Onkar Bhardwaj

Friendship and Stable Matching

We study stable matching problems in networks where players are embedded in a social context, and may incorporate friendship relations or altruism into their decisions. Each player is a node in a social network and strives to form a good match with a neighboring player. We consider the existence, computation, and inefficiency of stable matchings from which no pair of players wants to deviate. When the benefits from a match are the same for both players, we show that incorporating the well-being of other players into their matching decisions significantly decreases the price of stability, while the price of anarchy remains unaffected. Furthermore, a good stable matching achieving the price of stability bound always exists and can be reached in polynomial time. We extend these results to more general matching rewards, when players matched to each other may receive different utilities from the match. For this more general case, we show that incorporating social context (i.e., “caring about your friends”) can make an even larger difference, and greatly reduce the price of anarchy. We show a variety of existence results, and present upper and lower bounds on the prices of anarchy and stability for various matching utility structures.

Practical efficiency of asynchronous stochastic gradient descent

Stochastic gradient descent (SGD) and its distributed variants are essential to leverage modern computing resources for large-scale machine learning tasks. ASGD [1] is one of the most popular asynchronous distributed variant of SGD. Recent mathematical analyses have shown that with certain assumptions on the learning task (and ignoring communication cost), ASGD exhibits linear speed-up asymptotically. However, as practically observed, ASGD does not lead linear speed-up as we increase the number of learners. Motivated by this, we investigate finite time convergence properties of ASGD. We observe that the learning rate used by mathematical analyses to guarantee linear speed-up can be very small (and practically sub-optimal with respect to convergence speed) as opposed to practically chosen learning rates (for quick convergence) which exhibit sub-linear speed-up. We show that such an observation can in fact be supported by mathematical analysis, i.e., in the finite time regime, better convergence rate guarantees can be proven for ASGD with small number of learners, thus indicating lack of linear speed up as we increase the number of learners. Thus we conclude that even with ignoring communication cost, there is an inherent inefficiency in ASGD with respect to increasing the number of learners.

SymSig: A low latency interconnection topology for HPC clusters

This paper presents the underlying theory and the performance of a cluster using a new 2-hop network topology. This topology is constructed using a symmetric equation and Singer Difference Sets and is called SymSig. The degree of connections at each node with SymSig is about half compared to previous methods using Singer Difference Sets. A comparison with a cluster of Clos topology shows significant advantages. The worst case congestion in SymSig topology for unicast permutation is 2, where as in Clos it is proportional to the radix of the building block switches used. The number of switches required is smaller by about 25%, the size of the cluster is larger by about 15% and the worst bandwidth is better by about 50% for SymSig. These advantages are retained for peta and exascale systems. Its performance on a set of collectives like exchange-all, shift-all, broadcast-all and all-to-all send/receive shows improvements ranging from 39% to 83%. Its performance on a molecular dynamics application GROMMACS shows improvement of upto 33%. This network is particularly suitable for applications that require global all to all communications. The low latency of this network makes it scaleable and an attractive alternative for building peta and exascale systems.

Stable Matching with Network Externalities

We study the stable roommates problem in networks where players are embedded in a social context and may incorporate positive externalities into their decisions. Each player is a node in a social network and strives to form a good match with a neighboring player. We consider the existence, computation, and inefficiency of stable matchings from which no pair of players wants to deviate. We characterize prices of anarchy and stability, which capture the ratio of the total profit in the optimum matching over the total profit of the worst and best stable matching, respectively. When the benefit from a match (which we model by associating a reward with each edge) is the same for both players, we show that externalities can significantly improve the price of stability, while the price of anarchy remains unaffected. Furthermore, a good stable matching achieving the bound on the price of stability can be obtained in polynomial time. We extend these results to more general matching rewards, when players matched to each other may receive different benefits from the match. For this more general case, we show that network externalities (i.e., “caring about your friends”) can make an even larger difference and greatly reduce the price of anarchy. We show a variety of existence results and present upper and lower bounds on the prices of anarchy and stability for various structures of matching benefits. All our results on stable matchings immediately extend to the more general case of fractional stable matchings.

Friend of My Friend: Network Formation with Two-Hop Benefit

How and why people form ties is a critical issue for understanding the fabric of social networks. In contrast to most existing work, we are interested in settings where agents are neither so myopic as to consider only the benefit they derive from their immediate neighbors, nor do they consider the effects on the entire network when forming connections. Instead, we consider games on networks where a node tries to maximize its utility taking into account the benefit it gets from the nodes it is directly connected to (called direct benefit), as well as the benefit it gets from the nodes it is acquainted with via a two-hop connection (called two-hop benefit). We call such games Two-Hop Games. The decision to consider only two hops stems from the observation that human agents rarely consider “contacts of a contact of a contact” (3-hop contacts) or further while forming their relationships. We consider several versions of Two-Hop games which are extensions of well-studied games. While the addition of two-hop benefit changes the properties of these games significantly, we prove that in many important cases good equilibrium solutions still exist, and bound the change in the price of anarchy due to two-hop benefit both theoretically and in simulation.

Approximating optimal social choice under metric preferences

We examine the quality of social choice mechanisms using a utilitarian view, in which all of the agents have costs for each of the possible alternatives. While these underlying costs determine what the optimal alternative is, they may be unknown to the social choice mechanism; instead the mechanism must decide on a good alternative based only on the ordinal preferences of the agents which are induced by the underlying costs. Due to its limited information, such a social choice mechanism cannot simply select the alternative that minimizes the total social cost (or minimizes some other objective function). Thus, we seek to bound the distortion: the worst-case ratio between the social cost of the alternative selected and the optimal alternative. Distortion measures how good a mechanism is at approximating the alternative with minimum social cost, while using only ordinal preference information. The underlying costs can be arbitrary, implicit, and unknown; our only assumption is that the agent costs form a metric space, which is a natural assumption in many settings. We quantify the distortion of many well-known social choice mechanisms. We show that for both total social cost and median agent cost, many positional scoring rules have large distortion, while on the other hand Copeland and similar mechanisms perform optimally or near-optimally, always obtaining a distortion of at most 5. We also give lower bounds on the distortion that could be obtained by any deterministic social choice mechanism, and extend our results on median agent cost to more general objective functions.

Coalitionally stable pricing schemes for inter-domain forwarding

In this work, we model and analyze the problem of stable and efficient pricing for inter-domain traffic routing in the future Internet. We consider a general network topology with multiple sources and sinks of traffic, organized into separate domains managed by Internet Service Providers (ISPs) solely interested in maximizing their own profit. In this framework, we prove that there exists a pricing scheme that attains network-wide efficiency and is yet coalitionally stable, where the coalitions correspond to the ISPs that are acting in self-interest. This implies that this pricing scheme not only maximizes the overall utility of the resulting traffic flows, but is also such that ISPs cannot expect to improve their profit through deviation from it, even if multiple ISPs deviate at the same time. Through simulations on scale-free preferential attachment network topology models as well as actual inter-domain topologies obtained from the CAIDA database, we evaluate the convergence of best-response based simple price updates, and show that they quickly attain near-optimal network utility in these network topologies.