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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 Linear Response Estimators for Singular Statistical Models Statistical inference with belief functions: A survey Robust stochastic first order methods in heavy-tailed noise via medoid mini-batch gradient sampling Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity Adaptive auditing of AI systems with anytime-valid guarantees Locally Near Optimal Piecewise Linear Regression in High Dimensions via Difference of Max-Affine Functions Risk-Controlled Post-Processing of Decision Policies Covariate Balancing and Riesz Regression Should Be Guided by the Neyman Orthogonal Score in Debiased Machine Learning A Unified Pair-GRPO Family: From Implicit to Explicit Preference Constraints for Stable and General RL Alignment Time-Inhomogeneous Preconditioned Langevin Dynamics A Fine-Grained Understanding of Uniform Convergence for Halfspaces CITE: Anytime-Valid Statistical Inference in LLM Self-Consistency Ratio-based Loss Functions Optimal Confidence Band for Kernel Gradient Flow Estimator A renormalization-group inspired lattice-based framework for piecewise generalized linear models
Understanding Aggregations of Proper Learners in Multicla...
Julian Asilis, Mikael Møller Høgsgaard, Grigoris Velegkas · 2024-10-30 · via math.ST updates on arXiv.org

Multiclass learnability is known to exhibit a properness barrier: there are learnable classes which cannot be learned by any proper learner. Binary classification faces no such barrier for learnability, but a similar one for optimal learning, which can in general only be achieved by improper learners. Fortunately, recent advances in binary classification have demonstrated that this requirement can be satisfied using aggregations of proper learners, some of which are strikingly simple. This raises a natural question: to what extent can simple aggregations of proper learners overcome the properness barrier in multiclass classification? We give a positive answer to this question for classes which have finite Graph dimension, $d_G$. Namely, we demonstrate that the optimal binary learners of Hanneke, Larsen, and Aden-Ali et al. (appropriately generalized to the multiclass setting) achieve sample complexity $O\left(\frac{d_G + \ln(1 / δ)}ε\right)$. This forms a strict improvement upon the sample complexity of ERM. We complement this with a lower bound demonstrating that for certain classes of Graph dimension $d_G$, majorities of ERM learners require $Ω\left( \frac{d_G + \ln(1 / δ)}ε\right)$ samples. Furthermore, we show that a single ERM requires $Ω\left(\frac{d_G \ln(1 / ε) + \ln(1 / δ)}ε\right)$ samples on such classes, exceeding the lower bound of Daniely et al. (2015) by a factor of $\ln(1 / ε)$. For multiclass learning in full generality -- i.e., for classes of finite DS dimension but possibly infinite Graph dimension -- we give a strong refutation to these learning strategies, by exhibiting a learnable class which cannot be learned to constant error by any aggregation of a finite number of proper learners.