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We give near-optimal algorithms for computing an ellipsoidal rounding of a convex polytope whose vertices are given in a stream. The approximation factor is linear in the dimension (as in John's theorem) and only loses an excess logarithmic factor in the aspect ratio of the polytope. Our algorithms are nearly optimal in two senses: first, their runtimes nearly match those of the most efficient known algorithms for the offline version of the problem. Second, their approximation factors nearly match a lower bound we show against a natural class of geometric streaming algorithms. In contrast to existing works in the streaming setting that compute ellipsoidal roundings only for centrally symmetric convex polytopes, our algorithms apply to general convex polytopes. We also show how to use our algorithms to construct coresets from a stream of points that approximately preserve both the ellipsoidal rounding and the convex hull of the original set of points. 

A fundamental problem in robust learning is asymmetry: a learner needs to correctly classify every one of exponentially-many perturbations that an adversary might make to a test example, but the attacker only needs to find one successful perturbation. Xiang et al. [2022] proposed an algorithm for patch attacks that reduces the effective number of perturbations from an exponential to a polynomial, and learns using an ERM oracle. However, their guarantee requires the natural examples to be robustly realizable. In this work we consider the non-robustly-realizable case. Our first contribution is to give a guarantee for this setting by utilizing an approach of Feige, Mansour, and Schapire [2015]. Next, we extend our results to a multi-group setting and introduce a novel agnostic multi-robust learning problem where the goal is to learn a predictor that achieves low robust loss on a (potentially) rich collection of subgroups. 

We introduce and study the problem of dueling optimization with a monotone adversary, a generalization of (noiseless) dueling convex optimization. The goal is to design an online algorithm to find a minimizer x* for a function f:X→R, for X \subseteq R^d. In each round, the algorithm submits a pair of guesses x1 and x2, and the adversary responds with any point in the space that is at least as good as both guesses. The cost of each query is the suboptimality of the worst of the two guesses; i.e., max(f(x1) − f(x*),f(x2) − f(x*)). The goal is to minimize the number of iterations required to find an ε-optimal point and to minimize the total cost (regret) of the guesses over many rounds. Our main result is an efficient randomized algorithm for several natural choices of the function f and set X that incurs cost O(d) and iteration complexity O(d log(1/ε)^2). Moreover, our dependence on d is asymptotically optimal, as we show examples in which any randomized algorithm for this problem must incur Ω(d) cost and iteration complexity. 

Gameplay under various forms of uncertainty has been widely studied. Feldman et al. [Michal Feldman et al., 2010] studied a particularly low-information setting in which one observes the opponent’s actions but no payoffs, not even one’s own, and introduced an algorithm which guarantees one’s payoff nonetheless approaches the minimax optimal value (i.e., zero) in a symmetric zero-sum game. Against an opponent playing a minimax-optimal strategy, approaching the value of the game is the best one can hope to guarantee. However, a wealth of research in behavioral economics shows that people often do not make perfectly rational, optimal decisions. Here we consider whether it is possible to actually win in this setting if the opponent is behaviorally biased. We model several deterministic, biased opponents and show that even without knowing the game matrix in advance or observing any payoffs, it is possible to take advantage of each bias in order to win nearly every round (so long as the game has the property that each action beats and is beaten by at least one other action). We also provide a partial characterization of the kinds of biased strategies that can be exploited to win nearly every round, and provide algorithms for beating some kinds of biased strategies even when we don't know which strategy the opponent uses. 

The maximum bipartite matching problem is among the most fundamental and well-studied problems in combinatorial optimization. A beautiful and celebrated combinatorial algorithm of Hopcroft and Karp [26] shows that maximum bipartite matching can be solved in O(m√n) time on a graph with n vertices and m edges. For the case of very dense graphs, a different approach based on fast matrix multiplication was subsequently developed [27, 39], that achieves a running time of O(n2.371). For the next several decades, these results represented the fastest known algorithms for the problem until in 2013, a ground-breaking work of Madry [36] gave a significantly faster algorithm for sparse graphs. Subsequently, a sequence of works developed increasingly faster algorithms for solving maximum bipartite matching, and more generally directed maximum flow, culminating in a spectacular recent breakthrough [9] that gives an m1+o(1) time algorithm for maximum bipartite matching (and more generally, for min cost flows). These more recent developments collectively represented a departure from earlier combinatorial approaches: they all utilized continuous techniques based on interior-point methods for solving linear programs. This raises a natural question: are continuous techniques essential to obtaining fast algorithms for the bipartite matching problem? Our work makes progress on this question by presenting a new, purely combinatorial algorithm for bipartite matching, that, on moderately dense graphs outperforms both Hopcroft- Karp and the fast matrix multiplication based algorithms. Similar to the classical algorithms for bipartite matching, our approach is based on iteratively augmenting a current matching using augmenting paths in the (directed) residual flow network. A common method for designing fast algorithms for directed flow problems is via the multiplicative weights update (MWU) framework, that effectively reduces the flow problem to decremental single-source shortest paths (SSSP) in directed graphs. Our main observation is that a slight modification of this reduction results in a special case of SSSP that appears significantly easier than general decremental directed SSSP. Our main technical contribution is an efficient algorithm for this special case of SSSP, that outperforms the current state of the art algorithms for general decremental SSSP with adaptive adversary, leading to a deterministic algorithm for bipartite matching, whose running time is Õ(m1/3n5/3). This new algorithm thus starts to outperform the Hopcroft-Karp algorithm in graphs with m = ω(n7/4), and it also outperforms the fast matrix multiplication based algorithms on dense graphs. We believe that this framework for obtaining faster combinatorial algorithms for bipartite matching by exploiting the special properties of the resulting decremental SSSP instances is one of the main conceptual contributions of our work that we hope paves the way for even faster combinatorial algorithms for bipartite matching. Finally, using a standard reduction from the maximum vertex-capacitated s-t flow problem in directed graphs to maximum bipartite matching, we also obtain an O(m1/3n5/3) time deterministic algorithm for maximum vertex-capacitated s-t flow when all vertex capacities are identical. 

We present polylogarithmic approximation algorithms for variants of the Shortest Path, Group Steiner Tree, and Group ATSP problems with vector costs. In these problems, each edge e has a vector cost c_e ∈ ℝ_{≥0}^𝓁;. For a feasible solution - a path, subtree, or tour (respectively) - we find the total vector cost of all the edges in the solution and then compute the 𝓁_p-norm of the obtained cost vector (we assume that p ≥ 1 is an integer). Our algorithms for series-parallel graphs run in polynomial time and those for arbitrary graphs run in quasi-polynomial time. To obtain our results, we introduce and use new flow-based Sum-of-Squares relaxations. We also obtain a number of hardness results.