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This paper studies the convergence properties of a family of Relaxed $\ell$-Minimal Gradient Descent methods for quadratic optimization; the family includes the omnipresent Steepest Descent method, as well as the Minimal Gradient method. Simple proofs are provided that show, in an appropriately chosen norm, the gradient and the distance of the iterates from optimality converge linearly, for all members of the family. Moreover, the function values decrease linearly, and iteration complexity results are provided. All theoretical results hold when (fixed) relaxation is employed. It is also shown that, given a fixed overhead and storage budget, every Relaxed $\ell$-Minimal Gradient Descent method can be implemented using exactly one matrix vector product. Numerical experiments are presented that illustrate the benefits of relaxation across the family., Comment: 23 pages, 6 figures, 2 tables

This work shows that applying Gradient Descent (GD) with a fixed step size to minimize a (possibly nonconvex) quadratic function is equivalent to running the Power Method (PM) on the gradients. The connection between GD with a fixed step size and the PM, both with and without fixed momentum, is thus established. Consequently, valuable eigen-information is available via GD. Recent examples show that GD with a fixed step size, applied to locally quadratic nonconvex functions, can take exponential time to escape saddle points (Simon S. Du, Chi Jin, Jason D. Lee, Michael I. Jordan, Aarti Singh, and Barnabas Poczos: "Gradient descent can take exponential time to escape saddle points"; S. Paternain, A. Mokhtari, and A. Ribeiro: "A newton-based method for nonconvex optimization with fast evasion of saddle points"). Here, those examples are revisited and it is shown that eigenvalue information was missing, so that the examples may not provide a complete picture of the potential practical behaviour of GD. Thus, ongoing investigation of the behaviour of GD on nonconvex functions, possibly with an \emph{adaptive} or \emph{variable} step size, is warranted. It is shown that, in the special case of a quadratic in $R^2$, if an eigenvalue is known, then GD with a fixed step size will converge in two iterations, and a complete eigen-decomposition is available. By considering the dynamics of the gradients and iterates, new step size strategies are proposed to improve the practical performance of GD. Several numerical examples are presented, which demonstrate the advantages of exploiting the GD--PM connection.

Inspired by the recent work FedNL (Safaryan et al, FedNL: Making Newton-Type Methods Applicable to Federated Learning), we propose a new communication efficient second-order framework for Federated learning, namely FLECS. The proposed method reduces the high-memory requirements of FedNL by the usage of an L-SR1 type update for the Hessian approximation which is stored on the central server. A low dimensional `sketch' of the Hessian is all that is needed by each device to generate an update, so that memory costs as well as number of Hessian-vector products for the agent are low. Biased and unbiased compressions are utilized to make communication costs also low. Convergence guarantees for FLECS are provided in both the strongly convex, and nonconvex cases, and local linear convergence is also established under strong convexity. Numerical experiments confirm the practical benefits of this new FLECS algorithm.

This work considers the non-convex finite sum minimization problem. There are several algorithms for such problems, but existing methods often work poorly when the problem is badly scaled and/or ill-conditioned, and a primary goal of this work is to introduce methods that alleviate this issue. Thus, here we include a preconditioner based on Hutchinson's approach to approximating the diagonal of the Hessian, and couple it with several gradient-based methods to give new scaled algorithms: Scaled SARAH and Scaled L-SVRG. Theoretical complexity guarantees under smoothness assumptions are presented. We prove linear convergence when both smoothness and the PL condition are assumed. Our adaptively scaled methods use approximate partial second-order curvature information and, therefore, can better mitigate the impact of badly scaled problems. This improved practical performance is demonstrated in the numerical experiments also presented in this work., Comment: 40 pages, 2 new algorithms, 20 figures, 4 tables

This paper considers the problem of decentralized, personalized federated learning. For centralized personalized federated learning, a penalty that measures the deviation from the local model and its average, is often added to the objective function. However, in a decentralized setting this penalty is expensive in terms of communication costs, so here, a different penalty - one that is built to respect the structure of the underlying computational network - is used instead. We present lower bounds on the communication and local computation costs for this problem formulation and we also present provably optimal methods for decentralized personalized federated learning. Numerical experiments are presented to demonstrate the practical performance of our methods., Comment: New in v3: more detailed proofs, more experiments. 40 pages, 6 algorithms, 10 figures, 2 tables, 5 theorems

This work presents a new algorithm for empirical risk minimization. The algorithm bridges the gap between first- and second-order methods by computing a search direction that uses a second-order-type update in one subspace, coupled with a scaled steepest descent step in the orthogonal complement. To this end, partial curvature information is incorporated to help with ill-conditioning, while simultaneously allowing the algorithm to scale to the large problem dimensions often encountered in machine learning applications. Theoretical results are presented to confirm that the algorithm converges to a stationary point in both the strongly convex and nonconvex cases. A stochastic variant of the algorithm is also presented, along with corresponding theoretical guarantees. Numerical results confirm the strengths of the new approach on standard machine learning problems., Comment: 38 pages, 74 figures

This work investigates finite differences and the use of interpolation models to obtain approximations to the first and second derivatives of a function. Here, it is shown that if a particular set of points is used in the interpolation model, then the solution to the associated linear system (i.e., approximations to the gradient and diagonal of the Hessian) can be obtained in $\mathcal{O}(n)$ computations, which is the same cost as finite differences, and is a saving over the $\mathcal{O}(n^3)$ cost when solving a general unstructured linear system. Moreover, if the interpolation points are formed using a `regular minimal positive basis', then the error bound for the gradient approximation is the same as for a finite differences approximation. Numerical experiments are presented that show how the derivative estimates can be employed within an existing derivative free optimization algorithm, thus demonstrating one of the potential practical uses of these derivative approximations.

In this paper we develop an adaptive dual free Stochastic Dual Coordinate Ascent (adfSDCA) algorithm for regularized empirical risk minimization problems. This is motivated by the recent work on dual free SDCA of Shalev-Shwartz (2016). The novelty of our approach is that the coordinates to update at each iteration are selected non-uniformly from an adaptive probability distribution, and this extends the previously mentioned work which only allowed for a uniform selection of "dual" coordinates from a fixed probability distribution. We describe an efficient iterative procedure for generating the non-uniform samples, where the scheme selects the coordinate with the greatest potential to decrease the sub-optimality of the current iterate. We also propose a heuristic variant of adfSDCA that is more aggressive than the standard approach. Furthermore, in order to utilize multi-core machines we consider a mini-batch adfSDCA algorithm and develop complexity results that guarantee the algorithm's convergence. The work is concluded with several numerical experiments to demonstrate the practical benefits of the proposed approach.

In this work we introduce the concept of an Underestimate Sequence (UES), which is motivated by Nesterov's estimate sequence. Our definition of a UES utilizes three sequences, one of which is a lower bound (or under-estimator) of the objective function. The question of how to construct an appropriate sequence of lower bounds is addressed, and we present lower bounds for strongly convex smooth functions and for strongly convex composite functions, which adhere to the UES framework. Further, we propose several first order methods for minimizing strongly convex functions in both the smooth and composite cases. The algorithms, based on efficiently updating lower bounds on the objective functions, have natural stopping conditions that provide the user with a certificate of optimality. Convergence of all algorithms is guaranteed through the UES framework, and we show that all presented algorithms converge linearly, with the accelerated variants enjoying the optimal linear rate of convergence.

Simplex gradients are an essential feature of many derivative free optimization algorithms, and can be employed, for example, as part of the process of defining a direction of search, or as part of a termination criterion. The calculation of a general simplex gradient in $\mathbb{R}^n$ can be computationally expensive, and often requires an overhead operation count of $\mathcal{O}(n^3)$ and in some algorithms a storage overhead of $\mathcal{O}(n^2)$. In this work we demonstrate that the linear algebra overhead and storage costs can be reduced, both to $\mathcal{O}(n)$, when the simplex employed is regular and appropriately aligned. We also demonstrate that a second order gradient approximation can be obtained cheaply from a combination of two, first order (appropriately aligned) regular simplex gradients. Moreover, we show that, for an arbitrarily aligned regular simplex, the gradient can be computed in only $\mathcal{O}(n^2)$ operations., Comment: 28 pages, 1 figure

We present a novel randomized block coordinate descent method for the minimization of a convex composite objective function. The method uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems. We call the method Flexible Coordinate Descent (FCD). At each iteration of FCD, a block of coordinates is sampled randomly, a quadratic model is formed about that block and the model is minimized \emph{approximately/inexactly} to determine the search direction. An inexpensive line search is then employed to ensure a monotonic decrease in the objective function and acceptance of large step sizes. We present several high probability iteration complexity results to show that convergence of FCD is guaranteed theoretically. Finally, we present numerical results on large-scale problems to demonstrate the practical performance of the method., Comment: 31 pages, 24 figures

In this paper we generalize the framework of the feasible descent method (FDM) to a randomized (R-FDM) and a coordinate-wise random feasible descent method (RC-FDM) framework. We show that the famous SDCA algorithm for optimizing the SVM dual problem, or the stochastic coordinate descent method for the LASSO problem, fits into the framework of RC-FDM. We prove linear convergence for both R-FDM and RC-FDM under the weak strong convexity assumption. Moreover, we show that the duality gap converges linearly for RC-FDM, which implies that the duality gap also converges linearly for SDCA applied to the SVM dual problem.

In this work we study the parallel coordinate descent method (PCDM) proposed by Richt\'arik and Tak\'a\v{c} [26] for minimizing a regularized convex function. We adopt elements from the work of Xiao and Lu [39], and combine them with several new insights, to obtain sharper iteration complexity results for PCDM than those presented in [26]. Moreover, we show that PCDM is monotonic in expectation, which was not confirmed in [26], and we also derive the first high probability iteration complexity result where the initial levelset is unbounded., Comment: 26 pages; 2 figures

We present a flexible Alternating Direction Method of Multipliers (F-ADMM) algorithm for solving optimization problems involving a strongly convex objective function that is separable into $n \geq 2$ blocks, subject to (non-separable) linear equality constraints. The F-ADMM algorithm uses a Gauss-Seidel scheme to update blocks of variables, and a regularization term is added to each of the subproblems arising within F-ADMM. We prove, under common assumptions, that F-ADMM is globally convergent. We also present a special case of F-ADMM that is partially parallelizable, which makes it attractive in a big data setting. In particular, we partition the data into groups, so that each group consists of multiple blocks of variables. By applying F-ADMM to this partitioning of the data, and using a specific regularization matrix, we obtain a hybrid ADMM (H-ADMM) algorithm: the grouped data is updated in a Gauss-Seidel fashion, and the blocks within each group are updated in a Jacobi manner. Convergence of H-ADMM follows directly from the convergence properties of F-ADMM. Also, a special case of H-ADMM can be applied to functions that are convex, rather than strongly convex. We present numerical experiments to demonstrate the practical advantages of this algorithm., Comment: 20 pages

In this paper we present a novel randomized block coordinate descent method for the minimization of a convex composite objective function. The method uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems. We call the method Robust Coordinate Descent (RCD). At each iteration of RCD, a block of coordinates is sampled randomly, a quadratic model is formed about that block and the model is minimized approximately/inexactly to determine the search direction. An inexpensive line search is then employed to ensure a monotonic decrease in the objective function and acceptance of large step sizes. We prove global convergence of the RCD algorithm, and we also present several results on the local convergence of RCD for strongly convex functions. Finally, we present numerical results on large-scale problems to demonstrate the practical performance of the method., Comment: 23 pages, 6 figures

In this paper we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the Diagonal Quadratic Approximation Method (DQAM) of Mulvey and Ruszczy\'{n}ski and the Parallel Coordinate Descent Method (PCDM) of Richt\'arik and Tak\'a\v{c}. We show that the two methods are equivalent for feasibility problems up to the selection of a single step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least $8(L'/\bar{L})(\omega-1)^2$ times better than the best known bound for DQAM, where $\omega$ is the degree of partial separability and $L'$ and $\bar{L}$ are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian., Comment: 28 pages, 6 algorithms, 2 figures

In this paper we consider the problem of minimizing a convex function using a randomized block coordinate descent method. One of the key steps at each iteration of the algorithm is determining the update to a block of variables. Existing algorithms assume that in order to compute the update, a particular subproblem is solved exactly. In his work we relax this requirement, and allow for the subproblem to be solved inexactly, leading to an inexact block coordinate descent method. Our approach incorporates the best known results for exact updates as a special case. Moreover, these theoretical guarantees are complemented by practical considerations: the use of iterative techniques to determine the update as well as the use of preconditioning for further acceleration., Comment: 32 pages, 6 tables, 2 figures, 1 algorithm