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 #P-hard, even when the same language admits polynomial-time 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.

We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and no-regret 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.

Permutations are ubiquitous in many real-world 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 “low-frequency” 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 Fourier-based 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 camera-based multi-person tracking scenario.

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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

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 non-negative 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

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.

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 coordinate-wise 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

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 Fourier-theoretic frameworks which have been explored by a number of recent papers. 1

We analyze a sequential game between a Gambler and a Casino. The Gambler allocates bets from a limited budget over a fixed menu of gambling events that are offered at equal time intervals, and the Casino chooses a binary loss outcome for each of the events. We derive the optimal min-max strategies for both participants. We then prove that the minimum cumulative loss of the Gambler, assuming optimal play by the Casino, is exactly a well-known combinatorial quantity: the expected number of draws needed to complete a multiple set of “cards ” in the generalized Coupon Collector’s Problem. We show that this quantity and the optimal strategy of the Gambler can be efficiently estimated from a simple random walk. 1