Value-Gradient Hypothesis of RL for LLMs

Reinforcement learning substantially improves pretrained language models, but it remains understudied why critic-free methods such as PPO and GRPO work as well as they do, and when they should provide the largest gains. We develop a value-gradient perspective of critic-free RL for LLM post-training. First, under a differentiable rollout and additive-noise parameterization, we show that the actor update is value-gradient-like in expectation: the backward pass propagates costates whose conditional expectation equals the value gradient. Second, for discrete transformer policies, we show that autodifferentiation through attention produces empirical costates that approximate this value signal, with an error controlled by the sampling gap and policy entropy. These results motivate a decomposition of RL impact into value gradient signal and reachable reward headroom, yielding a criterion for when RL should be most effective along a pretraining trajectory.

A Granular Grassmannian Clustering Framework via the Schubert Variety of Best Fit

In many classification and clustering tasks, it is useful to compute a geometric representative for a dataset or a cluster, such as a mean or median. When datasets are represented by subspaces, these representatives become points on the Grassmann or flag manifold, with distances induced by their geometry, often via principal angles. We introduce a subspace clustering algorithm that replaces subspace means with a trainable prototype defined as a Schubert Variety of Best Fit (SVBF) - a subspace that comes as close as possible to intersecting each cluster member in at least one fixed direction. Integrated in the Linde-Buzo-Grey (LBG) pipeline, this SVBF-LBG scheme yields improved cluster purity on synthetic, image, spectral, and video action data, while retaining the mathematical structure required for downstream analysis.

A Multi-Domain Multi-Task Approach for Feature Selection from Bulk RNA Datasets

In this paper a multi-domain multi-task algorithm for feature selection in bulk RNAseq data is proposed. Two datasets are investigated arising from mouse host immune response to Salmonella infection. Data is collected from several strains of collaborative cross mice. Samples from the spleen and liver serve as the two domains. Several machine learning experiments are conducted and the small subset of discriminative across domains features have been extracted in each case. The algorithm proves viable and underlines the benefits of across domain feature selection by extracting new subset of discriminative features which couldn't be extracted only by one-domain approach.