We consider the problem of nonnegative submodular maximization in the online setting. At time step t, an algorithm selects a set St C 2V where C is a feasible family of sets. An adversary then reveals a submodular function ft. The goal is to design an efficient algorithm for minimizing the expected approximate regret. In this work, we give a general approach for improving regret bounds in online submodular maximization by exploiting "first-order" regret bounds for online linear optimization. For monotone submodular maximization subject to a matroid, we give an efficient algorithm which achieves a (1 c/e ε)-regret of O( p kTln(n/k)) where n is the size of the ground set, k is the rank of the matroid, ε > 0 is a constant, and cis the average curvature. Even without assuming any curvature (i.e., taking c = 1), this regret bound improves on previous results of Streeter et al. (2009) and Golovin et al. (2014). For nonmonotone, unconstrained submodular functions, we give an algorithm with 1/2-regret O( nT), improving on the results of Roughgarden and Wang (2018). Our approach is based on Blackwell approachability; in particular, we give a novel first-order regret bound for the Blackwell instances that arise in this setting.

We revisit the problem of Online Linear Optimization in case the set of feasible actions is accessible through an approximated linear optimization oracle with a factor $\alpha$ multiplicative approximation guarantee. This setting is in particular interesting since it captures natural online extensions of well-studied offline linear optimization problems which are NP-hard, yet admit efficient approximation algorithms. The goal here is to minimize the $\alpha$-regret which is the natural extension of the standard regret in online learning to this setting. We present new algorithms with significantly improved oracle complexity for both the full information and bandit variants of the problem. Mainly, for both variants, we present $\alpha$-regret bounds of $O(T^{-1/3})$, were $T$ is the number of prediction rounds, using only $O(\log(T))$ calls to the approximation oracle per iteration, on average. These are the first results to obtain both average oracle complexity of $O(\log(T))$ (or even poly-logarithmic in $T$) and $\alpha$-regret bound $O(T^{-c})$ for a positive constant $c$, for both variants.

In the recent years, a number of parameter-free algorithms have been developed for online linear optimization over Hilbert spaces and for learning with expert advice. These algorithms achieve optimal regret bounds that depend on the unknown competitors, without having to tune the learning rates with oracle choices. We present a new intuitive framework to design parameter-free algorithms for both online linear optimization over Hilbert spaces and for learning with expert advice, based on reductions to betting on outcomes of adversarial coins. We instantiate it using a betting algorithm based on the Krichevsky-Trofimov estimator. The resulting algorithms are simple, with no parameters to be tuned, and they improve or match previous results in terms of regret guarantee and per-round complexity.

We propose computationally efficient algorithms for \textit{online linear optimization with bandit feedback}, in which a player chooses an \textit{action vector} from a given (possibly infinite) set $\mathcal{A} \subseteq \mathbb{R}^d$, and then suffers a loss that can be expressed as a linear function in action vectors. Although existing algorithms achieve an optimal regret bound of $\tilde{O}(\sqrt{T})$ for $T$ rounds (ignoring factors of $\mathrm{poly} (d, \log T)$), computationally efficient ways of implementing them have not yet been specified, in particular when $|\mathcal{A}|$ is not bounded by a polynomial size in $d$. A standard way to pursue computational efficiency is to assume that we have an efficient algorithm referred to as \textit{oracle} that solves (offline) linear optimization problems over $\mathcal{A}$. Under this assumption, the computational efficiency of a bandit algorithm can then be measured in terms of \textit{oracle complexity}, i.e., the number of oracle calls. Our contribution is to propose algorithms that offer optimal regret bounds of $\tilde{O}(\sqrt{T})$ as well as low oracle complexity for both \textit{non-stochastic settings} and \textit{stochastic settings}. Our algorithm for non-stochastic settings has an oracle complexity of $\tilde{O}( T)$ and is the first algorithm that achieves both a regret bound of $\tilde{O}( \sqrt{T})$ and an oracle complexity of $\tilde{O} ( \mathrm{poly} ( T))$, given only linear optimization oracles. Our algorithm for stochastic settings calls the oracle only $O( \mathrm{poly} (d, \log T))$ times, which is smaller than the current best oracle complexity of $O( T)$ if $T$ is sufficiently large.

In the recent years, a number of parameter-free algorithms have been developed for online linear optimization over Hilbert spaces and for learning with expert advice. These algorithms achieve optimal regret bounds that depend on the unknown competitors, without having to tune the learning rates with oracle choices. We present a new intuitive framework to design parameter-free algorithms for both online linear optimization over Hilbert spaces and for learning with expert advice, based on reductions to betting on outcomes of adversarial coins. We instantiate it using a betting algorithm based on the Krichevsky-Trofimov estimator. The resulting algorithms are simple, with no parameters to be tuned, and they improve or match previous results in terms of regret guarantee and per-round complexity.

While there exist well known algorithms for choosing the decision maker's actions which guarantee

We study a variant of online linear optimization where the player receives a hint about the loss function at the beginning of each round. The hint is given in the form of a vector that is weakly correlated with the loss vector on that round.

This paper offers a nearly optimal algorithm for online linear optimization with delayed bandit feedback.