# Colloquium Details

## IBM Research NYU/Columbia Theory Day

Speaker:

Location: Warren Weaver Hall 109

Date: December 5, 2008, 9:30 a.m.

Host: Yevgeniy Dodis

Synopsis:

### Approximate Kernel Clustering

Prof. Assaf Naor
New York University

In the kernel clustering problem we are given a large n times n positive semi-definite matrix A=(a_{ij}) with \sum_{i,j=1}^n a_{ij}=0 and a small k times k positive semi-definite matrix B=(b_{ij}). The goal is to find a partition S_1,...,S_k of {1,...n} which maximizes the quantity \sum_{i,j=1}^k (\sum_{(p,q)\in S_i\times S_j} a_{pq}) b_{ij}.

We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song, Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp approximation threshold for this problem assuming the Unique Games Conjecture (UGC). In particular, when B is the 3 times 3 identity matrix the UGC hardness threshold of this problem is exactly 16*pi/27. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when B is the k times k identity matrix is (8*pi/9)*(1-1/k) for every k >= 3.

Joint work with Subhash Khot.

### Approximation Algorithms for some Geometric Coverage and Connectivity Problems

Prof. Joe Mitchell
Stony Brook University

We examine a variety of geometric optimization problems. We describe some recent progress in improved approximations algorithms for several of these problems, including the TSP with neighborhoods, relay placement in sensor networks, and visibility/sensor coverage. We highlight many open problems.

### A Truthful Mechanism for Offline Ad Slot Scheduling

Dr. Jonathan Feldman

Targeted advertising on web pages is an increasingly important advertising medium, attracting large numbers of advertisers and users. One popular method for assigning ads to various slots on a page (for example the slots along side web search results) is via a real-time auction among advertisers who have placed standing bids for clicks. These "position auctions" have been studied from a game-theoretic point of view and are now well understood as a single-shot game. However, since web pages are visited repeatedly over time, there are global phenomena at play such as supply estimates and budget constraints that are not modeled by this analysis.

We formulate the "Offline Ad Slot Scheduling" problem, where advertisers are scheduled beforehand to slots on a web page for portions of the day. Advertisers specify a daily budget constraint, as well as a per-click bid, and may not be assigned to more than one slot on the page during any given time period. We give a scheduling algorithm and a pricing method that amount to a truthful mechanism under the utility model where bidders try to maximize their clicks, subject to their personal constraints. In addition, we show that the revenue-maximizing schedule is not truthful, but has a Nash equilibrium whose outcome is identical to our mechanism. Our mechanism employs a descending-price auction that maintains a solution to a machine scheduling problem whose job lengths depend on the price.

Joint work with Muthu Muthukrishnan, Eddie Nikolova and Martin Pal.

### Regret Minimization for Global Cost Functions with Applications to Job Scheduling

Prof. Yishay Mansour

We consider standard regret minimization setting where at each time step the decision maker has to choose a distribution over $k$ alternatives, and then observes the loss of each alternative. The setting is very similar to the classical online job scheduling setting with three major differences:

(1) Information model: in the regret minimization setting losses are only observed after the actions (assigning the job to a machine) is performed and not observed before the action selection, as assumed in the classical online job scheduling setting,

(2) The comparison class: in regret minimization the comparison class is the best static algorithm (i.e., distribution over alternatives) and not the optimal offline solution.

(3) Performance measure: In regret minimization we measure the additive difference to the optimal solution in the comparison class, in contrast to the ratio used in online job scheduling setting.

Motivated by load balancing and job scheduling, we consider a global cost function (over the losses incur by each alternative/machine), rather than simply a summation of the instantaneous losses as done traditionally in regret minimization. Such global cost functions include the makespan (the maximum over the alternatives/machines) and the $L_d$ norm (over the alternatives/machines).

The major contribution of this work is to design a novel regret minimization algorithm based on calibration that guarantees a vanishing average regret, where the regret is measured with respect to the best static decision maker, who selects the same distribution over alternatives at every time step. Our results hold for a wide class of global cost functions. which include the makespan and the $L_d$ norms, for $d>1$. In contrast, we show that for concave global cost functions, such as $L_d$ norms for $d<1$, the worst-case average regret does not vanish.

In addition to the general calibration based algorithm, we provide simple and efficient algorithms for special interesting cases.

This is a joint work with Eyal Even-Dar and Shie Mannor.

Notes:

Refreshments will be offered starting 15 minutes prior to the scheduled start of the talk.

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