Two-budget Newton continuation for max-affine adapters

Integrate this as an outer calibration optimizer around a frozen or nearly frozen ReLU network. Let x be a small vector of trainable channel gains, LoRA scaling factors, activation offsets, clipping thresholds, or MoE routing thresholds; begin with n between 5 and 20. Define two objectives from held-out data, for example clean classification-margin violation and corruption classification-margin violation. Approximate each objective by a max of affine pieces in x. For a ReLU network, collect the affine coefficients for each fixed activation pattern; for the first MVP, use finite differences to fit affine pieces around the current x. Encode simple gain and coupling constraints as max-plus inequalities or directly retain the resulting halfspaces. Construct the bipartite graph from the finite entries of the paper's A and B matrices. For a candidate budget pair, run a small mean-payoff feasibility oracle using policy iteration or enumerate cycles when n is small. The oracle must return the currently active inequality coefficients k_1, k_2, and s. Start at a feasible budget point, select a direction w such as (1,-1), identify the active boundary just to the left using a small epsilon, and compute c_next with the displayed Newton formula. Repeat until the desired endpoint is reached. The cycle coefficients and Newton intersections are computed from the graph; affine objective pieces and initial feasible points are estimated empirically. First test on CIFAR-10 with a frozen ResNet-18 and 5-20 trainable channel gains, using clean error and CIFAR-10-C error as objectives. Compare against 20-100 scalarized training runs and coordinate bisection, equalizing backbone forward passes. Success is the same Pareto envelope with at least 5x fewer evaluations, or lower corruption error at matched clean accuracy.