In distributed second order optimization, a standard strategy is to average many local estimates, each of which is based on a small sketch or batch of the data. However, the local estimates on each machine are typically biased, relative to the full solution on all of the data, and this can limit the effectiveness of averaging. Here, we introduce a new technique for debiasing the local estimates, which leads to both theoretical and empirical improvements in the convergence rate of distributed second order methods. Our technique has two novel components: (1) modifying standard sketching techniques to obtain what we call a surrogate sketch; and (2) carefully scaling the global regularization parameter for local computations. Our surrogate sketches are based on determinantal point processes, a family of distributions for which the bias of an estimate of the inverse Hessian can be computed exactly. Based on this computation, we show that when the objective being minimized is l2-regularized with parameter ! and individual machines are each given a sketch of size m, then to eliminate the bias, local estimates should be computed using a shrunk regularization parameter given by (See PDF), where d(See PDF) is the (See PDF)-effective dimension of the Hessianmore »
Adaptive Newton Sketch: Linear-time Optimization with Quadratic Convergence and Effective Hessian Dimensionality
We propose a randomized algorithm with quadratic convergence rate for convex optimization problems with a self-concordant, composite, strongly convex objective function. Our method is based on performing an approximate Newton step using a random projection of the Hessian. Our first contribution is to show that, at each iteration, the embedding dimension (or sketch size) can be as small as the effective dimension of the Hessian matrix. Leveraging this novel fundamental result, we design an algorithm with a sketch size proportional to the effective dimension and which exhibits a quadratic rate of convergence. This result dramatically improves on the classical linear-quadratic convergence rates of state-of-theart sub-sampled Newton methods. However, in most practical cases, the effective dimension is not known beforehand, and this raises the question of how to pick a sketch size as small as the effective dimension while preserving a quadratic convergence rate. Our second and main contribution is thus to propose an adaptive sketch size algorithm with quadratic convergence rate and which does not require prior knowledge or estimation of the effective dimension: at each iteration, it starts with a small sketch size, and increases it until quadratic progress is achieved. Importantly, we show that the embedding dimension remains proportional more »
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- Preceedings of the 38th International Conference on Machine Learning
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- National Science Foundation
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