Convergence of adaptive algorithms for constrained weakly convex optimization

We analyze the adaptive first order algorithm AMSGrad, for solving a constrained stochastic optimization problem with a weakly convex objective. We prove the $\tilde O(t^{-1/2})$ rate of convergence for the squared norm of the gradient of Moreau envelope, which is the standard stationarity measure for this class of problems. It matches the known rates that adaptive algorithms enjoy for the specific case of unconstrained smooth nonconvex stochastic optimization. Our analysis works with mini-batch size of 1, constant first and second order moment parameters, and possibly unbounded optimization domains. Finally, we illustrate the applications and extensions of our results to specific problems and algorithms.

Decentralized Stochastic Optimization and Gossip Algorithms with Compressed Communication

Martin Jaggi, Anastasiia Koloskova, Sebastian Urban Stich

We consider decentralized stochastic optimization with the objective function (e.g. data samples for machine learning task) being distributed over n machines that can only communicate to their neighbors on a fixed communication graph. To reduce the communication bottleneck, the nodes compress (e.g. quantize or sparsify) their model updates. We cover both unbiased and biased compression operators with quality denoted by \omega 0. This is (up to our knowledge) the first gossip algorithm that supports arbitrary compressed messages for \omega > 0 and still exhibits linear convergence. We (iii) show in experiments that both of our algorithms do outperform the respective state-of-the-art baselines and CHOCO-SGD can reduce communication by at least two orders of magnitudes.

2019

Adaptation in Stochastic Algorithms: From Nonsmooth Optimization to Min-Max Problems and Beyond

Ahmet Alacaoglu

Stochastic gradient descent (SGD) and randomized coordinate descent (RCD) are two of the workhorses for training modern automated decision systems. Intriguingly, convergence properties of these methods are not well-established as we move away from the specific case of smooth minimization. In this dissertation, we focus on related problems of nonsmooth optimization and min-max optimization to improve the theoretical understanding of stochastic algorithms.First, we study SGD-based adaptive algorithms and propose a regret analysis framework overcoming the limitations of the existing ones in the convex case. In the nonconvex case, we prove convergence of an adaptive gradient algoritm for solving constrained weakly convex optimization, generalizing the previously known results on unconstrained smooth optimization. We also propose an algorithm combining Nesterov's smoothing with SGD to solve convex problems with infinitely many linear constraints, with optimal rates.Then, we move on to convex-concave min-max problems with bilinear coupling and analyze primal-dual coordinate descent (PDCD) algorithms. We obtain the first PDCD methods with the optimal

\mathcal{O}(1/k)

rate on the the standard optimality measure expected primal-dual gap, which was an open question since 2014. Our analysis also aims to explain the practical behavior of these algorithms by showing that the last iterate enjoys adaptive linear convergence without altering the parameters, depending on a certain error bound condition. Furthermore, we propose an algorithm combining the favorable properties of two branches of PDCD methods: the new method uses large step sizes with dense data and its per-iteration cost depends on the number of nonzeros of the data matrix. Thanks to these unique properties, this method enjoys compelling practical performance complementing its rigorous theoretical guarantees.Next, we consider monotone variational inequalities that generalize convex-concave min-max problems with nonbilinear coupling. We introduce variance reduced algorithms that converge under the same set of assumptions as their deterministic counterparts and improve the best-known complexities for solving convex-concave min-max problems with finite-sum structure. Optimality of our algorithms for this problem class is established in a recent work via matching lower bounds. Finally, we show our preliminary results on policy optimization methods for solving two player zero-sum Markov games for competitive reinforcement learning (RL). Even though this is a nonconvex-nonconcave min-max problem in general, thanks to the special structure, it is tractable to find an approximate Nash equilibrium. We introduce an algorithm that improves the best-known sample complexity of policy gradient methods. This development combines tools from RL and stochastic primal-dual optimization, showing the importance of techniques from convex-concave optimization.

EPFL2021

Trust-region methods based on radial basis functions with application to biomedical imaging

Rodrigue Oeuvray

We have developed a new derivative-free algorithm based on Radial Basis Functions (RBFs). Derivative-free optimization is an active field of research and several algorithms have been proposed recently. Problems of this nature in the industrial setting are quite frequent. The reason is that in a number of applications the optimization process contains simulation packages which are treated as black boxes. The development of our own algorithm was originally motivated by an application in biomedical imaging: the medical image registration problem. The particular characteristics of this problem have incited us to develop a new optimization algorithm based on trust-region methods. However it has been designed to be generic and to be applied to a wide range of problems. The main originality of our approach is the use of RBFs to build the models. In particular we have adapted the existing theory based on quadratic models to our own models and developed new procedures especially designed for models based on RBFs. We have tested our algorithm called BOOSTERS against state-of-the-art methods (UOBYQA, NEWUOA, DFO). On the medical image registration problem, BOOSTERS appears to be the method of choice. The tests on problems from the CUTEr collection show that BOOSTERS is comparable to, but not better than other methods on small problems (size 2-20). It is performing very well for medium size problems (20-80). Moreover, it is able to solve problems of dimension 200, which is considered very large in derivative-free optimization. We have also developed a new class of algorithms combining the robustness of derivative-free algorithms with the faster rate of convergence characterizing Newtonlike-methods. In fact, they define a new class of algorithms lying between derivative-free optimization and quasi-Newton methods. These algorithms are built on the skeleton of our derivative-free algorithm but they can incorporate the gradient when it is available. They can be interpreted as a way of doping derivative-free algorithms with derivatives. If the derivatives are available at each iteration, then our method can be seen as an alternative to quasi-Newton methods. At the opposite, if the derivatives are never evaluated, then the algorithm is totally similar to BOOSTERS. It is a very interesting alternative to existing methods for problems whose objective function is expensive to evaluate and when the derivatives are not available. In this situation, the gradient can be approximated by finite differences and its costs corresponds to n additional function evaluations assuming that Rn is the domain of definition of the objective function. We have compared our method with CFSQP and BTRA, two gradient-based algorithms, and the results show that our doped method performs best. We have also a theoretical analysis of the medical image registration problem based on maximization of mutual information. Most of the current research in this field is concentrated on registration based on nonlinear image transformation. However, little attention has been paid to the theoretical properties of the optimization problem. In our analysis, we focus on the continuity and the differentiability of the objective function. We show in particular that performing a registration without extension of the reference image may lead to discontinuities in the objective function. But we demonstrate that, under some mild assumptions, the function is differentiable almost everywhere. Our analysis is important from an optimization point of view and conditions the choice of a solver. The usual practice is to use generic optimization packages without worrying about the differentiability of the objective function. But the use of gradient-based methods when the objective function is not differentiable may result in poor performance or even in absence of convergence. One of our objectives with this analysis is also that practitioners become aware of these problems and to propose them new algorithms having a potential interest for their applications.

EPFL2005