Due to the massive size of the neural network models and training datasets used in machine learning today, it is imperative to distribute stochastic gradient descent (SGD) by splitting up tasks such as gradient evaluation across multiple worker nodes. However, running distributed SGD can be prohibitively expensive because it may require specialized computing resources such as GPUs for extended periods of time. We propose cost-effective strategies to exploit volatile cloud instances that are cheaper than standard instances, but may be interrupted by higher priority workloads. To the best of our knowledge, this work is the first to quantify how variations in the number of active worker nodes (as a result of preemption) affects SGD convergence and the time to train the model. By understanding these trade-offs between preemption probability of the instances, accuracy, and training time, we are able to derive practical strategies for configuring distributed SGD jobs on volatile instances such as Amazon EC2 spot instances and other preemptible cloud instances. Experimental results show that our strategies achieve good training performance at substantially lower cost.

Least mean squares (LMS) is a particular case of the backpropagation (BP) algorithm applied to single-layer neural networks with the mean squared error (MSE) loss. One drawback of the LMS is that the instantaneous weight update is proportional to the square of the norm of the input vector. Normalized least mean squares (NLMS) algorithm amends this drawback by dividing the weight changes by the square of the norm of the input vector. The affine projection algorithm (APA) improved the NLMS algorithm to weight update over a batch of recently seen samples. However, the application of NLMS and APA had been limited to single-layer networks and adaptive filters. In this paper, we consider a virtual target for each neuron of a multi-layer neural network and show that the BP algorithm is equivalent to training the weights of each layer using these virtual targets and the LMS algorithm. We also introduce a consequentialism interpretation of the NLMS and the APA algorithms that justifies their use in multi-layer neural networks. Given any optimization algorithm based on the BP over mini-batches, we propose a novel consequentialism method for updating the weights.Consequently, our proposed weight update can be applied both to plain stochastic gradient descent (SGD) and to momentum methods like RMSProp, Adam, and NAG. These ideas helped us to update the weights more carefully in such a way that minimization of the loss for one sample of the mini-batch does not interfere with other samples in that mini-batch. Our experiments show the usefulness of the proposed method in optimizing deep neural network architectures.

Critical aspects of computational imaging systems, such as experimental design and image priors, can be optimized through deep networks formed by the unrolled iterations of classical model-based reconstructions (termed physics-based networks). However, for real-world large-scale inverse problems, computing gradients via backpropagation is infeasible due to memory limitations of graphics processing units. In this work, we propose a memory-efficient learning procedure that exploits the reversibility of the network's layers to enable data-driven design for large-scale computational imaging systems. We demonstrate our method on a small-scale compressed sensing example, as well as two large-scale real-world systems: multi-channel magnetic resonance imaging and super-resolution optical microscopy.

A framework based on iterative coordinate minimization (CM) is developed for stochastic convex optimization. Given that exact coordinate minimization is impossible due to the unknown stochastic nature of the objective function, the crux of the proposed optimization algorithm is an optimal control of the minimization precision in each iteration. We establish the optimal precision control and the resulting order-optimal regret performance for strongly convex and separably nonsmooth functions. An interesting finding is that the optimal progression of precision across iterations is independent of the low-dimensional CM routine employed, suggesting a general framework for extending low-dimensional optimization routines to high-dimensional problems. The proposed algorithm is amenable to online implementation and inherits the scalability and parallelizability properties of CM for large-scale optimization. Requiring only a sublinear order of message exchanges, it also lends itself well to distributed computing as compared with the alternative approach of coordinate gradient descent.

The concept of a random process has been recently extended to graph signals, whereby random graph processes are a class of multivariate stochastic processes whose coefficients are matrices with a \textit{graph-topological} structure. The system identification problem of a random graph process therefore revolves around determining its underlying topology, or mathematically, the graph shift operators (GSOs) i.e. an adjacency matrix or a Laplacian matrix. In the same work that introduced random graph processes, a \textit{batch} optimization method to solve for the GSO was also proposed for the random graph process based on a \textit{causal} vertex-time autoregressive model. To this end, the online version of this optimization problem was proposed via the framework of adaptive filtering. The modified stochastic gradient projection method was employed on the regularized least squares objective to create the filter. The recursion is divided into 3 regularized sub-problems to address issues like multi-convexity, sparsity, commutativity and bias. A discussion on convergence analysis is also included. Finally, experiments are conducted to illustrate the performance of the proposed algorithm, from traditional MSE measure to successful recovery rate regardless correct values, all of which to shed light on the potential, the limit and the possible research attempt of this work.

Stochastic gradient descent (SGD) has been widely studied in the literature from different angles, and is commonly employed for solving many big data machine learning problems. However, the averaging technique, which combines all iterative solutions into a single solution, is still under-explored. While some increasingly weighted averaging schemes have been considered in the literature, existing works are mostly restricted to strongly convex objective functions and the convergence of optimization error. It remains unclear how these averaging schemes affect the convergence of {\it both optimization error and generalization error} (two equally important components of testing error) for {\bf non-strongly convex objectives, including non-convex problems}. In this paper, we {\it fill the gap} by comprehensively analyzing the increasingly weighted averaging on convex, strongly convex and non-convex objective functions in terms of both optimization error and generalization error. In particular, we analyze a family of increasingly weighted averaging, where the weight for the solution at iteration $t$ is proportional to $t^{\alpha}$ ($\alpha > 0$). We show how $\alpha$ affects the optimization error and the generalization error, and exhibit the trade-off caused by $\alpha$. Experiments have demonstrated this trade-off and the effectiveness of polynomially increased weighted averaging compared with other averaging schemes for a wide range of problems including deep learning.

When we enforce differential privacy in machine learning, the utility-privacy trade-off is different w.r.t. each group. Gradient clipping and random noise addition disproportionately affect underrepresented and complex classes and subgroups, which results in inequality in utility loss. In this work, we analyze the inequality in utility loss by differential privacy and propose a modified differentially private stochastic gradient descent (DPSGD), called DPSGD-F, to remove the potential disparate impact of differential privacy on the protected group. DPSGD-F adjusts the contribution of samples in a group depending on the group clipping bias such that differential privacy has no disparate impact on group utility. Our experimental evaluation shows how group sample size and group clipping bias affect the impact of differential privacy in DPSGD, and how adaptive clipping for each group helps to mitigate the disparate impact caused by differential privacy in DPSGD-F.

SGD performs frequent updates with a high variance, causing the objective function to fluctuate heavily. SGD's fluctuation enables it to jump from a local minima to a potentially better local minima, but complicates convergence to an exact minimum. Momentum is a parameter of SGD that can be added to assist SGD in ravines -- areas where the surface curves more steeply in one dimension than in another, common around optima. Momentum helps accelerate SGD in the correct direction, therefore dampening the redundant oscillations as seen in image 2. Nesterov momentum is an improvement over standard momentum -- a ball that blindly follows the slope is unsatisfactory. Ideally, the ball would know where it is going so it can slow down before the hill slopes up again.

Distributed training is useful to train complicated models to shorten the training time. As each of the workers only sees a small fraction of data, workers need to synchronize on the parameter updates. One of the central questions in distributed training is how to parsimoniously synchronize parameters while preserving model quality. To address this problem, we propose the \textbf{ShadowSync} framework, in which we isolate synchronization from training and run it in the background. In contrast to common strategies including synchronous stochastic gradient descent (SGD), asynchronous SGD, and model averaging on independently trained sub-models, where synchronization happens in the foreground, ShadowSync synchronization is neither part of the backward pass, nor happens every $k$ iterations. Our framework is generic to host various types of synchronization algorithms, and we propose 3 approaches under this theme. The superiority of ShadowSync is confirmed by experiments on training deep neural networks for click-through-rate prediction. Our methods all succeed in making the training throughput linearly scale with the number of trainers. Comparing to their foreground counterparts, our methods exhibit neutral to better model quality and better scalability when we keep the number of parameter servers the same. In our training system which expresses both replication and Hogwild parallelism, ShadowSync also accomplishes the highest example level parallelism number comparing to the prior arts.

We study the performance of decentralized stochastic gradient descent (DSGD) in a wireless network, where the nodes collaboratively optimize an objective function using their local datasets. Unlike the conventional setting, where the nodes communicate over error-free orthogonal communication links, we assume that transmissions are prone to additive noise and interference.We first consider a point-to-point (P2P) transmission strategy, termed the OAC-P2P scheme, in which the node pairs are scheduled in an orthogonal fashion to minimize interference. Since in the DSGD framework, each node requires a linear combination of the neighboring models at the consensus step, we then propose the OAC-MAC scheme, which utilizes the signal superposition property of the wireless medium to achieve over-the-air computation (OAC). For both schemes, we cast the scheduling problem as a graph coloring problem. We numerically evaluate the performance of these two schemes for the MNIST image classification task under various network conditions. We show that the OAC-MAC scheme attains better convergence performance with a fewer communication rounds.