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We propose an accurate data-driven numerical scheme to solve Stochastic Differential Equations (SDEs), by taking large time steps. The SDE discretization is built up by means of a polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Error analysis confirms that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression-decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. Numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.

Edge computing has been an efficient way to provide prompt and near-data computing services for resource-and-delay sensitive IoT applications via computation offloading. Effective computation offloading strategies need to comprehensively cope with several major issues, including the allocation of dynamic communication and computational resources, the deadline constraints of heterogeneous tasks, and the requirements for computationally inexpensive and distributed algorithms. However, most of the existing works mainly focus on part of these issues, which would not suffice to achieve expected performance in complex and practical scenarios. To tackle this challenge, in this paper, we systematically study a distributed computation offloading problem with hard delay constraints, where heterogeneous computational tasks require continually offloading to a set of edge servers via a limiting number of stochastic communication channels. The task offloading problem is then cast as a delay-constrained long-term stochastic optimization problem under unknown priori statistical knowledge. To resolve this problem, we first provide a technical path to transform and decompose it into several slot-level subproblems, then we develop a distributed online algorithm, namely TODG, to efficiently allocate the resources and schedule the offloading tasks with delay guarantees. Further, we present a comprehensive analysis for TODG, in terms of the optimality gap, the delay guarantees, and the impact of system parameters. Extensive simulation results demonstrate the effectiveness and efficiency of TODG.

Many Markov Chain Monte Carlo (MCMC) methods leverage gradient information of the potential function of target distribution to explore sample space efficiently. However, computing gradients can often be computationally expensive for large scale applications, such as those in contemporary machine learning. Stochastic Gradient (SG-)MCMC methods approximate gradients by stochastic ones, commonly via uniformly subsampled data points, and achieve improved computational efficiency, however at the price of introducing sampling error. We propose a non-uniform subsampling scheme to improve the sampling accuracy. The proposed exponentially weighted stochastic gradient (EWSG) is designed so that a non-uniform-SG-MCMC method mimics the statistical behavior of a batch-gradient-MCMC method, and hence the inaccuracy due to SG approximation is reduced. EWSG differs from classical variance reduction (VR) techniques as it focuses on the entire distribution instead of just the variance; nevertheless, its reduced local variance is also proved. EWSG can also be viewed as an extension of the importance sampling idea, successful for stochastic-gradient-based optimizations, to sampling tasks. In our practical implementation of EWSG, the non-uniform subsampling is performed efficiently via a Metropolis-Hastings chain on the data index, which is coupled to the MCMC algorithm. Numerical experiments are provided, not only to demonstrate EWSG's effectiveness, but also to guide hyperparameter choices, and validate our \emph{non-asymptotic global error bound} despite of approximations in the implementation. Notably, while statistical accuracy is improved, convergence speed can be comparable to the uniform version, which renders EWSG a practical alternative to VR (but EWSG and VR can be combined too).

Principal Subspace Analysis (PSA) is one of the most popular approaches for dimensionality reduction in signal processing and machine learning. But centralized PSA solutions are fast becoming irrelevant in the modern era of big data, in which the number of samples and/or the dimensionality of samples often exceed the storage and/or computational capabilities of individual machines. This has led to study of distributed PSA solutions, in which the data are partitioned across multiple machines and an estimate of the principal subspace is obtained through collaboration among the machines. It is in this vein that this paper revisits the problem of distributed PSA under the general framework of an arbitrarily connected network of machines that lacks a central server. The main contributions of the paper in this regard are threefold. First, two algorithms are proposed in the paper that can be used for distributed PSA in the case of data that are partitioned across either samples or (raw) features. Second, in the case of sample-wise partitioned data, the proposed algorithm and a variant of it are analyzed, and their convergence to the true subspace at linear rates is established. Third, extensive experiments on both synthetic and real-world data are carried out to validate the usefulness of the proposed algorithms. In particular, in the case of sample-wise partitioned data, an MPI-based distributed implementation is carried out to study the interplay between network topology and communications cost as well as to study of effect of straggler machines on the proposed algorithms.

The training process of neural networks usually optimize weights and bias parameters of linear transformations, while nonlinear activation functions are pre-specified and fixed. This work develops a systematic approach to constructing matrix activation functions whose entries are generalized from ReLU. The activation is based on matrix-vector multiplications using only scalar multiplications and comparisons. The proposed activation functions depend on parameters that are trained along with the weights and bias vectors. Neural networks based on this approach are simple and efficient and are shown to be robust in numerical experiments.

This paper studies a stochastic variant of the vehicle routing problem (VRP) where both customer locations and demands are uncertain. In particular, potential customers are not restricted to a predefined customer set but are continuously spatially distributed in a given service area. The objective is to maximize the served demands while fulfilling vehicle capacities and time restrictions. We call this problem the VRP with stochastic customers and demands (VRPSCD). For this problem, we first propose a Markov Decision Process (MDP) formulation representing the classical centralized decision-making perspective where one decision-maker establishes the routes of all vehicles. While the resulting formulation turns out to be intractable, it provides us with the ground to develop a new MDP formulation of the VRPSCD representing a decentralized decision-making framework, where vehicles autonomously establish their own routes. This new formulation allows us to develop several strategies to reduce the dimension of the state and action spaces, resulting in a considerably more tractable problem. We solve the decentralized problem via Reinforcement Learning, and in particular, we develop a Q-learning algorithm featuring state-of-the-art acceleration techniques such as Replay Memory and Double Q Network. Computational results show that our method considerably outperforms two commonly adopted benchmark policies (random and heuristic). Moreover, when comparing with existing literature, we show that our approach can compete with specialized methods developed for the particular case of the VRPSCD where customer locations and expected demands are known in advance. Finally, we show that the value functions and policies obtained by our algorithm can be easily embedded in Rollout algorithms, thus further improving their performances.

Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) is one of the most popular algorithms in distributed machine learning. However, its convergence properties for these complicated nonconvex problems is still largely unknown, because of the current technical limit. Therefore, in this paper, we propose to analyze the algorithm through a simpler but nontrivial nonconvex problem - streaming PCA, which helps us to understand Aync-MSGD better even for more general problems. Specifically, we establish the asymptotic rate of convergence of Async-MSGD for streaming PCA by diffusion approximation. Our results indicate a fundamental tradeoff between asynchrony and momentum: To ensure convergence and acceleration through asynchrony, we have to reduce the momentum (compared with Sync-MSGD). To the best of our knowledge, this is the first theoretical attempt on understanding Async-MSGD for distributed nonconvex stochastic optimization. Numerical experiments on both streaming PCA and training deep neural networks are provided to support our findings for Async-MSGD.

We propose accelerated randomized coordinate descent algorithms for stochastic optimization and online learning. Our algorithms have significantly less per-iteration complexity than the known accelerated gradient algorithms. The proposed algorithms for online learning have better regret performance than the known randomized online coordinate descent algorithms. Furthermore, the proposed algorithms for stochastic optimization exhibit as good convergence rates as the best known randomized coordinate descent algorithms. We also show simulation results to demonstrate performance of the proposed algorithms.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.