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Register a new userOnline Submodular Maximization under a Matroid Constraint with Application to Learning Assignments
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Daniel Golovin
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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Which ads should we display in sponsored search in order to maximize our revenue? How should we dynamically rank information sources to maximize the value of the ranking? These applications exhibit strong diminishing returns: Redundancy decreases the marginal utility of each ad or information source. We show that these and other problems can be formalized as repeatedly selecting an assignment of items to positions to maximize a sequence of monotone submodular functions that arrive one by one. We present an efficient algorithm for this general problem and analyze it in the no-regret model. Our algorithm possesses strong theoretical guarantees, such as a performance ratio that converges to the optimal constant of 1 - 1/e. We empirically evaluate our algorithm on two real-world online optimization problems on the web: ad allocation with submodular utilities, and dynamically ranking blogs to detect information cascades. Finally, we present a second algorithm that handles the more general case in which the feasible sets are given by a matroid constraint, while still maintaining a 1 - 1/e asymptotic performance ratio.
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We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a emph{general matroid constraint} with a $1 - 1/e - epsilon$ approximation that achieves a fast running time provided we have a fast data structure for maintaining a maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the emph{first algorithms} for these classes of matroids that achieve a nearly-optimal, $1 - 1/e - epsilon$ approximation, using a nearly-linear number of function evaluations and arithmetic operations.
In this work we consider the problem of online submodular maximization under a cardinality constraint with differential privacy (DP). A stream of $T$ submodular functions over a common finite ground set $U$ arrives online, and at each time-step the decision maker must choose at most $k$ elements of $U$ before observing the function. The decision maker obtains a payoff equal to the function evaluated on the chosen set, and aims to learn a sequence of sets that achieves low expected regret. In the full-information setting, we develop an $(varepsilon,delta)$-DP algorithm with expected $(1-1/e)$-regret bound of $mathcal{O}left( frac{k^2log |U|sqrt{T log k/delta}}{varepsilon} right)$. This algorithm contains $k$ ordered experts that learn the best marginal increments for each item over the whole time horizon while maintaining privacy of the functions. In the bandit setting, we provide an $(varepsilon,delta+ O(e^{-T^{1/3}}))$-DP algorithm with expected $(1-1/e)$-regret bound of $mathcal{O}left( frac{sqrt{log k/delta}}{varepsilon} (k (|U| log |U|)^{1/3})^2 T^{2/3} right)$. Our algorithms contains $k$ ordered experts that learn the best marginal item to select given the items chosen her predecessors, while maintaining privacy of the functions. One challenge for privacy in this setting is that the payoff and feedback of expert $i$ depends on the actions taken by her $i-1$ predecessors. This particular type of information leakage is not covered by post-processing, and new analysis is required. Our techniques for maintaining privacy with feedforward may be of independent interest.
We propose a cumulative oversampling (CO) method for online learning. Our key idea is to sample parameter estimations from the updated belief space once in each round (similar to Thompson Sampling), and utilize the cumulative samples up to the current round to construct optimistic parameter estimations that asymptotically concentrate around the true parameters as tighter upper confidence bounds compared to the ones constructed with standard UCB methods. We apply CO to a novel budgeted variant of the Influence Maximization (IM) semi-bandits with linear generalization of edge weights, whose offline problem is NP-hard. Combining CO with the oracle we design for the offline problem, our online learning algorithm simultaneously tackles budget allocation, parameter learning, and reward maximization. We show that for IM semi-bandits, our CO-based algorithm achieves a scaled regret comparable to that of the UCB-based algorithms in theory, and performs on par with Thompson Sampling in numerical experiments.
We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-epsilon$ approximation for monotone functions and a $1/e-epsilon$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(log^2{n}/epsilon^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-epsilon$ approximation using $O(log(n/epsilon) log(1/epsilon) log(n+m)/ epsilon^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-epsilon$ approximation in $O(log(n/epsilon)log(m)/epsilon^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.
We present a polynomial-time online algorithm for maximizing the conditional value at risk (CVaR) of a monotone stochastic submodular function. Given $T$ i.i.d. samples from an underlying distribution arriving online, our algorithm produces a sequence of solutions that converges to a ($1-1/e$)-approximate solution with a convergence rate of $O(T^{-1/4})$ for monotone continuous DR-submodular functions. Compared with previous offline algorithms, which require $Omega(T)$ space, our online algorithm only requires $O(sqrt{T})$ space. We extend our online algorithm to portfolio optimization for monotone submodular set functions under a matroid constraint. Experiments conducted on real-world datasets demonstrate that our algorithm can rapidly achieve CVaRs that are comparable to those obtained by existing offline algorithms.
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