More Like this

1. Balancing Notions of Equity: Trade-offs Between Fair Portfolio Sizes and Achievable Guarantees

   https://doi.org/10.1137/1.9781611978322.33  

   Gupta, Swati; Moondra, Jai; Singh, Mohit (January 2025, Society for Industrial and Applied Mathematics)

   Motivated by fairness concerns, we study existence and computation of portfolios, defined as: given anoptimization problem with feasible solutions D, a class C of fairness objective functions, a set X of feasible solutions is an α-approximate portfolio if for each objective f in C, there is an α-approximation for f in X. We study the trade-off between the size |X| of the portfolio and its approximation factor αfor various combinatorial problems, such as scheduling, covering, and facility location, and choices of C as top-k, ordered and symmetric monotonic norms. Our results include: (i) an α-approximate portfolio of size O(log d*log(α/4))for ordered norms and lower bounds of size \Omega( log dlog α+log log d)for the problem of scheduling identical jobs on d unidentical machines, (ii) O(log n)-approximate O(log n)-sized portfolios for facility location on n points for symmetric monotonic norms, and (iii) logO(d)^(r^2)-size O(1)-approximate portfolios for ordered norms and O(log d)-approximate for symmetric monotonic norms for covering polyhedra with a constant rnumber of constraints. The latter result uses our novel OrderAndCount framework that obtains an exponentialimprovement in portfolio sizes compared to current state-of-the-art, which may be of independent interest. 

   more » « less
2. Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

   Ghaffari, Mohsen; Gouleakis, Themis; Konrad, Christian; Mitrovic, Slobodan; Rubinfeld, Ronitt (July 2018, Proceedings of the 37th ACM Principles of Distributed Computing (PODC 2018))

   We present O(log logn)-round algorithms in the Massively Parallel Computation (MPC) model, with ˜O(n) memory per machine, that compute a maximal independent set, a 1 + ε approximation of maximum matching, and a 2 + ε approximation of minimum vertex cover, for any n-vertex graph and any constant ε > 0. These improve the state of the art as follows: • Our MIS algorithm leads to a simple O(log log Δ)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on the ˜O (plog Δ)-round algorithm of Ghaffari [PODC’17]. • OurO(log logn)-round (1+ε)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log2 logn)-round (1 + ε)-approximation algorithm of Czumaj et al. [STOC’18] and O(log logn)-round (1 + ε)- approximation algorithm of Assadi et al. [arXiv’17]. • Our O(log logn)-round (2+ε)-approximate minimum vertex cover algorithm improves on an O(log logn)-round O(1)- approximation of Assadi et al. [arXiv’17]. 

   more » « less
3. Logarithmic Regret from Sublinear Hints

   Bhaskara, Aditya; Cutkosky, Ashok; Kumar, Ravi; Purohit, Manish (December 2021, Advances in Neural Information Processing Systems 34 (NeurIPS 2021))

   We consider the online linear optimization problem, where at every step the algorithm plays a point x_t in the unit ball, and suffers loss for some cost vector c_t that is then revealed to the algorithm. Recent work showed that if an algorithm receives a "hint" h_t that has non-trivial correlation with c_t before it plays x_t, then it can achieve a logarithmic regret guarantee, improving on the classical sqrt(T) bound. In this work, we study the question of whether an algorithm really requires a hint at every time step. Somewhat surprisingly, we show that an algorithm can obtain logarithmic regret with just O(sqrt(T)) hints under a natural query model. We give two applications of our result, to the well-studied setting of optimistic regret bounds and to the problem of online learning with abstention. 

   more » « less
4. Online Convex Optimization with Stochastic Constraints

   Yu, Hao; Neely, Michael J (January 2017, Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.)

   This paper considers online convex optimization (OCO) with stochastic constraints, which generalizes Zinkevich’s OCO over a known simple fixed set by introducing multiple stochastic functional constraints that are i.i.d. generated at each round and are disclosed to the decision maker only after the decision is made. This formulation arises naturally when decisions are restricted by stochastic environ- ments or deterministic environments with noisy observations. It also includes many important problems as special case, such as OCO with long term constraints, stochastic constrained convex optimization, and deterministic constrained con- vex optimization. To solve this problem, this paper proposes a new algorithm that achieves O(√T ) expected regret and constraint violations and O(√T log(T )) high probability regret and constraint violations. Experiments on a real-world data center scheduling problem further verify the performance of the new algorithm. 

   more » « less
5. Online Convex Optimization with Stochastic Constraints

   Yu, Hao; Neely, Michael J; Wei, Xiaohan (January 2017, 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.)

   This paper considers online convex optimization (OCO) with stochastic constraints, which generalizes Zinkevich’s OCO over a known simple fixed set by introducing multiple stochastic functional constraints that are i.i.d. generated at each round and are disclosed to the decision maker only after the decision is made. This formulation arises naturally when decisions are restricted by stochastic environ- ments or deterministic environments with noisy observations. It also includes many important problems as special case, such as OCO with long term constraints, stochastic constrained convex optimization, and deterministic constrained con- vex optimization. To solve this problem, this paper proposes a new algorithm that achieves O(√T ) expected regret and constraint violations and O(√T log(T )) high probability regret and constraint violations. Experiments on a real-world data center scheduling problem further verify the performance of the new algorithm. 

   more » « less