Complexity Theory, Game Theory, and Economics: The Barbados Lectures

This document collects the lecture notes from my mini-course "Complexity Theory, Game Theory, and Economics," taught at the Bellairs Research Institute of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th McGill Invitational Workshop on Computational Complexity. The goal of this mini-course is twofold: (i) to explain how complexity theory has helped illuminate several barriers in economics and game theory; and (ii) to illustrate how game-theoretic questions have led to new and interesting complexity theory, including recent several breakthroughs. It consists of two five-lecture sequences: the Solar Lectures, focusing on the communication and computational complexity of computing equilibria; and the Lunar Lectures, focusing on applications of complexity theory in game theory and economics. No background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some recent citations to v1 Revised v3 corrects a few typos in v

Rendezvous of Distance-aware Mobile Agents in Unknown Graphs

We study the problem of rendezvous of two mobile agents starting at distinct locations in an unknown graph. The agents have distinct labels and walk in synchronous steps. However the graph is unlabelled and the agents have no means of marking the nodes of the graph and cannot communicate with or see each other until they meet at a node. When the graph is very large we want the time to rendezvous to be independent of the graph size and to depend only on the initial distance between the agents and some local parameters such as the degree of the vertices, and the size of the agent's label. It is well known that even for simple graphs of degree $\Delta$, the rendezvous time can be exponential in $\Delta$ in the worst case. In this paper, we introduce a new version of the rendezvous problem where the agents are equipped with a device that measures its distance to the other agent after every step. We show that these \emph{distance-aware} agents are able to rendezvous in any unknown graph, in time polynomial in all the local parameters such the degree of the nodes, the initial distance $D$ and the size of the smaller of the two agent labels $l = \min(l_1, l_2)$. Our algorithm has a time complexity of $O(\Delta(D+\log{l}))$ and we show an almost matching lower bound of $\Omega(\Delta(D+\log{l}/\log{\Delta}))$ on the time complexity of any rendezvous algorithm in our scenario. Further, this lower bound extends existing lower bounds for the general rendezvous problem without distance awareness

Blowfish Privacy: Tuning Privacy-Utility Trade-offs using Policies

Privacy definitions provide ways for trading-off the privacy of individuals in a statistical database for the utility of downstream analysis of the data. In this paper, we present Blowfish, a class of privacy definitions inspired by the Pufferfish framework, that provides a rich interface for this trade-off. In particular, we allow data publishers to extend differential privacy using a policy, which specifies (a) secrets, or information that must be kept secret, and (b) constraints that may be known about the data. While the secret specification allows increased utility by lessening protection for certain individual properties, the constraint specification provides added protection against an adversary who knows correlations in the data (arising from constraints). We formalize policies and present novel algorithms that can handle general specifications of sensitive information and certain count constraints. We show that there are reasonable policies under which our privacy mechanisms for k-means clustering, histograms and range queries introduce significantly lesser noise than their differentially private counterparts. We quantify the privacy-utility trade-offs for various policies analytically and empirically on real datasets.Comment: Full version of the paper at SIGMOD'14 Snowbird, Utah US

Structure-Aware Sampling: Flexible and Accurate Summarization

In processing large quantities of data, a fundamental problem is to obtain a summary which supports approximate query answering. Random sampling yields flexible summaries which naturally support subset-sum queries with unbiased estimators and well-understood confidence bounds. Classic sample-based summaries, however, are designed for arbitrary subset queries and are oblivious to the structure in the set of keys. The particular structure, such as hierarchy, order, or product space (multi-dimensional), makes range queries much more relevant for most analysis of the data. Dedicated summarization algorithms for range-sum queries have also been extensively studied. They can outperform existing sampling schemes in terms of accuracy on range queries per summary size. Their accuracy, however, rapidly degrades when, as is often the case, the query spans multiple ranges. They are also less flexible - being targeted for range sum queries alone - and are often quite costly to build and use. In this paper we propose and evaluate variance optimal sampling schemes that are structure-aware. These summaries improve over the accuracy of existing structure-oblivious sampling schemes on range queries while retaining the benefits of sample-based summaries: flexible summaries, with high accuracy on both range queries and arbitrary subset queries

Efficient Diverse Ensemble for Discriminative Co-Tracking

Ensemble discriminative tracking utilizes a committee of classifiers, to label data samples, which are in turn, used for retraining the tracker to localize the target using the collective knowledge of the committee. Committee members could vary in their features, memory update schemes, or training data, however, it is inevitable to have committee members that excessively agree because of large overlaps in their version space. To remove this redundancy and have an effective ensemble learning, it is critical for the committee to include consistent hypotheses that differ from one-another, covering the version space with minimum overlaps. In this study, we propose an online ensemble tracker that directly generates a diverse committee by generating an efficient set of artificial training. The artificial data is sampled from the empirical distribution of the samples taken from both target and background, whereas the process is governed by query-by-committee to shrink the overlap between classifiers. The experimental results demonstrate that the proposed scheme outperforms conventional ensemble trackers on public benchmarks.Comment: CVPR 2018 Submissio

Rethinking Location Privacy for Unknown Mobility Behaviors

Location Privacy-Preserving Mechanisms (LPPMs) in the literature largely consider that users' data available for training wholly characterizes their mobility patterns. Thus, they hardwire this information in their designs and evaluate their privacy properties with these same data. In this paper, we aim to understand the impact of this decision on the level of privacy these LPPMs may offer in real life when the users' mobility data may be different from the data used in the design phase. Our results show that, in many cases, training data does not capture users' behavior accurately and, thus, the level of privacy provided by the LPPM is often overestimated. To address this gap between theory and practice, we propose to use blank-slate models for LPPM design. Contrary to the hardwired approach, that assumes known users' behavior, blank-slate models learn the users' behavior from the queries to the service provider. We leverage this blank-slate approach to develop a new family of LPPMs, that we call Profile Estimation-Based LPPMs. Using real data, we empirically show that our proposal outperforms optimal state-of-the-art mechanisms designed on sporadic hardwired models. On non-sporadic location privacy scenarios, our method is only better if the usage of the location privacy service is not continuous. It is our hope that eliminating the need to bootstrap the mechanisms with training data and ensuring that the mechanisms are lightweight and easy to compute help fostering the integration of location privacy protections in deployed systems