We study the problem of approximating the eigenspectrum of a symmetric matrix $A in mathbb{R}^{n times n}$ with bounded entries (i.e., $|A|_{infty} leq 1$). We present a simple sublinear time algorithm that approximates all eigenvalues of $A$ up to a
dditive error $pm epsilon n$ using those of a randomly sampled $tilde{O}(frac{1}{epsilon^4}) times tilde O(frac{1}{epsilon^4})$ principal submatrix. Our result can be viewed as a concentration bound on the full eigenspectrum of a random principal submatrix. It significantly extends existing work which shows concentration of just the spectral norm [Tro08]. It also extends work on sublinear time algorithms for testing the presence of large negative eigenvalues in the spectrum [BCJ20]. To complement our theoretical results, we provide numerical simulations, which demonstrate the effectiveness of our algorithm in approximating the eigenvalues of a wide range of matrices.
We study a multi-user downlink scheduling problem for optimizing the freshness of information available to users roaming across multiple cells. We consider both adversarial and stochastic settings and design scheduling policies that optimize two dist
inct information freshness metrics, namely the average age-of-information and the peak age-of-information. We show that a natural greedy scheduling policy is competitive with the optimal offline policy in the adversarial setting. We also derive fundamental lower bounds to the competitive ratio achievable by any online policy. In the stochastic environment, we show that a Max-Weight scheduling policy that takes into account the channel statistics achieves an approximation factor of $2$ for minimizing the average age of information in two extreme mobility scenarios. We conclude the paper by establishing a large-deviation optimality result achieved by the greedy policy for minimizing the peak age of information for static users situated at a single cell.
In this short paper, we consider the problem of designing a near-optimal competitive scheduling policy for $N$ mobile users, to maximize the freshness of available information uniformly across all users. Prompted by the unreliability and non-stationa
rity of the emerging 5G-mmWave channels for high-speed users, we forego of any statistical assumptions of the wireless channels and user-mobility. Instead, we allow the channel states and the mobility patterns to be dictated by an omniscient adversary. It is not difficult to see that no competitive scheduling policy can exist for the corresponding throughput-maximization problem in this adversarial model. Surprisingly, we show that there exists a simple online distributed scheduling policy with a finite competitive ratio for maximizing the freshness of information in this adversarial model. Moreover, we also prove that the proposed policy is competitively optimal up to an $O(ln N)$ factor.
Optimal caching of files in a content distribution network (CDN) is a problem of fundamental and growing commercial interest. Although many different caching algorithms are in use today, the fundamental performance limits of network caching algorithm
s from an online learning point-of-view remain poorly understood to date. In this paper, we resolve this question in the following two settings: (1) a single user connected to a single cache, and (2) a set of users and a set of caches interconnected through a bipartite network. Recently, an online gradient-based coded caching policy was shown to enjoy sub-linear regret. However, due to the lack of known regret lower bounds, the question of the optimality of the proposed policy was left open. In this paper, we settle this question by deriving tight non-asymptotic regret lower bounds in both of the above settings. In addition to that, we propose a new Follow-the-Perturbed-Leader-based uncoded caching policy with near-optimal regret. Technically, the lower-bounds are obtained by relating the online caching problem to the classic probabilistic paradigm of balls-into-bins. Our proofs make extensive use of a new result on the expected load in the most populated half of the bins, which might also be of independent interest. We evaluate the performance of the caching policies by experimenting with the popular MovieLens dataset and conclude the paper with design recommendations and a list of open problems.
We study the multi-user scheduling problem for minimizing the Age of Information (AoI) in cellular wireless networks under stationary and non-stationary regimes. We derive fundamental lower bounds for the scheduling problem and design efficient onlin
e policies with provable performance guarantees. In the stationary setting, we consider the AoI optimization problem for a set of mobile users travelling around multiple cells. In this setting, we propose a scheduling policy and show that it is $2$-optimal. Next, we propose a new adversarial channel model for studying the scheduling problem in non-stationary environments. For $N$ users, we show that the competitive ratio of any online scheduling policy in this setting is at least $Omega(N)$. We then propose an online policy and show that it achieves a competitive ratio of $O(N^2)$. Finally, we introduce a relaxed adversarial model with channel state estimations for the immediate future. We propose a heuristic model predictive control policy that exploits this feature and compare its performance through numerical simulations.
In this paper, we propose online algorithms for multiclass classification using partial labels. We propose two variants of Perceptron called Avg Perceptron and Max Perceptron to deal with the partial labeled data. We also propose Avg Pegasos and Max
Pegasos, which are extensions of Pegasos algorithm. We also provide mistake bounds for Avg Perceptron and regret bound for Avg Pegasos. We show the effectiveness of the proposed approaches by experimenting on various datasets and comparing them with the standard Perceptron and Pegasos.
