Fully Unconstrained Online Learning (2405.20540v1)

Abstract: We provide an online learning algorithm that obtains regret $G|w_\star|\sqrt{T\log(|w_\star|G\sqrt{T})} + |w_\star|^2 + G^2$ on $G$-Lipschitz convex losses for any comparison point $w_\star$ without knowing either $G$ or $|w_\star|$. Importantly, this matches the optimal bound $G|w_\star|\sqrt{T}$ available with such knowledge (up to logarithmic factors), unless either $|w_\star|$ or $G$ is so large that even $G|w_\star|\sqrt{T}$ is roughly linear in $T$. Thus, it matches the optimal bound in all cases in which one can achieve sublinear regret, which arguably most "interesting" scenarios.

• The paper presents an online learning algorithm that achieves sublinear regret without relying on predefined gradient or comparator bounds.
• It leverages a reduction to 1D learning and adaptive magnitude hints to handle unknown loss sequence characteristics in high-dimensional settings.
• Regularization via epigraph constraints refines the regret bounds, highlighting both theoretical advances and practical robustness in unconstrained optimization.

Fully Unconstrained Online Learning: An Academic Overview

Ashok Cutkosky and Zakaria Mhammedi's paper, "Fully Unconstrained Online Learning," explores the development and analysis of new algorithms designed for online learning under the framework of online convex optimization (OCO). This document provides both novel theoretical insights and practical techniques for achieving low regret bounds without prior knowledge of loss sequence characteristics.

Introduction to Unconstrained Online Learning

Online learning is often formalized as a sequential game where a learner aims to minimize cumulative losses against an adversarial environment. A significant challenge in this setting is to develop algorithms that do not rely on predefined constraints, such as bounds on gradients or comparator norms. Cutkosky and Mhammedi tackle this challenge head-on by proposing algorithms that attain regret bounds comparable to optimal bounds that assume prior knowledge of these constraints.

Key Contributions

The authors present an online learning algorithm that achieves sublinear regret: $$\text{Regret}_T(w_\star)\le G\|w_\star\|\sqrt{T\log(\|w_\star\|G\sqrt{T})} + \|w_\star\|^2 + G^2 + \epsilon \sqrt{V}$$ where $G$ is the Lipschitz constant, $V$ incorporates cumulative gradient norms, $w_\star$ represents any comparator, and $\epsilon$ is a tuning parameter. Notably, this method does not require a priori knowledge of $G$ or $\|w_\star\|$.

Algorithmic Development

1. Reduction to 1D Learning: The problem in $R^d$ is reduced to $R$ by projecting vectors onto a distinguished direction and then applying the developed 1-dimensional unconstrained learning algorithms. This reduction allows for bounding the regret in high-dimensional settings by leveraging low-dimensional techniques.
2. Magnitude Hints: The algorithm adapts by incorporating "magnitude hints," which are estimates of gradient magnitudes that improve with each iteration. This ensures the algorithm remains robust even when the gradient norms are unknown.
3. Regularized Learning and Epigraph Constraints: By considering the epigraph of convex functions, the authors incorporate regularization terms seamlessly. This allows bounding the regret in scenarios where the loss functions are not only linear but also subject to convex regularization.

Performance Analysis

Theoretical results assert that the proposed algorithm provides a regret bound: $$\widetilde{O}\left(\max_{t\in[T]}\|g_t\|^2/\gamma+ \gamma \|w_\star\|^2 +\|w_\star\|\sqrt{\sum_{t=1}^T \|g_t\|^2}\right)$$ This bound is significant as it tightens previous results by improving the dependency on $\sum_{t=1}^T \|g_t\|^2$, making the method applicable across a broader range of scenarios without resorting to conservative constraints. The introduction of regularization parameters $a_t=\gamma \cdot (h_{t+1} - h_t)/h_{t+1}$ facilitates this bound by effectively penalizing large deviations.

Practical and Theoretical Implications

Practical Implications

• Implementation Without Prior Knowledge: The algorithm's structure ensures it remains effective without needing explicit bounds on gradient norms, making it suitable for real-world applications where such information is often unavailable.
• Versatility and Robustness: The theory is supported by practical algorithms, such as those utilizing Follow-The-Regularized-Leader (FTRL) principles, demonstrating robustness and versatility across different types of loss functions and adversarial sequences.

Theoretical Implications

• Advancements in Online Learning: The results advance the understanding of regret minimization by presenting a paradigm that works without prior knowledge constraints. This contributes to the broader theory of unconstrained optimization.
• Logarithmic Dependence: By refining the likelihood of achieving bounds dependent only on logarithms of key quantities (like gradient norms), the paper sets the stage for more refined future analyses and tighter regret bounds.

Conclusion and Future Directions

Cutkosky and Mhammedi present pioneering work in developing fully unconstrained online learning algorithms, pushing the boundaries of what's achievable without prior gradient and comparator knowledge. Their approach intertwines advanced regularization techniques and optimally exploits magnitude hints to enforce near-optimal regret bounds. Future work may focus on practical implementations in high-dimensional spaces and further empirical analyses to explore the potential of these algorithms in non-convex optimization scenarios, especially in the field of deep learning and beyond.