Safeguarded Enlarged-BB Optimizer

1. Integration point: modify the scalar learning-rate line of SGD or momentum-free large-batch training. Start with full-batch or deterministic minibatch training so that objective comparisons are meaningful. Flatten all trainable parameters into one vector for curvature statistics, while applying the scalar step to the original parameter tensors. 2. Pseudocode: initialize t=1e-3, x_old=None, g_old=None. At each update compute g=grad(loss(x)). If old values exist, set s=x-x_old and y=g-g_old. If s dot y is positive, choose BB1 or BB2; otherwise retain the previous step. Compute Lhat=norm(y)/(norm(s)+1e-12), u=2*(1-delta)/(Lhat+1e-12), and project t into [0.01*u,u]. Propose x_try=x-t*g. Re-evaluate the same loss on the same batch; while loss(x_try)>loss(x)-c*t*norm(g)^2, set t=t/2 and retry, up to a fixed maximum number of trials. Apply the accepted update and store x_old=x and g_old=g. Every 20 accepted steps, optionally multiply the next candidate by 1.05, still clipping it by u. 3. Computed from the mathematics: BB secant steps, the local Lipschitz proxy, the 2/L-style enlarged cap, projection, and sufficient-decrease rejection. Estimated empirically: whether the secant curvature predicts minibatch curvature and which delta, c, and BB variant work best. 4. First experiment: train a 2-4 layer MLP on MNIST or CIFAR-10 with batch size 4096, comparing SGD with tuned constant learning rate, cosine-decay SGD, ordinary clipped BB, and this safeguarded method. Track loss versus optimizer steps, extra loss evaluations, wall-clock time, accepted-step fraction, and validation accuracy. Success means faster loss decrease at equal gradient evaluations, fewer catastrophic loss spikes than unconstrained BB, and equal or better final accuracy. Repeat with batch size 256 to quantify stochastic-noise degradation.