Self-Directed Apple Tasting

We study how switching costs affect minimax regret under partial feedback. Which features of the feedback preserve the $\sqrt T$ rate, and which force the $T^{2/3}$ rate? We address this question in the framework of feedback graphs, which interpolate between full information and bandits: playing an action reveals the losses of its out-neighbors in a known directed graph. Prior work on feedback graphs with switching costs gave algorithms with regret of order $T^{2/3}$, but matching lower bounds only for restricted classes of graphs, leaving the question open. We establish a complete characterization of the minimax regret with switching costs for all feedback graphs: $\widetildeΘ(\sqrt T)$ if the graph is strongly revealing, meaning that every pair of distinct actions has an endpoint that observes the losses of both; $\widetildeΘ(γ^{1/3}T^{2/3})$ for every other observable graph, where $γ$ is the domination number; and linear otherwise. In particular, tournaments, in which every action observes its own loss and each pair of actions is joined by one directed edge, sit at the phase transition: they behave like full information, but deleting the edge of any pair forces the rate $T^{2/3}$. Hence an action that observes all losses is neither necessary nor sufficient for the rate $\sqrt T$. Our algorithm TournamentSteering for the strongly revealing case rests on a new principle: each observed comparison of two actions serves both to estimate losses and to steer play, and the loss reduction from steering pays for the variance of the estimates. For every strongly revealing graph, distributing the comparisons among observing actions keeps this variance, and hence the regret, polylogarithmic in the number of actions, by an argument similar to the greedy construction of logarithmic dominating sets in tournaments.

We consider the problem of learning an unknown subset $N_\text{target}$ of a domain in an online setting. In each round $t$, the learner predicts a set of items ${N}_t$ and receives one of two types of feedback, each with equal probability: precision feedback, in which a randomly chosen item from the predicted set $N_t$ is revealed and the learner is told whether it belongs to $N_\text{target}$ (incurring a reward if it does), or recall feedback, in which a randomly chosen item from the target set $N_\text{target}$ is revealed and the learner is told whether it belongs to $N_t$ (incurring a reward if it does). The goal is to maximize the cumulative reward over time. This simple online set learning problem abstracts a variety of learning scenarios with precision- and recall-type feedback. We show that a hypothesis class (a family of subsets of the domain) is learnable in this setting if and only if it has finite Vapnik-Chervonenkis (VC) dimension, mirroring the classical PAC characterization. However, the resulting algorithmic structure is markedly more intricate: in contrast to standard Probably Approximately Correct (PAC) learning -- where the algorithmic landscape is governed by the simple principle of Empirical Risk Minimization (ERM) -- our partial feedback model can invalidate ERM and even all proper learning rules. We develop algorithms to address the dependencies induced by the feedback, obtaining regret guarantees in both the realizable and agnostic settings. Our results provide a qualitative characterization of learnability in this model, addressing its most basic question, while pointing to a range of natural and intriguing open questions, including the determination of optimal regret rates.

We study mistake bounds for differentially private online learning and online prediction under oblivious realisable adversaries. Online learning requires the learner to release a hypothesis at each time step whereas in online prediction, the learner only needs to make predictions without releasing a hypothesis. Using a novel lower bound for private online learning and an upper bound for private prediction, we show that the sample complexity of these two problems are separated by a factor that grows with the time horizon for every class of finite Littlestone dimension $d$. First, we prove that every $(ε,δ)$-private online learner has a deterministic realisable stream of length $T$ on which the mistake bound is at least $\mathbb E [M_T]= O\left(\frac dε\log^{2/3} T\right)$. In particular, this is the first non-trivial lower in the range $1/T<δ<1/\log T)$ left open in earlier works[SR22,DSS24,LWY24]. Second, we prove that for every class of of Littlestone dimension $d$, there exists an $(ε,δ)$-jointly private predictor with at most $2^{2^{cd^2}}ε^{-2}\log^2(2/(εδ))$ expected mistakes, independently of $T$, for some absolute constant $c>0$. Thus, for every fixed class of finite Littlestone dimension when $δ=Θ(1/\log T)$, private learning requires $O\left(\log^{2/3} T\right)$ expected mistakes, whereas private prediction admits $O(\log\log T^2)$.