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Tag: bounds

  • Towards A Unified PAC-Bayesian Framework for Norm-based Generalization Bounds

    Towards A Unified PAC-Bayesian Framework for Norm-based Generalization Bounds arXiv:2601.08100v1 Announce Type: new Abstract: Understanding the generalization behavior of deep neural networks remains a fundamental challenge in modern statistical learning theory. Among existing approaches, PAC-Bayesian norm-based bounds have demonstrated particular promise due to their data-dependent nature and their ability to capture algorithmic and geometric properties…

  • A Gapped Scale-Sensitive Dimension and Lower Bounds for Offset Rademacher Complexity

    A Gapped Scale-Sensitive Dimension and Lower Bounds for Offset Rademacher Complexity arXiv:2509.20618v1 Announce Type: new Abstract: We study gapped scale-sensitive dimensions of a function class in both sequential and non-sequential settings. We demonstrate that covering numbers for any uniformly bounded class are controlled above by these gapped dimensions, generalizing the results of cite{anthony2000function,alon1997scale}. Moreover, we…

  • Dimension-Free Bounds for Generalized First-Order Methods via Gaussian Coupling

    Dimension-Free Bounds for Generalized First-Order Methods via Gaussian Coupling arXiv:2508.10782v1 Announce Type: new Abstract: We establish non-asymptotic bounds on the finite-sample behavior of generalized first-order iterative algorithms — including gradient-based optimization methods and approximate message passing (AMP) — with Gaussian data matrices and full-memory, non-separable nonlinearities. The central result constructs an explicit coupling between the…

  • Stochastic Trace Optimization of Parameter Dependent Matrices Based on Statistical Learning Theory

    Stochastic Trace Optimization of Parameter Dependent Matrices Based on Statistical Learning Theory arXiv:2508.05764v1 Announce Type: new Abstract: We consider matrices $boldsymbol{A}(boldsymboltheta)inmathbb{R}^{mtimes m}$ that depend, possibly nonlinearly, on a parameter $boldsymboltheta$ from a compact parameter space $Theta$. We present a Monte Carlo estimator for minimizing $text{trace}(boldsymbol{A}(boldsymboltheta))$ over all $boldsymbolthetainTheta$, and determine the sampling amount so that…

  • A General-Purpose Theorem for High-Probability Bounds of Stochastic Approximation with Polyak Averaging

    A General-Purpose Theorem for High-Probability Bounds of Stochastic Approximation with Polyak Averaging arXiv:2505.21796v1 Announce Type: new Abstract: Polyak-Ruppert averaging is a widely used technique to achieve the optimal asymptotic variance of stochastic approximation (SA) algorithms, yet its high-probability performance guarantees remain underexplored in general settings. In this paper, we present a general framework for establishing…

  • Risk Bounds For Distributional Regression

    Risk Bounds For Distributional Regression arXiv:2505.09075v1 Announce Type: new Abstract: This work examines risk bounds for nonparametric distributional regression estimators. For convex-constrained distributional regression, general upper bounds are established for the continuous ranked probability score (CRPS) and the worst-case mean squared error (MSE) across the domain. These theoretical results are applied to isotonic and trend…

  • Beyond Worst-Case Online Classification: VC-Based Regret Bounds for Relaxed Benchmarks

    Beyond Worst-Case Online Classification: VC-Based Regret Bounds for Relaxed Benchmarks arXiv:2504.10598v1 Announce Type: new Abstract: We revisit online binary classification by shifting the focus from competing with the best-in-class binary loss to competing against relaxed benchmarks that capture smoothed notions of optimality. Instead of measuring regret relative to the exact minimal binary error — a…

  • Quantum Reservoir Computing and Risk Bounds

    Quantum Reservoir Computing and Risk Bounds arXiv:2501.08640v1 Announce Type: cross Abstract: We propose a way to bound the generalisation errors of several classes of quantum reservoirs using the Rademacher complexity. We give specific, parameter-dependent bounds for two particular quantum reservoir classes. We analyse how the generalisation bounds scale with growing numbers of qubits. Applying our…