Online learning of deep neural networks suffers from challenges such as hysteretic non-incremental updating, increasing memory usage, past retrospective retraining, and catastrophic forgetting. To alleviate these drawbacks and achieve progressive immediate decision-making, we propose a novel Incremental Online Learning (IOL) process of Randomized Neural Networks (Randomized NN), a framework facilitating continuous improvements to Randomized NN performance in restrictive online scenarios. Within the framework, we further introduce IOL with ridge regularization (-R) and IOL with forward regularization (-F). -R generates stepwise incremental updates without retrospective retraining and avoids catastrophic forgetting. Moreover, we substituted -R with -F as it enhanced precognition learning ability using semi-supervision and realized better online regrets to offline global experts compared to -R during IOL. The algorithms of IOL for Randomized NN with -R/-F on non-stationary batch stream were derived respectively, featuring recursive weight updates and variable learning rates. Additionally, we conducted a detailed analysis and theoretically derived relative cumulative regret bounds of the Randomized NN learners with -R/-F in IOL under adversarial assumptions using a novel methodology and presented several corollaries, from which we observed the superiority on online learning acceleration and regret bounds of employing -F in IOL. Finally, our proposed methods were rigorously examined across regression and classification tasks on diverse datasets, which distinctly validated the efficacy of IOL frameworks of Randomized NN and the advantages of forward regularization.

Multi-arm bandit (MAB) is a classic online learning framework that studies the sequential decision-making in an uncertain environment. The MAB framework, however, overlooks the scenario where the decision-maker cannot take actions (e.g., pulling arms) directly. It is a practically important scenario in many applications such as spectrum sharing, crowdsensing, and edge computing. In these applications, the decision-maker would incentivize other selfish agents to carry out desired actions (i.e., pulling arms on the decision-maker's behalf). This paper establishes the incentivized online learning (IOL) framework for this scenario. The key challenge to design the IOL framework lies in the tight coupling of the unknown environment learning and asymmetric information revelation. To address this, we construct a special Lagrangian function based on which we propose a socially-optimal mechanism for the IOL framework. Our mechanism satisfies various desirable properties such as agent fairness, incentive compatibility, and voluntary participation. It achieves the same asymptotic performance as the state-of-art benchmark that requires extra information. Our analysis also unveils the power of crowd in the IOL framework: a larger agent crowd enables our mechanism to approach more closely the theoretical upper bound of social performance. Numerical results demonstrate the advantages of our mechanism in large-scale edge computing.