# Same Loss, Different Gradients

arXiv: https://arxiv.org/abs/2609.38786 · HTML: https://mlsanity.com/paper/2609.38786/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 35 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Learning Imbalanced Datasets with Label-Distribution-Aware Margin Loss](https://mlsanity.com/paper/1906.07413/) (2019)
- [Deep Anomaly Detection with Outlier Exposure](https://mlsanity.com/paper/1812.04606/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Decoupling Representation and Classifier for Long-Tailed Recognition](https://mlsanity.com/paper/1910.09217/) (2019)
- [Long-tail learning via logit adjustment](https://mlsanity.com/paper/2007.07314/) (2020)
- [Understanding Straight-Through Estimator in Training Activation Quantized Neural Nets](https://mlsanity.com/paper/1903.05662/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Semantically Coherent Out-of-Distribution Detection](https://mlsanity.com/paper/2108.11941/) (2021)
- [Implicit Reparameterization Gradients](https://mlsanity.com/paper/1805.08498/) (2018)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [A mathematical model for automatic differentiation in machine learning](https://mlsanity.com/paper/2006.02080/) (2020)
- [Partial and Asymmetric Contrastive Learning for Out-of-Distribution Detection in Long-Tailed Recognition](https://mlsanity.com/paper/2207.01160/) (2022)
- [von Mises-Fisher Loss: An Exploration of Embedding Geometries for Supervised Learning](https://mlsanity.com/paper/2103.15718/) (2021)
- [Probabilistic Contrastive Learning for Long-Tailed Visual Recognition](https://mlsanity.com/paper/2403.06726/) (2024)
- [Out-of-Distribution Detection in Long-Tailed Recognition with Calibrated Outlier Class Learning](https://mlsanity.com/paper/2312.10686/) (2023)
- [EAT: Towards Long-Tailed Out-of-Distribution Detection](https://mlsanity.com/paper/2312.08939/) (2023)
- [Long-Tailed Out-of-Distribution Detection: Prioritizing Attention to Tail](https://mlsanity.com/paper/2408.06742/) (2024)
- [Long-Tailed Out-of-Distribution Detection via Normalized Outlier Distribution Adaptation](https://mlsanity.com/paper/2410.20807/) (2024)
- [Accurate Computation of the Logarithm of Modified Bessel Functions on GPUs](https://mlsanity.com/paper/2409.08729/) (2024)
- [Exploiting Inter-Sample Information for Long-tailed Out-of-Distribution Detection](https://mlsanity.com/paper/2511.16015/) (2025)

## Abstract (arXiv)

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 bound relative to the exact Bessel ratio over the entire nonnegative concentration axis and propagate this guarantee to learning scores and objectives. As representation dimension increases, the original finite recurrence becomes sequentially deeper, whereas the worst-case AR/FR error guarantee tightens cubically, jointly providing coherence, certified fidelity, and fixed-depth computation. These results suggest that a differentiable numerical primitive is defined by both the values it realizes and the derivatives it actually supplies to the optimizer; together, they constitute the numerical realization of the learning algorithm.