Multiplicative Bidding in Online Advertising

In this paper, we initiate the study of the multiplicative bidding language adopted by major Internet search companies. In multiplicative bidding, the effective bid on a particular search auction is the product of a base bid and bid adjustments that are dependent on features of the search (for example, the geographic location of the user, or the platform on which the search is conducted). We consider the task faced by the advertiser when setting these bid adjustments, and establish a foundational optimization problem that captures the core difficulty of bidding under this language. We give matching algorithmic and approximation hardness results for this problem; these results are against an information-theoretic bound, and thus have implications on the power of the multiplicative bidding language itself. Inspired by empirical studies of search engine price data, we then codify the relevant restrictions of the problem, and give further algorithmic and hardness results. Our main technical contribution is an $O(\log n)$-approximation for the case of multiplicative prices and monotone values. We also provide empirical validations of our problem restrictions, and test our algorithms on real data against natural benchmarks. Our experiments show that they perform favorably compared with the baseline.Comment: 25 pages; accepted to EC'1

An Uncertainty Aided Framework for Learning based Liver T1ρT_1\rho Mapping and Analysis

Objective: Quantitative $T_1\rho$ imaging has potential for assessment of biochemical alterations of liver pathologies. Deep learning methods have been employed to accelerate quantitative $T_1\rho$ imaging. To employ artificial intelligence-based quantitative imaging methods in complicated clinical environment, it is valuable to estimate the uncertainty of the predicated $T_1\rho$ values to provide the confidence level of the quantification results. The uncertainty should also be utilized to aid the post-hoc quantitative analysis and model learning tasks. Approach: To address this need, we propose a parametric map refinement approach for learning-based $T_1\rho$ mapping and train the model in a probabilistic way to model the uncertainty. We also propose to utilize the uncertainty map to spatially weight the training of an improved $T_1\rho$ mapping network to further improve the mapping performance and to remove pixels with unreliable $T_1\rho$ values in the region of interest. The framework was tested on a dataset of 51 patients with different liver fibrosis stages. Main results: Our results indicate that the learning-based map refinement method leads to a relative mapping error of less than 3% and provides uncertainty estimation simultaneously. The estimated uncertainty reflects the actual error level, and it can be used to further reduce relative $T_1\rho$ mapping error to 2.60% as well as removing unreliable pixels in the region of interest effectively. Significance: Our studies demonstrate the proposed approach has potential to provide a learning-based quantitative MRI system for trustworthy $T_1\rho$ mapping of the liver

Competitive algorithms for online matching and vertex cover problems

Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 73-75).The past decade has witnessed an explosion of research on the online bipartite matching problem. Surprisingly, its dual problem, online bipartite vertex cover, has never been explicitly studied before. One of the motivation for studying this problem is that it significantly generalizes the classical ski rental problem. An instance of such problems specifies a bipartite graph G = (L, R, E) whose left vertices L are offline and right vertices arrive online one at a time. An algorithm must maintain a valid vertex cover from which no vertex can ever be removed. The objective is to minimize the size of the cover. In this thesis, we introduce a charging-based algorithmic framework for this problem as well as its generalizations. One immediate outcome is a simple analysis of an optimal 1/1-1/e- competitive algorithm for online bipartite vertex cover. By extending the charging-based analysis in various nontrivial ways, we also obtain optimal l_1 e-competitive algorithms for the edge-weighted and submodular versions of online bipartite vertex cover, which all match the best performance of ski rental. As an application, we show that by analyzing our algorithm in the primal-dual framework, our result on submodular vertex cover implies an optimal (1/1-1/e)-competitive algorithm for its dual, online bipartite submodular matching. This problem is a generalization of online bipartite matching and may have applications in display ad allocation. We consider also the more general scenario where all the vertices are online and the graph is not necessarily bipartite, which is known as the online fractional vertex cover and matching problems. Our contribution in this direction is a primal-dual 1.901-competitive (or 1/1.901 ~~ 0.526) algorithm for these problems. Previously, it was only known that they admit a simple well-known 2-competitive (or 1/2) greedy algorithm. Our result is the first successful attempt to beat the greedy algorithm for these two problems. Moreover, our algorithm for the online matching problem significantly generalizes the traditional online bipartite graph matching problem, where vertices from only one side of the bipartite graph arrive online. In particular, our algorithm improves upon the result of the fractional version of the online edge-selection problem in Blum et. al. (JACM '06). Finally, on the hardness side, we show that no randomized online algorithm can achieve a competitive ratio better than 1.753 and 0.625 for the online fractional vertex cover problem and the online fractional matching problem respectively, even for bipartite graphs.by Chiu Wai Wong.M. Eng