#01Nov 9, 2026
math.OC
Learning Rate Decay Can Exponentially Accelerate SGD for Global Optimization of Nonconvex Functions
Guneykan Ozgul, Dylan Herman, Junhyung Lyle Kim and 2 more
Stochastic Gradient Descent (SGD) is a workhorse algorithm for continuous optimization. It is a folk belief among deep learning practitioners that learning rate decay in SGD substantially improves performance. However, it has remained unclear whether learning rate decay offers an asymptotic advantage for global nonconvex optimization. We answer this question in the affirmative by demonstrating a natural class of nonconvex functions for which, we prove that SGD with learning rate decay requires exponentially fewer gradient queries for global optimization than SGD with any fixed learning rate in the small-step size regime. To the best of our knowledge, this is the first exponential advantage that has been demonstrated for learning rate decay. The class of functions we consider includes many popular benchmark nonconvex functions. Our results provide a robust explanatory theory for the empirically observed benefits of learning rate decay for global optimization. We show that annealing performs the non-logconcave sampling task associated to the optimization problem in polynomial time, via a novel application of weak Poincaré inequalities.
#02Nov 9, 2026
cs.CV
2D Gum-Net: Unsupervised Dense Elastic Image Registration via Symmetric Correlation and Spectral Pooling
Aarya Bhave, Anushka Ajay Bhave
Dense non-rigid registration, the recovery of a pixel-wise deformation field that warps one image into geometric agreement with another, is fundamental across medical imaging, structural biology, and correspondence estimation, yet learning-based approaches face a severe bottleneck: ground-truth deformation fields are notoriously difficult to obtain for real data. We present 2D Gum-Net, an end-to-end architecture that predicts dense elastic deformation fields fully unsupervised. We transfer the Geometric Unsupervised Matching Network (Gum-Net), originally developed for 3D subtomogram alignment in cryo-electron tomography, to 2D elastic registration: all three modules are redesigned in 2D, and the rigid transformation head is replaced by a Dense Spatial Transformer that regresses a sparse control-point grid, lifted to a full-resolution field $Φ$ by bicubic interpolation. Downsampling uses DCT spectral pooling, implemented as frozen-basis matrix multiplications ($\approx 50\times$ GPU speedup), preserving the fine, quasi-periodic structure that max and average pooling destroy. We evaluate on contact-based fingerprints, a challenging testbed for this task: locally self-similar ridge patterns make correspondence maximally ambiguous, skin pressed against a sensor generates authentic large-displacement elastic distortion, and minutiae act as precise landmarks through which an independent matcher externally verifies the topological correctness of predicted fields. Trained purely on synthetic impressions and evaluated zero-shot on real benchmarks, our method improves structural correlation by up to 15.7% on FVC2004 and raises genuine match scores under an independent matcher while leaving imposter scores essentially unchanged. This suggests that the predicted deformations restore true structure without fabricating correspondences.
#03Nov 9, 2026
cs.AI
Shortcuts and Boundaries: Verifiable RL Misalignment and Contextual Safety Learning
Xinjie Shen
Reinforcement learning (RL) can improve an agent's task capability while reinforcing behavior that violates contextual execution requirements. We study this reward-driven misalignment, including reward hacking, through verifiable reward--context conflicts, where compliant task completion remains feasible while some executable shortcuts earn more reward than every compliant strategy. Encoding these conflicts in task dynamics lets us generate diverse interactive environments at scale and derive compliance feedback from execution histories. In resource and reporting workflows, task optimization increases real productivity through unauthorized resource use or raises reported scores while actual work declines. Controlled changes in the training distribution show that minority conflict exposure can still lead to widespread shortcut acquisition, reaching 97.9\% certified profitable violations on an API task under a 25\% conflict mixture after 128 updates. The same contextual checks supply a simple binary penalty that supports joint capability and compliance gains and partially mitigates acquired violations. Learned safety adapts to expanded authorization and transfers across mechanisms into workflows with evolving evidence and approval requirements. On 32 unseen changing-evidence tasks, resource-trained safety increases safe completion from 53.1\% to 89.1\% relative to task-only training, while both policies achieve full real work. These results connect a verifiable account of reward-driven misalignment with scalable task construction and safety learning that remains useful under changing execution requirements.
#04Nov 9, 2026
cs.LG
Where Does Alignment Emerge and What Does It Mean? A Closer Look at the Platonic Representation Hypothesis
Julia Herbst, Simon Roschmann, Quentin Bouniot and 1 more
The Platonic Representation Hypothesis (PRH) suggests that neural networks are converging toward similar representation spaces. As language models become more capable, their representations are aligning with those of vision models. However, recent work has questioned whether this trend continues to hold for newer language models. Moreover, it has been shown that representation similarity metrics can be confounded by model depth and width, requiring appropriate baselines for interpretation. In this work, we first investigate where alignment emerges between unimodal models across model layers and representation dimensions. Next, we find that the PRH capability-alignment trend persists for newer models as long as the captions are sufficiently clean and informative. Finally, we contextualize cross-modal alignment, introducing unimodal and multimodal baselines on various datasets. Our results show that the cross-modal representational alignment of unimodal models can surpass that of explicitly aligned vision-language models, but remains substantially behind the alignment observed for models of the same modality. Code is available at https://github.com/ExplainableML/prh-closer-look.
#05Nov 9, 2026
cs.CV
What kills $v$-prediction? A Patch-wise PCA Perspective on Pixel-Space Flow Matching
Tianze Luo, Haofeng Huang, Yuan Yao
Latent Diffusion Models dominate generative modeling but face two-stage training complexity and the inherent reconstruction limits of pre-trained autoencoders. Pixel-space Flow Matching offers a simplified end-to-end alternative yet often encounters training difficulties when using the standard $v$-prediction objective. Recent research attributes this difficulty to the off-manifold nature of noised targets, motivating a shift toward $x$-prediction. In this paper, we investigate this explanation through patch-wise Principal Component Analysis (PCA) and identify a bandwidth bottleneck. The velocity in minor components is largely determined by the raw noisy input, requiring deep backbones to preserve information that competes with principal-component modeling. Through controlled experiments and representation analysis, we further examine why $x$-prediction remains effective: its parameterization provides a raw-input bypass, while its learned input representation can spontaneously concentrate on principal directions. Under low-rank observations, our analysis of the induced flow characterizes $x$-prediction as an implicit conditional-mean decoder, reconstructing discarded components from the retained information. These observations motivate the Raw-injected Transformer (RiT), which combines a lightweight raw-conditioned decoder with an output-side shortcut to separate semantic modeling from raw-state propagation. Our method restores effective $v$-prediction and achieves competitive performance on ImageNet $256\times256$, providing a perspective on the interaction between prediction objectives and architectural bandwidth.