Reinforcement Learning with Verifiable Rewards has shown great promise in enhancing mathematical reasoning in large language models. However, existing approaches provide limited intermediate supervision for complex, multi-step reasoning tasks. To bridge this gap, we propose \textbf{Hierarchical Hint-based Reward}, a framework that leverages the intrinsic sub-question structure of mathematical problems as structured intermediate conditioning signals during RLVR training. Unlike chain-of-thought prompting, these hierarchical hints serve as logically grounded intermediate milestones that effectively reduce the reasoning horizon toward the final answer. To validate this framework, we introduce \textbf{HiMath}, a benchmark of hierarchically structured mathematical problems. Across diverse LLMs, hierarchical hints consistently improve reasoning performance even without task-specific training. When integrated into RL training via H$^2$R, they boost Qwen2.5-Math-7B accuracy on HiMath from 30.4\% to 39.1\% using hierarchical hints, and up to 45.6\% with Oracle Hierarchical Hints. This pronounced improvement demonstrates that incorporating reliable intermediate checkpoints into reward design is essential for sustaining complex mathematical reasoning.
Kernel techniques are among the most powerful methods in data science and statistics. One of the key properties of kernels contributing to their numerous successful applications is that the associated reproducing kernel Hilbert space is expressive enough to encode probability distributions without loss of information or to capture the joint independence of $M\ge 2$ random variables, under mild conditions. Computationally, these measures (i) rely on the so-called Gram matrix which contains the pairwise similarity of sample points, and (ii) converge at a rate $O \left(n^{-\frac{1}{2}}\right)$, where $n$ denotes the sample size. We focus on measuring the independence of latent position random graphs (LPRGs), a model widely-applied for instance in neuroscience, sociology, and epidemiology. In LPRGs a latent representation is associated to each node. The similarity of these representations (meant via a kernel) determines whether two nodes are connected, making the Gram matrices only implicitly observable and the estimation challenging. Despite the fundamental importance of quantifying the independence of $M$ LPRGs, very little is known about the solution of this problem; existing techniques are restricted to $M=2$ components and only guarantee consistency without rates. In this work, (i) we define an independence measure called graph Hilbert-Schmidt independence criterion (GHSIC) for $M\ge 2$ LPRGs, (ii) we prove that it can be estimated at a rate $\tilde{O}\left(n^{-\frac{1}{3}}\right)$ relying on spectral decomposition of adjacency matrices, (iii) without the need to choose kernels and their hyperparameters. Numerical experiments demonstrate the effectiveness of the proposed GHSIC estimator both on synthetic examples and in the analysis of brain functional connectivity subnetworks in natural language perception.