Existing gradient coding schemes introduce identical redundancy across the coordinates of gradients and hence cannot fully utilize the computation results from partial stragglers. This motivates the introduction of diverse redundancies across the coordinates of gradients. This paper considers a distributed computation system consisting of one master and$N$workers characterized by a general partial straggler model and focuses on solving a general large-scale machine learning problem with$L$model parameters. We show that it is sufficient to provide at most$N$levels of redundancies for tolerating$0,1, \cdots, N-1$stragglers, respectively. Consequently, we propose an optimal block coordinate gradient coding scheme based on a stochastic optimization problem that optimizes the partition of the$L$coordinates into$N$blocks, each with identical redundancy, to minimize the expected overall runtime for collaboratively computing the gradient. We obtain an optimal solution using a stochastic projected subgradient method and propose two low-complexity approximate solutions with closed-from expressions, for the stochastic optimization problem. We also show that under a shifted-exponential distribution, for any$L$, the expected overall runtimes of the two approximate solutions and the minimum overall runtime have sub-linear multiplicative gaps in$N$. To the best of our knowledge, this is the first work that optimizes the redundancies of gradient coding introduced across the coordinates of gradients.

Existing decentralized coded caching solutions cannot guarantee small loads in the general scenario with arbitrary file sizes and cache sizes. In this paper, we propose an optimization framework for decentralized coded caching in the general scenario to minimize the worst-case load and average load (under an arbitrary file popularity), respectively. Specifically, we first propose a class of decentralized coded caching schemes for the general scenario, which are specified by a general caching parameter and include several known schemes as special cases. Then, we optimize the caching parameter to minimize the worst-case load and average load, respectively. Each of the two optimization problems is a challenging nonconvex problem with a nondifferentiable objective function. For each optimization problem, we develop an iterative algorithm to obtain a stationary point using techniques for solving Complementary Geometric Programming (GP). We also obtain a low-complexity approximate solution by solving an approximate problem with a differentiable objective function which is an upper bound on the original nondifferentiable one, and characterize the performance loss caused by the approximation. Finally, we present two information-theoretic converse bounds on the worst-case load and average load (under an arbitrary file popularity) in the general scenario, respectively. To the best of our knowledge, this is the first work that provides optimization-based decentralized coded caching schemes and information-theoretic converse bounds for the general scenario.

The high-definition and panorama characteristics of 360-degree video make its transmission in existing networks more challenging. To alleviate this problem, this paper leverages transcoding-enabled edge caching in tile-granularity and makes joint caching, transcoding and delivery decisions among multiple edge servers for adaptive 360-degree video streaming. We formulate the joint caching, transcoding and delivery decision-making problem as a multivariate nonlinear integer programming problem, aiming at minimizing the aggregate network operational cost and balancing the server load, subject to storage and computation capacity constraints of the edge servers. Through decomposition and relaxation approach, the original intractable optimization problem is solved by an effective iterative algorithm. Finally, simulation results show that the proposed iterative algorithm outperforms the existing schemes in terms of resource utilization, cost reduction and server coordination.