arXiv subjects

Tim Roughgarden

Publications and source records attributed to Tim Roughgarden.

At least 55 records · Page 3Linked to original sources

arXiv2020

Distribution-Free Models of Social Networks

The structure of large-scale social networks has predominantly been articulated using generative models, a form of average-case analysis. This chapter surveys recent proposals of more robust models of such networks. These models posit deterministic and empirically supported combinatorial structure rather than a specific probability distribution. We discuss the formal definitions of these models and how they relate to empirical observations in social networks, as well as the known structural and algorithmic results for the corresponding graph classes.

cs.DS↗

arXiv2020

Resource Augmentation

This chapter introduces resource augmentation, in which the performance of an algorithm is compared to the best-possible solution that is handicapped by less resources. We consider three case studies: online paging, with cache size as the resource; selfish routing, with capacity as the resource; and scheduling, with processor speed as the resource. Resource augmentation bounds also imply "loosely competitive" bounds, which show that an algorithm's performance is near-optimal for most resource levels.

cs.DS↗

arXiv2020

Distributional Analysis

In distributional or average-case analysis, the goal is to design an algorithm with good-on-average performance with respect to a specific probability distribution. Distributional analysis can be useful for the study of general-purpose algorithms on "non-pathological" inputs, and for the design of specialized algorithms in applications in which there is detailed understanding of the relevant input distribution. For some problems, however, pure distributional analysis encourages "overfitting" an algorithmic solution to a particular distributional assumption and a more robust analysis framework is called for. This chapter presents numerous examples of the pros and cons of distributional analysis, highlighting some of its greatest hits while also setting the stage for the hybrids of worst- and average-case analysis studied in later chapters.

cs.DS↗

arXiv2020

Beyond the Worst-Case Analysis of Algorithms (Introduction)

One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the "best" for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its worst performance on any input of a given size, implicitly advocating for the algorithm with the best-possible worst-case performance. Strong worst-case guarantees are the holy grail of algorithm design, providing an application-agnostic certification of an algorithm's robustly good performance. However, for many fundamental problems and performance measures, such guarantees are impossible and a more nuanced analysis approach is called for. This chapter surveys several alternatives to worst-case analysis that are discussed in detail later in the book.

cs.DS↗

arXiv2020

Mathematical Foundations for Social Computing

Social computing encompasses the mechanisms through which people interact with computational systems: crowdsourcing systems, ranking and recommendation systems, online prediction markets, citizen science projects, and collaboratively edited wikis, to name a few. These systems share the common feature that humans are active participants, making choices that determine the input to, and therefore the output of, the system. The output of these systems can be viewed as a joint computation between machine and human, and can be richer than what either could produce alone. The term social computing is often used as a synonym for several related areas, such as "human computation" and subsets of "collective intelligence"; we use it in its broadest sense to encompass all of these things. Social computing is blossoming into a rich research area of its own, with contributions from diverse disciplines including computer science, economics, and other social sciences. Yet a broad mathematical foundation for social computing is yet to be established, with a plethora of under-explored opportunities for mathematical research to impact social computing. As in other fields, there is great potential for mathematical work to influence and shape the future of social computing. However, we are far from having the systematic and principled understanding of the advantages, limitations, and potentials of social computing required to match the impact on applications that has occurred in other fields. In June 2015, we brought together roughly 25 experts in related fields to discuss the promise and challenges of establishing mathematical foundations for social computing. This document captures several of the key ideas discussed.

cs.CY↗

arXiv2020

Resource Pools and the CAP Theorem

Blockchain protocols differ in fundamental ways, including the mechanics of selecting users to produce blocks (e.g., proof-of-work vs. proof-of-stake) and the method to establish consensus (e.g., longest chain rules vs. BFT-inspired protocols). These fundamental differences have hindered "apples-to-apples" comparisons between different categories of blockchain protocols and, in turn, the development of theory to formally discuss their relative merits. This paper presents a parsimonious abstraction sufficient for capturing and comparing properties of many well-known permissionless blockchain protocols, simultaneously capturing essential properties of both proof-of-work and proof-of-stake protocols, and of both longest-chain-type and BFT-type protocols. Our framework blackboxes the precise mechanics of the user selection process, allowing us to isolate the properties of the selection process which are significant for protocol design. We illustrate our framework's utility with two results. First, we prove an analog of the CAP theorem from distributed computing for our framework in a partially synchronous setting. This theorem shows that a fundamental dichotomy holds between protocols (such as Bitcoin) that are adaptive, in the sense that they can function given unpredictable levels of participation, and protocols (such as Algorand) that have certain finality properties. Second, we formalize the idea that proof-of-work (PoW) protocols and non-PoW protocols can be distinguished by the forms of permission that users are given to carry out updates to the state.

cs.DC↗

arXiv2020

Smoothed Analysis of Online and Differentially Private Learning

Practical and pervasive needs for robustness and privacy in algorithms have inspired the design of online adversarial and differentially private learning algorithms. The primary quantity that characterizes learnability in these settings is the Littlestone dimension of the class of hypotheses [Ben-David et al., 2009, Alon et al., 2019]. This characterization is often interpreted as an impossibility result because classes such as linear thresholds and neural networks have infinite Littlestone dimension. In this paper, we apply the framework of smoothed analysis [Spielman and Teng, 2004], in which adversarially chosen inputs are perturbed slightly by nature. We show that fundamentally stronger regret and error guarantees are possible with smoothed adversaries than with worst-case adversaries. In particular, we obtain regret and privacy error bounds that depend only on the VC dimension and the bracketing number of a hypothesis class, and on the magnitudes of the perturbations.

cs.LG↗

arXiv2020

The Complexity of Contracts

We initiate the study of computing (near-)optimal contracts in succinctly representable principal-agent settings. Here optimality means maximizing the principal's expected payoff over all incentive-compatible contracts---known in economics as "second-best" solutions. We also study a natural relaxation to approximately incentive-compatible contracts. We focus on principal-agent settings with succinctly described (and exponentially large) outcome spaces. We show that the computational complexity of computing a near-optimal contract depends fundamentally on the number of agent actions. For settings with a constant number of actions, we present a fully polynomial-time approximation scheme (FPTAS) for the separation oracle of the dual of the problem of minimizing the principal's payment to the agent, and use this subroutine to efficiently compute a delta-incentive-compatible (delta-IC) contract whose expected payoff matches or surpasses that of the optimal IC contract. With an arbitrary number of actions, we prove that the problem is hard to approximate within any constant c. This inapproximability result holds even for delta-IC contracts where delta is a sufficiently rapidly-decaying function of c. On the positive side, we show that simple linear delta-IC contracts with constant delta are sufficient to achieve a constant-factor approximation of the "first-best" (full-welfare-extracting) solution, and that such a contract can be computed in polynomial time.

cs.DS↗

arXiv2020

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.

cs.CC↗

arXiv2019

An Axiomatic Approach to Block Rewards

Proof-of-work blockchains reward each miner for one completed block by an amount that is, in expectation, proportional to the number of hashes the miner contributed to the mining of the block. Is this proportional allocation rule optimal? And in what sense? And what other rules are possible? In particular, what are the desirable properties that any "good" allocation rule should satisfy? To answer these questions, we embark on an axiomatic theory of incentives in proof-of-work blockchains at the time scale of a single block. We consider desirable properties of allocation rules including: symmetry; budget balance (weak or strong); sybil-proofness; and various grades of collusion-proofness. We show that Bitcoin's proportional allocation rule is the unique allocation rule satisfying a certain system of properties, but this does not hold for slightly weaker sets of properties, or when the miners are not risk-neutral. We also point out that a rich class of allocation rules can be approximately implemented in a proof-of-work blockchain.

cs.GT↗

arXiv2019

On the Computational Power of Online Gradient Descent

We prove that the evolution of weight vectors in online gradient descent can encode arbitrary polynomial-space computations, even in very simple learning settings. Our results imply that, under weak complexity-theoretic assumptions, it is impossible to reason efficiently about the fine-grained behavior of online gradient descent.

cs.LG↗

arXiv2019

The idemetric property: when most distances are (almost) the same

We introduce the \emph{idemetric} property, which formalises the idea that most nodes in a graph have similar distances between them, and which turns out to be quite standard amongst small-world network models. Modulo reasonable sparsity assumptions, we are then able to show that a strong form of idemetricity is actually equivalent to a very weak expander condition (PUMP). This provides a direct way of providing short proofs that small-world network models such as the Watts-Strogatz model are strongly idemetric (for a wide range of parameters), and also provides further evidence that being idemetric is a common property. We then consider how satisfaction of the idemetric property is relevant to algorithm design. For idemetric graphs we observe, for example, that a single breadth-first search provides a solution to the all-pairs shortest paths problem, so long as one is prepared to accept paths which are of stretch close to 2 with high probability. Since we are able to show that Kleinberg's model is idemetric, these results contrast nicely with the well known negative results of Kleinberg concerning efficient decentralised algorithms for finding short paths: for precisely the same model as Kleinberg's negative results hold, we are able to show that very efficient (and decentralised) algorithms exist if one allows for reasonable preprocessing. For deterministic distributed routing algorithms we are also able to obtain results proving that less routing information is required for idemetric graphs than the worst case in order to achieve stretch less than 3 with high probability: while $Ω(n^2)$ routing information is required in the worst case for stretch strictly less than 3 on almost all pairs, for idemetric graphs the total routing information required is $O(nlog(n))$.

cs.SI↗

arXiv2018

Communication Complexity of Discrete Fair Division

We initiate the study of the communication complexity of fair division with indivisible goods. We focus on some of the most well-studied fairness notions (envy-freeness, proportionality, and approximations thereof) and valuation classes (submodular, subadditive and unrestricted). Within these parameters, our results completely resolve whether the communication complexity of computing a fair allocation (or determining that none exist) is polynomial or exponential (in the number of goods), for every combination of fairness notion, valuation class, and number of players, for both deterministic and randomized protocols.

cs.GT↗

arXiv2018

Beyond Worst-Case Analysis

In the worst-case analysis of algorithms, the overall performance of an algorithm is summarized by its worst performance on any input. This approach has countless success stories, but there are also important computational problems --- like linear programming, clustering, online caching, and neural network training --- where the worst-case analysis framework does not provide any helpful advice on how to solve the problem. This article covers a number of modeling methods for going beyond worst-case analysis and articulating which inputs are the most relevant.

cs.DS↗

arXiv2018

An Optimal Algorithm for Online Unconstrained Submodular Maximization

We consider a basic problem at the interface of two fundamental fields: submodular optimization and online learning. In the online unconstrained submodular maximization (online USM) problem, there is a universe $[n]=\{1,2,...,n\}$ and a sequence of $T$ nonnegative (not necessarily monotone) submodular functions arrive over time. The goal is to design a computationally efficient online algorithm, which chooses a subset of $[n]$ at each time step as a function only of the past, such that the accumulated value of the chosen subsets is as close as possible to the maximum total value of a fixed subset in hindsight. Our main result is a polynomial-time no-$1/2$-regret algorithm for this problem, meaning that for every sequence of nonnegative submodular functions, the algorithm's expected total value is at least $1/2$ times that of the best subset in hindsight, up to an error term sublinear in $T$. The factor of $1/2$ cannot be improved upon by any polynomial-time online algorithm when the submodular functions are presented as value oracles. Previous work on the offline problem implies that picking a subset uniformly at random in each time step achieves zero $1/4$-regret. A byproduct of our techniques is an explicit subroutine for the two-experts problem that has an unusually strong regret guarantee: the total value of its choices is comparable to twice the total value of either expert on rounds it did not pick that expert. This subroutine may be of independent interest.

cs.LG↗

arXiv2018

Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization

In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first $\frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $\frac{1}{2}$ is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular functions is also coordinate wise concave along all coordinates, we provide a different $\frac{1}{2}$-approximation algorithm that runs in quasilinear time. Both of these results improve upon prior work [Bian et al, 2017, Soma and Yoshida, 2017]. Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.

cs.DS↗

arXiv2018

Finding Cliques in Social Networks: A New Distribution-Free Model

We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure---the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a "$c$-closed" graph, where for every pair of vertices $u,v$ with at least $c$ common neighbors, $u$ and $v$ are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to $c$ on $c$-closed graphs. Our results carry over to "weakly $c$-closed graphs", which only require a vertex deletion ordering that avoids pairs of non-adjacent vertices with $c$ common neighbors. Numerical experiments show that well-studied social networks tend to be weakly $c$-closed for modest values of $c$.

math.CO↗

arXiv2018

Pricing Multi-Unit Markets

We study the power and limitations of posted prices in multi-unit markets, where agents arrive sequentially in an arbitrary order. We prove upper and lower bounds on the largest fraction of the optimal social welfare that can be guaranteed with posted prices, under a range of assumptions about the designer's information and agents' valuations. Our results provide insights about the relative power of uniform and non-uniform prices, the relative difficulty of different valuation classes, and the implications of different informational assumptions. Among other results, we prove constant-factor guarantees for agents with (symmetric) subadditive valuations, even in an incomplete-information setting and with uniform prices.

cs.GT↗