Hédi Hadiji

Raymond Zhang, Hédi Hadiji, Richard Combes

We consider the maximization of $x^\top θ$ over $(x,θ) \in \mathcal{X} \times Θ$, with $\mathcal{X} \subset \mathbb{R}^d$ convex and $Θ\subset \mathbb{R}^d$ an ellipsoid. This problem is fundamental in linear bandits, as the learner must solve it at every time step using optimistic algorithms. We first show that for some sets $\mathcal{X}$ e.g. $\ell_p$ balls with $p>2$, no efficient algorithms exist unless $\mathcal{P} = \mathcal{NP}$. We then provide two novel algorithms solving this problem efficiently when $\mathcal{X}$ is a centered ellipsoid. Our findings provide the first known method to implement optimistic algorithms for linear bandits in high dimensions.

Hédi Hadiji, Sarah Sachs, Cristóbal Guzmán

Tracking the solution of time-varying variational inequalities is an important problem with applications in game theory, optimization, and machine learning. Existing work considers time-varying games or time-varying optimization problems. For strongly convex optimization problems or strongly monotone games, these results provide tracking guarantees under the assumption that the variation of the time-varying problem is restrained, that is, problems with a sublinear solution path. In this work we extend existing results in two ways: In our first result, we provide tracking bounds for (1) variational inequalities with a sublinear solution path but not necessarily monotone functions, and (2) for periodic time-varying variational inequalities that do not necessarily have a sublinear solution path-length. Our second main contribution is an extensive study of the convergence behavior and trajectory of discrete dynamical systems of periodic time-varying VI. We show that these systems can exhibit provably chaotic behavior or can converge to the solution. Finally, we illustrate our theoretical results with experiments.

Sarah Sachs, Hédi Hadiji, Tim van Erven et al.

Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previously known only for linear losses. In addition, they weaken the i.i.d. assumption by allowing, for example, adversarially poisoned rounds, which were previously considered in the expert and bandit setting. Our results extend this to the online convex optimization framework. In the fully i.i.d. case, our bounds match the rates one would expect from results in stochastic acceleration, and in the fully adversarial case they gracefully deteriorate to match the minimax regret. We further provide lower bounds showing that our regret upper bounds are tight for all intermediate regimes in terms of the stochastic variance and the adversarial variation of the loss gradients.

Jack J. Mayo, Hédi Hadiji, Tim van Erven

A sequence of works in unconstrained online convex optimisation have investigated the possibility of adapting simultaneously to the norm $U$ of the comparator and the maximum norm $G$ of the gradients. In full generality, matching upper and lower bounds are known which show that this comes at the unavoidable cost of an additive $G U^3$, which is not needed when either $G$ or $U$ is known in advance. Surprisingly, recent results by Kempka et al. (2019) show that no such price for adaptivity is needed in the specific case of $1$-Lipschitz losses like the hinge loss. We follow up on this observation by showing that there is in fact never a price to pay for adaptivity if we specialise to any of the other common supervised online learning losses: our results cover log loss, (linear and non-parametric) logistic regression, square loss prediction, and (linear and non-parametric) least-squares regression. We also fill in several gaps in the literature by providing matching lower bounds with an explicit dependence on $U$. In all cases we obtain scale-free algorithms, which are suitably invariant under rescaling of the data. Our general goal is to establish achievable rates without concern for computational efficiency, but for linear logistic regression we also provide an adaptive method that is as efficient as the recent non-adaptive algorithm by Agarwal et al. (2021).

Dirk van der Hoeven, Hédi Hadiji, Tim van Erven

We consider distributed online learning for joint regret with communication constraints. In this setting, there are multiple agents that are connected in a graph. Each round, an adversary first activates one of the agents to issue a prediction and provides a corresponding gradient, and then the agents are allowed to send a $b$-bit message to their neighbors in the graph. All agents cooperate to control the joint regret, which is the sum of the losses of the activated agents minus the losses evaluated at the best fixed common comparator parameters $u$. We observe that it is suboptimal for agents to wait for gradients that take too long to arrive. Instead, the graph should be partitioned into local clusters that communicate among themselves. Our main result is a new method that can adapt to the optimal graph partition for the adversarial activations and gradients, where the graph partition is selected from a set of candidate partitions. A crucial building block along the way is a new algorithm for online convex optimization with delayed gradient information that is comparator-adaptive, meaning that its joint regret scales with the norm of the comparator $||u||$. We further provide near-optimal gradient compression schemes depending on the ratio of $b$ and the dimension times the diameter of the graph.

Hédi Hadiji, Sébastien Gerchinovitz, Jean-Michel Loubes et al.

We consider the bandit-based framework for diversity-preserving recommendations introduced by Celis et al. (2019), who approached it in the case of a polytope mainly by a reduction to the setting of linear bandits. We design a UCB algorithm using the specific structure of the setting and show that it enjoys a bounded distribution-dependent regret in the natural cases when the optimal mixed actions put some probability mass on all actions (i.e., when diversity is desirable). The regret lower bounds provided show that otherwise, at least when the model is mean-unbounded, a $\ln T$ regret is suffered. We also discuss an example beyond the special case of polytopes.

Hédi Hadiji, Gilles Stoltz

We consider stochastic bandit problems with $K$ arms, each associated with a bounded distribution supported on the range $[m,M]$. We do not assume that the range $[m,M]$ is known and show that there is a cost for learning this range. Indeed, a new trade-off between distribution-dependent and distribution-free regret bounds arises, which prevents from simultaneously achieving the typical $\ln T$ and $\sqrt{T}$ bounds. For instance, a $\sqrt{T}$}distribution-free regret bound may only be achieved if the distribution-dependent regret bounds are at least of order $\sqrt{T}$. We exhibit a strategy achieving the rates for regret indicated by the new trade-off.

Hédi Hadiji

In the context of stochastic continuum-armed bandits, we present an algorithm that adapts to the unknown smoothness of the objective function. We exhibit and compute a polynomial cost of adaptation to the H{ö}lder regularity for regret minimization. To do this, we first reconsider the recent lower bound of Locatelli and Carpentier [20], and define and characterize admissible rate functions. Our new algorithm matches any of these minimal rate functions. We provide a finite-time analysis and a thorough discussion about asymptotic optimality.

Aurélien Garivier, Hédi Hadiji, Pierre Menard et al.

We consider $K$-armed stochastic bandits and consider cumulative regret bounds up to time $T$. We are interested in strategies achieving simultaneously a distribution-free regret bound of optimal order $\sqrt{KT}$ and a distribution-dependent regret that is asymptotically optimal, that is, matching the $κ\ln T$ lower bound by Lai and Robbins (1985) and Burnetas and Katehakis (1996), where $κ$ is the optimal problem-dependent constant. This constant $κ$ depends on the model $\mathcal{D}$ considered (the family of possible distributions over the arms). Ménard and Garivier (2017) provided strategies achieving such a bi-optimality in the parametric case of models given by one-dimensional exponential families, while Lattimore (2016, 2018) did so for the family of (sub)Gaussian distributions with variance less than $1$. We extend this result to the non-parametric case of all distributions over $[0,1]$. We do so by combining the MOSS strategy by Audibert and Bubeck (2009), which enjoys a distribution-free regret bound of optimal order $\sqrt{KT}$, and the KL-UCB strategy by Cappé et al. (2013), for which we provide in passing the first analysis of an optimal distribution-dependent $κ\ln T$ regret bound in the model of all distributions over $[0,1]$. We were able to obtain this non-parametric bi-optimality result while working hard to streamline the proofs (of previously known regret bounds and thus of the new analyses carried out); a second merit of the present contribution is therefore to provide a review of proofs of classical regret bounds for index-based strategies for $K$-armed stochastic bandits.