Optimal Dimension-Free Sampling for Regularized Classification

New theoretical work establishes optimal sampling bounds for regularized classification, proving that the number of training samples required to achieve a given error rate can be sharply characterized across common loss functions and regularization strategies [1]. The paper, posted to arXiv on 22 May 2026, proves upper and lower bounds achieving \(1\pm\varepsilon\)-relative error for a broad class of Lipschitz continuous classification loss functions under various regularization terms [1][2]. The analysis covers widely used functions including logistic loss, sigmoid loss, hinge loss, and ReLU loss [2]. Regularization, a process that adds a penalty term to an optimization problem to prevent overfitting, is a foundational technique in machine learning [3]. Sample complexity — the number of training examples an algorithm needs to learn a target function within a specified error — is a core metric for evaluating learning algorithms [4]. For \(\|\cdot\|_2/k\) regularization, the work establishes matching upper and lower bounds of \(k^2/\varepsilon^2\) [1][2]. For \(\|\cdot\|_1/k\) regularization, the bounds tighten to \(k/\varepsilon^2\) [1][2]. The case of \(\|\cdot\|_2^2/k\) regularization proves more nuanced: when the loss function satisfies a bounded derivative property — specifically \(|g'(x)|\leq g(x)\), with \(g(0)>0\) and \(g\) monotonic or convex — the sampling complexity is linear in \(k\); otherwise the general bound remains \(k^2/\varepsilon^2\) [2]. A notable finding concerns loss functions where \(g(0)=0\). Under this condition, the results indicate that no dimension-free bounds are possible, and even sublinear bounds are ruled out [2]. All upper bounds are complemented by matching lower bounds up to polylogarithmic terms [2]. The technical approach relies on uniform or squared-norm sampling rather than more complex sensitivity sampling frameworks. This yields an improvement over the cubic \(k^3/\varepsilon^2\) bounds previously established by Alishahi and Phillips at ICML 2024 [2]. The advance is achieved through refined arguments involving higher moment bounds and empirical process analyses, which avoid the overcounting inherent in the standard VC-dimension and sensitivity framework [2]. Machine learning, a subfield of artificial intelligence, involves constructing algorithms that build models from training data to make predictions or decisions [5]. The sample complexity of such algorithms is often analyzed through the lens of VC dimension, which measures the capacity of a function class [4]. The new bounds provide tighter theoretical guarantees on the data requirements for regularized classifiers, offering a more precise understanding of the trade-offs between model complexity, regularization type, and the number of samples needed for reliable learning.