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Register a new userOnline Risk-Averse Submodular Maximization
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Added by
Tasuku Soma
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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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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 study the recently introduced idea of worst-case sensitivity for monotone submodular maximization with cardinality constraint $k$, which captures the degree to which the output argument changes on deletion of an element in the input. We find that for large classes of algorithms that non-trivial sensitivity of $o(k)$ is not possible, even with bounded curvature, and that these results also hold in the distributed framework. However, we also show that in the regime $k = Omega(n)$ that we can obtain $O(1)$ sensitivity for sufficiently low curvature.
In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first $frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $frac{1}{2}$ is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular functions is also coordinate wise concave along all coordinates, we provide a different $frac{1}{2}$-approximation algorithm that runs in quasilinear time. Both of these results improve upon prior work [Bian et al, 2017, Soma and Yoshida, 2017]. Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.
In this work, we give a new parallel algorithm for the problem of maximizing a non-monotone diminishing returns submodular function subject to a cardinality constraint. For any desired accuracy $epsilon$, our algorithm achieves a $1/e - epsilon$ approximation using $O(log{n} log(1/epsilon) / epsilon^3)$ parallel rounds of function evaluations. The approximation guarantee nearly matches the best approximation guarantee known for the problem in the sequential setting and the number of parallel rounds is nearly-optimal for any constant $epsilon$. Previous algorithms achieve worse approximation guarantees using $Omega(log^2{n})$ parallel rounds. Our experimental evaluation suggests that our algorithm obtains solutions whose objective value nearly matches the value obtained by the state of the art sequential algorithms, and it outperforms previous parallel algorithms in number of parallel rounds, iterations, and solution quality.
As the scales of data sets expand rapidly in some application scenarios, increasing efforts have been made to develop fast submodular maximization algorithms. This paper presents a currently the most efficient algorithm for maximizing general non-negative submodular objective functions subject to $k$-extendible system constraints. Combining the sampling process and the decreasing threshold strategy, our algorithm Sample Decreasing Threshold Greedy Algorithm (SDTGA) obtains an expected approximation guarantee of ($p-epsilon$) for monotone submodular functions and of ($p(1-p)-epsilon$) for non-monotone cases with expected computational complexity of only $O(frac{pn}{epsilon}lnfrac{r}{epsilon})$, where $r$ is the largest size of the feasible solutions, $0<p leq frac{1}{1+k}$ is the sampling probability and $0< epsilon < p$. If we fix the sampling probability $p$ as $frac{1}{1+k}$, we get the best approximation ratios for both monotone and non-monotone submodular functions which are $(frac{1}{1+k}-epsilon)$ and $(frac{k}{(1+k)^2}-epsilon)$ respectively. While the parameter $epsilon$ exists for the trade-off between the approximation ratio and the time complexity. Therefore, our algorithm can handle larger scale of submodular maximization problems than existing algorithms.
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