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1. Differentially Private Monotone Submodular Maximization Under Matroid and Knapsack Constraints

   Sadeghi, Omid ; Fazel, Maryam ( April 2021 , Proceedings of Machine Learning Research)

   Banerjee, Arindam ; Fukumizu, Kenji (Ed.)

   Numerous tasks in machine learning and artificial intelligence have been modeled as submodular maximization problems. These problems usually involve sensitive data about individuals, and in addition to maximizing the utility, privacy concerns should be considered. In this paper, we study the general framework of non-negative monotone submodular maximization subject to matroid or knapsack constraints in both offline and online settings. For the offline setting, we propose a differentially private $(1-\frac{\kappa}{e})$-approximation algorithm, where $\kappa\in[0,1]$ is the total curvature of the submodular set function, which improves upon prior works in terms of approximation guarantee and query complexity under the same privacy budget. In the online setting, we propose the first differentially private algorithm, and we specify the conditions under which the regret bound scales as $Ø(\sqrt{T})$, i.e., privacy could be ensured while maintaining the same regret bound as the optimal regret guarantee in the non-private setting. 

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2. Privacy-Preserving Dynamic Personalized Pricing with Demand Learning

   https://doi.org/10.1287/mnsc.2021.4129  

   Chen, Xi ; Simchi-Levi, David ; Wang, Yining ( October 2021 , Management Science)

   null (Ed.)

   The prevalence of e-commerce has made customers’ detailed personal information readily accessible to retailers, and this information has been widely used in pricing decisions. When using personalized information, the question of how to protect the privacy of such information becomes a critical issue in practice. In this paper, we consider a dynamic pricing problem over T time periods with an unknown demand function of posted price and personalized information. At each time t, the retailer observes an arriving customer’s personal information and offers a price. The customer then makes the purchase decision, which will be utilized by the retailer to learn the underlying demand function. There is potentially a serious privacy concern during this process: a third-party agent might infer the personalized information and purchase decisions from price changes in the pricing system. Using the fundamental framework of differential privacy from computer science, we develop a privacy-preserving dynamic pricing policy, which tries to maximize the retailer revenue while avoiding information leakage of individual customer’s information and purchasing decisions. To this end, we first introduce a notion of anticipating [Formula: see text]-differential privacy that is tailored to the dynamic pricing problem. Our policy achieves both the privacy guarantee and the performance guarantee in terms of regret. Roughly speaking, for d-dimensional personalized information, our algorithm achieves the expected regret at the order of [Formula: see text] when the customers’ information is adversarially chosen. For stochastic personalized information, the regret bound can be further improved to [Formula: see text]. This paper was accepted by J. George Shanthikumar, big data analytics. 

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3. Online Continuous DR-Submodular Maximization with Long-Term Budget Constraints

   Sadeghi, Omid ; Fazel, Maryam ( August 2020 , Proceedings of Machine Learning Research)

   In this paper, we study a class of online optimization problems with long-term budget constraints where the objective functions are not necessarily concave (nor convex), but they instead satisfy the Diminishing Returns (DR) property. In this online setting, a sequence of monotone DR-submodular objective functions and linear budget functions arrive over time and assuming a limited total budget, the goal is to take actions at each time, before observing the utility and budget function arriving at that round, to achieve sub-linear regret bound while the total budget violation is sub-linear as well. Prior work has shown that achieving sub-linear regret and total budget violation simultaneously is impossible if the utility and budget functions are chosen adversarially. Therefore, we modify the notion of regret by comparing the agent against the best fixed decision in hindsight which satisfies the budget constraint proportionally over any window of length W. We propose the Online Saddle Point Hybrid Gradient (OSPHG) algorithm to solve this class of online problems. For W = T, we recover the aforementioned impossibility result. However, if W is sublinear in T, we show that it is possible to obtain sub-linear bounds for both the regret and the total budget violation. 

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4. Sharper Model-free Reinforcement Learning for Average-reward Markov Decision Processes

   Zhang, Zihan ; Xie, Qiaomin ( July 2023 , Conference on Learning Theory)

   We study model-free reinforcement learning (RL) algorithms for infinite-horizon average-reward Markov decision process (MDP), which is more appropriate for applications that involve continuing operations not divided into episodes. In contrast to episodic/discounted MDPs, theoretical understanding of model-free RL algorithms is relatively inadequate for the average-reward setting. In this paper, we consider both the online setting and the setting with access to a simulator. We develop computationally efficient model-free algorithms that achieve sharper guarantees on regret/sample complexity compared with existing results. In the online setting, we design an algorithm, UCB-AVG, based on an optimistic variant of variance-reduced Q-learning. We show that UCB-AVG achieves a regret bound $\widetilde{O}(S^5A^2sp(h^*)\sqrt{T})$ after $T$ steps, where $S\times A$ is the size of state-action space, and $sp(h^*)$ the span of the optimal bias function. Our result provides the first computationally efficient model-free algorithm that achieves the optimal dependence in $T$ (up to log factors) for weakly communicating MDPs, which is necessary for low regret. In contrast, prior results either are suboptimal in $T$ or require strong assumptions of ergodicity or uniformly mixing of MDPs. In the simulator setting, we adapt the idea of UCB-AVG to develop a model-free algorithm that finds an $\epsilon$-optimal policy with sample complexity $\widetilde{O}(SAsp^2(h^*)\epsilon^{-2} + S^2Asp(h^*)\epsilon^{-1}).$ This sample complexity is near-optimal for weakly communicating MDPs, in view of the minimax lower bound $\Omega(SAsp(^*)\epsilon^{-2})$. Existing work mainly focuses on ergodic MDPs and the results typically depend on $t_{mix},$ the worst-case mixing time induced by a policy. We remark that the diameter $D$ and mixing time $t_{mix}$ are both lower bounded by $sp(h^*)$, and $t_{mix}$ can be arbitrarily large for certain MDPs. On the technical side, our approach integrates two key ideas: learning an $\gamma$-discounted MDP as an approximation, and leveraging reference-advantage decomposition for variance in optimistic Q-learning. As recognized in prior work, a naive approximation by discounted MDPs results in suboptimal guarantees. A distinguishing feature of our method is maintaining estimates of value-difference between state pairs to provide a sharper bound on the variance of reference advantage. We also crucially use a careful choice of the discounted factor $\gamma$ to balance approximation error due to discounting and the statistical learning error, and we are able to maintain a good-quality reference value function with $O(SA)$ space complexity. 

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5. Algorithms for covering multiple submodular constraints and applications

   https://doi.org/10.1007/s10878-022-00874-x  

   Chekuri, Chandra ; Inamdar, Tanmay ; Quanrud, Kent ; Varadarajan, Kasturi ; Zhang, Zhao ( June 2022 , Journal of Combinatorial Optimization)

   We consider the problem of covering multiple submodular constraints. Given a finite ground setN, a weight function$$w: N \rightarrow \mathbb {R}_+$$$w:N\to R_{+}$,rmonotone submodular functions$$f_1,f_2,\ldots ,f_r$$$f_1,f_2,\dots ,f_r$overNand requirements$$k_1,k_2,\ldots ,k_r$$$k_1,k_2,\dots ,k_r$the goal is to find a minimum weight subset$$S \subseteq N$$$S\subseteq N$such that$$f_i(S) \ge k_i$$$f_i\left(S\right)\ge k_i$for$$1 \le i \le r$$$1\le i\le r$. We refer to this problem asMulti-Submod-Coverand it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$$r=1$Multi-Submod-Covergeneralizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced toSubmod-SC. A simple greedy algorithm gives an$$O(\log (kr))$$$O(log(kr\left)\right)$approximation where$$k = \sum _i k_i$$$k={\sum }_i{k}_i$and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm forMulti-Submod-Coverthat covers each constraint to within a factor of$$(1-1/e-\varepsilon )$$$(1-1/e-\epsilon )$while incurring an approximation of$$O(\frac{1}{\epsilon }\log r)$$$O(\frac{1}{ϵ}logr)$in the cost. Second, we consider the special case when each$$f_i$$$f_i$is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (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.

    

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