#01Nov 5, 2026
cs.LG
Spectral and Subspace Dynamics of LoRA in Supervised Fine-Tuning and Reinforcement Learning
Chenruo Liu, Zihan Wang, Yijun Dong and 1 more
Low-rank adaptation (LoRA) is commonly used in both supervised fine-tuning (SFT) and reinforcement learning (RL), often under similar optimizer settings. Yet, how the different dynamics of the two stages manifest in the low-rank increment that LoRA optimizes remains unclear. Tracking the singular spectrum and subspaces of LoRA updates on mathematical reasoning tasks, we find that RL concentrates the spectrum at every learning rate, while SFT expands it at moderate learning rates; moreover, RL rotates the singular subspaces much faster than SFT at small learning rates. Interpolating between SFT and RL one ingredient at a time, we pinpoint group-mean baseline subtraction as the source of both contrasts. A quadratic multi-index model updated via LoRA unveils the underlying mechanism: group-mean baseline subtraction tilts the balance between two competing forces, coherent fitting and gradient noise, toward spectral concentration and faster subspace rotation, while forcing the risk-favorable learning rate to be low. Guided by these dynamics, we tailor LoRA updates to each stage: in SFT, we flatten the spectrum of the full update to spread adaptation across singular directions, whereas in RL, we flatten the part of the update outside the current LoRA increment span to promote subspace rotation. Experiments on mathematical reasoning tasks across multiple models show that the respective configurations for SFT and RL yield the best performance among the compared methods in most evaluated settings.
#02Nov 5, 2026
cs.LG
Data-driven discovery of intrinsic Lie point symmetries of differential equation systems
Ilker Gurcan, Kadir Ustun, Ramazan Gokberk Cinbis
Discovering continuous symmetries directly from differential-equation data can reveal geometric structure for model reduction and scientific machine learning without symbolic determining equations or a closed-form representation of the governing system. We formulate intrinsic Lie point-symmetry discovery as an operator-theoretic problem on sampled jet-space data. Admissible generators are identified from a sampled determining operator and separated from redundant, solution-dependent, and geometrically trivial directions. The framework combines prolongation, sequential subspace restrictions, and a fixed nonlinear dictionary linear in trainable coefficients. A rank-stabilization preflight test diagnoses systems whose represented intrinsic symmetry algebra does not stabilize under dictionary refinement. On a controlled differential-equation benchmark with an analytically known 11-dimensional represented intrinsic symmetry algebra, the method recovers the complete represented generator set, reproduces its Lie-bracket structure to numerical precision, and approximates a known generator beyond the training domain. Here we show that explicit intrinsic continuous-symmetry generators can be recovered from data obtained from exact or approximate differential systems.
#03Nov 5, 2026
cs.AI
Scientist World: Grow a Research World for AI-Scientists
Taiming Lu, Sachin Konan, Zhuang Liu
AI-Scientists arrived, but a world for them to grow in did not. Human scientists take theirs for granted $-$ centuries of open problems and shared ideas, waiting in the scientific literature. To AI-Scientists, that world is readable but not runnable, so we build Scientist World for them: every research paper becomes an executable environment. Our autonomous pipeline extracts each paper's open-ended research question and reproduces its proposed method as the baseline; an AI-Scientist entering the environment is evaluated against what the human authors achieved. We release the first 1,000 verified environments. With the world as an evaluation, we found that frontier AI-Scientists can already surpass the human researchers 74% of the time. As a record of how AI does science, it contains 6,400 hours of research traces, with every idea, experiment, failure, and success. As a training ground, we are able to raise (+158%) a small model to match frontier models in research ideation. At last, AI-Scientists' own world has arrived, grown from and growing with science itself.
#04Nov 5, 2026
cs.LG
Adversarial Training with Large Step Sizes: Implicit Bias, Optimization, and Sharpness
Yi Feng, Andrea Paudice
Adversarial Training (AT) is an important method for improving model robustness. We study the dynamics of deterministic Gradient Descent (GD) in AT. Most existing works in this direction focus on the small or vanishing step sizes regime, where GD resembles a continuous-time flow. However, such step sizes can make training inefficient and are rarely used in practice. In this paper, we study the AT dynamics of GD with large constant step sizes. We first study GD in adversarial linear logistic regression, focusing on two aspects: where the model parameters converge (implicit bias) and how quickly the losses decrease (optimization). We then turn to neural networks training and study how the sharpness of losses evolve during AT. Our results advance the understanding of the behaviors of GD with large step sizes in AT and highlight three key differences from its behavior in Standard Training (ST): (i) the implicit bias gap between AT and ST, previously established only for small step sizes, persists for arbitrarily large constant step sizes, (ii) different from ST, there is no uniform rate of convergence for the optimization error in AT, and (iii) sharpness can exhibit two-stage dynamics that differ from the Edge-of-Stability behavior observed in ST.
#05Nov 5, 2026
cs.LG
A Hidden Markov Preference Model for Online Prediction of Adversary Actions
Angel Daruna, Michael Cogswell, Tixiao Shan and 3 more
A defender anticipating an attacker needs a probability distribution over the attacker's next action from information available before that action occurs. Two structures complicate this: the attacker's behavioral regime persists and can switch within an episode, and each action combines an operation with state-dependent arguments whose feasible values change over time. We capture both with a hidden Markov model whose latent state is the active regime and whose regime-conditioned action law is defined on a feasibility-masked tree of executable actions. The same likelihood forms the next-action distribution and updates the regime belief once the action is seen, and a partially observed prefix contributes its resolved local factors. Estimation separates into convex logistic fits that Baum-Welch reuses when regimes are unlabeled. We derive a reach-weighted nodewise KL expression for regime information, a sequence-wise log-loss bound for switching regime paths, and a uniform fallback guarantee. On the CAGE 2 cyber defense benchmark the filtered latent state is worth 0.8 nats per step, and after a controller switch the loss begins to fall in three steps against ten for a recurrent baseline.