#01Oct 29, 2026
cs.LG
Where, When, and How Many Bits? Scaling Laws for SGD with Stochastic Rounding
Zhichao Wang, Michael W. Mahoney
Low-precision training reduces computation but introduces additional rounding error. We theoretically study how stochastic rounding (SR) affects prediction risk as model size, training time, and precision change. We consider a two-layer linear network with a fixed random first layer, trained by mini-batch SGD on fresh Gaussian data with power-law spectra. At each SGD step, the features used in forward and backward computations are rounded independently. We prove matched upper and lower bounds on prediction error for the final model and for uniform parameter averaging, with general learning-rate and precision schedules chosen before training. For fixed precisions and constant learning rates with parameter averaging, we derive how the smallest prediction error decreases with compute, and we identify model sizes and training times that achieve this optimal-compute scaling. Rounding only the forward features preserves the compute scaling of exact arithmetic, while rounding backward features changes it. Averaging parameters can change which computation needs higher precision and when higher precision is useful. Our results also give bit requirements for a given accuracy tolerance and show how backward rounding can change learning-rate schedule design. At a fixed compute budget, lower precision can improve prediction when the benefit of a larger model or longer training outweighs the additional SR-induced error. Model simulations illustrate these tradeoffs, and experiments with an MLP and a small transformer examine rounding choices beyond the theoretical model.
#02Oct 30, 2026
cs.RO
Mechanistic Interpretability for End-to-End Self-Driving
Manav Gagvani, Benjamin Namikas, Sivamurugan Velmurugan and 6 more
End-to-end (E2E) autonomous driving models achieve strong planning performance but remain largely opaque, limiting observability and trust. We apply mechanistic interpretability to E2E driving by training Sparse Autoencoders (SAEs) on the internal activations of a simple, single-camera end-to-end path planning model. Unsupervised clustering of SAE latents reveals interpretable structure aligned with route intent, vehicle speed, and braking behavior, recovering route intent with 98.3% accuracy without supervision. Through paired counterfactual image edits, we further show that specific SAE features are visually grounded and systematically respond to scene changes such as traffic light state and stop sign presence. Steering individual SAE latents produces predictable, monotonic changes in planner outputs including lateral displacement, curvature, and braking. Finally, we demonstrate a practical application of these representations: we build a lightweight adapter trained on SAE latents that reduces trajectory selection error by 3.44%, outperforming an equivalent adapter that omits SAE latents, confirming that the sparse representation captures useful information for interpretability. Code and visualizations are available at https://mgagvani.github.io/mechanistic-interpretability-for-e2e-driving/
#03Oct 29, 2026
cs.LG
bf16 Rounding Silently Discards Most RL Policy Updates: A Hidden Training-Inference Mismatch and an Unbiased One-Line Fix
Andrew Gao, William Gao
Reinforcement learning (RL) post-training of language models keeps fp32 master weights in the optimizer and serves rollouts from a bf16 copy produced by round-to-nearest conversion. At the learning rates in common use, most per-step updates are smaller than half a bf16 spacing and the conversion discards them: on Qwen3-1.7B only 0.4-0.5% of served weights change per step, and the policy that generates the training data lags the optimizer by 50-150 steps. Existing mismatch corrections cannot see this lag because they compare two copies downstream of the same rounding. We propose Fair Cast: export the master with an unbiased rounding rule (stochastic rounding or error feedback) and compute the training loss on the served copy. The change is one line at the export call, adds no hyperparameters and costs 22-38 ms per export of 1.7B parameters. Across 25 matched seed-pairs on Qwen3-0.6B to 4B with AdamW and Muon, Fair Cast is ahead on 18, tied on 6 and behind on 1, with a mean gain of 6.1 points; on zebra puzzles it reaches 0.57 +/- 0.16 over four runs where round-to-nearest collapses to 0.33 +/- 0.10. It also makes the saved fp32 checkpoint agree with the served model, and it exposes the implicit lag of round-to-nearest as a regularizer that can be restored explicitly when a task needs it.
#04Oct 29, 2026
cs.LG
Before We Adapt: The Zero-Shot Generalization Bottleneck in Continual World Models
Amir Mesbah, Di Wu
Continual reinforcement learning (CRL) requires agents to operate under environmental changes while reusing prior knowledge whenever possible. We argue that model-based reinforcement learning provides a promising framework for this setting because learned world models (WMs) can, in principle, support change detection, knowledge reuse, adaptation, and acquisition of new knowledge. However, such benefits depend heavily on whether the learned model can generalize beyond the contexts in which it was trained. In this work, we propose to formulate this problem with a contextual Markov decision process in which context is decomposed into behavioral and physical components. This view organizes continual WM learning into reuse, adaptation, and acquisition of new knowledge. As a first step, we study direct reuse when behavioral rules remain unchanged. Using Dyna-Q and TD-MPC in controlled FourRooms tasks, we find that zero-shot reuse can degrade substantially under physical-context changes and that this behavior depends strongly on the observation representation used by the WM. Additional ablation studies and latent-space diagnostics suggest that learned representations do not automatically provide the invariances needed for reuse. These results identify representation generalization as an important bottleneck for reliable world-model reuse in CRL.
#05Oct 29, 2026
cs.LG
Nonlinear Maps Between Irreducible Representations in Graph Neural Networks
Mani Shayestehfar
Equivariant linear maps cannot mix non-isomorphic irreducible representations, whereas nonlinear activations can. We develop a degree-resolved version of this nonlinear interaction picture and use the resulting block inventory to construct Specht Triple, a residual on unordered graph triples. On a synthetic structural benchmark with a frozen GIN backbone, Specht Triple reduces aggregate standardised test MSE by 11.05% relative to a parameter-matched Projector Control using the same irreducible projectors and complete linear equivariant space. The results support irreducible decomposition as an inductive bias for nonlinear higher-order equivariant processing.