# Large Distant Gradients Need Not Be Reliable: reliability-weighted credit assignment for long-horizon autoregressive forecasting

arXiv: https://arxiv.org/abs/2609.12890 · HTML: https://mlsanity.com/paper/2609.12890/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 35 of its 62 references that are in the map.
- First posted on arXiv 11 Sep 2026.

## Builds on (most cited first, 30 of 35)

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- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
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- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Decoupled Neural Interfaces using Synthetic Gradients](https://mlsanity.com/paper/1608.05343/) (2016)
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- [Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces](https://mlsanity.com/paper/1905.07357/) (2019)
- [On the difficulty of learning chaotic dynamics with RNNs](https://mlsanity.com/paper/2110.07238/) (2021)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [PIPPS: Flexible Model-Based Policy Search Robust to the Curse of Chaos](https://mlsanity.com/paper/1902.01240/) (2019)
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- [Unbiasing Truncated Backpropagation Through Time](https://mlsanity.com/paper/1705.08209/) (2017)
- [Approximating Real-Time Recurrent Learning with Random Kronecker Factors](https://mlsanity.com/paper/1805.10842/) (2018)
- [Synthetic Returns for Long-Term Credit Assignment](https://mlsanity.com/paper/2102.12425/) (2021)
- [Differentiability in Unrolled Training of Neural Physics Simulators on Transient Dynamics](https://mlsanity.com/paper/2402.12971/) (2024)
- [Optimal Kronecker-Sum Approximation of Real Time Recurrent Learning](https://mlsanity.com/paper/1902.03993/) (2019)
- [Recurrent Neural Operators: Stable Long-Term PDE Prediction](https://mlsanity.com/paper/2505.20721/) (2025)
- [TRACE: Turn-level Reward Assignment via Credit Estimation for Long-Horizon Agents](https://mlsanity.com/paper/2607.13988/) (2026)
- [Credit Assignment Techniques in Stochastic Computation Graphs](https://mlsanity.com/paper/1901.01761/) (2019)
- [Do Transformer World Models Give Better Policy Gradients?](https://mlsanity.com/paper/2402.05290/) (2024)

## Abstract (arXiv)

In autoregressive forecasting, long prediction rollouts provide distant supervision, but backpropagation through time (BPTT) carries gradients from those losses through many autoregressive steps. Repeated Jacobian products can make distant gradients dominate the update while amplifying predictable signal and unpredictable innovation together; a large distant gradient therefore need not carry reliable learning signal. Motivated by this, we introduce Internal Dual-Wiener routing (Internal-DW), a backward-only intervention that preserves the full forward rollout and all step losses while reliability-weighting internal gradient routes. At each residual block, we derive bounded Wiener gains for the identity and nonlinear routes that balance preserving predictable learning signal against suppressing unpredictable variation, and estimate them from route-level gradient statistics and an explicit noise model. In a controlled system with known gradient signal-to-noise ratio (SNR), we show that distant gradients can grow even as their SNR falls, and that Internal-DW reduces error in recovering predictable gradient signals and improves forecasting. On four history-dominated, weak-drive testbeds, Internal-DW reduces forecast error by 5.2%-13.8% relative to full BPTT, outperforms gradient clipping and Jacobian regularization on three testbeds, with similar performance on shear flow, and outperforms validation-selected truncated BPTT (TBPTT) on three. It also extends or preserves the fitted optimal training-horizon range across these four testbeds. Across benchmarks, the current Internal-DW estimator has a clear applicability boundary: its benefit diminishes or reverses when usable history is limited or when the selected sampler fails to represent dominant drive-dependent variation. The results show that retaining long-horizon supervision does not require trusting every backward contribution equally.