We give a unified way of testing and learning finite Markov chains from a single Markovian trajectory, using the idea of $k$-cover time introduced here. The $k$-cover time is the expected length of a random walk to cover every state at least $k$ times. This generalizes the notion of cover time in the literature. The error metric in the testing and learning problems is the infinity matrix norm between the transition matrices, as considered by Wolfer and Kontorovich. Specifically, we show that if we can learn or test discrete distributions using $k$ samples, then we can learn or test Markov chains using a number of samples equal to the $k$-cover time of the chain, up to constant factors. We then derive asymptotic bounds on the $k$-cover time in terms of the number of states, minimum stationary probability and the cover time of the chain. Our bounds are tight for reversible Markov chains and almost tight (up to logarithmic factors) for irreducible ones. Our results on $k$-cover time yield sample complexity bounds for a wider range of learning and testing tasks (including learning, uniformity testing, identity testing, closeness testing and their tolerant versions) over Markov chains, and can be applied to a broader family of Markov chains (irreducible and reversible ones) than previous results which only applies to ergodic ones.

We generalize a convexity result due to Wyner and Ziv to channels with binary inputs and arbitrary outputs. This results in a convex reformulation of some non-convex optimization problems that arise naturally in multi-user information theory.

Riemannian gradient descent (RGD) is a simple, popular and efficient algorithm for leading eigenvector computation [AMS08]. However, the existing analysis of RGD for eigenproblem is still not tight, which is O(log(n/epsilon)/Delta^2) due to [Xu et al., 2018]. In this paper, we show that RGD in fact converges at rate O(log(n/epsilon)/Delta), and give instances to shows the tightness of our result. This improves the best prior analysis by a quadratic factor. Besides, we also give tight convergence analysis of a deterministic variant of Oja's rule due to [Oja, 1982]. We show that it also enjoys fast convergence rate of O(log(n/epsilon)/Delta). Previous papers only gave asymptotic characterizations [Oja, 1982; Oja, 1989; Yi et al., 2005]. Our tools for proving convergence results include an innovative reduction and chaining technique, and a noisy fixed point iteration argument. Besides, we also give empirical justifications of our convergence rates over synthetic and real data.

This work proposes a novel momentum technique, the Amortized Nesterov’s Momentum, for stochastic convex optimization. The proposed method can be regarded as a smooth transition between Nesterov’s method and mirror descent. By tuning only a single parameter, users can trade Nesterov’s acceleration for robustness, that is, the variance control of the stochastic noise. Motivated by the recent success of using momentum in deep learning, we conducted extensive experiments to evaluate this new momentum in deep learning tasks. The results suggest that it can serve as a favorable alternative for Nesterov’s momentum.

Variance reduction is a simple and effective technique that accelerates convex (or non-convex) stochastic optimization. Among existing variance reduction methods, SVRG and SAGA adopt unbiased gradient estimators and are the most popular variance reduction methods in recent years. Although various accelerated variants of SVRG (e.g., Katyusha and Acc-Prox-SVRG) have been proposed, the direct acceleration of SAGA still remains unknown. In this paper, we propose a directly accelerated variant of SAGA using a novel Sampled Negative Momentum (SSNM), which achieves the best known oracle complexity for strongly convex problems (with known strong convexity parameter). Consequently, our work fills the void of directly accelerated SAGA.

Crowdsourced live streaming (CLS), such as Twitch.tv and Inke.tv, has emerged as an important multimedia application in recent years. Video delivery in such CLS service involves two steps: 1) video uploading-video streaming (i.e., a live channel) generated from a broadcaster is uploaded to the server, which we call the “first mile” network and 2) video distribution-the video streaming is then delivered to viewers in the channel. Today's CLS services usually use conventional content delivery network solutions to address the video distribution problem, while little attention has been paid to improve the video uploading quality. Our measurement study shows that the first mile network causes 17% viewer rebuffers, and some viewers quit the channel once encountering rebuffer. In this paper, we propose a content harvest network (CHN) architecture to address the uploading problem in the CLS service. Specifically, the CHN architecture employs edge devices in the network as relays to receive the streaming uploaded by broadcasters and then forward to the central servers. On one hand, we need to reduce the latency since it is live streaming; on the other hand, we must provide sustainable upload bandwidth. It is challenging to achieve both at the same time, especially in such high dynamic system as CLS. In order to provide global optimal and real-time assignment, we propose a hybrid solution, i.e., centralized and distributed assignment. Specifically, we formulate the centralized relay assignment problem as an optimization problem to achieve both low latency and sustainable bandwidth. To cope with the frequent channel establishments, we use a multi-armed bandit method to characterize the time-variant network condition. Experiment results on a large-scale trace provided by Inke.tv show that our solution can reduce the overall viewer cost by 40% compared to state-of-the-art solutions. The viewers' rebuffer can also be reduced by 50%.

Edge caching is a key technology to accelerate the content delivery and to mitigate the network congestion. As a real-world example, Microsoft has been providing the edge caching service with a usage-based pricing mechanism since 2009. However, the challenge of designing a better pricing mechanism lies in balancing the finer grain size of the mechanism and the complexity of business operations it required. Our work proposes a novel location dependent pricing mechanism for the edge caching market which both improves the Internet service provider's (ISP) revenue and the social welfare. First, we introduce the location dependent pricing mechanism (LDP) that offers location-specific price, considering the popularity differences between the edge nodes(e.g. base stations, WiFi access points) at different locations. We capture the interactions between the content providers (CPs) and the ISP as a two-stage Stackelberg game and derive the CP's best responses and the ISP's optimal pricing strategy. Then we provide a partial location dependent pricing scheme (PLDP) which limits the number of price tiers to provide a balance between the optimality and economic viability. Finally, through extensive numerical simulation, we find that the LDP (or PLDP) improves the revenue of the ISP by 31%↑50% (23%↑%35) and also improves the social welfare by 46% (%32) on average, compared to the uniform pricing scheme.

Cellular data sponsoring (CDS) is a traditional data sponsor scheme widely used in cellular video delivery networks, where content providers (CPs) bear the cellular data downloading cost for mobile video users (MUs), so as to attract more MUs and achieve higher revenue (e.g., via more attached advertisements). Edge caching sponsoring (ECS) is a novel data sponsor scheme recently introduced in the emerging 5G network, where CPs cache popular video contents on the edge network in advance and deliver them to local MUs directly. Thus, it can not only achieve the benefits of CDS (i.e., attracting more MUs and achieving higher revenue), but also reduce the congestion of backhaul network. In this work, we will perform a competitive analysis of CDS and ECS for mobile video streaming. Specifically, we consider a mobile video delivery network with two CPs who adopt CDS and ECS, respectively. MUs can choose one or neither of these two sponsor schemes (from the corresponding CPs) for his video content requests. We formulate the interaction of CPs and MUs as a two-stage Stackelberg game, where CPs act as leaders determining the efforts of their adopted sponsor schemes in the first stage, and MUs act as followers choosing the best sponsor schemes for their content requests in the second stage. We analyze the sub-game perfect equilibrium systematically for both cooperative and competitive scenarios (depending on whether two CPs cooperate or compete with each other). Numerical results show that in the competitive scenario, the joint sponsor of ECS and CDS can increase the total MU payoff by 36% ~ 140%, comparing with that with only one sponsor scheme. Moreover, the CPs can benefit more from ECS than from CDS when the revenue is higher.

Data sponsoring is a widely-used incentive method in today's cellular networks, where video content providers (CPs) cover part or all of the cellular data cost for mobile users so as to attract more video users and increase data traffic. In the forthcoming 5G cellular networks, edge caching is emerging as a promising technique to deliver videos with lower cost and higher quality. The key idea is to cache video contents on edge networks (e.g., femtocells and WiFi access points) in advance and deliver the cached contents to local video users directly (without involving cellular data cost for users). In this work, we aim to study how the edge caching will affect the CP's data sponsoring strategy as well as the users' behaviors and the data market. Specifically, we consider a single CP who offers both the edge caching service and the data sponsoring service to a set of heterogeneous mobile video users (with different mobility and video request patterns). We formulate the interactions of the CP and the users as a two-stage Stackelberg game, where the CP (leader) determines the budgets (efforts) for both services in Stage I, and the users (followers) decide whether and which service(s) they would like to subscribe to. We analyze the sub-game perfect equilibrium (SPE) of the proposed game systematically. Our analysis and experimental results show that by introducing the edge caching, the CP can increase his revenue by 105%.

Edge caching technique is becoming more and more important in 5G network architecture, which shows great performance in terms of mitigating the backbone network congestion and reducing the delivery latency. The key idea of this technique is to cache popular contents in the small base stations(SBSs) which locate at the edge of the wireless network, and deliver these contents to nearby mobile users(MUs) from the cache directly. The collaborative in-cluster prefetching is improving the network performance even more. In this paper, we study the case where one monopolistic state-owned mobile network operator (MNO) offers edge caching service to multiple Content Providers(CPs), and focus on the mechanism design for the cache space allocation in this market. As the caching policy on each edge node is coupled with each other due to in-cluster collaboration, one time auction or flat pricing will both harm the social welfare. We propose a 2-optimal sequential auction mechanism (SAM) to solve this problem, which improves the social welfare by 20%~70% compared to the traditional Vickery-Clarke-Groves(VCG) mechanism.