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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. 

We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph G = (V, E). The buffered expansion of a set S ⊆ V with a buffer B ⊆ V∖S is the edge expansion of S after removing all the edges from set S to its buffer B. An ε-buffered k-partitioning is a partitioning of a graph into disjoint components P_i and buffers B_i, in which the size of buffer B_i for P_i is small relative to the size of P_i: |B_i| ≤ ε|P_i|. The buffered expansion of a buffered partition is the maximum of buffered expansions of the k sets P_i with buffers B_i. Let h^{k,ε}_G be the buffered expansion of the optimal ε-buffered k-partitioning, then for every δ>0, h^{k,ε}_G ≤ O(1)⋅(log k) ⋅λ_{⌊(1+δ)k⌋} / ε, where λ_{⌊(1+δ)k⌋} is the ⌊(1+δ)k⌋-th smallest eigenvalue of the normalized Laplacian of G. Our inequality is constructive and avoids the ``square-root loss'' that is present in the standard Cheeger inequalities (even for k=2). We also provide a complementary lower bound, and a novel generalization to the setting with arbitrary vertex weights and edge costs. Moreover our result implies and generalizes the standard higher-order Cheeger inequalities and another recent Cheeger-type inequality by Kwok, Lau, and Lee (2017) involving robust vertex expansion. 

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. 

In this work, we propose a multi-objective decision making framework that accommodates different user preferences over objectives, where preferences are learned via policy comparisons. Our model consists of a known Markov decision process with a vector-valued reward function, with each user having an unknown preference vector that expresses the relative importance of each objective. The goal is to efficiently compute a near-optimal policy for a given user. We consider two user feedback models. We first address the case where a user is provided with two policies and returns their preferred policy as feedback. We then move to a different user feedback model, where a user is instead provided with two small weighted sets of representative trajectories and selects the preferred one. In both cases, we suggest an algorithm that finds a nearly optimal policy for the user using a number of comparison queries that scales quasilinearly in the number of objectives. 

We introduce a framework for performing vector-valued regression in finite-dimensional Hilbert spaces. Using Lipschitz smoothness as our regularizer, we leverage Kirszbraun’s extension theorem for off-data prediction. We analyze the statistical and computational aspects of this method—to our knowledge, its first application to supervised learning. We decompose this task into two stages: training (which corresponds operationally to smoothing/regularization) and prediction (which is achieved via Kirszbraun extension). Both are solved algorithmically via a novel multiplicative weight updates (MWU) scheme, which, for our problem formulation, achieves significant runtime speedups over generic interior point methods. Our empirical results indicate a dramatic advantage over standard off-the-shelf solvers in our regression setting. 

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph G into k parts so as to separate k given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced l_p-norm Multiway Cut, a generalization of the problem, in which the goal is to minimize the l_p norm of the edge boundaries of k parts. We provide an O(log^{1/2} n log^{1/2 + 1/p} k) approximation algorithm for this problem, improving upon the approximation guarantee of O(log^{3/2} n log^{1/2} k) due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an O(log^{1/2} n log^{7/2} k) approximation algorithm with a weaker oracle and an O(log^{1/2} n log^{5/2} k) approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no n^{1/4-ε} approximation algorithm for every ε > 0 assuming the Hypergraph Dense-vs-Random Conjecture. 

We study the problem of online binary classification where strategic agents can manipulate their observable features in predefined ways, modeled by a manipulation graph, in order to receive a positive classification. We show this setting differs in fundamental ways from classic (non-strategic) online classification. For instance, whereas in the non-strategic case, a mistake bound of ln |H| is achievable via the halving algorithm when the target function belongs to a known class H, we show that no deterministic algorithm can achieve a mistake bound o(Δ) in the strategic setting, where Δ is the maximum degree of the manipulation graph (even when |H| = O(Δ)). We complement this with a general algorithm achieving mistake bound O(Δ ln |H|). We also extend this to the agnostic setting, and show that this algorithm achieves a Δ multiplicative regret (mistake bound of O(Δ · OPT + Δ · ln |H|)), and that no deterministic algorithm can achieve o(Δ) multiplicative regret. Next, we study two randomized models based on whether the random choices are made before or after agents respond, and show they exhibit fundamental differences. In the first, fractional model, at each round the learner deterministically chooses a probability distribution over classifiers inducing expected values on each vertex (probabilities of being classified as positive), which the strategic agents respond to. We show that any learner in this model has to suffer linear regret. On the other hand, in the second randomized algorithms model, while the adversary who selects the next agent must respond to the learner's probability distribution over classifiers, the agent then responds to the actual hypothesis classifier drawn from this distribution. Surprisingly, we show this model is more advantageous to the learner, and we design randomized algorithms that achieve sublinear regret bounds against both oblivious and adaptive adversaries.