Natural Spectral Fusion: p-Exponent Cyclic Scheduling and Early Decision-Boundary Alignment in First-Order Optimization

Spectral behaviors have been widely discussed in machine learning, yet the optimizer's own spectral bias remains unclear. We argue that first-order optimizers exhibit an intrinsic frequency preference that significantly reshapes the optimization path. To address this, we propose Natural Spectral Fusion (NSF): reframing training as controllable spectral coverage and information fusion rather than merely scaling step sizes. NSF has two core principles: treating the optimizer as a spectral controller that dynamically balances low- and high-frequency information; and periodically reweighting frequency bands at negligible cost, without modifying the model, data, or training pipeline. We realize NSF via a p-exponent extension of the second-moment term, enabling both positive and negative exponents, and implement it through cyclic scheduling. Theory and experiments show that adaptive methods emphasize low frequencies, SGD is near-neutral, and negative exponents amplify high-frequency information. Cyclic scheduling broadens spectral coverage, improves cross-band fusion, and induces early decision-boundary alignment, where accuracy improves even while loss remains high. Across multiple benchmarks, with identical learning-rate strategies and fixed hyperparameters, p-exponent cyclic scheduling consistently reduces test error and demonstrates distinct convergence behavior; on some tasks, it matches baseline accuracy with only one-quarter of the training cost. Overall, NSF reveals the optimizer's role as an active spectral controller and provides a unified, controllable, and efficient framework for first-order optimization.

The Unified Balance Theory of Second-Moment Exponential Scaling Optimizers in Visual Tasks

We have identified a potential method for unifying first-order optimizers through the use of variable Second-Moment Exponential Scaling(SMES). We begin with back propagation, addressing classic phenomena such as gradient vanishing and explosion, as well as issues related to dataset sparsity, and introduce the theory of balance in optimization. Through this theory, we suggest that SGD and adaptive optimizers can be unified under a broader inference, employing variable moving exponential scaling to achieve a balanced approach within a generalized formula for first-order optimizers. We conducted tests on some classic datasets and networks to confirm the impact of different balance coefficients on the overall training process.

A Simple Asymmetric Momentum Make SGD Greatest Again

We propose the simplest SGD enhanced method ever, Loss-Controlled Asymmetric Momentum(LCAM), aimed directly at the Saddle Point problem. Compared to the traditional SGD with Momentum, there's no increase in computational demand, yet it outperforms all current optimizers. We use the concepts of weight conjugation and traction effect to explain this phenomenon. We designed experiments to rapidly reduce the learning rate at specified epochs to trap parameters more easily at saddle points. We selected WRN28-10 as the test network and chose cifar10 and cifar100 as test datasets, an identical group to the original paper of WRN and Cosine Annealing Scheduling(CAS). We compared the ability to bypass saddle points of Asymmetric Momentum with different priorities. Finally, using WRN28-10 on Cifar100, we achieved a peak average test accuracy of 80.78\% around 120 epoch. For comparison, the original WRN paper reported 80.75\%, while CAS was at 80.42\%, all at 200 epoch. This means that while potentially increasing accuracy, we use nearly half convergence time. Our demonstration code is available at\\ https://github.com/hakumaicc/Asymmetric-Momentum-LCAM