This work studies and develop projection-free algorithms for online learning with linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, this work (i) provides an improved (optimized) variant of an online Frank-Wolfe algorithm along with its conceptually simple potential-based proof, and (ii) shows how to leverage semidefinite programming to jointly design and analyze online Frank-Wolfe-type algorithms numerically in a variety of settings-that include the design of the variant (i). Based on the semidefinite technique, we conclude with strong numerical evidence suggesting that no pure online Frank-Wolfe algorithm within our model class can have a regret guarantee better than O(T^3/4) (T is the time horizon) without additional assumptions, that the current algorithms do not have optimal constants, that the algorithm benefits from similar anytime properties O(t^3/4) not requiring to know T in advance, and that multiple linear optimization rounds do not generally help to obtain better regret bounds.

In the first-order query model for zero-sum $K\times K$ matrix games, players observe the expected pay-offs for all their possible actions under the randomized action played by their opponent. This classical model has received renewed interest after the discovery by Rakhlin and Sridharan that $\epsilon$-approximate Nash equilibria can be computed efficiently from $O(\frac{\ln K}{\epsilon})$ instead of $O(\frac{\ln K}{\epsilon^2})$ queries. Surprisingly, the optimal number of such queries, as a function of both $\epsilon$ and $K$, is not known. We make progress on this question on two fronts. First, we fully characterise the query complexity of learning exact equilibria ($\epsilon=0$), by showing that they require a number of queries that is linear in $K$, which means that it is essentially as hard as querying the whole matrix, which can also be done with $K$ queries. Second, for $\epsilon > 0$, the current query complexity upper bound stands at $O(\min(\frac{\ln(K)}{\epsilon} , K))$. We argue that, unfortunately, obtaining a matching lower bound is not possible with existing techniques: we prove that no lower bound can be derived by constructing hard matrices whose entries take values in a known countable set, because such matrices can be fully identified by a single query. This rules out, for instance, reducing to an optimization problem over the hypercube by encoding it as a binary payoff matrix. We then introduce a new technique for lower bounds, which allows us to obtain lower bounds of order $\tilde\Omega(\log(\frac{1}{K\epsilon})$ for any $\epsilon \leq 1 / (cK^4)$, where $c$ is a constant independent of $K$. We further discuss possible future directions to improve on our techniques in order to close the gap with the upper bounds.

We provide a new adaptive method for online convex optimization, MetaGrad, that is robust to general convex losses but achieves faster rates for a broad class of special functions, including exp-concave and strongly convex functions, but also various types of stochastic and non-stochastic functions without any curvature. We prove this by drawing a connection to the Bernstein condition, which is known to imply fast rates in offline statistical learning. MetaGrad further adapts automatically to the size of the gradients. Its main feature is that it simultaneously considers multiple learning rates, which are weighted directly proportional to their empirical performance on the data using a new meta-algorithm. We provide three versions of MetaGrad. The full matrix version maintains a full covariance matrix and is applicable to learning tasks for which we can afford update time quadratic in the dimension. The other two versions provide speed-ups for high-dimensional learning tasks with an update time that is linear in the dimension: one is based on sketching, the other on running a separate copy of the basic algorithm per coordinate. We evaluate all versions of MetaGrad on benchmark online classification and regression tasks, on which they consistently outperform both online gradient descent and AdaGrad.

We revisit the classic regret-minimization problem in the stochastic multi-armed bandit setting when the arm-distributions are allowed to be heavy-tailed. Regret minimization has been well studied in simpler settings of either bounded support reward distributions or distributions that belong to a single parameter exponential family. We work under the much weaker assumption that the moments of order $(1+\epsilon)$ are uniformly bounded by a known constant B, for some given $\epsilon > 0$. We propose an optimal algorithm that matches the lower bound exactly in the first-order term. We also give a finite-time bound on its regret. We show that our index concentrates faster than the well known truncated or trimmed empirical mean estimators for the mean of heavy-tailed distributions. Computing our index can be computationally demanding. To address this, we develop a batch-based algorithm that is optimal up to a multiplicative constant depending on the batch size. We hence provide a controlled trade-off between statistical optimality and computational cost.

Conditional value-at-risk (CVaR) and value-at-risk (VaR) are popular tail-risk measures in finance and insurance industries where often the underlying probability distributions are heavy-tailed. We use the multi-armed bandit best-arm identification framework and consider the problem of identifying the arm-distribution from amongst finitely many that has the smallest CVaR or VaR. We first show that in the special case of arm-distributions belonging to a single-parameter exponential family, both these problems are equivalent to the best mean-arm identification problem, which is widely studied in the literature. This equivalence however is not true in general. We then propose optimal $\delta$-correct algorithms that act on general arm-distributions, including heavy-tailed distributions, that match the lower bound on the expected number of samples needed, asymptotically (as $ \delta$ approaches $0$). En-route, we also develop new non-asymptotic concentration inequalities for certain functions of these risk measures for the empirical distribution, that may have wider applicability.

We study reward maximisation in a wide class of structured stochastic multi-armed bandit problems, where the mean rewards of arms satisfy some given structural constraints, e.g. linear, unimodal, sparse, etc. Our aim is to develop methods that are flexible (in that they easily adapt to different structures), powerful (in that they perform well empirically and/or provably match instance-dependent lower bounds) and efficient in that the per-round computational burden is small. We develop asymptotically optimal algorithms from instance-dependent lower-bounds using iterative saddle-point solvers. Our approach generalises recent iterative methods for pure exploration to reward maximisation, where a major challenge arises from the estimation of the sub-optimality gaps and their reciprocals. Still we manage to achieve all the above desiderata. Notably, our technique avoids the computational cost of the full-blown saddle point oracle employed by previous work, while at the same time enabling finite-time regret bounds. Our experiments reveal that our method successfully leverages the structural assumptions, while its regret is at worst comparable to that of vanilla UCB.

Pure exploration (aka active testing) is the fundamental task of sequentially gathering information to answer a query about a stochastic environment. Good algorithms make few mistakes and take few samples. Lower bounds (for multi-armed bandit models with arms in an exponential family) reveal that the sample complexity is determined by the solution to an optimisation problem. The existing state of the art algorithms achieve asymptotic optimality by solving a plug-in estimate of that optimisation problem at each step. We interpret the optimisation problem as an unknown game, and propose sampling rules based on iterative strategies to estimate and converge to its saddle point.

We study Online Convex Optimization in the unbounded setting where neither predictions nor gradient are constrained. The goal is to simultaneously adapt to both the sequence of gradients and the comparator. We first develop parameter-free and scale-free algorithms for a simplified setting with hints. We present two versions: the first adapts to the squared norms of both comparator and gradients separately using $O(d)$ time per round, the second adapts to their squared inner products (which measure variance only in the comparator direction) in time $O(d^3)$ per round. We then generalize two prior reductions to the unbounded setting; one to not need hints, and a second to deal with the range ratio problem (which already arises in prior work). We discuss their optimality in light of prior and new lower bounds. We apply our methods to obtain sharper regret bounds for scale-invariant online prediction with linear models.

Most methods for decision-theoretic online learning are based on the Hedge algorithm, which takes a parameter called the learning rate. In most previous analyses the learning rate was carefully tuned to obtain optimal worst-case performance, leading to suboptimal performance on easy instances, for example when there exists an action that is significantly better than all others. We propose a new way of setting the learning rate, which adapts to the difficulty of the learning problem: in the worst case our procedure still guarantees optimal performance, but on easy instances it achieves much smaller regret. In particular, our adaptive method achieves constant regret in a probabilistic setting, when there exists an action that on average obtains strictly smaller loss than all other actions. We also provide a simulation study comparing our approach to existing methods.

We consider sequential prediction algorithms that are given the predictions from a set of models as inputs. If the nature of the data is changing over time in that different models predict well on different segments of the data, then adaptivity is typically achieved by mixing into the weights in each round a bit of the initial prior (kind of like a weak restart). However, what if the favored models in each segment are from a small subset, i.e. the data is likely to be predicted well by models that predicted well before? Curiously, fitting such ''sparse composite models'' is achieved by mixing in a bit of all the past posteriors. This self-referential updating method is rather peculiar, but it is efficient and gives superior performance on many natural data sets.