We study learning problems on correlated stochastic block models with two balanced communities.

Understanding the fundamental principles behind the success of deep neural networks is one of the most important open questions in the current literature.

While one side of the coin is a boost of interest and investment on deep learning research, theother isanemergent need foritsrobustness, sample efficiency,security,and interpretability.

It is known that Binary Segmentation is consistent but not optimal (Venkatraman (1992)). As an improvement, Fryzlewicz (2014) propose WildBinary Segmentation andshowthatithasabetter localization rate.

In the BAI, the learner needs to pull arms efficiently for better identification.

A.2 Proof of proposition 1 Let Pœ{B,D,E}, k be a valid kernel (assumptions of theorem 1) with K Inversion of conditional with Bayes rule gives: 'W œS As a complement, we now explicit the simple forms taken by the posterior limit graph in each case. A.3 Proof of theorem 2 We consider the following hierarchical model, for Nonetheless it can be simplified as we now show. We focus on finding the optimal eigenvectors first. Only the left term in (18) depends on R. The optimization problem for eigenvectors writes: min tr! Note that the identity permutation i.e. for i œ [n], (i) =i is optimal in this case as the ( We will choose this U in what follows as the sign of the axes do not influence the characterization of the final result in Z as a PCA embedding. Note that this solution is not unique if there are repeated eigenvalues.

Non-negative matrix factorization with fixed row and columnsums.

Kernel ridge regression (KRR) is a foundational tool in machine learning, with recent work emphasizing its connections to neural networks. However, existing theory primarily addresses the i.i.d. setting, while real-world data often exhibits structured dependencies - particularly in applications like denoising score learning where multiple noisy observations derive from shared underlying signals. We present the first systematic study of KRR generalization for non-i.i.d. data with signal-noise causal structure, where observations represent different noisy views of common signals. By developing a novel blockwise decomposition method that enables precise concentration analysis for dependent data, we derive excess risk bounds for KRR that explicitly depend on: (1) the kernel spectrum, (2) causal structure parameters, and (3) sampling mechanisms (including relative sample sizes for signals and noises). We further apply our results to denoising score learning, establishing generalization guarantees and providing principled guidance for sampling noisy data points. This work advances KRR theory while providing practical tools for analyzing dependent data in modern machine learning applications.