A Modern Introduction to Online Learning (1912.13213v6)

Abstract: In this monograph, I introduce the basic concepts of Online Learning through a modern view of Online Convex Optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I present first-order and second-order algorithms for online learning with convex losses, in Euclidean and non-Euclidean settings. All the algorithms are clearly presented as instantiation of Online Mirror Descent or Follow-The-Regularized-Leader and their variants. Particular attention is given to the issue of tuning the parameters of the algorithms and learning in unbounded domains, through adaptive and parameter-free online learning algorithms. Non-convex losses are dealt through convex surrogate losses and through randomization. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. These notes do not require prior knowledge of convex analysis and all the required mathematical tools are rigorously explained. Moreover, all the included proofs have been carefully chosen to be as simple and as short as possible.

• The paper introduces online learning principles via convex optimization, focusing on regret minimization under worst-case conditions with both first and second order algorithms.
• It details methodologies like Online Mirror Descent and Follow-The-Regularized-Leader, showcasing adaptable parameter tuning and mirror map techniques for diverse loss settings.
• The document contrasts algorithm performances—including Exponentiated Gradient, AdaHedge, and Online Newton Step—to provide theoretical and practical insights for robust online learning applications.

An Exploration of Online Learning Principles

The document presents an elaboration on the fundamental aspects of online learning, articulated through the lens of online convex optimization (OCO). This exposition serves as a modern introduction to the main ideas prevalent in this evolving field of paper. Within the online learning framework, the central theme is regret minimization under stringent worst-case conditions. The manuscript delineates first-order and second-order algorithms designed for tackling online learning dilemmas, where losses are convex, set within both Euclidean and non-Euclidean spaces.

The text meticulously describes how online learning algorithms, especially those based on Online Mirror Descent (OMD) and its variants like Follow-The-Regularized-Leader (FTRL), promote adaptability through parameter tuning and facilitate learning in unbounded domains. The author extends the discussion, incorporating the challenge of optimizing with non-convex losses via surrogate convex losses and randomization. The notion of bandit settings is likewise touched upon, highlighting adversarial and stochastic multi-armed bandits to underline the breadth of online learning applications.

Implementation and Theoretical Underpinnings

A distinctive aspect of this text is its methodical presentation of the mathematical tools requisite for understanding online learning without necessitating prior convex analysis knowledge. Through careful selection of proofs, the explanations remain concise yet comprehensive, striving for simplicity without sacrificing rigor.

The core strategy emphasized is that of OMD, which generalizes the Online Subgradient Descent (OSD) algorithm to handle non-Euclidean loss settings. OMD embodies a pivotal advancement, enabling practitioners to minimize the regret over a series of loss functions by leveraging mirror maps grounded in Bregman divergences instead of traditional Euclidean projections. Similarly, the FTRL approach is expanded with strategies such as linearized and proximal losses, pointing towards a flexible adaptation to varying problem characteristics and complex composite losses.

Key Numerical Insights and Contrast of Techniques

The document elaborates on several algorithms, such as Exponentiated Gradient (EG), AdaHedge, and the Online Newton Step (ONS), providing crucial insights into their performance dynamics across different problem scenarios. Each algorithm is meticulously crafted to exploit specific properties of the loss functions—convexity, strong convexity, exp-concavity—ensuring optimal or near-optimal regret bounds in diverse settings.

The shift from OSD to variants of FTRL and their symbiosis with local norm analysis articulate a narrative where theoretical bounds are tightly intertwined with empirical success. This interplay is especially notable in scenarios involving exponential families or group norms, where strategic regularization curbs overfitting, while offering improved regret bounds.

Practical and Theoretical Implications

The practical implications range from enhancing prediction algorithms for learning with expert advice to integrating stochastic optimization with online methods. Such adaptations are crucial for real-time applications spanning recommendation systems, dynamic resource allocation, and iterative decision making in adversarial environments.

From a theoretical standpoint, the text speculates on the future trajectory of AI, advocating for an alignment between online learning principles and broader AI methodologies. The document posits that innovations in understanding and exploiting data structures (via kernel methods or hierarchical models, for instance) will continue to mature the online learning landscape.

Future Directions

The discourse on online learning posits challenges that lie ahead, particularly in parameter-free algorithmic designs and widening scope into novel, complex structures, potentially merging with aspects of deep learning architectures or adaptive systems in broader AI applications.

In essence, this document functions as a bridge linking robust theoretical foundations with profound practical implications, anchoring the discussion firmly in the online learning field while opening portals to future exploration and interdisciplinary convergence in AI research.