אותו האובייקטיב, גרדיאנטים שונים

תקציר מקורי באנגליתarXiv:2609.38786v1 Announce Type: new Abstract: Differentiable learning typically assumes that the scalar objective evaluated in the forward pass and the gradient supplied to the optimizer in the backward pass describe the same mathematical object. We show that this correspondence can fail when probabilistic objectives rely on finite special-function recurrences, custom backward rules, and numerical clipping. In high-dimensional von Mises-Fisher learning, real numerical implementations can produce identical forward scores and losses at the same learning state while supplying different gradients and following different optimization trajectories. We characterize the structure of this mismatch in finite-start Bessel recurrence and show that classwise radial mismatch can compose through probab