We study online learning of feedforward neural networks with the sign activation function that implement functions from the unit ball in Rd to a finite label set Y = {1,...,Y}. First, we characterize a margin condition that is sufficient and in some cases necessary for online learnability of a neural network: Every neuron in the first hidden layer classifies all instances with some margin γ bounded away from zero. Quantitatively, we prove that for any net, the optimal mistake bound is at most approximately TS(d,γ), which is the (d,γ)-totally-separablepacking number, a more restricted variation of the standard (d,γ)-packing number. We complement this result by constructing a net on which any learner makes TS(d,γ) many mistakes. We also give a quantitative lower bound of approximately TS(d,γ) max{1/(γ d)d,d} when γ 1/2, implying that for some nets and input sequences every learner will err for exp(d) many times, and that a dimension-free mistake bound is almost always impossible.

The multi-armed bandit (MAB) is a fundamental online decision-making framework that has been extensively studied over the past two decades. To mitigate the high cost and slow convergence of purely online learning, modern MAB approaches have explored paradigms that leverage offline data to warm-start online learning. However, existing approaches face a significant limitation by assuming that the offline and online data are homogeneous--they share the same feedback structure and are drawn from the same underlying distribution. This assumption is often violated in practice, where offline data often originate from diverse sources and evolving environments, resulting in feedback heterogeneity and distributional shifts. In this work, we tackle the challenge of learning across this offline-online gap by developing a general hybrid bandit framework that incorporates heterogeneous offline data to improve online performance. We study two hybrid settings: (1) using reward-based offline data to accelerate online learning in preference-based bandits (i.e., dueling bandits), and (2) using preference-based offline data to improve online standard MAB algorithms. For both settings, we design novel algorithms and derive tight regret bounds that match or improve upon existing benchmarks despite heterogeneity. Empirical evaluations on both synthetic and real-world datasets show that our proposed methods significantly outperform baseline algorithms.

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.

In Orabona and Pál [2016], we introduced the shifted KT potentials, to remove the $\ln \ln T$ factor in the parameter-free learning with expert bound. In this short technical note, I show that this is equivalent to changing the prior in the Krichevsky--Trofimov algorithm. Then, I show how to use the same idea to remove the $\ln \ln T$ factor in the data-independent bound for the Squint algorithm.

We study the design of computationally efficient online learning algorithms under smoothed analysis. In this setting, at every step an adversary generates a sample from an adaptively chosen distribution whose density is upper bounded by 1/ times the uniform density. Given access to an offline optimization (ERM) oracle, we give the first computationally efficient online algorithms whose sublinear regret depends only on the pseudo/VC dimension dof the class and the smoothness parameter .

We introduce an efficient algorithmic framework for model selection in online learning, also known as parameter-free online learning. Departing from previous work, which has focused on highly structured function classes such as nested balls in Hilbert space, we propose a generic meta-algorithm framework that achieves online model selection oracle inequalities under minimal structural assumptions. We give the first computationally efficient parameter-free algorithms that work in arbitrary Banach spaces under mild smoothness assumptions; previous results applied only to Hilbert spaces. We further derive new oracle inequalities for matrix classes, non-nested convex sets, and $\mathbb{R}^{d}$ with generic regularizers. Finally, we generalize these results by providing oracle inequalities for arbitrary non-linear classes in the online supervised learning model. These results are all derived through a unified meta-algorithm scheme using a novel multi-scale algorithm for prediction with expert advice based on random playout, which may be of independent interest.

Some recent results do consider residual-like elements (see discussion of related work below),butgenerallydonotapply tostandard architectures.