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1. Optimal robustness-consistency tradeoffs for learning-augmented metrical task systems

   Nicolas Christianson, Junxuan Shen (April 2023, roceedings of The 26th International Conference on Artificial Intelligence and Statistics)

   We examine the problem of designing learning-augmented algorithms for metrical task systems (MTS) that exploit machine-learned advice while maintaining rigorous, worst-case guarantees on performance. We propose an algorithm, DART, that achieves this dual objective, providing cost within a multiplicative factor (1+ϵ) of the machine-learned advice (i.e., consistency) while ensuring cost within a multiplicative factor 2O(1/ϵ) of a baseline robust algorithm (i.e., robustness) for any ϵ>0 . We show that this exponential tradeoff between consistency and robustness is unavoidable in general, but that in important subclasses of MTS, such as when the metric space has bounded diameter and in the k -server problem, our algorithm achieves improved, polynomial tradeoffs between consistency and robustness. 

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2. Circulation Control for Faster Minimum Cost Flow in Unit-Capacity Graphs

   Axiotis, Kyriakos; Madry, Aleksander; Vladu, Adrian (November 2020, IEEE Symposium on Foundations of Computer Science)

   null (Ed.)

   We present an m^{4/3+o(1)} log W -time algorithm for solving the minimum cost flow problem in graphs with unit capacity, where W is the maximum absolute value of any edge weight. For sparse graphs, this improves over the best known running time for this problem and, by well-known reductions, also implies improved running times for the shortest path problem with negative weights, minimum cost bipartite b-matching when |b|_1 = O(m), and recovers the running time of the currently fastest algorithm for maximum flow in graphs with unit capacities (Liu-Sidford, 2020). Our algorithm relies on developing an interior point method–based framework which acts on the space of circulations in the underlying graph. From the combinatorial point of view, this framework can be viewed as iteratively improving the cost of a suboptimal solution by pushing flow around circulations. These circulations are derived by computing a regularized version of the standard Newton step, which is partially inspired by previous work on the unit-capacity maximum flow problem (Liu-Sidford, 2019), and subsequently refined based on the very recent progress on this problem (Liu-Sidford, 2020). The resulting step problem can then be computed efficiently using the recent work on l_p-norm minimizing flows (Kyng-Peng-Sachdeva-Wang, 2019). We obtain our faster algorithm by combining this new step primitive with a customized preconditioning method, which aims to ensure that the graph on which these circulations are computed has sufficiently large conductance. 

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3. Circulation Control for Faster Minimum Cost Flow in Unit-Capacity Graphs

   Axiotis, Kyriakos; Madry, Aleksander; Vladu, Adrian (November 2020, IEEE Symposium on Foundations of Computer Science)

   null (Ed.)

   We present an $$m^{4/3}+o(1) \log W$$ -time algorithm for solving the minimum cost flow problem in graphs with unit capacity, where W is the maximum absolute value of any edge weight. For sparse graphs, this improves over the best known running time for this problem and, by well-known reductions, also implies improved running times for the shortest path problem with negative weights, minimum cost bipartite $$b$$-matching when $$\|b\|_1 = O(m)$$, and recovers the running time of the currently fastest algorithm for maximum flow in graphs with unit capacities (Liu-Sidford, 2020). Our algorithm relies on developing an interior point method–based framework which acts on the space of circulations in the underlying graph. From the combinatorial point of view, this framework can be viewed as iteratively improving the cost of a suboptimal solution by pushing flow around circulations. These circulations are derived by computing a regularized version of the standard Newton step, which is partially inspired by previous work on the unit-capacity maximum flow problem (Liu-Sidford, 2019), and subsequently refined based on the very re- cent progress on this problem (Liu-Sidford, 2020). The resulting step problem can then be computed efficiently using the recent work on $$l_p$$-norm minimizing flows (Kyng-Peng-Sachdeva- Wang, 2019). We obtain our faster algorithm by combining this new step primitive with a customized preconditioning method, which aims to ensure that the graph on which these circulations are computed has sufficiently large conductance. 

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4. Power of Hints for Online Learning with Movement Costs

   Bhaskara, Aditya; Cutkosky, Ashok; Kumar, Ravi; Purohit, Manish (January 2021, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics)

   We consider the online linear optimization problem with movement costs, a variant of online learning in which the learner must not only respond to cost vectors c_t with points x_t in order to maintain low regret, but is also penalized for movement by an additional cost. Classically, simple algorithms that obtain the optimal sqrt(T) regret already are very stable and do not incur a significant movement cost. However, recent work has shown that when the learning algorithm is provided with weak "hint" vectors that have a positive correlation with the costs, the regret can be significantly improved to log(T). In this work, we study the stability of such algorithms, and provide matching upper and lower bounds showing that incorporating movement costs results in intricate tradeoffs logarithmic and sqrt(T) regret. 

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5. Data Structures for Density Estimation

   Aamand, Anders; Andoni, Alexandr; Chen, Justin; Indyk, Piotr; Narayanan, Shyam; Silwal, Sandeep (January 2023, International Conference on Machine Learning)

   We study statistical/computational tradeoffs for the following density estimation problem: given kdistributionsv1,...,vk overadiscretedomain of size n, and sampling access to a distribution p, identify vi that is “close” to p. Our main result is the first data structure that, given a sublinear (in n) number of samples from p, identifies vi in time sublinear in k. We also give an improved version of the algorithm of (Acharya et al., 2018) that reports vi in time linear in k. The experimental evaluation of the latter algorithm shows that it achieves a significant reduction in the number of operations needed to achieve a given accuracy compared to prior work. 

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