Directional Consistency as a Complementary Optimization Signal: The GONO Framework

Despite the central role of optimization in deep learning, most optimizers rely on update structures whose functional form is fixed before training begins. This static design can limit their ability to respond to changing gradient behavior across the loss landscape, where training may shift between stable, noisy, and inconsistent regimes. This study proposes PILOT (Policy-Informed Learned OpTimizer), an online optimizer that adapts its update behavior during training. Rather than using a fixed balance between momentum, normalization, and sign-based updates, PILOT uses gradient-direction agreement as a signal of local training stability. Conditioning the update rule on this agreement signal allows the optimizer to adjust its behavior when gradients become stable, noisy, or inconsistent. Experiments on FashionMNIST and CIFAR-10 show that PILOT consistently achieves the highest accuracy among the evaluated optimizers across convolutional settings. On the CNN architecture, PILOT reaches 94.13% on FashionMNIST and 81.94% on CIFAR-10. On ResNet-18, it further improves performance, reaching 95.71% on FashionMNIST and 93.42% on CIFAR-10. These results suggest that learning how to adapt the update structure during training can improve performance across both compact and deeper convolutional models while preserving a simple first-order optimization framework. The implementation of PILOT is publicly available at https://github.com/SattamAltwaim/PILOT.git

Single-optimizer training is a poor fit for the distinct phases of deep network optimization: adaptive methods handle noisy early gradients well but overshoot flat minima, while SGD with momentum generalizes better in the late phase but converges slowly early on. We introduce AOS-R (Adaptive Optimizer Switching, Rule-Based), a lightweight controller that monitors six online gradient-space signals -- gradient noise scale (GNS), Hutchinson curvature trace, loss stagnation, update stability ratio, gradient stability index (GSI), and loss improvement ratio (LIR) -- and switches among AdamW, SGD-M, and Lion as the optimization landscape evolves. State-preserving momentum transfer and a 400-step learning-rate bridge prevent accuracy degradation at every transition point. On CIFAR-100/WRN-28x10, AOS-R reaches 78% top-1 in 81 epochs -- 26% fewer than AdamW (109), 43% fewer than SGD-M (143), and 16% fewer than Lion (96). Across eight model-dataset benchmarks, AOS-R achieves best accuracy on 6 of 8 combinations with a mean +0.4 pp gain and 0.80x convergence speedup over AdamW under a single shared hyperparameter configuration.

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\barΘ= Θ/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.