The following game is played on a weighted graph: Alice selects a matching @math and Bob selects a number @math . Alice's payoff is the ratio of the weight of the @math heaviest edges of @math to the maximum weight of a matching of size at most @math . If @math guarantees a payoff of at least @math then it is called @math -robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a @math -robust matching, which is best possible. We show that Alice can improve her payoff to @math by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound.
Results on the impact of randomization in robust optimization are much scarcer. Bertsimas, Nasrabadi, and Orlin @cite_5 show that the randomized version of network flow interdiction is equivalent to the maximum robust flow problem and use this insight to obtain an approximation algorithm for both problems. Mastin, Jaillet, and Chin @cite_10 consider a randomized version of the minmax (additive) regret model. They show that if an optimization problem can be solved efficiently, then also the corresponding randomized minmax regret version can be solved efficiently. Inspired by a preliminary version of our work, Kobayashi and Takazawa @cite_19 provide an analysis of the randomized robustness of general independence systems. They obtain a robustness guarantee of @math for general systems and a guarantee of @math for systems induced by instances of the knapsack problem (here @math is again the extendibility of the system, while @math is the ratio of the largest size of an independent set to the smallest size of a non-independent set minus @math ). They also construct instances that do not allow for a better than @math -robust or @math -robust randomized independent set, respectively.
@cite_19 @cite_5 @cite_10
2153675120 2547508527 2787223504 2007104311
The minmax regret problem for combinatorial optimization under uncertainty can be viewed as a zero-sum game played between an optimizing player and an adversary, where the optimizing player selects a solution and the adversary selects costs with the intention of maximizing the regret of the player. The conventional minmax regret model considers only deterministic solutions strategies, and minmax regret versions of most polynomial solvable problems are NP-hard. In this paper, we consider a randomized model where the optimizing player selects a probability distribution (corresponding to a mixed strategy) over solutions and the adversary selects costs with knowledge of the player’s distribution, but not its realization. We show that under this randomized model, the minmax regret version of any polynomial solvable combinatorial problem becomes polynomial solvable. This holds true for both interval and discrete scenario representations of uncertainty. Using the randomized model, we show new proofs of existing approximation algorithms for the deterministic model based on primal-dual approaches. We also determine integrality gaps of minmax regret formulations, giving tight bounds on the limits of performance gains from randomization. Finally, we prove that minmax regret problems are NP-hard under general convex uncertainty. We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. We propose in this paper a new approach to train the Generative Adversarial Nets (GANs) with a mixture of generators to overcome the mode collapsing problem. The main intuition is to employ multiple generators, instead of using a single one as in the original GAN. The idea is simple, yet proven to be extremely effective at covering diverse data modes, easily overcoming the mode collapsing problem and delivering state-of-the-art results. A minimax formulation was able to establish among a classifier, a discriminator, and a set of generators in a similar spirit with GAN. Generators create samples that are intended to come from the same distribution as the training data, whilst the discriminator determines whether samples are true data or generated by generators, and the classifier specifies which generator a sample comes from. The distinguishing feature is that internal samples are created from multiple generators, and then one of them will be randomly selected as final output similar to the mechanism of a probabilistic mixture model. We term our method Mixture Generative Adversarial Nets (MGAN). We develop theoretical analysis to prove that, at the equilibrium, the Jensen-Shannon divergence (JSD) between the mixture of generators’ distributions and the empirical data distribution is minimal, whilst the JSD among generators’ distributions is maximal, hence effectively avoiding the mode collapsing problem. By utilizing parameter sharing, our proposed model adds minimal computational cost to the standard GAN, and thus can also efficiently scale to large-scale datasets. We conduct extensive experiments on synthetic 2D data and natural image databases (CIFAR-10, STL-10 and ImageNet) to demonstrate the superior performance of our MGAN in achieving state-of-the-art Inception scores over latest baselines, generating diverse and appealing recognizable objects at different resolutions, and specializing in capturing different types of objects by the generators. In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson J. Comput. System Sci., 68 (2004), pp. 442-470]. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an @math -approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of @math , which is better than the ratio of @math for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349.