Same Loss, Different Gradients

Ningkang Peng, Xiaoqian Peng, Yifan He, Anjie Hu, Chao Tan +2

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 probabilities into a locally nonconservative update field. Evaluating the accuracy of special-function values and derivatives separately is therefore insufficient to characterize the realized learning objective. Motivated by this observation, we introduce AR/FR, a fixed-depth analytic realization that constructs a potential and its derivative jointly, ensuring forward-backward coherence by construction. We establish a uniform cubic-order error…