Robbins and Monro 1951 founded stochastic approximation — the method family SGD belongs to — as root-finding from noisy experiments

"A Stochastic Approximation Method" (Robbins & Monro, Ann. Math. Statist. 1951) — now read directly, discharging the capture's note that its body text had never been quoted in this vault's research chain — frames the problem as statistical root-finding, not optimization: "Let M(x) denote the expected value at level x of the response to a certain experiment. M(x) is assumed to be a monotone function of x but is unknown to the experimenter, and it is desired to find the solution x = θ of the equation M(x) = α … We give a method for making successive experiments at levels x₁, x₂, … in such a way that xₙ will tend to θ in probability."

The iterate-on-noisy-observations scheme with diminishing step sizes is the mathematical ancestor of stochastic gradient descent (set M = ∇L and α = 0 and root-finding becomes stationary-point-finding). Wikipedia's SGD History section credits exactly this ("the earliest stochastic approximation methods, preceding stochastic gradient descent") — and, notably, moves from Robbins–Monro straight to Rosenblatt and then RHW 1986, with no Amari mention: the Amari-1967 claim is not part of SGD's own mainstream Wikipedia narrative, only of the Schmidhuber-derived history-of-ANN pages (claim-wikipedia-amari-sgd-citogenesis).

This note anchors the "SGD" half of myth-amari-first-sgd-mlp: when the circulating claim says Amari trained networks "by SGD, a method proposed in 1951," this paper is the 1951 object. Amari's own 確率的降下法 ("stochastic descent method") in the 1968 book is the same family applied to discriminant learning — claim-amari-1968-saito-experiment-primary-read. Cluster: moc-backpropagation-origins.