This technical note studies the reliability limits of LLM-based multi-agent planning as a delegated decision problem. We model the LLM-based multi-agent architecture as a finite acyclic decision network in which multiple stages process shared model-context information, communicate through language interfaces with limited capacity, and may invoke human review. We show that, without new exogenous signals, any delegated network is decision-theoretically dominated by a centralized Bayes decision maker with access to the same information. In the common-evidence regime, this implies that optimizing over multi-agent directed acyclic graphs under a finite communication budget can be recast as choosing a budget-constrained stochastic experiment on the shared signal. We also characterize the loss induced by communication and information compression. Under proper scoring rules, the gap between the centralized Bayes value and the value after communication admits an expected posterior divergence representation, which reduces to conditional mutual information under logarithmic loss and to expected squared posterior error under the Brier score. These results characterize the fundamental reliability limits of delegated LLM planning. Experiments with LLMs on a controlled problem set further demonstrate these characterizations.

We study the effects of center initialization on the performance of a family of distributed gradient-based clustering algorithms introduced in [1], that work over connected networks of users. In the considered scenario, each user contains a local dataset and communicates only with its immediate neighbours, with the aim of finding a global clustering of the joint data. We perform extensive numerical experiments, evaluating the effects of center initialization on the performance of our family of methods, demonstrating that our methods are more resilient to the effects of initialization, compared to centralized gradient clustering [2]. Next, inspired by the $K$-means++ initialization [3], we propose a novel distributed center initialization scheme, which is shown to improve the performance of our methods, compared to the baseline random initialization.

We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Moving beyond static or heuristic approaches, we formulate the problem as a population dynamics game governed by a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system. Driven by a variational cost functional rather than predefined statistical shapes, this continuous-time formulation provides a flexible basis for non-parametric cluster evolution. To validate the framework, we analyze the setting of time-dependent Gaussian mixtures, showing that the MFG dynamics recover the trajectories of the classical Expectation-Maximization (EM) algorithm while ensuring mass conservation. Furthermore, we introduce time-averaged log-likelihood functionals to regularize temporal fluctuations. Numerical experiments illustrate the stability of our approach and suggest a path toward more general non-parametric clustering applications where traditional EM methods may face limitations.

We revisit the standard perturbation-based approach of Abernethy et al. (2008) in the context of unconstrained Bandit Linear Optimization (uBLO). We show the surprising result that in the unconstrained setting, this approach effectively reduces Bandit Linear Optimization (BLO) to a standard Online Linear Optimization (OLO) problem. Our framework improves on prior work in several ways. First, we derive expected-regret guarantees when our perturbation scheme is combined with comparator-adaptive OLO algorithms, leading to new insights about the impact of different adversarial models on the resulting comparator-adaptive rates. We also extend our analysis to dynamic regret, obtaining the optimal $\sqrt{P_T}$ path-length dependencies without prior knowledge of $P_T$. We then develop the first high-probability guarantees for both static and dynamic regret in uBLO. Finally, we discuss lower bounds on the static regret, and prove the folklore $Ω(\sqrt{dT})$ rate for adversarial linear bandits on the unit Euclidean ball, which is of independent interest.

Although standard Convolutional Neural Networks (CNNs) can be mathematically reinterpreted as Self-Explainable Models (SEMs), their built-in prototypes do not on their own accurately represent the data. Replacing the final linear layer with a $k$-means-based classifier addresses this limitation without compromising performance. This work introduces a common formalization of $k$-means-based post-hoc explanations for the classifier, the encoder's final output (B4), and combinations of intermediate feature activations. The latter approach leverages the spatial consistency of convolutional receptive fields to generate concept-based explanation maps, which are supported by gradient-free feature attribution maps. Empirical evaluation with a ResNet34 shows that using shallower, less compressed feature activations, such as those from the last three blocks (B234), results in a trade-off between semantic fidelity and a slight reduction in predictive performance.

We propose a topological framework for the detection of Hopf bifurcations directly from time series, based on persistent homology applied to phase space reconstructions via Takens embedding within the framework of Topological Data Analysis. The central idea is that changes in the dynamical regime are reflected in the emergence or disappearance of a dominant one-dimensional homological features in the reconstructed attractor. To quantify this behavior, we introduce a simple and interpretable scalar topological functional defined as the maximum persistence of homology classes in dimension one. This functional is used to construct a computable criterion for identifying critical parameters in families of dynamical systems without requiring knowledge of the underlying equations. The proposed approach is validated on representative systems of increasing complexity, showing consistent detection of the bifurcation point. The results support the interpretation of dynamical transitions as topological phase transitions and demonstrate the potential of topological data analysis as a model-free tool for the quantitative analysis of nonlinear time series.

Preference-based many-objective optimization faces two obstacles: an expanding space of trade-offs and heterogeneous, context-dependent human value structures. Towards this, we propose a Bayesian framework that learns a small set of latent preference archetypes rather than assuming a single fixed utility function, modelling them as components of a Dirichlet-process mixture with uncertainty over both archetypes and their weights. To query efficiently, we designing hybrid queries that target information about (i) mode identity and (ii) within-mode trade-offs. Under mild assumptions, we provide a simple regret guarantee for the resulting mixture-aware Bayesian optimization procedure. Empirically, our method outperforms standard baselines on synthetic and real-world many-objective benchmarks, and mixture-aware diagnostics reveal structure that regret alone fails to capture.

Efficient estimation of high-dimensional matrices--including covariance and precision matrices--is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators (e.g., consistency and sparsity), while largely overlooking the computational challenges inherent in high-dimensional settings. Theoretically, we first prove the convergence of LADMM, and then establish the convergence, convergence rate, and monotonicity of its reparameterized counterpart; importantly, we show that the reparameterized LADMM enjoys a faster convergence rate. Notably, the proposed reparameterization theory and methodology are applicable to the estimation of both high-dimensional covariance and precision matrices. Keywords: ADMM; High-dimensional; Learning-based optimization; Matrix estimation. 1. Introduction High-dimensional matrix estimation--covering both covariance and precision matrix estimation--constitutes a cornerstone of modern statistics and data science [1, 2, 3]. Accurate covariance estimation enables the characterization of dependence structures among a large number of variables [4, 5, 6], which is indispensable in diverse domains such as genomics [7, 8], neuroscience [9], finance [10, 11, 12], and climate science [13, 14]. Over the past two decades, substantial progress has been made in the statistical theory of high-dimensional matrix estimation, particularly with respect to the accuracy of estimators, including properties such as sparsistency and consistency [5, 15, 16]. However, in empirical studies, the dimensionality is often only on the order of tens to hundreds, and in many cases is comparable to the sample size [21, 22, 23, 24]. This observation highlights a notable gap between the statistical theory of estimators and the practical challenges of their computational implementation.

Test-time adaptation (TTA) enables neural forecasters to adapt to distribution shifts in streaming time series, but existing methods apply the same adaptation intensity regardless of the nature of the shift. We propose Regime-Guided Test-Time Adaptation (RG-TTA), a meta-controller that continuously modulates adaptation intensity based on distributional similarity to previously-seen regimes. Using an ensemble of Kolmogorov-Smirnov, Wasserstein-1, feature-distance, and variance-ratio metrics, RG-TTA computes a similarity score for each incoming batch and uses it to (i) smoothly scale the learning rate -- more aggressive for novel distributions, conservative for familiar ones -- and (ii) control gradient effort via loss-driven early stopping rather than fixed budgets, allowing the system to allocate exactly the effort each batch requires. As a supplementary mechanism, RG-TTA gates checkpoint reuse from a regime memory, loading stored specialist models only when they demonstrably outperform the current model (loss improvement >= 30%). RG-TTA is model-agnostic and strategy-composable: it wraps any forecaster exposing train/predict/save/load interfaces and enhances any gradient-based TTA method. We demonstrate three compositions -- RG-TTA, RG-EWC, and RG-DynaTTA -- and evaluate 6 update policies (3 baselines + 3 regime-guided variants) across 4 compact architectures (GRU, iTransformer, PatchTST, DLinear), 14 datasets (6 real-world multivariate benchmarks + 8 synthetic regime scenarios), and 4 forecast horizons (96, 192, 336, 720) under a streaming evaluation protocol with 3 random seeds (672 experiments total). Regime-guided policies achieve the lowest MSE in 156 of 224 seed-averaged experiments (69.6%), with RG-EWC winning 30.4% and RG-TTA winning 29.0%. Overall, RG-TTA reduces MSE by 5.7% vs TTA while running 5.5% faster; RG-EWC reduces MSE by 14.1% vs standalone EWC.

We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = λ_{k+1} - λ_k. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/π) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n^{-0.6}. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio \langle r \rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which \langle r \rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.