Stochastic gradient algorithms are often unstable when applied to functions that do not have Lipschitz-continuous and/or bounded gradients. Gradient clipping is a simple and effective technique to stabilize the training process for problems that are prone to the exploding gradient problem. Despite its widespread popularity, the convergence properties of the gradient clipping heuristic are poorly understood, especially for stochastic problems. This paper establishes both qualitative and quantitative convergence results of the clipped stochastic (sub)gradient method (SGD) for non-smooth convex functions with rapidly growing subgradients. Our analyses show that clipping enhances the stability of SGD and that the clipped SGD algorithm enjoys finite convergence rates in many cases. We also study the convergence of a clipped method with momentum, which includes clipped SGD as a special case, for weakly convex problems under standard assumptions. With a novel Lyapunov analysis, we show that the proposed method achieves the best-known rate for the considered class of problems, demonstrating the effectiveness of clipped methods also in this regime. Numerical results confirm our theoretical developments.

Anderson acceleration is a well-established and simple technique for speeding up fixed-point computations with countless applications. This work introduces novel methods for adapting Anderson acceleration to proximal gradient algorithms. Under some technical conditions, we extend existing local convergence results of Anderson acceleration for smooth fixed-point mappings to the proposed non-smooth setting. We also prove analytically that it is in general, impossible to guarantee global convergence of native Anderson acceleration. We therefore propose a simple scheme for stabilization that combines the global worst-case guarantees of proximal gradient methods with the local adaptation and practical speed-up of Anderson acceleration. Finally, we provide the first applications of Anderson acceleration to non-Euclidean geometry.

Stochastic gradient methods with momentum are widely used in applications and at the core of optimization subroutines in many popular machine learning libraries. However, their sample complexities have not been obtained for problems beyond those that are convex or smooth. This paper establishes the convergence rate of a stochastic subgradient method with a momentum term of Polyak type for a broad class of non-smooth, non-convex, and constrained optimization problems. Our key innovation is the construction of a special Lyapunov function for which the proven complexity can be achieved without any tuning of the momentum parameter. For smooth problems, we extend the known complexity bound to the constrained case and demonstrate how the unconstrained case can be analyzed under weaker assumptions than the state-of-the-art. Numerical results confirm our theoretical developments.

This paper introduces a novel technique for nonlinear acceleration of first-order methods for constrained convex optimization. Previous studies of nonlinear acceleration have only been able to provide convergence guarantees for unconstrained convex optimization. In contrast, our method is able to avoid infeasibility of the accelerated iterates and retains the theoretical performance guarantees of the unconstrained case. We focus on Anderson acceleration of the classical projected gradient descent (PGD) method, but our techniques can easily be extended to more sophisticated algorithms, such as mirror descent. Due to the presence of a constraint set, the relevant fixed-point mapping for PGD is not differentiable. However, we show that the convergence results for Anderson acceleration of smooth fixed-point iterations can be extended to the non-smooth case under certain technical conditions.

This paper introduces an efficient second-order method for solving the elastic net problem. Its key innovation is a computationally efficient technique for injecting curvature information in the optimization process which admits a strong theoretical performance guarantee. In particular, we show improved run time over popular first-order methods and quantify the speed-up in terms of statistical measures of the data matrix. The improved time complexity is the result of an extensive exploitation of the problem structure and a careful combination of second-order information, variance reduction techniques, and momentum acceleration. Beside theoretical speed-up, experimental results demonstrate great practical performance benefits of curvature information, especially for ill-conditioned data sets.

Cochannel interference is one of the inevitable deleterious components in designing and analyzing of a wireless network. The use of multiple antennas at both transmitting and receiving nodes is a promising technique to suppress and/or alleviate the effect of cochannel interference on capacity. In this paper, we assess the effects of both antenna correlation and cochannel interference on the ergodic capacity of multiple-input multiple-output channels with covariance feedback. In particular, we consider a general family of spatial fading correlation model-called unitary-independent-unitary-which encompasses most of zero-mean channels with arbitrary fading profiles including the popular separable correlation channel models. We derive the average minimum mean-square error and signal-to-interference-plus-noise ratio of the parallel spatial streams using Berezin's supermathematics. We then put forth the structure of optimal input covariance matrix maximizing the mutual information connected with the necessary and sufficient conditions as a generalization of the noise-limited case, which is tested by a simple iterative algorithm. Together with the powerful supermathematical framework, the result in the paper enables us to quantify the multiuser MIMO interference effects on the capacity in terms of spatial correlation and interference power heterogeneity.

We introduce a new achievability scheme, termed opportunistic network decoupling (OND), operating in virtual full-duplex mode. In the scheme, a novel relay scheduling strategy is utilized in the K × N × K channel with interfering relays, consisting of K source-destination pairs and N half-duplex relays in-between them. A subset of relays using alternate relaying is opportunistically selected in terms of producing the minimum total interference level, thereby resulting in network decoupling. As our main result, it is shown that under a certain relay scaling condition, the OND protocol achieves K degrees of freedom even in the presence of interfering links among relays. Numerical evaluation is also shown to validate the performance of the proposed OND. Our protocol basically operates in a fully distributed fashion along with local channel state information, thereby resulting in relatively easy implementation.