Phase-HDC: Replacing Optimizer History with Gradient Thresholds in Discrete Phase Learning
Training a compact model often needs far more memory than storing it, because the optimizer keeps its own records of past gradients. For a hyperdimensional classifier whose learned parameters are low-bit angles, which we call a \emph{phase memory}, these records take several times more memory than the model itself. We ask whether such a model can be trained while storing nothing but the model. The proposed method, Phase-HDC, turns each stored angle by at most one step per update, against the sign of its current gradient, and only when that gradient is large enough. We show that this simple rule is the exact solution of a first-order loss model in which every changed parameter pays a fixed cost. When everything except the update rule is held fixed, Phase-HDC matches the accuracy of Adam with 6-bit moments while storing three times less. Across eleven image, tabular, and text datasets, it stores 16--23 less than standard float32 Adam and 4--6 less than 8-bit Adam. The price is an average loss of about five accuracy points against float32 Adam, while Phase-HDC is more accurate than 8-bit Adam on six of the eleven datasets, including byte-level text prediction, where 8-bit Adam collapses. Instrumented training runs explain these outcomes. Once parameters must sit on a discrete grid, Adam's moments mainly decide whether a parameter moves at all, a decision that a threshold on the current gradient can make without memory, and coarse quantization of the moments breaks this decision for inputs that the data rarely contain. The storage savings are logical state rather than measured hardware memory.