In this paper, we consider non-convex optimization problems under \textit{unknown} yet safety-critical constraints. Such problems naturally arise in a variety of domains including robotics, manufacturing, and medical procedures, where it is infeasible to know or identify all the constraints. Therefore, the parameter space should be explored in a conservative way to ensure that none of the constraints are violated during the optimization process once we start from a safe initialization point. To this end, we develop an algorithm called Reliable Frank-Wolfe (Reliable-FW). Given a general non-convex function and an unknown polytope constraint, Reliable-FW simultaneously learns the landscape of the objective function and the boundary of the safety polytope. More precisely, by assuming that Reliable-FW has access to a (stochastic) gradient oracle of the objective function and a noisy feasibility oracle of the safety polytope, it finds an $\epsilon$-approximate first-order stationary point with the optimal ${\mathcal{O}}({1}/{\epsilon^2})$ gradient oracle complexity (resp. $\tilde{\mathcal{O}}({1}/{\epsilon^3})$ (also optimal) in the stochastic gradient setting), while ensuring the safety of all the iterates. Rather surprisingly, Reliable-FW only makes $\tilde{\mathcal{O}}(({d^2}/{\epsilon^2})\log 1/\delta)$ queries to the noisy feasibility oracle (resp. $\tilde{\mathcal{O}}(({d^2}/{\epsilon^4})\log 1/\delta)$ in the stochastic gradient setting) where $d$ is the dimension and $\delta$ is the reliability parameter, tightening the existing bounds even for safe minimization of convex functions. We further specialize our results to the case that the objective function is convex. A crucial component of our analysis is to introduce and apply a technique called geometric shrinkage in the context of safe optimization.

Mobile network operators store an enormous amount of information like log files that describe various events and users' activities. Analysis of these logs might be used in many critical applications such as detecting cyber-attacks, finding behavioral patterns of users, security incident response, network forensics, etc. In a cellular network Call Detail Records (CDR) is one type of such logs containing metadata of calls and usually includes valuable information about contact such as the phone numbers of originating and receiving subscribers, call duration, the area of activity, type of call (SMS or voice call) and a timestamp. With anomaly detection, it is possible to determine abnormal reduction or increment of network traffic in an area or for a particular person. This paper's primary goal is to study subscribers' behavior in a cellular network, mainly predicting the number of calls in a region and detecting anomalies in the network traffic. In this paper, a new hybrid method is proposed based on various anomaly detection methods such as GARCH, K-means, and Neural Network to determine the anomalous data. Moreover, we have discussed the possible causes of such anomalies.

We consider Model-Agnostic Meta-Learning (MAML) methods for Reinforcement Learning (RL) problems where the goal is to find a policy (using data from several tasks represented by Markov Decision Processes (MDPs)) that can be updated by one step of stochastic policy gradient for the realized MDP. In particular, using stochastic gradients in MAML update step is crucial for RL problems since computation of exact gradients requires access to a large number of possible trajectories. For this formulation, we propose a variant of the MAML method, named Stochastic Gradient Meta-Reinforcement Learning (SG-MRL), and study its convergence properties. We derive the iteration and sample complexity of SG-MRL to find an $\epsilon$-first-order stationary point, which, to the best of our knowledge, provides the first convergence guarantee for model-agnostic meta-reinforcement learning algorithms. We further show how our results extend to the case where more than one step of stochastic policy gradient method is used in the update during the test time.

Federated Learning is a novel paradigm that aims to train a statistical model at the "edge" nodes as opposed to the traditional distributed computing systems such as data centers [Konečn y et al., 2016, Li et al., 2019a]. The main objective of federated learning is to fit a model to data generated from network devices without continuous transfer of the massive amount of collected data from edge of the network to back-end servers for processing. Federated learning has been deployed by major technology companies with the goal of providing privacy-preserving services using users' data [Bonawitz et al., 2019]. Examples of such applications are learning from wearable devices [Huang et al., 2018], learning sentiment [Smith et al., 2017], and location-based services [Samarakoon et al., 2018]. While federated learning is a promising paradigm for such applications, there are several challenges that remain to be resolved. In this paper, we focus on two significant challenges of federated learning, and propose a novel federated learning algorithm that addresses the following two challenges: (i) Communication bottleneck. Communication bandwidth is a major bottleneck in federated learning as a large number of devices attempt to communicate their local updates to a central parameter server. Thus, at a high level, for a communication-efficient federated learning algorithm, it is crucial that such updates are sent in a compressed manner and infrequently.

One of the beauties of the projected gradient descent method lies in its rather simple mechanism and yet stable behavior with inexact, stochastic gradients, which has led to its wide-spread use in many machine learning applications. However, once we replace the projection operator with a simpler linear program, as is done in the Frank-Wolfe method, both simplicity and stability take a serious hit. The aim of this paper is to bring them back without sacrificing the efficiency. In this paper, we propose the first one-sample stochastic Frank-Wolfe algorithm, called 1-SFW, that avoids the need to carefully tune the batch size, step size, learning rate, and other complicated hyper parameters. In particular, 1-SFW achieves the optimal convergence rate of $\mathcal{O}(1/\epsilon^2)$ for reaching an $\epsilon$-suboptimal solution in the stochastic convex setting, and a $(1-1/e)-\epsilon$ approximate solution for a stochastic monotone DR-submodular maximization problem. Moreover, in a general non-convex setting, 1-SFW finds an $\epsilon$-first-order stationary point after at most $\mathcal{O}(1/\epsilon^3)$ iterations, achieving the current best known convergence rate. All of this is possible by designing a novel unbiased momentum estimator that governs the stability of the optimization process while using a single sample at each iteration.

In this paper, we study the convergence of a class of gradient-based Model-Agnostic Meta-Learning (MAML) methods and characterize their overall computational complexity as well as their best achievable level of accuracy in terms of gradient norm for nonconvex loss functions. In particular, we start with the MAML algorithm and its first order approximation (FO-MAML) and highlight the challenges that emerge in their analysis. By overcoming these challenges not only we provide the first theoretical guarantees for MAML and FO-MAML in nonconvex settings, but also we answer some of the unanswered questions for the implementation of these algorithms including how to choose their learning rate (stepsize) and the batch size for both tasks and datasets corresponding to tasks. In particular, we show that MAML can find an $\epsilon$-first-order stationary point for any positive $\epsilon$ after at most $\mathcal{O}(1/\epsilon^2)$ iterations at the expense of requiring second-order information. We also show that the FO-MAML method which ignores the second-order information required in the update of MAML cannot achieve any small desired level of accuracy, i.e, FO-MAML cannot find an $\epsilon$-first-order stationary point for any positive $\epsilon$. We further propose a new variant of the MAML algorithm called Hessian-free MAML (HF-MAML) which preserves all theoretical guarantees of MAML, without requiring access to the second-order information of loss functions.

We consider a decentralized learning problem, where a set of computing nodes aim at solving a non-convex optimization problem collaboratively. It is well-known that decentralized optimization schemes face two major system bottlenecks: stragglers' delay and communication overhead. In this paper, we tackle these bottlenecks by proposing a novel decentralized and gradient-based optimization algorithm named as QuanTimed-DSGD. Our algorithm stands on two main ideas: (i) we impose a deadline on the local gradient computations of each node at each iteration of the algorithm, and (ii) the nodes exchange quantized versions of their local models. The first idea robustifies to straggling nodes and the second alleviates communication efficiency. The key technical contribution of our work is to prove that with non-vanishing noises for quantization and stochastic gradients, the proposed method exactly converges to the global optimal for convex loss functions, and finds a first-order stationary point in non-convex scenarios. Our numerical evaluations of the QuanTimed-DSGD on training benchmark datasets, MNIST and CIFAR-10, demonstrate speedups of up to 3x in run-time, compared to state-of-the-art decentralized optimization methods.

In this paper we analyze the iteration complexity of the optimistic gradient descent-ascent (OGDA) method as well as the extra-gradient (EG) method for finding a saddle point of a convex-concave unconstrained min-max problem. To do so, we first show that both OGDA and EG can be interpreted as approximate variants of the proximal point method. We then exploit this interpretation to show that both of these algorithms achieve a convergence rate of $\mathcal{O}(1/k)$ for smooth convex-concave saddle point problems. Our theoretical analysis is of interest as it provides a simple convergence analysis for the EG algorithm in terms of objective function value without using compactness assumption. Moreover, it provides the first convergence guarantee for OGDA in the general convex-concave setting.

How can we efficiently mitigate the overhead of gradient communications in distributed optimization? This problem is at the heart of training scalable machine learning models and has been mainly studied in the unconstrained setting. In this paper, we propose Quantized Frank-Wolfe (QFW), the first projection-free and communication-efficient algorithm for solving constrained optimization problems at scale. We consider both convex and non-convex objective functions, expressed as a finite-sum or more generally a stochastic optimization problem, and provide strong theoretical guarantees on the convergence rate of QFW. This is done by proposing quantization schemes that efficiently compress gradients while controlling the variance introduced during this process. Finally, we empirically validate the efficiency of QFW in terms of communication and the quality of returned solution against natural baselines.

We consider solving convex-concave saddle point problems. We focus on two variants of gradient decent-ascent algorithms, Extra-gradient (EG) and Optimistic Gradient (OGDA) methods, and show that they admit a unified analysis as approximations of the classical proximal point method for solving saddle-point problems. This viewpoint enables us to generalize EG (in terms of extrapolation steps) and OGDA (in terms of parameters) and obtain new convergence rate results for these algorithms for the bilinear case as well as the strongly convex-concave case.