🤖 AI Summary
This work studies agnostic smooth online learning under the challenging setting where the underlying probability measure μ is unknown—a more general and difficult scenario than the conventional assumption of known μ. To address this, we propose the R-Cover algorithm, the first method that operates without prior knowledge of μ, leveraging a recursive covering construction and analysis based on VC dimension and fat-shattering dimension to achieve robust online prediction. Theoretically, for classification tasks, R-Cover attains the optimal adaptive regret bound $ ilde{O}(sqrt{dT/sigma})$, where $d$ is the VC dimension and $sigma$ the smoothness parameter; for regression, it achieves sublinear oblivious regret under the mild condition that the fat-shattering dimension grows polynomially. This work breaks the long-standing dependency on prior knowledge of the base measure μ, providing the first sublinear regret guarantee for agnostic, non-stationary smooth online learning.
📝 Abstract
Classical results in statistical learning typically consider two extreme data-generating models: i.i.d. instances from an unknown distribution, or fully adversarial instances, often much more challenging statistically. To bridge the gap between these models, recent work introduced the smoothed framework, in which at each iteration an adversary generates instances from a distribution constrained to have density bounded by $sigma^{-1}$ compared to some fixed base measure $mu$. This framework interpolates between the i.i.d. and adversarial cases, depending on the value of $sigma$. For the classical online prediction problem, most prior results in smoothed online learning rely on the arguably strong assumption that the base measure $mu$ is known to the learner, contrasting with standard settings in the PAC learning or consistency literature. We consider the general agnostic problem in which the base measure is unknown and values are arbitrary. Along this direction, Block et al. showed that empirical risk minimization has sublinear regret under the well-specified assumption. We propose an algorithm R-Cover based on recursive coverings which is the first to guarantee sublinear regret for agnostic smoothed online learning without prior knowledge of $mu$. For classification, we prove that R-Cover has adaptive regret $ ilde O(sqrt{dT/sigma})$ for function classes with VC dimension $d$, which is optimal up to logarithmic factors. For regression, we establish that R-Cover has sublinear oblivious regret for function classes with polynomial fat-shattering dimension growth.