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This work considers the decentralized successive convex approximation (SCA) method for minimizing stochastic non-convex objectives subject to convex constraints, along with possibly non-smooth convex regularizers. Although SCA has been widely applied in decentralized settings, its stochastic first order (SFO) complexity is unknown, and it is thought to be slower than the centralized momentum-enhanced SCA variants. In this work, we advance the state-of-the-art for SCA methods by proposing an accelerated variant, namely the \textbf{D}ecentralized \textbf{M}omentum-based \textbf{S}tochastic \textbf{SCA} (\textbf{D-MSSCA}) and analyze its SFO complexity. The proposed algorithm entails creating a stochastic surrogate of the objective at every iteration, which is minimized at each node separately. Remarkably, the D-MSSCA achieves an SFO complexity of $\mathcal{O}(\epsilon^{-3/2})$ to reach an $\epsilon$-stationary point, which is at par with the SFO complexity lower bound for unconstrained stochastic non-convex optimization in centralized setting.

We consider stochastic optimization problems with functional constraints, such as those arising in trajectory generation, sparse approximation, and robust classification. To this end, we put forth a recursive momentum-based accelerated successive convex approximation (SCA) algorithm. At each iteration, the proposed algorithm entails constructing convex surrogates of the stochastic objective and the constraint functions, and solving the resulting convex optimization problem. A recursive update rule is employed to track the gradient of the stochastic objective function, which contributes to variance reduction and hence accelerates the algorithm convergence. A key ingredient of the proof is a new parameterized version of the standard Mangasarian-Fromowitz Constraints Qualification, that allows us to bound the dual variables and hence obtain problem-dependent bounds on the rate at which the iterates approach an $\epsilon$-stationary point. Remarkably, the proposed algorithm achieves near-optimal stochastic first order (SFO) complexity, almost at par with that achieved by state-of-the-art stochastic optimization algorithms for solving unconstrained problems. As an example, we detail a obstacle-avoiding trajectory optimization problem that can be solved using the proposed algorithm and show that its performance is superior to that of the existing algorithms used for trajectory optimization. The performance of the proposed algorithm is also shown to be comparable to that of a specialized sparse classification algorithm applied to a binary classification problem., Comment: 32 pages, 4 figures, journal submission

This work considers the problem of decentralized online learning, where the goal is to track the optimum of the sum of time-varying functions, distributed across several nodes in a network. The local availability of the functions and their gradients necessitates coordination and consensus among the nodes. We put forth the Generalized Gradient Tracking (GGT) framework that unifies a number of existing approaches, including the state-of-the-art ones. The performance of the proposed GGT algorithm is theoretically analyzed using a novel semidefinite programming-based analysis that yields the desired regret bounds under very general conditions and without requiring the gradient boundedness assumption. The results are applicable to the special cases of GGT, which include various state-of-the-art algorithms as well as new dynamic versions of various classical decentralized algorithms. To further minimize the regret, we consider a condensed version of GGT with only four free parameters. A procedure for offline tuning of these parameters using only the problem parameters is also detailed. The resulting optimized GGT (oGGT) algorithm not only achieves improved dynamic regret bounds, but also outperforms all state-of-the-art algorithms on both synthetic and real-world datasets., Comment: 30 pages, 6 Figures

This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix. The resulting updates avoid the costly matrix operations like matrix exponentiation and dense matrix multiplication, which are generally required in almost all other Riemannian optimization algorithms on SPD manifold. We further identify a broad class of functions, arising in diverse applications, such as kernel matrix learning, covariance estimation of Gaussian distributions, maximum likelihood parameter estimation of elliptically contoured distributions, and parameter estimation in Gaussian mixture model problems, over which the Riemannian gradients can be calculated efficiently. The proposed uni-directional and multi-directional Riemannian subspace descent variants incur per-iteration complexities of $O(n)$ and $O(n^2)$ respectively, as compared to the $O(n^3)$ or higher complexity incurred by all existing Riemannian gradient descent variants. The superior runtime and low per-iteration complexity of the proposed algorithms is also demonstrated via numerical tests on large-scale covariance estimation and matrix square root problems. MATLAB code implementation is publicly available on GitHub : https://github.com/yogeshd-iitk/subspace_descent_over_SPD_manifold

Missing data is a common problem in real-world sensor data collection. The performance of various approaches to impute data degrade rapidly in the extreme scenarios of low data sampling and noisy sampling, a case present in many real-world problems in the field of traffic sensing and environment monitoring, etc. However, jointly exploiting the spatiotemporal and periodic structure, which is generally not captured by classical matrix completion approaches, can improve the imputation performance of sensor data in such real-world conditions. We present a Bayesian approach towards spatiotemporal matrix completion wherein we estimate the underlying temporarily varying subspace using a Variational Bayesian technique. We jointly couple the low-rank matrix completion with the state space autoregressive framework along with a penalty function on the slowly varying subspace to model the temporal and periodic evolution in the data. A major advantage of our method is that a critical parameter like the rank of the model is automatically tuned using the automatic relevance determination (ARD) approach, unlike most matrix/tensor completion techniques. We also propose a robust version of the above formulation, which improves the performance of imputation in the presence of outliers. We evaluate the proposed Variational Bayesian Filtering with Subspace Information (VBFSI) method to impute matrices in real-world traffic and air pollution data. Simulation results demonstrate that the proposed method outperforms the recent state-of-the-art methods and provides a sufficiently accurate imputation for different sampling rates. In particular, we demonstrate that fusing the subspace evolution over days can improve the imputation performance with even 15% of the data sampling.

This paper considers stochastic convex optimization problems with two sets of constraints: (a) deterministic constraints on the domain of the optimization variable, which are difficult to project onto; and (b) deterministic or stochastic constraints that admit efficient projection. Problems of this form arise frequently in the context of semidefinite programming as well as when various NP-hard problems are solved approximately via semidefinite relaxation. Since projection onto the first set of constraints is difficult, it becomes necessary to explore projection-free algorithms, such as the stochastic Frank-Wolfe (FW) algorithm. On the other hand, the second set of constraints cannot be handled in the same way, and must be incorporated as an indicator function within the objective function, thereby complicating the application of FW methods. Similar problems have been studied before; however, they suffer from slow convergence rates. This work, equipped with momentum based gradient tracking technique, guarantees fast convergence rates on par with the best-known rates for problems without the second set of constraints. Zeroth-order variants of the proposed algorithms are also developed and again improve upon the state-of-the-art rate results. We further propose the novel trimmed FW variants that enjoy the same convergence rates as their classical counterparts, but are empirically shown to require significantly fewer calls to the linear minimization oracle speeding up the overall algorithm. The efficacy of the proposed algorithms is tested on relevant applications of sparse matrix estimation, clustering via semidefinite relaxation, and uniform sparsest cut problem.

We consider stochastic optimization of a smooth non-convex loss function with a convex non-smooth regularizer. In the online setting, where a single sample of the stochastic gradient of the loss is available at every iteration, the problem can be solved using the proximal stochastic gradient descent (SGD) algorithm and its variants. However in many problems, especially those arising in communications and signal processing, information beyond the stochastic gradient may be available thanks to the structure of the loss function. Such extra-gradient information is not used by SGD, but has been shown to be useful, for instance in the context of stochastic expectation-maximization, stochastic majorization-minimization, and stochastic successive convex approximation (SCA) approaches. By constructing a stochastic strongly convex surrogates of the loss function at every iteration, the stochastic SCA algorithms can exploit the structural properties of the loss function and achieve superior empirical performance as compared to the SGD. In this work, we take a closer look at the stochastic SCA algorithm and develop its asynchronous variant which can be used for resource allocation in wireless networks. While the stochastic SCA algorithm is known to converge asymptotically, its iteration complexity has not been well-studied, and is the focus of the current work. The insights obtained from the non-asymptotic analysis allow us to develop a more practical asynchronous variant of the stochastic SCA algorithm which allows the use of surrogates calculated in earlier iterations. We characterize precise bound on the maximum delay the algorithm can tolerate, while still achieving the same convergence rate. We apply the algorithm to the problem of linear precoding in wireless sensor networks, where it can be implemented at low complexity but is shown to perform well in practice.

This paper considers stochastic convex optimization problems where the objective and constraint functions involve expectations with respect to the data indices or environmental variables, in addition to deterministic convex constraints on the domain of the variables. Although the setting is generic and arises in different machine learning applications, online and efficient approaches for solving such problems have not been widely studied. Since the underlying data distribution is unknown a priori, a closed-form solution is generally not available, and classical deterministic optimization paradigms are not applicable. State-of-the-art approaches, such as those using the saddle point framework, can ensure that the optimality gap as well as the constraint violation decay as $\O\left(T^{-\frac{1}{2}}\right)$ where $T$ is the number of stochastic gradients. The domain constraints are assumed simple and handled via projection at every iteration. In this work, we propose a novel conservative stochastic optimization algorithm (CSOA) that achieves zero constraint violation and $\O\left(T^{-\frac{1}{2}}\right)$ optimality gap. Further, the projection operation (for scenarios when calculating projection is expensive) in the proposed algorithm can be avoided by considering the conditional gradient or Frank-Wolfe (FW) variant of the algorithm. The state-of-the-art stochastic FW variants achieve an optimality gap of $\O\left(T^{-\frac{1}{3}}\right)$ after $T$ iterations, though these algorithms have not been applied to problems with functional expectation constraints. In this work, we propose the FW-CSOA algorithm that is not only projection-free but also achieves zero constraint violation with $\O\left(T^{-\frac{1}{4}}\right)$ decay of the optimality gap. The efficacy of the proposed algorithms is tested on two relevant problems: fair classification and structured matrix completion.

Batch training of machine learning models based on neural networks is now well established, whereas to date streaming methods are largely based on linear models. To go beyond linear in the online setting, nonparametric methods are of interest due to their universality and ability to stably incorporate new information via convexity or Bayes' Rule. Unfortunately, when used online, nonparametric methods suffer a "curse of dimensionality" which precludes their use: their complexity scales at least with the time index. We survey online compression tools which bring their memory under control and attain approximate convergence. The asymptotic bias depends on a compression parameter that trades off memory and accuracy. Further, the applications to robotics, communications, economics, and power are discussed, as well as extensions to multi-agent systems.

An open challenge in supervised learning is \emph{conceptual drift}: a data point begins as classified according to one label, but over time the notion of that label changes. Beyond linear autoregressive models, transfer and meta learning address drift, but require data that is representative of disparate domains at the outset of training. To relax this requirement, we propose a memory-efficient \emph{online} universal function approximator based on compressed kernel methods. Our approach hinges upon viewing non-stationary learning as online convex optimization with dynamic comparators, for which performance is quantified by dynamic regret. Prior works control dynamic regret growth only for linear models. In contrast, we hypothesize actions belong to reproducing kernel Hilbert spaces (RKHS). We propose a functional variant of online gradient descent (OGD) operating in tandem with greedy subspace projections. Projections are necessary to surmount the fact that RKHS functions have complexity proportional to time. For this scheme, we establish sublinear dynamic regret growth in terms of both loss variation and functional path length, and that the memory of the function sequence remains moderate. Experiments demonstrate the usefulness of the proposed technique for online nonlinear regression and classification problems with non-stationary data.

We consider learning in decentralized heterogeneous networks: agents seek to minimize a convex functional that aggregates data across the network, while only having access to their local data streams. We focus on the case where agents seek to estimate a regression \emph{function} that belongs to a reproducing kernel Hilbert space (RKHS). To incentivize coordination while respecting network heterogeneity, we impose nonlinear proximity constraints. To solve the constrained stochastic program, we propose applying a functional variant of stochastic primal-dual (Arrow-Hurwicz) method which yields a decentralized algorithm. To handle the fact that agents' functions have complexity proportional to time (owing to the RKHS parameterization), we project the primal iterates onto subspaces greedily constructed from kernel evaluations of agents' local observations. The resulting scheme, dubbed Heterogeneous Adaptive Learning with Kernels (HALK), when used with constant step-sizes, yields $\mathcal{O}(\sqrt{T})$ attenuation in sub-optimality and exactly satisfies the constraints in the long run, which improves upon the state of the art rates for vector-valued problems.

Matrix completion and robust principal component analysis have been widely used for the recovery of data suffering from missing entries or outliers. In many real-world applications however, the data is also time-varying, and the naive approach of per-snapshot recovery is both expensive and sub-optimal. This paper develops generative Bayesian models that fit sequential multivariate measurements arising from a low-dimensional time-varying subspace. A variational Bayesian subspace filtering approach is proposed that learns the underlying subspace and its state-transition matrix. Different from the plethora of deterministic counterparts, the proposed approach utilizes automatic relevance determination priors that obviate the need to tune key parameters such as rank and noise power. We also propose a forward-backward algorithm that allows the updates to be carried out at low complexity. Extensive tests over traffic and electricity data demonstrate the superior imputation, outlier rejection, and temporal prediction prowess of the proposed algorithm over the state-of-the-art matrix/tensor completion algorithms., Comment: 13 Pages

We consider the problem of tracking the minimum of a time-varying convex optimization problem over a dynamic graph. Motivated by target tracking and parameter estimation problems in intermittently connected robotic and sensor networks, the goal is to design a distributed algorithm capable of handling non-differentiable regularization penalties. The proposed proximal online gradient descent algorithm is built to run in a fully decentralized manner and utilizes consensus updates over possibly disconnected graphs. The performance of the proposed algorithm is analyzed by developing bounds on its dynamic regret in terms of the cumulative path length of the time-varying optimum. It is shown that as compared to the centralized case, the dynamic regret incurred by the proposed algorithm over $T$ time slots is worse by a factor of $\log(T)$ only, despite the disconnected and time-varying network topology. The empirical performance of the proposed algorithm is tested on the distributed dynamic sparse recovery problem, where it is shown to incur a dynamic regret that is close to that of the centralized algorithm.