We investigate the problem of unconstrained combinatorial multi-armed bandits with fullbandit feedback and stochastic rewards for submodular maximization. Previous works investigate the same problem assuming a submodular and monotone reward function. In this work, we study a more general problem, i.e., when the reward function is not necessarily monotone, and the submodularity is assumed only in expectation. We propose Randomized Greedy Learning (RGL) algorithm and theoretically prove that it achieves a 1 2 -regret upper bound of O˜(nT 2 3 ) for horizon T and number of arms n. We also show in experiments that RGL empirically outperforms other full-bandit variants in submodular and non-submodular settings.
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On the Unreasonable Effectiveness of the Greedy Algorithm: Greedy Adapts to Sharpness
It is well known that the standard greedy algorithm guarantees a worst-case approximation factor of 1 − 1/e when maximizing a monotone submodular function under a cardinality constraint. However, empirical studies show that its performance is substantially better in practice. This raises a natural question of explaining this improved performance of the greedy algorithm. In this work, we define sharpness for submodular functions as a candidate explanation for this phenomenon. We show that the greedy algorithm provably performs better as the sharpness of the submodular function increases. This improvement ties in closely with the faster convergence rates of first order methods for sharp functions in convex optimization.
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- Award ID(s):
- 1717947
- NSF-PAR ID:
- 10178884
- Date Published:
- Journal Name:
- Thirty-seventh International Conference on Machine Learning
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We investigate the problem of unconstrained combinatorial multi-armed bandits with full-bandit feedback and stochastic rewards for submodular maximization. Previous works investigate the same problem assuming a submodular and monotone reward function. In this work, we study a more general problem, i.e., when the reward function is not necessarily monotone, and the submodularity is assumed only in expectation. We propose Randomized Greedy Learning (RGL) algorithm and theoretically prove that it achieves a $\frac{1}{2}$-regret upper bound of $\Tilde{\mathcal{O}}(n T^{\frac{2}{3}})$ for horizon $T$ and number of arms $n$. We also show in experiments that RGL empirically outperforms other full-bandit variants in submodular and non-submodular settings.more » « less
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Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function$$w: N \rightarrow \mathbb {R}_+$$ r monotone submodular functions$$f_1,f_2,\ldots ,f_r$$ N and requirements$$k_1,k_2,\ldots ,k_r$$ $$S \subseteq N$$ $$f_i(S) \ge k_i$$ $$1 \le i \le r$$ Multi-Submod-Cover and it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ Multi-Submod-Cover generalizes the well-known Submodular Set Cover problem (Submod-SC ), and it can also be easily reduced toSubmod-SC . A simple greedy algorithm gives an$$O(\log (kr))$$ $$k = \sum _i k_i$$ Multi-Submod-Cover that covers each constraint to within a factor of$$(1-1/e-\varepsilon )$$ $$O(\frac{1}{\epsilon }\log r)$$ $$f_i$$ Partial-SC ), covering integer programs (CIPs ) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems. -
Mobile edge computing pushes computationally-intensive services closer to the user to provide reduced delay due to physical proximity. This has led many to consider deploying deep learning models on the edge – commonly known as edge intelligence (EI). EI services can have many model implementations that provide different QoS. For instance, one model can perform inference faster than another (thus reducing latency) while achieving less accuracy when evaluated. In this paper, we study joint service placement and model scheduling of EI services with the goal to maximize Quality-of-Servcice (QoS) for end users where EI services have multiple implementations to serve user requests, each with varying costs and QoS benefits. We cast the problem as an integer linear program and prove that it is NP-hard. We then prove the objective is equivalent to maximizing a monotone increasing, submodular set function and thus can be solved greedily while maintaining a (1 – 1/e)-approximation guarantee. We then propose two greedy algorithms: one that theoretically guarantees this approximation and another that empirically matches its performance with greater efficiency. Finally, we thoroughly evaluate the proposed algorithm for making placement and scheduling decisions in both synthetic and real-world scenarios against the optimal solution and some baselines. In the real-world case, we consider real machine learning models using the ImageNet 2012 data-set for requests. Our numerical experiments empirically show that our more efficient greedy algorithm is able to approximate the optimal solution with a 0.904 approximation on average, while the next closest baseline achieves a 0.607 approximation on average.more » « less
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We study the problem of approximating maximum Nash social welfare (NSW) when allocating
m indivisible items amongn asymmetric agents with submodular valuations. TheNSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents’ valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with a factor independent ofm was known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations before this work.In this article, we extend our understanding of the
NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high-valued items are done separately by un-matching certain items and re-matching them by different processes in both algorithms. We show that these approaches achieve approximation factors ofO (n ) andO (n logn ) for additive and submodular cases, independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1 ).Furthermore, we show that the
NSW problem under submodular valuations is strictly harder than all currently known settings with an\(\frac{\mathrm{e}}{\mathrm{e}-1}\) \(\frac{\mathrm{e}}{\mathrm{e}-1}\)
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Accepted Manuscript1.0
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