Online learning from a stream of data is a defining feature of intelligence, yet modern machine learning systems often struggle in this setting, especially under distributional shift. To understand its basic properties, we study the relationship between online and offline learning in the context of kernel regression. We derive a closed-form expression for the function learned by online kernel regression, revealing that online kernel regression is equivalent to offline regression with shifted, inaccurate target outputs. Conversely, we show that by compensating for this effective shift in the teaching signal through target correction, online kernel-based learning can provably learn the same predictor as its offline counterpart. We derive both a closed-form expression for this target correction and an iterative form that can be applied sequentially. Applying this framework to image classification tasks on CIFAR-10 and CORe50, we show that online stochastic gradient descent with iteratively corrected targets outperforms learning with the true targets in continual learning settings. This work therefore provides a basic framework for analyzing and improving online learning in non-stationary environments.

翻译：从数据流中进行在线学习是智能的一个决定性特征，然而现代机器学习系统在这种设置中，尤其是在分布偏移下，常常表现不佳。为理解其基本属性，我们在核回归背景下研究了在线学习与离线学习之间的关系。我们推导出在线核回归所学习函数的闭式表达式，揭示出在线核回归等价于使用偏移、不准确的目标输出进行离线回归。相反，我们证明通过目标校正补偿教学信号中的这种有效偏移，基于核的在线学习能够可靠地学习与离线对应方法相同的预测器。我们推导出这种目标校正的闭式表达式以及可顺序应用的迭代形式。将该框架应用于CIFAR-10和CORe50上的图像分类任务，我们展示了在持续学习设置中，使用迭代校正目标的在线随机梯度下降法优于使用真实目标的学习。因此，这项工作为分析和改进非平稳环境下的在线学习提供了一个基本框架。