A Unified Approach to Adaptive Regularization in Online and Stochastic Optimization

We describe a framework for deriving and analyzing online optimization algorithms that incorporate adaptive, data-dependent regularization, also termed preconditioning. Such algorithms have been proven useful in stochastic optimization by reshaping the gradients according to the geometry of the data. Our framework captures and unifies much of the existing literature on adaptive online methods, including the AdaGrad and Online Newton Step algorithms as well as their diagonal versions. As a result, we obtain new convergence proofs for these algorithms that are substantially simpler than previous analyses. Our framework also exposes the rationale for the different preconditioned updates used in common stochastic optimization methods.

Tight Bounds for Bandit Combinatorial Optimization

We revisit the study of optimal regret rates in bandit combinatorial optimization---a fundamental framework for sequential decision making under uncertainty that abstracts numerous combinatorial prediction problems. We prove that the attainable regret in this setting grows as $\widetilde{\Theta}(k^{3/2}\sqrt{dT})$ where $d$ is the dimension of the problem and $k$ is a bound over the maximal instantaneous loss, disproving a conjecture of Audibert, Bubeck, and Lugosi (2013) who argued that the optimal rate should be of the form $\widetilde{\Theta}(k\sqrt{dT})$. Our bounds apply to several important instances of the framework, and in particular, imply a tight bound for the well-studied bandit shortest path problem. By that, we also resolve an open problem posed by Cesa-Bianchi and Lugosi (2012).

Bandits with Movement Costs and Adaptive Pricing

We extend the model of Multi-armed Bandit with unit switching cost to incorporate a metric between the actions. We consider the case where the metric over the actions can be modeled by a complete binary tree, and the distance between two leaves is the size of the subtree of their least common ancestor, which abstracts the case that the actions are points on the continuous interval $[0,1]$ and the switching cost is their distance. In this setting, we give a new algorithm that establishes a regret of $\widetilde{O}(\sqrt{kT} + T/k)$, where $k$ is the number of actions and $T$ is the time horizon. When the set of actions corresponds to whole $[0,1]$ interval we can exploit our method for the task of bandit learning with Lipschitz loss functions, where our algorithm achieves an optimal regret rate of $\widetilde{\Theta}(T^{2/3})$, which is the same rate one obtains when there is no penalty for movements. As our main application, we use our new algorithm to solve an adaptive pricing problem. Specifically, we consider the case of a single seller faced with a stream of patient buyers. Each buyer has a private value and a window of time in which they are interested in buying, and they buy at the lowest price in the window, if it is below their value. We show that with an appropriate discretization of the prices, the seller can achieve a regret of $\widetilde{O}(T^{2/3})$ compared to the best fixed price in hindsight, which outperform the previous regret bound of $\widetilde{O}(T^{3/4})$ for the problem.

Online Learning with Feedback Graphs Without the Graphs

We study an online learning framework introduced by Mannor and Shamir (2011) in which the feedback is specified by a graph, in a setting where the graph may vary from round to round and is \emph{never fully revealed} to the learner. We show a large gap between the adversarial and the stochastic cases. In the adversarial case, we prove that even for dense feedback graphs, the learner cannot improve upon a trivial regret bound obtained by ignoring any additional feedback besides her own loss. In contrast, in the stochastic case we give an algorithm that achieves $\widetilde \Theta(\sqrt{\alpha T})$ regret over $T$ rounds, provided that the independence numbers of the hidden feedback graphs are at most $\alpha$. We also extend our results to a more general feedback model, in which the learner does not necessarily observe her own loss, and show that, even in simple cases, concealing the feedback graphs might render a learnable problem unlearnable.

Online Learning with Low Rank Experts

We consider the problem of prediction with expert advice when the losses of the experts have low-dimensional structure: they are restricted to an unknown $d$-dimensional subspace. We devise algorithms with regret bounds that are independent of the number of experts and depend only on the rank $d$. For the stochastic model we show a tight bound of $\Theta(\sqrt{dT})$, and extend it to a setting of an approximate $d$ subspace. For the adversarial model we show an upper bound of $O(d\sqrt{T})$ and a lower bound of $\Omega(\sqrt{dT})$.

The Computational Power of Optimization in Online Learning

We consider the fundamental problem of prediction with expert advice where the experts are "optimizable": there is a black-box optimization oracle that can be used to compute, in constant time, the leading expert in retrospect at any point in time. In this setting, we give a novel online algorithm that attains vanishing regret with respect to $N$ experts in total $\widetilde{O}(\sqrt{N})$ computation time. We also give a lower bound showing that this running time cannot be improved (up to log factors) in the oracle model, thereby exhibiting a quadratic speedup as compared to the standard, oracle-free setting where the required time for vanishing regret is $\widetilde{\Theta}(N)$. These results demonstrate an exponential gap between the power of optimization in online learning and its power in statistical learning: in the latter, an optimization oracle---i.e., an efficient empirical risk minimizer---allows to learn a finite hypothesis class of size $N$ in time $O(\log{N})$. We also study the implications of our results to learning in repeated zero-sum games, in a setting where the players have access to oracles that compute, in constant time, their best-response to any mixed strategy of their opponent. We show that the runtime required for approximating the minimax value of the game in this setting is $\widetilde{\Theta}(\sqrt{N})$, yielding again a quadratic improvement upon the oracle-free setting, where $\widetilde{\Theta}(N)$ is known to be tight.

Online Learning with Feedback Graphs: Beyond Bandits

We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced $T$-round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first class induces learning problems with $\widetilde\Theta(\alpha^{1/2} T^{1/2})$ minimax regret, where $\alpha$ is the independence number of the underlying graph; the second class induces problems with $\widetilde\Theta(\delta^{1/3}T^{2/3})$ minimax regret, where $\delta$ is the domination number of a certain portion of the graph; and the third class induces problems with linear minimax regret. Our results subsume much of the previous work on learning with feedback graphs and reveal new connections to partial monitoring games. We also show how the regret is affected if the graphs are allowed to vary with time.

Bandit Convex Optimization: sqrt{T} Regret in One Dimension

We analyze the minimax regret of the adversarial bandit convex optimization problem. Focusing on the one-dimensional case, we prove that the minimax regret is $\widetilde\Theta(\sqrt{T})$ and partially resolve a decade-old open problem. Our analysis is non-constructive, as we do not present a concrete algorithm that attains this regret rate. Instead, we use minimax duality to reduce the problem to a Bayesian setting, where the convex loss functions are drawn from a worst-case distribution, and then we solve the Bayesian version of the problem with a variant of Thompson Sampling. Our analysis features a novel use of convexity, formalized as a "local-to-global" property of convex functions, that may be of independent interest.

Chasing Ghosts: Competing with Stateful Policies

We consider sequential decision making in a setting where regret is measured with respect to a set of stateful reference policies, and feedback is limited to observing the rewards of the actions performed (the so called "bandit" setting). If either the reference policies are stateless rather than stateful, or the feedback includes the rewards of all actions (the so called "expert" setting), previous work shows that the optimal regret grows like $\Theta(\sqrt{T})$ in terms of the number of decision rounds $T$. The difficulty in our setting is that the decision maker unavoidably loses track of the internal states of the reference policies, and thus cannot reliably attribute rewards observed in a certain round to any of the reference policies. In fact, in this setting it is impossible for the algorithm to estimate which policy gives the highest (or even approximately highest) total reward. Nevertheless, we design an algorithm that achieves expected regret that is sublinear in $T$, of the form $O( T/\log^{1/4}{T})$. Our algorithm is based on a certain local repetition lemma that may be of independent interest. We also show that no algorithm can guarantee expected regret better than $O( T/\log^{3/2} T)$.

Online Learning with Composite Loss Functions

We study a new class of online learning problems where each of the online algorithm's actions is assigned an adversarial value, and the loss of the algorithm at each step is a known and deterministic function of the values assigned to its recent actions. This class includes problems where the algorithm's loss is the minimum over the recent adversarial values, the maximum over the recent values, or a linear combination of the recent values. We analyze the minimax regret of this class of problems when the algorithm receives bandit feedback, and prove that when the minimum or maximum functions are used, the minimax regret is $\tilde \Omega(T^{2/3})$ (so called hard online learning problems), and when a linear function is used, the minimax regret is $\tilde O(\sqrt{T})$ (so called easy learning problems). Previously, the only online learning problem that was known to be provably hard was the multi-armed bandit with switching costs.