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What is Learnable in Valiant's Theory of the Learnable? Learning Perturbations to Extrapolate Your LLM Byzantine-Robust Distributed Sparse Learning Revisited The Sample Complexity of Multiple Change Point Identification under Bandit Feedback A proximal gradient algorithm for composite log-concave sampling Model-based Bootstrap of Controlled Markov Chains Approximation of Maximally Monotone Operators : A Graph Convergence Perspective Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces MIST: Reliable Streaming Decision Trees for Online Class-Incremental Learning via McDiarmid Bound A Spectral Framework for Closed-Form Relative Density Estimation Fast Rates for Offline Contextual Bandits with Forward-KL Regularization under Single-Policy Concentrability Higher-Order Equilibrium Tracking for EM-Compressible Online Estimation Scaling Limits of Long-Context Transformers A Note on Non-Negative $L_1$-Approximating Polynomials Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning 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Phase Transitions in Approximate Ranking
Chao Gao · 2017-11-30 · via math.ST updates on arXiv.org

We study the problem of approximate ranking from observations of pairwise interactions. The goal is to estimate the underlying ranks of $n$ objects from data through interactions of comparison or collaboration. Under a general framework of approximate ranking models, we characterize the exact optimal statistical error rates of estimating the underlying ranks. We discover important phase transition boundaries of the optimal error rates. Depending on the value of the signal-to-noise ratio (SNR) parameter, the optimal rate, as a function of SNR, is either trivial, polynomial, exponential or zero. The four corresponding regimes thus have completely different error behaviors. To the best of our knowledge, this phenomenon, especially the phase transition between the polynomial and the exponential rates, has not been discovered before.

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