Proper Agnostic Learning of Arbitrarily Biased ReLUs under Log-Concave Marginals

A sample-only proper constant-factor agnostic learner for one arbitrarily biased ReLU under every isotropic log-concave marginal. For arbitrary real labels, the learner returns a single affine ReLU with squared loss C OPT_W + epsilon, where comparator slopes have norm at most supplied W and comparator bias is unrestricted. Sample and real-arithmetic operation counts are polynomial jointly in dimension, W, inverse accuracy, and log inverse failure probability. The proof combines an output-scale threshold witness, robust cap normalization, positive-intercept regression, a squared-loss cap-transfer inequality, and median-centered unrestricted-bias validation. Output slopes need not satisfy the comparator bound. No finite-bit, multiple-neuron, or general monotone-single-index theorem is asserted.