We revisit the sequential variants of linear regression with the squared loss, classification problems with hinge loss, and logistic regression, all characterized by unbounded losses in the setup where no assumptions are made on the magnitude of design vectors and the norm of the optimal vector of parameters. The key distinction from existing results lies in our assumption that the set of design vectors is known in advance (though their order is not), a setup sometimes referred to as transductive online learning. While this assumption seems similar to fixed design regression or denoising, we demonstrate that the sequential nature of our algorithms allows us to convert our bounds into statistical ones with random design without making any additional assumptions about the distribution of the design vectors--an impossibility for standard denoising results. Our key tools are based on the exponential weights algorithm with carefully chosen transductive (design-dependent) priors, which exploit the full horizon of the design vectors. Our classification regret bounds have a feature that is only attributed to bounded losses in the literature: they depend solely on the dimension of the parameter space and on the number of rounds, independent of the design vectors or the norm of the optimal solution. For linear regression with squared loss, we further extend our analysis to the sparse case, providing sparsity regret bounds that additionally depend on the magnitude of the response variables. We argue that these improved bounds are specific to the transductive setting and unattainable in the worst-case sequential setup. Our algorithms, in several cases, have polynomial time approximations and reduce to sampling with respect to log-concave measures instead of aggregating over hard-to-construct $\varepsilon$-covers of classes.

本研究针对线性回归及分类问题中的无界损失，提出了一种在已知设计向量集的条件下的新方法。通过结合传导在线学习的思路，研究展示了如何在无额外假设的情况下，将序列算法的界限转化为随机设计下的统计界限。研究的关键发现是，这些改进的界限针对传导设置是特有的，且在最坏情况下的序列设置中无法实现。

通过传导先验优化无界损失的风险界限