Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions

Given a non-convex twice differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.Comment: 2 figure

Informational Substitutes

We propose definitions of substitutes and complements for pieces of information ("signals") in the context of a decision or optimization problem, with game-theoretic and algorithmic applications. In a game-theoretic context, substitutes capture diminishing marginal value of information to a rational decision maker. We use the definitions to address the question of how and when information is aggregated in prediction markets. Substitutes characterize "best-possible" equilibria with immediate information aggregation, while complements characterize "worst-possible", delayed aggregation. Game-theoretic applications also include settings such as crowdsourcing contests and Q\&A forums. In an algorithmic context, where substitutes capture diminishing marginal improvement of information to an optimization problem, substitutes imply efficient approximation algorithms for a very general class of (adaptive) information acquisition problems. In tandem with these broad applications, we examine the structure and design of informational substitutes and complements. They have equivalent, intuitive definitions from disparate perspectives: submodularity, geometry, and information theory. We also consider the design of scoring rules or optimization problems so as to encourage substitutability or complementarity, with positive and negative results. Taken as a whole, the results give some evidence that, in parallel with substitutable items, informational substitutes play a natural conceptual and formal role in game theory and algorithms.Comment: Full version of FOCS 2016 paper. Single-column, 61 pages (48 main text, 13 references and appendix

Behavioral Mechanism Design: Optimal Contests for Simple Agents

Incentives are more likely to elicit desired outcomes when they are designed based on accurate models of agents' strategic behavior. A growing literature, however, suggests that people do not quite behave like standard economic agents in a variety of environments, both online and offline. What consequences might such differences have for the optimal design of mechanisms in these environments? In this paper, we explore this question in the context of optimal contest design for simple agents---agents who strategically reason about whether or not to participate in a system, but not about the input they provide to it. Specifically, consider a contest where $n$ potential contestants with types $(q_i,c_i)$ each choose between participating and producing a submission of quality $q_i$ at cost $c_i$, versus not participating at all, to maximize their utilities. How should a principal distribute a total prize $V$ amongst the $n$ ranks to maximize some increasing function of the qualities of elicited submissions in a contest with such simple agents? We first solve the optimal contest design problem for settings with homogenous participation costs $c_i = c$. Here, the optimal contest is always a simple contest, awarding equal prizes to the top $j^*$ contestants for a suitable choice of $j^*$. (In comparable models with strategic effort choices, the optimal contest is either a winner-take-all contest or awards possibly unequal prizes, depending on the curvature of agents' effort cost functions.) We next address the general case with heterogeneous costs where agents' types are inherently two-dimensional, significantly complicating equilibrium analysis. Our main result here is that the winner-take-all contest is a 3-approximation of the optimal contest when the principal's objective is to maximize the quality of the best elicited contribution.Comment: This is the full version of a paper in the ACM Conference on Economics and Computation (ACM-EC), 201

Crowdsourcing atop blockchains

Traditional crowdsourcing systems, such as Amazon\u27s Mechanical Turk (MTurk), though once acquiring great economic successes, have to fully rely on third-party platforms to serve between the requesters and the workers for basic utilities. These third-parties have to be fully trusted to assist payments, resolve disputes, protect data privacy, manage user authentications, maintain service online, etc. Nevertheless, tremendous real-world incidents indicate how elusive it is to completely trust these platforms in reality, and the reduction of such over-reliance becomes desirable. In contrast to the arguably vulnerable centralized approaches, a public blockchain is a distributed and transparent global consensus computer that is highly robust. The blockchain is usually managed and replicated by a large-scale peer-to-peer network collectively, thus being much more robust to be fully trusted for correctness and availability. It, therefore, becomes enticing to build novel crowdsourcing applications atop blockchains to reduce the over-trust on third-party platforms. However, this new fascinating technology also brings about new challenges, which were never that severe in the conventional centralized setting. The most serious issue is that the blockchain is usually maintained in the public Internet environment with a broader attack surface open to anyone. This not only causes serious privacy and security issues, but also allows the adversaries to exploit the attack surface to hamper more basic utilities. Worse still, most existing blockchains support only light on-chain computations, and the smart contract executed atop the decentralized consensus computer must be simple, which incurs serious feasibility problems. In reality, the privacy/security issue and the feasibility problem even restrain each other and create serious tensions to hinder the broader adoption of blockchain. The dissertation goes through the non-trivial challenges to realize secure yet still practical decentralization (for urgent crowdsourcing use-cases), and lay down the foundation for this line of research. In sum, it makes the next major contributions. First, it identifies the needed security requirements in decentralized knowledge crowdsourcing (e.g., data privacy), and initiates the research of private decentralized crowdsourcing. In particular, the confidentiality of solicited data is indispensable to prevent free-riders from pirating the others\u27 submissions, thus ensuring the quality of solicited knowledge. To this end, a generic private decentralized crowdsourcing framework is dedicatedly designed, analyzed, and implemented. Furthermore, this dissertation leverages concretely efficient cryptographic design to reduce the cost of the above generic framework. It focuses on decentralizing the special use-case of Amazon MTurk, and conducts multiple specific-purpose optimizations to remove needless generality to squeeze performance. The implementation atop Ethereum demonstrates a handling cost even lower than MTurk. In addition, it focuses on decentralized crowdsourcing of computing power for specific machine learning tasks. It lets a requester place deposits in the blockchain to recruit some workers for a designated (randomized) programs. If and only if these workers contribute their resources to compute correctly, they would earn well-deserved payments. For these goals, a simple yet still useful incentive mechanism is developed atop the blockchain to deter rational workers from cheating. Finally, the research initiates the first systematic study on crowdsourcing blockchains\u27 full nodes to assist superlight clients (e.g., mobile phones and IoT devices) to read the blockchain\u27s records. This dissertation presents a novel generic solution through the powerful lens of game-theoretic treatments, which solves the long-standing open problem of designing generic superlight clients for all blockchains

Incentivizing Exploration with Heterogeneous Value of Money

Recently, Frazier et al. proposed a natural model for crowdsourced exploration of different a priori unknown options: a principal is interested in the long-term welfare of a population of agents who arrive one by one in a multi-armed bandit setting. However, each agent is myopic, so in order to incentivize him to explore options with better long-term prospects, the principal must offer the agent money. Frazier et al. showed that a simple class of policies called time-expanded are optimal in the worst case, and characterized their budget-reward tradeoff. The previous work assumed that all agents are equally and uniformly susceptible to financial incentives. In reality, agents may have different utility for money. We therefore extend the model of Frazier et al. to allow agents that have heterogeneous and non-linear utilities for money. The principal is informed of the agent's tradeoff via a signal that could be more or less informative. Our main result is to show that a convex program can be used to derive a signal-dependent time-expanded policy which achieves the best possible Lagrangian reward in the worst case. The worst-case guarantee is matched by so-called "Diamonds in the Rough" instances; the proof that the guarantees match is based on showing that two different convex programs have the same optimal solution for these specific instances. These results also extend to the budgeted case as in Frazier et al. We also show that the optimal policy is monotone with respect to information, i.e., the approximation ratio of the optimal policy improves as the signals become more informative.Comment: WINE 201

Incentive Mechanisms for Participatory Sensing: Survey and Research Challenges

Participatory sensing is a powerful paradigm which takes advantage of smartphones to collect and analyze data beyond the scale of what was previously possible. Given that participatory sensing systems rely completely on the users' willingness to submit up-to-date and accurate information, it is paramount to effectively incentivize users' active and reliable participation. In this paper, we survey existing literature on incentive mechanisms for participatory sensing systems. In particular, we present a taxonomy of existing incentive mechanisms for participatory sensing systems, which are subsequently discussed in depth by comparing and contrasting different approaches. Finally, we discuss an agenda of open research challenges in incentivizing users in participatory sensing.Comment: Updated version, 4/25/201

Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design

We study a type of reverse (procurement) auction problems in the presence of budget constraints. The general algorithmic problem is to purchase a set of resources, which come at a cost, so as not to exceed a given budget and at the same time maximize a given valuation function. This framework captures the budgeted version of several well known optimization problems, and when the resources are owned by strategic agents the goal is to design truthful and budget feasible mechanisms, i.e. elicit the true cost of the resources and ensure the payments of the mechanism do not exceed the budget. Budget feasibility introduces more challenges in mechanism design, and we study instantiations of this problem for certain classes of submodular and XOS valuation functions. We first obtain mechanisms with an improved approximation ratio for weighted coverage valuations, a special class of submodular functions that has already attracted attention in previous works. We then provide a general scheme for designing randomized and deterministic polynomial time mechanisms for a class of XOS problems. This class contains problems whose feasible set forms an independence system (a more general structure than matroids), and some representative problems include, among others, finding maximum weighted matchings, maximum weighted matroid members, and maximum weighted 3D-matchings. For most of these problems, only randomized mechanisms with very high approximation ratios were known prior to our results

Truthful Multi-unit Procurements with Budgets

We study procurement games where each seller supplies multiple units of his item, with a cost per unit known only to him. The buyer can purchase any number of units from each seller, values different combinations of the items differently, and has a budget for his total payment. For a special class of procurement games, the {\em bounded knapsack} problem, we show that no universally truthful budget-feasible mechanism can approximate the optimal value of the buyer within $\ln n$, where $n$ is the total number of units of all items available. We then construct a polynomial-time mechanism that gives a $4(1+\ln n)$-approximation for procurement games with {\em concave additive valuations}, which include bounded knapsack as a special case. Our mechanism is thus optimal up to a constant factor. Moreover, for the bounded knapsack problem, given the well-known FPTAS, our results imply there is a provable gap between the optimization domain and the mechanism design domain. Finally, for procurement games with {\em sub-additive valuations}, we construct a universally truthful budget-feasible mechanism that gives an $O(\frac{\log^2 n}{\log \log n})$-approximation in polynomial time with a demand oracle.Comment: To appear at WINE 201