The vanishing gradient problem is a chain-rule pathology — early layers stop learning as gradient magnitudes diverge across depth
The vanishing gradient problem is "the problem of greatly diverging gradient magnitudes between earlier and later layers encountered when training neural networks with backpropagation": as the loss signal traverses many layers backward, early-layer weight updates shrink toward zero while later layers keep updating — the layers that learn basic representations effectively stop learning. The inverse pathology (exploding gradients) arises from the same structure: repeated chain-rule multiplication of Jacobians across depth.
This is the failure mode inside the algorithm the origins cluster traces: the backward pass (claim-backpropagation-special-case-of-reverse-mode-ad) multiplies per-layer derivatives, so whatever is slightly less than or greater than one compounds exponentially with depth — in the authors' own words, error signals "tend to either (1) blow up or (2) vanish: the temporal evolution of the backpropagated error exponentially depends on the size of the weights" (Hochreiter, Bengio, Frasconi & Schmidhuber 2001, §1, Tier 1, read directly 2026-09-11; the paper derives this for backpropagation through time, where each time step is a layer). It is a large part of why deep networks stalled for a decade after 1986 and why the field's fixes (ReLU, normalization, residual connections, LSTM gating) all target the multiplication chain.
Attribution boundary, honestly held: the field's canonical peer-reviewed characterization is Bengio, Simard & Frasconi (1994, bibliographic record Tier-2-confirmed); the claim that Hochreiter's 1991 Diplom thesis was the first formal identification circulates widely but the capture found no independent Tier 1–2 source asserting primacy — that sub-claim stays in the capture under its [unverified-historical] flag. See moc-backpropagation-origins, claim-rhw-1986-demonstration-not-invention.