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1. Optimal Convergence and Adaptation for Utility Optimal Opportunistic Scheduling

   Neely, Michael J (January 2017, Proc. Allerton Conference on Communication, Control, and Computing)

   This paper considers the fundamental convergence time for opportunistic scheduling over time-varying channels. The channel state probabilities are unknown and algorithms must perform some type of estimation and learning while they make decisions to optimize network utility. Existing schemes can achieve a utility within ε of optimality, for any desired ε > 0, with convergence and adaptation times of O(1/ε^2). This paper shows that if the utility function is concave and smooth, then O(log(1/ε)/ε) convergence time is possible via an existing stochastic variation on the Frank-Wolfe algorithm, called the RUN algorithm. Next, a converse result is proven to show it is impossible for any algorithm to have convergence time better than O(1/ε), provided the algorithm has no a- priori knowledge of channel state probabilities. Hence, RUN is within a logarithmic factor of convergence time optimality. However, RUN has a vanishing stepsize and hence has an infinite adaptation time. Using stochastic Frank-Wolfe with a fixed step- size yields improved O(1/ε^2) adaptation time, but convergence time increases to O(1/ε^2), similar to existing drift-plus-penalty based algorithms. This raises important open questions regarding optimal adaptation. 

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2. A Converse Result on Convergence Time for Opportunistic Wireless Scheduling

   Neely, Michael J. (January 2020, IEEE INFOCOM---International Conference on Computer Communications)

   This paper proves an impossibility result for stochastic network utility maximization for multi-user wireless systems, including multi-access and broadcast systems. Every time slot an access point observes the current channel states and opportunistically selects a vector of transmission rates. Channel state vectors are assumed to be independent and identically distributed with an unknown probability distribution. The goal is to learn to make decisions over time that maximize a concave utility function of the running time average transmission rate of each user. Recently it was shown that a stochastic Frank-Wolfe algorithm converges to utility-optimality with an error of O(log(T)/T ), where T is the time the algorithm has been running. An existing O(1/T ) converse is known. The current paper improves the converse to O(log(T)/T), which matches the known achievability result. The proof uses a reduction from the opportunistic scheduling problem to a Bernoulli estimation problem. Along the way, it refines a result on Bernoulli estimation. 

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3. Communication-Efficient Projection-Free Algorithm for Nonconvex Constrained Learning Models

   Wenhan Xian, Feihu Huang (February 2021, 35th AAAI Conference on Artificial Intelligence (AAAI 2021))

   null (Ed.)

   Recently decentralized optimization attracts much attention in machine learning because it is more communication-efficient than the centralized fashion. Quantization is a promising method to reduce the communication cost via cutting down the budget of each single communication using the gradient compression. To further improve the communication efficiency, more recently, some quantized decentralized algorithms have been studied. However, the quantized decentralized algorithm for nonconvex constrained machine learning problems is still limited. Frank-Wolfe (a.k.a., conditional gradient or projection-free) method is very efficient to solve many constrained optimization tasks, such as low-rank or sparsity-constrained models training. In this paper, to fill the gap of decentralized quantized constrained optimization, we propose a novel communication-efficient Decentralized Quantized Stochastic Frank-Wolfe (DQSFW) algorithm for non-convex constrained learning models. We first design a new counterexample to show that the vanilla decentralized quantized stochastic Frank-Wolfe algorithm usually diverges. Thus, we propose DQSFW algorithm with the gradient tracking technique to guarantee the method will converge to the stationary point of non-convex optimization safely. In our theoretical analysis, we prove that to achieve the stationary point our DQSFW algorithm achieves the same gradient complexity as the standard stochastic Frank-Wolfe and centralized Frank-Wolfe algorithms, but has much less communication cost. Experiments on matrix completion and model compression applications demonstrate the efficiency of our new algorithm. 

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4. Solving Non-smooth Constrained Programs with Lower Complexity than O(1/ε): A Primal-Dual Homotopy Smoothing Approach

   Wei, Xiaohan; Yu, Hao; Ling, Qing; Neely, Michael J. (December 2018, 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.)

   We propose a new primal-dual homotopy smoothing algorithm for a linearly constrained convex program, where neither the primal nor the dual function has to be smooth or strongly convex. The best known iteration complexity solving such a non-smooth problem is O(ε−1). In this paper, we show that by leveraging a local error bound condition on the dual function, the proposed algorithm can achieve a better primal convergence time of O 􏰕ε−2/(2+β) log2(ε−1)􏰖, where β ∈ (0, 1] is a local error bound parameter. As an example application of the general algorithm, we show that the distributed geometric median problem, which can be formulated as a constrained convex program, has its dual function non-smooth but satisfying the aforementioned local error bound condition with β = 1/2, therefore enjoying a convergence time of O 􏰕ε−4/5 log2(ε−1)􏰖. This result improves upon the O(ε−1) convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm. 

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5. Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing

   https://doi.org/10.4230/LIPICS.ESA.2023.56  

   Harb, Elfarouk; Quanrud, Kent; Chekuri, Chandra (January 2023, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands)

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J.; Herman, Grzegorz (Ed.)

   Boob et al. [Boob et al., 2020] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Qaunrud and Torres [Chandra Chekuri et al., 2022] extended the algorithm to supermodular density problems (of which DSG is a special case) and proved that the resulting algorithm Super-Greedy++ (and hence also Greedy++) converges. In this paper we revisit the convergence proof and provide a different perspective. This is done via a connection to Fujishige’s quadratic program for finding a lexicographically optimal base in a (contra) polymatroid [Satoru Fujishige, 1980], and a noisy version of the Frank-Wolfe method from convex optimization [Frank and Wolfe, 1956; Jaggi, 2013]. This yields a simpler convergence proof, and also shows a stronger property that Super-Greedy++ converges to the optimal dense decomposition vector, answering a question raised in Harb et al. [Harb et al., 2022]. A second contribution of the paper is to understand Thorup’s work on ideal tree packing and greedy tree packing [Thorup, 2007; Thorup, 2008] via the Frank-Wolfe algorithm applied to find a lexicographically optimum base in the graphic matroid. This yields a simpler and transparent proof. The two results appear disparate but are unified via Fujishige’s result and convex optimization. 

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