Agnostic Learning of Positive Homogeneous ReLU Networks by Sequential Moment Contractions

We study squared-loss agnostic learning of positive, bias-free, one-hidden-layer ReLU networks under standard Gaussian inputs. The comparator has at most k units and total output weight at most one, with no separation or conditioning assumptions. For arbitrary real labels with a supplied second-moment bound E[Y^2] <= B, we give an ordinary-sample algorithm with sample and real-arithmetic operation counts polynomial in dimension d, width k, B, inverse accuracy, and log inverse failure probability. Its output is a positive homogeneous ReLU network of width O(k log(2/epsilon)) and total output weight at most one, with squared risk at most C k^3 log^5(2k/epsilon) OPT + epsilon. The output is continuous, convex, nonnegative, and 1-Lipschitz. Sequential Gaussian contractions of compressed positive moment states yield a robust multiscale direction dictionary. A cone-constrained convex fit with a scale-weighted penalty converts local tangent approximations into a positive network while preserving the agnostic bound. This version strengthens the output representation of the preceding version in the same research line. The approximation factor remains width-dependent, the output width may exceed k, and the computational statement is in exact real arithmetic. No width-independent approximation factor, strict width-k proper learner, finite-bit complexity, or practical polynomial exponents are asserted.