Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2512.14686

Bias-Variance Trade-off for Clipped Stochastic First-Order Methods: From Bounded Variance to Infinite Mean

Chuan He
View original source ↗Read source document ↗Export metadata ↓

Abstract

Stochastic optimization is fundamental to modern machine learning. Recent research has extended the study of stochastic first-order methods (SFOMs) from light-tailed to heavy-tailed noise, which frequently arises in practice, with clipping emerging as a key technique for controlling heavy-tailed gradients. Extensive theoretical advances have further shown that the oracle complexity of SFOMs depends on the tail index $α$ of the noise. Nonetheless, existing complexity results often cover only the case $α\in (1,2]$, that is, the regime where the noise has a finite mean, while the complexity bounds tend to infinity as $α$ approaches $1$. This paper tackles the general case of noise with tail index $α\in(0,2]$, covering regimes ranging from noise with bounded variance to noise with an infinite mean, where the latter case has been scarcely studied. Through a novel analysis of the bias-variance trade-off in gradient clipping, we show that when a symmetry measure of the noise tail is controlled, clipped SFOMs achieve improved complexity guarantees in the presence of heavy-tailed noise for any tail index $α\in (0,2]$. Our analysis of the bias-variance trade-off not only yields new unified complexity guarantees for clipped SFOMs across this full range of tail indices, but is also straightforward to apply and can be combined with classical analyses under light-tailed noise to establish oracle complexity guarantees under heavy-tailed noise. Finally, numerical experiments validate our theoretical findings.

Explore related subjects

cs.LGcs.AImath.OCstat.COstat.ML

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chuan He. 2025-12-16. Bias-Variance Trade-off for Clipped Stochastic First-Order Methods: From Bounded Variance to Infinite Mean. https://arxiv.org/abs/2512.14686

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Analysis of Regularized Learning in Banach Spaces for Linear-functional Data

This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized learning is designed to minimize regularized empirical risks over a Banach space. The empirical risks are calculated by utilizing training data and multi-loss functions. The input training data are composed of linear functionals in a predual space of the Banach space to capture discrete local information from multimodal data and multiscale models. Through the regularized learning, approximations of the exact solution to an unidentified or uncertain original problem are globally achieved. In the convergence theorems, the convergence of the approximate solutions to the exact solution is established through the utilization of the weak* topology of the Banach space. The theorems of regularized learning are utilized in the interpretation of classical machine learning, such as support vector machines and artificial neural networks.

cs.LG↗
arXiv2026

On Minimal Depth in Neural Networks

Understanding the relationship between the depth of a neural network and its representational capacity is a central problem in deep learning theory. In this work, we develop a geometric framework to analyze the expressivity of ReLU networks with the notion of depth complexity for convex polytopes. The depth of a polytope recursively quantifies the number of alternating convex hull and Minkowski sum operations required to construct it. This geometric perspective serves as a rigorous tool for deriving depth lower bounds and understanding the structural limits of deep neural architectures. We establish lower and upper bounds on the depth of polytopes, as well as tight bounds for classical families. These results yield two main consequences. First, we provide a purely geometric proof of the expressivity bound by Arora et al. (2018), confirming that $\lceil \log_2(n+1)\rceil$ hidden layers suffice to represent any continuous piecewise linear (CPWL) function. Second, we prove that, unlike general ReLU networks, convex polytopes do not admit a universal depth bound. Specifically, the depth of cyclic polytopes in dimensions $n \geq 4$ grows unboundedly with the number of vertices. This result implies that Input Convex Neural Networks (ICNNs) cannot represent all convex CPWL functions with a fixed depth, revealing a sharp separation in expressivity between ICNNs and standard ReLU networks.

cs.LG↗
arXiv2026

ELEMENT: Episodic and Lifelong Exploration via Maximum Entropy

Reinforcement learning agents depend on reward signals whose density is rarely under the designer's control, and when such signals are absent, an agent must generate its own drive to explore. State entropy maximization offers a principled objective for this, but existing methods break down at scale in two ways: the intrinsic reward vanishes once a state has been visited, discouraging revisits to the very gateways that lead onward, and estimating entropy over millions of accumulated observations becomes computationally prohibitive. We address both with Episodic and Lifelong Exploration via Maximum Entropy (ELEMENT), a multiscale intrinsically motivated framework for reward-free exploration that transfers to downstream tasks. ELEMENT couples lifelong entropy maximization with a complementary episodic term acting on a faster timescale. For the episodic term, we derive average episodic state entropy, an intrinsic reward that is the exact minimizer of a tractable upper bound on the reward-decomposition objective; for the lifelong term, we propose a $k$NN graph-based estimator that keeps entropy tractable without forgetting. ELEMENT consistently outperforms state-of-the-art intrinsic reward baselines on state coverage and unsupervised pre-training. Videos, code, and supplementary material: https://sites.google.com/view/element-rl.

cs.LG↗