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

Workload Equity in Vehicle Routing Problems: A Survey and Analysis

Over the past two decades, equity aspects have been considered in a growing number of models and methods for vehicle routing problems (VRPs). Equity concerns most often relate to fairly allocating workloads and to balancing the utilization of resources, and many practical applications have been reported in the literature. However, there has been only limited discussion about how workload equity should be modeled in VRPs, and various measures for optimizing such objectives have been proposed and implemented without a critical evaluation of their respective merits and consequences. This article addresses this gap with an analysis of classical and alternative equity functions for biobjective VRP models. In our survey, we review and categorize the existing literature on equitable VRPs. In the analysis, we identify a set of axiomatic properties that an ideal equity measure should satisfy, collect six common measures, and point out important connections between their properties and those of the resulting Pareto-optimal solutions. To gauge the extent of these implications, we also conduct a numerical study on small biobjective VRP instances solvable to optimality. Our study reveals two undesirable consequences when optimizing equity with nonmonotonic functions: Pareto-optimal solutions can consist of non-TSP-optimal tours, and even if all tours are TSP optimal, Pareto-optimal solutions can be workload inconsistent, i.e. composed of tours whose workloads are all equal to or longer than those of other Pareto-optimal solutions. We show that the extent of these phenomena should not be underestimated. The results of our biobjective analysis are valid also for weighted sum, constraint-based, or single-objective models. Based on this analysis, we conclude that monotonic equity functions are more appropriate for certain types of VRP models, and suggest promising avenues for further research.Comment: Accepted Manuscrip

Energy Sharing for Multiple Sensor Nodes with Finite Buffers

We consider the problem of finding optimal energy sharing policies that maximize the network performance of a system comprising of multiple sensor nodes and a single energy harvesting (EH) source. Sensor nodes periodically sense the random field and generate data, which is stored in the corresponding data queues. The EH source harnesses energy from ambient energy sources and the generated energy is stored in an energy buffer. Sensor nodes receive energy for data transmission from the EH source. The EH source has to efficiently share the stored energy among the nodes in order to minimize the long-run average delay in data transmission. We formulate the problem of energy sharing between the nodes in the framework of average cost infinite-horizon Markov decision processes (MDPs). We develop efficient energy sharing algorithms, namely Q-learning algorithm with exploration mechanisms based on the $\epsilon$-greedy method as well as upper confidence bound (UCB). We extend these algorithms by incorporating state and action space aggregation to tackle state-action space explosion in the MDP. We also develop a cross entropy based method that incorporates policy parameterization in order to find near optimal energy sharing policies. Through simulations, we show that our algorithms yield energy sharing policies that outperform the heuristic greedy method.Comment: 38 pages, 10 figure

Clustering by compression

We present a new method for clustering based on compression. The method doesn't use subject-specific features or background knowledge, and works as follows: First, we determine a universal similarity distance, the normalized compression distance or NCD, computed from the lengths of compressed data files (singly and in pairwise concatenation). Second, we apply a hierarchical clustering method. The NCD is universal in that it is not restricted to a specific application area, and works across application area boundaries. A theoretical precursor, the normalized information distance, co-developed by one of the authors, is provably optimal but uses the non-computable notion of Kolmogorov complexity. We propose precise notions of similarity metric, normal compressor, and show that the NCD based on a normal compressor is a similarity metric that approximates universality. To extract a hierarchy of clusters from the distance matrix, we determine a dendrogram (binary tree) by a new quartet method and a fast heuristic to implement it. The method is implemented and available as public software, and is robust under choice of different compressors. To substantiate our claims of universality and robustness, we report evidence of successful application in areas as diverse as genomics, virology, languages, literature, music, handwritten digits, astronomy, and combinations of objects from completely different domains, using statistical, dictionary, and block sorting compressors. In genomics we presented new evidence for major questions in Mammalian evolution, based on whole-mitochondrial genomic analysis: the Eutherian orders and the Marsupionta hypothesis against the Theria hypothesis.Comment: LaTeX, 27 pages, 20 figure

Addressing selection bias in cluster randomized experiments via weighting

In cluster randomized experiments, units are often recruited after the random cluster assignment, and data are only available for the recruited sample. Post-randomization recruitment can lead to selection bias, inducing systematic differences between the overall and the recruited populations, and between the recruited intervention and control arms. In this setting, we define causal estimands for the overall and the recruited populations. We first show that if units select their cluster independently of the treatment assignment, cluster randomization implies individual randomization in the overall population. We then prove that under the assumption of ignorable recruitment, the average treatment effect on the recruited population can be consistently estimated from the recruited sample using inverse probability weighting. Generally we cannot identify the average treatment effect on the overall population. Nonetheless, we show, via a principal stratification formulation, that one can use weighting of the recruited sample to identify treatment effects on two meaningful subpopulations of the overall population: units who would be recruited into the study regardless of the assignment, and units who would be recruited in the study under treatment but not under control. We develop a corresponding estimation strategy and a sensitivity analysis method for checking the ignorable recruitment assumption