beta}L(\beta_k).$$

$$\text{For }S=U\operatorname{diag}(s_i)U^{\top},\quad \beta_t=U\operatorname{diag}\!\left(g_{\nu,t}(s_i)\right)U^{\top}b;\quad f_{\nu,t}(s_i)=s_i g_{\nu,t}(s_i).$$

Mathematical statement

For squared loss with design matrix X, sample count n, labels y, parameters beta, empirical covariance S=\widehat{\Sigma}=X^ op X/n, and gradient vector b=X^ op y/n, the paper's negative-shifted gradient flow is \dot{\beta}_t=b-(S-
u I)\beta_t with initialization \beta_0=0 and shift
u\ge 0. If S has eigenpair (u,s), then the coefficient along u is multiplied by g_{
u,t}(s)=(1-e^{-(s-
u)t})/(s-
u), with removable value g_{
u,t}(
u)=t; the corresponding prediction filter is f_{
u,t}(s)=s g_{
u,t}(s), whose removable value is f_{
u,t}(
u)=
u t. Thus the apparent pole of the endpoint is harmless at finite time, while the sign of f_{
u,t}(s)-1 can change across the spectrum, allowing a high-eigenvalue prefix to be anti-shrunk and weaker directions to remain shrunk. The discrete neural update is an Euler discretization: beta_{k+1}=beta_k-eta(
abla L(beta_k)-
u beta_k), where eta is the step size and K is the stopping iteration. Stability is intentionally supplied by finite K, validation selection, gradient clipping, and a cap on eta(nu-s_max), rather than by requiring S-nu I to be positive semidefinite.

Implementation notes

1. Integration point: use this optimizer only on an approximately linear parameter block, initially a frozen-backbone linear classifier, a LoRA matrix, or the final transformer block; leave the rest of the network on AdamW. Center the block at its pretrained value beta0 and apply the shift to the displacement delta=beta-beta0, so the update does not encourage raw weights to grow away from a useful initialization. For squared or locally quadratic loss, set delta <- (1+eta*nu)*delta - eta*grad_delta. For a general loss, use the same update with the minibatch gradient and optionally clip delta's norm.

2. Pseudocode: choose a grid of nu values, for example {0, 0.1, 0.3, 1.0} times an estimated largest Hessian eigenvalue, and stopping horizons K in {100, 300, 1000}. Initialize delta=0. At every minibatch, compute g=gradient(loss(theta0+delta)); set delta=(1+eta*nu)*delta-eta*g; apply global clipping if ||delta|| exceeds a preset multiple of the baseline delta norm; evaluate validation loss at each checkpoint and retain the best (nu,K) pair. The continuous-time spectral prediction is g_nu,t(s)=(1-exp(-(s-nu)t))/(s-nu), with value t when s=nu.

3. Quantities computed from the mathematics versus estimated: eta, nu, and K are hyperparameters in the first MVP. Estimate the top curvature scale s_max with 5-10 Hessian-vector power iterations, using Hessian-vector products through the selected block. Optionally estimate a few eigenvalues s_i and compare observed parameter or prediction amplification against f_nu,t(s_i). The paper supplies the finite-time filter and removable-pole mechanism; the nonlinear network's changing Hessian and minibatch noise must be measured empirically.

4. First experiment: use CIFAR-10 with a frozen pretrained small ViT or ResNet and a trainable linear head, comparing SGD, SGD with positive weight decay, and this method at matched learning-rate schedules and FLOPs. Run a second test on LoRA fine-tuning of a small language model. Record validation accuracy versus optimizer steps, final accuracy, norm growth, and divergence rate. Success is lower validation loss or higher accuracy than SGD or AdamW at equal steps, particularly in low-data fine-tuning, without more than 2x norm growth or materially higher instability. An additional success signal is faster early loss descent caused by amplification of high-curvature, signal-bearing directions.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Inconclusive

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 readable spectral NS-GD MVP and an anisotropic quadratic-regression comparison in experiment.py. The removable-pole limit matched f(nu)=nu*t exactly, off-pole error scaled linearly, mixed-sign crossover roots agreed with dense sweeps within 2.2e-5, and the Euler stability boundary matched eta=2/(s-nu) within 2.6e-4. The regression run did not show an accuracy win: NS-GD test MSE was 0.6523 versus 0.5875 for GD and 0.5859 for positive ridge, with 2.04x the GD parameter norm.

Agent confidence: 9/10

Baseline: GD: test MSE 0.5875058, train MSE 0.4680532, parameter norm 0.3604186; positive ridge: test MSE 0.5859458
Idea: NS-GD: test MSE 0.6522977, train MSE 0.3875421, parameter norm 0.7347119; removable-pole error 0, near-pole slope 0.999997, crossover error <=2.2e-5, Euler-boundary error 2.6e-4

Limitations: Only a deterministic full-batch linear regression toy was tested; no CIFAR-10, neural-network or LoRA block, minibatch noise, validation-based nu/K selection, Hessian-vector estimation, clipping, or wall-clock/FLOP comparison was run.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.22474) Sep 2, 2026

Ahead of the baseline (−5.8%) but not statistically significant (p=0.1835, wins 6 / 8 paired seeds).

Benchmark: Tabular regression (Friedman)
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 8.207
Idea mean: 7.7338
Effect (Δ): -0.4732 (−5.8%; negative = idea better)
Wins: 6 / 8 paired seeds
p-value: 0.1835 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±9.2%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 6 configs

Limitations:

Only the registered tabular Friedman regression track with mlp_tiny was tested. The benchmark used minibatch Adam with a displacement-centered negative term rather than exact full-batch Euler dynamics; the harness provided no validation split; and the mechanism signature compared a one-step predicted factor with a whole-run observed displacement ratio.

How to run: python3 stage2_negative_shift_bench.py

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

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 9/10

The registered tabular Friedman regression benchmark used mlp_tiny, a tuned Adam baseline sweep, shared learning-rate parity, and 8 paired seeds. The negative-shift idea had lower mean MSE but was not a significant win (delta_mean=-0.47317224740982056, p_value=0.1835); the selected configuration used nu=0 and all nonzero shifts were worse. The trained-model mechanism signature was not confirmed.

Baseline: Adam best cfg lr=0.01, weight_decay=0.0001; mean test MSE=8.206986248493195.
Idea: Negative-shift Adam best cfg lr=0.01, weight_decay=0.0001, nu_mult=0.0; mean test MSE=7.733814001083374; paired delta=-0.47317224740982056; p=0.1835.

Benchmark result