Division-free adaptive learning-rate ladder

(1) Integration point: modify optimizer hyperparameter initialization and the outer training scheduler, not the neural-network layers. For each worker, use SGD or momentum SGD with a learning rate selected from a power-of-two ladder eta0, eta0/2, eta0/4, and so on. Optionally use a second power-of-two ladder for gradient-clipping thresholds. (2) Pseudocode: choose eta0 from a deliberately aggressive upper bound and exponents r=0,...,R. For each worker j and stage i, assign (r[j,i],q[j,i]) from a rotating grid; set eta=eta0*(2**(-r)), budget=T0*(2**q), initialize the model from the same checkpoint, and execute for t in range(budget): g=grad(loss(model,batch)); model=model-eta*g. At the stage boundary compute an exponential moving average of training loss and a fixed validation loss. Keep the best checkpoint and allocate the next worker to the uncovered (r,q) pair. Use integer exponent changes rather than adaptive floating-point divisions. (3) Compute every power of two and cumulative work exactly. Estimate the unknown useful horizon by the first budget crossing a fixed validation-loss target, and estimate the useful learning-rate exponent by the best validation loss among workers. Record gradient norms, update norms, loss variance, divergence events, and validation loss to test whether the ladder is adapting to stochastic noise rather than merely benefiting from extra trials. (4) First cheap experiment: use a 4-layer MLP and a small CNN on MNIST or CIFAR-10, with p=4 workers, R=5 learning-rate exponents, and budgets from 256 to 8192 updates. Compare against SGD with one tuned learning rate, AdamW, and ordinary logarithmic random search under the same total update budget. The desired signal is lower median updates to reach 90% accuracy, fewer divergent runs at high initial learning rates, and performance close to the oracle learning rate without any smoothness or variance estimate.