## Optimization (Stochastic)

Moderator: Lam Nguyen

Abstract:

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Tue 20 July 18:00 - 18:20 PDT
##### The Power of Adaptivity for Stochastic Submodular Cover

Rohan Ghuge · Anupam Gupta · viswanath nagarajan

In the stochastic submodular cover problem, the goal is to select a subset of stochastic items of minimum expected cost to cover a submodular function. Solutions in this setting correspond to a sequential decision process that selects items one by one adaptively'' (depending on prior observations). While such adaptive solutions achieve the best objective, the inherently sequential nature makes them undesirable in many applications. We ask: \emph{how well can solutions with only a few adaptive rounds approximate fully-adaptive solutions?} We consider both cases where the stochastic items are independent, and where they are correlated. For both situations, we obtain nearly tight answers, establishing smooth tradeoffs between the number of adaptive rounds and the solution quality, relative to fully adaptive solutions. Experiments on synthetic and real datasets validate the practical performance of our algorithms, showing qualitative improvements in the solutions as we allow more rounds of adaptivity; in practice, solutions using just a few rounds of adaptivity are nearly as good as fully adaptive solutions.

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Tue 20 July 18:20 - 18:25 PDT
##### The Heavy-Tail Phenomenon in SGD

Mert Gurbuzbalaban · Umut Simsekli · Lingjiong Zhu

In recent years, various notions of capacity and complexity have been proposed for characterizing the generalization properties of stochastic gradient descent (SGD) in deep learning. Some of the popular notions that correlate well with the performance on unseen data are (i) the flatness' of the local minimum found by SGD, which is related to the eigenvalues of the Hessian, (ii) the ratio of the stepsize $\eta$ to the batch-size $b$, which essentially controls the magnitude of the stochastic gradient noise, and (iii) the tail-index', which measures the heaviness of the tails of the network weights at convergence. In this paper, we argue that these three seemingly unrelated perspectives for generalization are deeply linked to each other. We claim that depending on the structure of the Hessian of the loss at the minimum, and the choices of the algorithm parameters $\eta$ and $b$, the SGD iterates will converge to a \emph{heavy-tailed} stationary distribution. We rigorously prove this claim in the setting of quadratic optimization: we show that even in a simple linear regression problem with independent and identically distributed data whose distribution has finite moments of all order, the iterates can be heavy-tailed with infinite variance. We further characterize the behavior of the tails with respect to algorithm parameters, the dimension, and the curvature. We then translate our results into insights about the behavior of SGD in deep learning. We support our theory with experiments conducted on synthetic data, fully connected, and convolutional neural networks.

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Tue 20 July 18:25 - 18:30 PDT
##### Federated Composite Optimization

Honglin Yuan · Manzil Zaheer · Sashank Jakkam Reddi

Federated Learning (FL) is a distributed learning paradigm that scales on-device learning collaboratively and privately. Standard FL algorithms such as FᴇᴅAᴠɢ are primarily geared towards smooth unconstrained settings. In this paper, we study the Federated Composite Optimization (FCO) problem, in which the loss function contains a non-smooth regularizer. Such problems arise naturally in FL applications that involve sparsity, low-rank, monotonicity, or more general constraints. We first show that straightforward extensions of primal algorithms such as FedAvg are not well-suited for FCO since they suffer from the "curse of primal averaging," resulting in poor convergence. As a solution, we propose a new primal-dual algorithm, Federated Dual Averaging (FedDualAvg), which by employing a novel server dual averaging procedure circumvents the curse of primal averaging. Our theoretical analysis and empirical experiments demonstrate that FedDualAvg outperforms the other baselines.

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Tue 20 July 18:30 - 18:35 PDT
##### On Estimation in Latent Variable Models

Guanhua Fang · Ping Li

Latent variable models have been playing a central role in statistics, econometrics, machine learning with applications to repeated observation study, panel data inference, user behavior analysis, etc. In many modern applications, the inference based on latent variable models involves one or several of the following features: the presence of complex latent structure, the observed and latent variables being continuous or discrete, constraints on parameters, and data size being large. Therefore, solving an estimation problem for general latent variable models is highly non-trivial. In this paper, we consider a gradient based method via using variance reduction technique to accelerate estimation procedure. Theoretically, we show the convergence results for the proposed method under general and mild model assumptions. The algorithm has better computational complexity compared with the classical gradient methods and maintains nice statistical properties. Various numerical results corroborate our theory.

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Tue 20 July 18:35 - 18:40 PDT
##### Asynchronous Distributed Learning : Adapting to Gradient Delays without Prior Knowledge

Rotem Zamir Aviv · Ido Hakimi · Assaf Schuster · Kfir Levy

We consider stochastic convex optimization problems, where several machines act asynchronously in parallel while sharing a common memory. We propose a robust training method for the constrained setting and derive non asymptotic convergence guarantees that do not depend on prior knowledge of update delays, objective smoothness, and gradient variance. Conversely, existing methods for this setting crucially rely on this prior knowledge, which render them unsuitable for essentially all shared-resources computational environments, such as clouds and data centers. Concretely, existing approaches are unable to accommodate changes in the delays which result from dynamic allocation of the machines, while our method implicitly adapts to such changes.

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Tue 20 July 18:40 - 18:45 PDT
##### Randomized Algorithms for Submodular Function Maximization with a $k$-System Constraint

Shuang Cui · Kai Han · Tianshuai Zhu · Jing Tang · Benwei Wu · He Huang

Submodular optimization has numerous applications such as crowdsourcing and viral marketing. In this paper, we study the problem of non-negative submodular function maximization subject to a $k$-system constraint, which generalizes many other important constraints in submodular optimization such as cardinality constraint, matroid constraint, and $k$-extendible system constraint. The existing approaches for this problem are all based on deterministic algorithmic frameworks, and the best approximation ratio achieved by these algorithms (for a general submodular function) is $k+2\sqrt{k+2}+3$. We propose a randomized algorithm with an improved approximation ratio of $(1+\sqrt{k})^2$, while achieving nearly-linear time complexity significantly lower than that of the state-of-the-art algorithm. We also show that our algorithm can be further generalized to address a stochastic case where the elements can be adaptively selected, and propose an approximation ratio of $(1+\sqrt{k+1})^2$ for the adaptive optimization case. The empirical performance of our algorithms is extensively evaluated in several applications related to data mining and social computing, and the experimental results demonstrate the superiorities of our algorithms in terms of both utility and efficiency.

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Tue 20 July 18:45 - 18:50 PDT
##### BASGD: Buffered Asynchronous SGD for Byzantine Learning

Yi-Rui Yang · Wu-Jun Li

Distributed learning has become a hot research topic due to its wide application in cluster-based large-scale learning, federated learning, edge computing and so on. Most traditional distributed learning methods typically assume no failure or attack. However, many unexpected cases, such as communication failure and even malicious attack, may happen in real applications. Hence, Byzantine learning (BL), which refers to distributed learning with failure or attack, has recently attracted much attention. Most existing BL methods are synchronous, which are impractical in some applications due to heterogeneous or offline workers. In these cases, asynchronous BL (ABL) is usually preferred. In this paper, we propose a novel method, called buffered asynchronous stochastic gradient descent (BASGD), for ABL. To the best of our knowledge, BASGD is the first ABL method that can resist malicious attack without storing any instances on server. Compared with those methods which need to store instances on server, BASGD has a wider scope of application. BASGD is proved to be convergent, and be able to resist failure or attack. Empirical results show that BASGD significantly outperforms vanilla asynchronous stochastic gradient descent (ASGD) and other ABL baselines when there exists failure or attack on workers.

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Tue 20 July 18:50 - 18:55 PDT