Results 1  10
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23
Complexity of Combinatorial Market Makers ∗
"... We analyze the computational complexity of market maker pricing algorithms for combinatorial prediction markets. We focus on Hanson’s popular logarithmic market scoring rule market maker (LMSR). Our goal is to implicitly maintain correct LMSR prices across an exponentially large outcome space. We ex ..."
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Cited by 30 (17 self)
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We analyze the computational complexity of market maker pricing algorithms for combinatorial prediction markets. We focus on Hanson’s popular logarithmic market scoring rule market maker (LMSR). Our goal is to implicitly maintain correct LMSR prices across an exponentially large outcome space. We examine both permutation combinatorics, where outcomes are permutations of objects, and Boolean combinatorics, where outcomes are combinations of binary events. We look at three restrictive languages that limit what traders can bet on. Even with severely limited languages, we find that LMSR pricing is #Phard, even when the same language admits polynomialtime matching without the market maker. We then propose an approximation technique for pricing permutation markets based on a recent algorithm for online permutation learning. The connections we draw between LMSR pricing and the vast literature on online learning with expert advice may be of independent interest.
A new understanding of prediction markets via noregret learning
 In ACM EC
, 2010
"... We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and noregret learning. We first show that any cost function based prediction market can be interpreted as an algorithm for the commonly studied problem of learning from ..."
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Cited by 30 (10 self)
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We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and noregret learning. We first show that any cost function based prediction market can be interpreted as an algorithm for the commonly studied problem of learning from expert advice by equating the set of outcomes on which bets are placed in the market with the set of experts in the learning setting, and equating trades made in the market with losses observed by the learning algorithm. If the loss of the market organizer is bounded, this bound can be used to derive an O ( √ T) regret bound for the corresponding learning algorithm. We then show that the class of markets with convex cost functions exactly corresponds to the class of Follow the Regularized Leader learning algorithms, with the choice of a cost function in the market corresponding to the choice of a regularizer in the learning problem. Finally, we show an equivalence between market scoring rules and prediction markets with convex cost functions. This implies both that any market scoring rule can be implemented as a cost function based market maker, and that market scoring rules can be interpreted naturally as Follow the Regularized Leader algorithms. These connections provide new insight into how it is that commonly studied markets, such as the Logarithmic Market Scoring Rule, can aggregate opinions into accurate estimates of the likelihood of future events.
Fourier Theoretic Probabilistic Inference over Permutations
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2009
"... Permutations are ubiquitous in many realworld problems, such as voting, ranking, and data association. Representing uncertainty over permutations is challenging, since there are n! possibilities, and typical compact and factorized probability distribution representations, such as graphical models, ..."
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Cited by 19 (8 self)
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Permutations are ubiquitous in many realworld problems, such as voting, ranking, and data association. Representing uncertainty over permutations is challenging, since there are n! possibilities, and typical compact and factorized probability distribution representations, such as graphical models, cannot capture the mutual exclusivity constraints associated with permutations. In this paper, we use the “lowfrequency” terms of a Fourier decomposition to represent distributions over permutations compactly. We present Kronecker conditioning, a novel approach for maintaining and updating these distributions directly in the Fourier domain, allowing for polynomial time bandlimited approximations. Low order Fourierbased approximations, however, may lead to functions that do not correspond to valid distributions. To address this problem, we present a quadratic program defined directly in the Fourier domain for projecting the approximation onto a relaxation of the polytope of legal marginal distributions. We demonstrate the effectiveness of our approach on a real camerabased multiperson tracking scenario.
Combinatorial Bandits
"... We study sequential prediction problems in which, at each time instance, the forecaster chooses a binary vector from a certain fixed set S ⊆ {0, 1} d and suffers a loss that is the sum of the losses of those vector components that equal to one. The goal of the forecaster is to achieve that, in the l ..."
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Cited by 16 (4 self)
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We study sequential prediction problems in which, at each time instance, the forecaster chooses a binary vector from a certain fixed set S ⊆ {0, 1} d and suffers a loss that is the sum of the losses of those vector components that equal to one. The goal of the forecaster is to achieve that, in the long run, the accumulated loss is not much larger than that of the best possible vector in the class. We consider the “bandit ” setting in which the forecaster has only access to the losses of the chosen vectors. We introduce a new general forecaster achieving a regret bound that, for a variety of concrete choices of S, is of order √ nd ln S  where n is the time horizon. This is not improvable in general and is better than previously known bounds. We also point out that computationally efficient implementations for various interesting choices of S exist. 1
Hedging structured concepts
 In COLT
, 2010
"... We develop an online algorithm called Component Hedge for learning structured concept classes when the loss of a structured concept sums over its components. Example classes include paths through a graph (composed of edges) and partial permutations (composed of assignments). The algorithm maintains ..."
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Cited by 13 (3 self)
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We develop an online algorithm called Component Hedge for learning structured concept classes when the loss of a structured concept sums over its components. Example classes include paths through a graph (composed of edges) and partial permutations (composed of assignments). The algorithm maintains a parameter vector with one nonnegative weight per component, which always lies in the convex hull of the structured concept class. The algorithm predicts by decomposing the current parameter vector into a convex combination of concepts and choosing one of those concepts at random. The parameters are updated by first performing a multiplicative update and then projecting back into the convex hull. We show that Component Hedge has optimal regret bounds for a large variety of structured concept classes. 1
Riffled Independence for Ranked Data
"... Representing distributions over permutations can be a daunting task due to the fact that the number of permutations of n objects scales factorially in n. One recent way that has been used to reduce storage complexity has been to exploit probabilistic independence, but as we argue, full independence ..."
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Cited by 9 (3 self)
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Representing distributions over permutations can be a daunting task due to the fact that the number of permutations of n objects scales factorially in n. One recent way that has been used to reduce storage complexity has been to exploit probabilistic independence, but as we argue, full independence assumptions impose strong sparsity constraints on distributions and are unsuitable for modeling rankings. We identify a novel class of independence structures, called riffled independence, which encompasses a more expressive family of distributions while retaining many of the properties necessary for performing efficient inference and reducing sample complexity. In riffled independence, one draws two permutations independently, then performs the riffle shuffle, common in card games, to combine the two permutations to form a single permutation. In ranking, riffled independence corresponds to ranking disjoint sets of objects independently, then interleaving those rankings. We provide a formal introduction and present algorithms for using riffled independence within Fouriertheoretic frameworks which have been explored by a number of recent papers. 1
Randomized Online PCA Algorithms with Regret Bounds that are Logarithmic in the Dimension
, 2007
"... We design an online algorithm for Principal Component Analysis. In each trial the current instance is centered and projected into a probabilistically chosen low dimensional subspace. The regret of our online algorithm, i.e. the total expected quadratic compression loss of the online algorithm minus ..."
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Cited by 8 (1 self)
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We design an online algorithm for Principal Component Analysis. In each trial the current instance is centered and projected into a probabilistically chosen low dimensional subspace. The regret of our online algorithm, i.e. the total expected quadratic compression loss of the online algorithm minus the total quadratic compression loss of the batch algorithm, is bounded by a term whose dependence on the dimension of the instances is only logarithmic. We first develop our methodology in the expert setting of online learning by giving an algorithm for learning as well as the best subset of experts of a certain size. This algorithm is then lifted to the matrix setting where the subsets of experts correspond to subspaces. The algorithm represents the uncertainty over the best subspace as a density matrix whose eigenvalues are bounded. The running time is O(n²) per trial, where n is the dimension of the instances.
Supplement: NonStochastic Bandit Slate Problems
"... Recall our special variant of Hedge: we are allowed to uses only distributions p(t) from some fixed convex subset P of the simplex of all distributions. The goal then is to minimize regret relative to an arbitrary distribution p ∈ P. Such a version of Hedge is given in Figure 1, and a statement of i ..."
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Cited by 8 (0 self)
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Recall our special variant of Hedge: we are allowed to uses only distributions p(t) from some fixed convex subset P of the simplex of all distributions. The goal then is to minimize regret relative to an arbitrary distribution p ∈ P. Such a version of Hedge is given in Figure 1, and a statement of its performance below. This algorithm is implicit in the work of [4, 6]. Algorithm MW(P) Initialization: An arbitrary probability distribution p(1) ∈ P on the experts, and some η> 0. For t = 1, 2,..., T: 1. Choose distribution p(t) over experts, and observe the cost vector ℓ(t). 2. Compute the probability vector ˆp(t + 1) using the following multiplicative update rule: for every expert i, ˆpi(t + 1) = pi(t) exp(−ηℓi(t))/Z(t) (1) where Z(t) = ∑ i pi(t) exp(−ηℓi(t)) is the normalization factor. 3. Set p(t + 1) to be the projection of ˆp(t + 1) on the set P using the RE as a distance function, i.e. p(t + 1) = arg minp∈P RE(p ‖ ˆp(t + 1)). Figure 1: The Multiplicative Weights Algorithm with Restricted Distributions Theorem 1.1. Assume that η> 0 is chosen so that for all t and i, ηℓi(t) ≥ −1. Then algorithm MW(P) generates distributions p(1),..., p(T) ∈ P, such that for any p ∈ P, T∑ T∑ ℓ(t) · p(t) − ℓ(t) · p ≤ η (ℓ(t)) 2 · p(t) + t=1 Here, (ℓ(t)) 2 is the vector that is the coordinatewise square of ℓ(t). t=1 RE(p ‖ p(1)) η Proof. We use the relative entropy between p and p(t), RE(p ‖ p(t)): = ∑ i pi ln(pi/pi(t)) as a “potential ” function. We have RE(p ‖ ˆp t+1) − RE(p ‖ p(t)) = ∑ pi(t) pi ln
Efficient market making via convex optimization, and a connection to online learning
 ACM Transactions on Economics and Computation. To Appear
, 2012
"... We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. The framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any ..."
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Cited by 5 (2 self)
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We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. The framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any market satisfying a set of intuitive conditions must price securities via a convex cost function, which is constructed via conjugate duality. Rather than deal with an exponentially large or infinite outcome space directly, our framework only requires optimization over a convex hull. By reducing the problem of automated market making to convex optimization, where many efficient algorithms exist, we arrive at a range of new polynomialtime pricing mechanisms for various problems. We demonstrate the advantages of this framework with the design of some particular markets. We also show that by relaxing the convex hull we can gain computational tractability without compromising the market institution’s bounded budget. Although our framework was designed with the goal of deriving efficient automated market makers for markets with very large outcome spaces, this framework also provides new insights into the relationship between market design and machine learning, and into the complete market setting. Using our framework, we illustrate the mathematical parallels between cost function based markets and online learning and establish a correspondence between cost function based markets and market scoring rules for complete markets. 1