We consider three central problems in optimization: the restricted assignment load-balancing problem, the Steiner tree network design problem, and facility location clustering. We consider the online setting, where the input arrives over time, and irrevocable decisions must be made without knowledge of the future. For all these problems, any online algorithm must incur a cost that is approximately $\log |I|$ times the optimal cost in the worst-case, where $|I|$ is the length of the input. But can we go beyond the worst-case? In this work we give algorithms that perform substantially better when a $p$-fraction of the input is given as a sample: the algorithm use this sample to \emph{learn} a good strategy to use for the rest of the input.

We study the problem of solving Packing Integer Programs (PIPs) in the online setting, where columns in [0, 1]d of the constraint matrix are revealed sequentially, and the goal is to pick a subset of the columns that sum to at most B in each coordinate while maximizing the objective. Excellent results are known in the secretary setting, where the columns are adversarially chosen, but presented in a uniformly random order. However, these existing algorithms are susceptible to adversarial attacks: they try to “learn” characteristics of a good solution, but tend to over-fit to the model, and hence a small number of adversarial corruptions can cause the algorithm to fail. In this paper, we give the first robust algorithms for Packing Integer Programs, specifically in the recently proposed Byzantine Secretary framework [BGSZ20]. Our techniques are based on a two-level use of online learning, to robustly learn an approximation to the optimal value, and then to use this robust estimate to pick a good solution. These techniques are general and we use them to design robust algorithms for PIPs in the prophet model as well, specifically in the Prophet-with-Augmentations framework [ISW20]. We also improve known results in the Byzantine Secretary framework: we make the non-constructive results algorithmic and improve the existing bounds for single-item and matroid constraints.

The problem of chasing convex functions is easy to state: faced with a sequence of convex functions f t over d-dimensional Euclidean spaces, the goal of the algorithm is to output a point x t at each time, so that the sum of the function costs f t (x t ), plus the movement costs ||x t − x t − 1 || is minimized. This problem generalizes questions in online algorithms such as caching and the k-server problem. In 1994, Friedman and Linial posed the question of getting an algorithm with a competitive ratio that depends only on the dimension d. In this talk we give an O (d)-competitive algorithm, based on the notion of the Steiner point of a convex body.

We study the problem of chasing convex bodies online: given a sequence of convex bodies the algorithm must respond with points in an online fashion (i.e., is chosen before is revealed). The objective is to minimize the sum of distances between successive points in this sequence. Bubeck et al. (STOC 2019) gave a -competitive algorithm for this problem. We give an algorithm that is -competitive for any sequence of length .

We consider the problem of chasing convex functions, where functions arrive over time. The player takes actions after seeing the function, and the goal is to achieve a small function cost for these actions, as well as a small cost for moving between actions. While the general problem requires a polynomial dependence on the dimension, we show how to get dimension-independent bounds for well-behaved functions. In particular, we consider the case where the convex functions are $\kappa$-well-conditioned, and give an algorithm that achieves an $O(\sqrt \kappa)$-competitiveness. Moreover, when the functions are supported on $k$-dimensional affine subspaces—e.g., when the function are the indicators of some affine subspaces—we get $O(\min(k, \sqrt{k \log T}))$-competitive algorithms for request sequences of length $T$. We also show some lower bounds, that well-conditioned functions require $\Omega(\kappa^{1/3})$-competitiveness, and $k$-dimensional functions require $\Omega(\sqrt{k})$-competitiveness.

We study the problem of chasing convex bodies online: given a sequence of convex bodies Kt ⊆ ℝd the algorithm must respond with points xt ϵ Kt in an on-line fashion (i.e., xt is chosen before Kt+1 is revealed). The objective is to minimize the total distance between successive points in this sequence. Recently, Bubeck et al. (STOC 2019) gave a 2O(d)-competitive algorithm for this problem. We give an algorithm that is -competitive for any sequence of length T.

Friedman and Linial [8] introduced the convex body chasing problem to explore the interplay between geometry and competitive ratio in metrical task systems. In convex body chasing, at each time step t ∊ ℕ, the online algorithm receives a request in the form of a convex body Kt ⊂ ℝ and must output a point xt ∊ Kt. The goal is to minimize the total movement between consecutive output points, where the distance is measured in some given norm. This problem is still far from being understood. Recently Bansal et al. [4] gave an 6d(d!)2-competitive algorithm for the nested version, where each convex body is contained within the previous one. We propose a different strategy which is O(d log d)-competitive algorithm for this nested convex body chasing problem. Our algorithm works for any norm. This result is almost tight, given an Ω(d) lower bound for the ℓ∞ norm [8].