Zheng Qu

Xu Qian, Zheng Qu, Peter Richtárik

We study the problem of minimizing the average of a very large number of smooth functions, which is of key importance in training supervised learning models. One of the most celebrated methods in this context is the SAGA algorithm. Despite years of research on the topic, a general-purpose version of SAGA---one that would include arbitrary importance sampling and minibatching schemes---does not exist. We remedy this situation and propose a general and flexible variant of SAGA following the {\em arbitrary sampling} paradigm. We perform an iteration complexity analysis of the method, largely possible due to the construction of new stochastic Lyapunov functions. We establish linear convergence rates in the smooth and strongly convex regime, and under a quadratic functional growth condition (i.e., in a regime not assuming strong convexity). Our rates match those of the primal-dual method Quartz for which an arbitrary sampling analysis is available, which makes a significant step towards closing the gap in our understanding of complexity of primal and dual methods for finite sum problems.

Zeyuan Allen-Zhu, Zheng Qu, Peter Richtárik et al.

Accelerated coordinate descent is widely used in optimization due to its cheap per-iteration cost and scalability to large-scale problems. Up to a primal-dual transformation, it is also the same as accelerated stochastic gradient descent that is one of the central methods used in machine learning. In this paper, we improve the best known running time of accelerated coordinate descent by a factor up to $\sqrt{n}$. Our improvement is based on a clean, novel non-uniform sampling that selects each coordinate with a probability proportional to the square root of its smoothness parameter. Our proof technique also deviates from the classical estimation sequence technique used in prior work. Our speed-up applies to important problems such as empirical risk minimization and solving linear systems, both in theory and in practice.

Dominik Csiba, Zheng Qu, Peter Richtárik

This paper introduces AdaSDCA: an adaptive variant of stochastic dual coordinate ascent (SDCA) for solving the regularized empirical risk minimization problems. Our modification consists in allowing the method adaptively change the probability distribution over the dual variables throughout the iterative process. AdaSDCA achieves provably better complexity bound than SDCA with the best fixed probability distribution, known as importance sampling. However, it is of a theoretical character as it is expensive to implement. We also propose AdaSDCA+: a practical variant which in our experiments outperforms existing non-adaptive methods.

Zheng Qu, Peter Richtárik, Martin Takáč et al.

We propose a new algorithm for minimizing regularized empirical loss: Stochastic Dual Newton Ascent (SDNA). Our method is dual in nature: in each iteration we update a random subset of the dual variables. However, unlike existing methods such as stochastic dual coordinate ascent, SDNA is capable of utilizing all curvature information contained in the examples, which leads to striking improvements in both theory and practice - sometimes by orders of magnitude. In the special case when an L2-regularizer is used in the primal, the dual problem is a concave quadratic maximization problem plus a separable term. In this regime, SDNA in each step solves a proximal subproblem involving a random principal submatrix of the Hessian of the quadratic function; whence the name of the method. If, in addition, the loss functions are quadratic, our method can be interpreted as a novel variant of the recently introduced Iterative Hessian Sketch.

Zheng Qu, Peter Richtárik

The design and complexity analysis of randomized coordinate descent methods, and in particular of variants which update a random subset (sampling) of coordinates in each iteration, depends on the notion of expected separable overapproximation (ESO). This refers to an inequality involving the objective function and the sampling, capturing in a compact way certain smoothness properties of the function in a random subspace spanned by the sampled coordinates. ESO inequalities were previously established for special classes of samplings only, almost invariably for uniform samplings. In this paper we develop a systematic technique for deriving these inequalities for a large class of functions and for arbitrary samplings. We demonstrate that one can recover existing ESO results using our general approach, which is based on the study of eigenvalues associated with samplings and the data describing the function.

Zheng Qu, Peter Richtárik

We study the problem of minimizing the sum of a smooth convex function and a convex block-separable regularizer and propose a new randomized coordinate descent method, which we call ALPHA. Our method at every iteration updates a random subset of coordinates, following an arbitrary distribution. No coordinate descent methods capable to handle an arbitrary sampling have been studied in the literature before for this problem. ALPHA is a remarkably flexible algorithm: in special cases, it reduces to deterministic and randomized methods such as gradient descent, coordinate descent, parallel coordinate descent and distributed coordinate descent -- both in nonaccelerated and accelerated variants. The variants with arbitrary (or importance) sampling are new. We provide a complexity analysis of ALPHA, from which we deduce as a direct corollary complexity bounds for its many variants, all matching or improving best known bounds.

Zheng Qu, Peter Richtárik, Tong Zhang

We study the problem of minimizing the average of a large number of smooth convex functions penalized with a strongly convex regularizer. We propose and analyze a novel primal-dual method (Quartz) which at every iteration samples and updates a random subset of the dual variables, chosen according to an arbitrary distribution. In contrast to typical analysis, we directly bound the decrease of the primal-dual error (in expectation), without the need to first analyze the dual error. Depending on the choice of the sampling, we obtain efficient serial, parallel and distributed variants of the method. In the serial case, our bounds match the best known bounds for SDCA (both with uniform and importance sampling). With standard mini-batching, our bounds predict initial data-independent speedup as well as additional data-driven speedup which depends on spectral and sparsity properties of the data. We calculate theoretical speedup factors and find that they are excellent predictors of actual speedup in practice. Moreover, we illustrate that it is possible to design an efficient mini-batch importance sampling. The distributed variant of Quartz is the first distributed SDCA-like method with an analysis for non-separable data.

Jakub Konečný, Zheng Qu, Peter Richtárik

We propose a novel stochastic gradient method---semi-stochastic coordinate descent (S2CD)---for the problem of minimizing a strongly convex function represented as the average of a large number of smooth convex functions: $f(x)=\tfrac{1}{n}\sum_i f_i(x)$. Our method first performs a deterministic step (computation of the gradient of $f$ at the starting point), followed by a large number of stochastic steps. The process is repeated a few times, with the last stochastic iterate becoming the new starting point where the deterministic step is taken. The novelty of our method is in how the stochastic steps are performed. In each such step, we pick a random function $f_i$ and a random coordinate $j$---both using nonuniform distributions---and update a single coordinate of the decision vector only, based on the computation of the $j^{th}$ partial derivative of $f_i$ at two different points. Each random step of the method constitutes an unbiased estimate of the gradient of $f$ and moreover, the squared norm of the steps goes to zero in expectation, meaning that the stochastic estimate of the gradient progressively improves. The complexity of the method is the sum of two terms: $O(n\log(1/ε))$ evaluations of gradients $\nabla f_i$ and $O(\hatκ\log(1/ε))$ evaluations of partial derivatives $\nabla_j f_i$, where $\hatκ$ is a novel condition number.

Olivier Fercoq, Zheng Qu, Peter Richtárik et al.

We propose an efficient distributed randomized coordinate descent method for minimizing regularized non-strongly convex loss functions. The method attains the optimal $O(1/k^2)$ convergence rate, where $k$ is the iteration counter. The core of the work is the theoretical study of stepsize parameters. We have implemented the method on Archer - the largest supercomputer in the UK - and show that the method is capable of solving a (synthetic) LASSO optimization problem with 50 billion variables.