22,275 research outputs found
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
A quasi-Newton proximal splitting method
A new result in convex analysis on the calculation of proximity operators in
certain scaled norms is derived. We describe efficient implementations of the
proximity calculation for a useful class of functions; the implementations
exploit the piece-wise linear nature of the dual problem. The second part of
the paper applies the previous result to acceleration of convex minimization
problems, and leads to an elegant quasi-Newton method. The optimization method
compares favorably against state-of-the-art alternatives. The algorithm has
extensive applications including signal processing, sparse recovery and machine
learning and classification
Adaptive Momentum for Neural Network Optimization
In this thesis, we develop a novel and efficient algorithm for optimizing neural networks inspired by a recently proposed geodesic optimization algorithm. Our algorithm, which we call Stochastic Geodesic Optimization (SGeO), utilizes an adaptive coefficient on top of Polyaks Heavy Ball method effectively controlling the amount of weight put on the previous update to the parameters based on the change of direction in the optimization path. Experimental results on strongly convex functions with Lipschitz gradients and deep Autoencoder benchmarks show that SGeO reaches lower errors than established first-order methods and competes well with lower or similar errors to a recent second-order method called K-FAC (Kronecker-Factored Approximate Curvature). We also incorporate Nesterov style lookahead gradient into our algorithm (SGeO-N) and observe notable improvements. We believe that our research will open up new directions for high-dimensional neural network optimization where combining the efficiency of first-order methods and the effectiveness of second-order methods proves a promising avenue to explore
Breaking the Nonsmooth Barrier: A Scalable Parallel Method for Composite Optimization
Due to their simplicity and excellent performance, parallel asynchronous
variants of stochastic gradient descent have become popular methods to solve a
wide range of large-scale optimization problems on multi-core architectures.
Yet, despite their practical success, support for nonsmooth objectives is still
lacking, making them unsuitable for many problems of interest in machine
learning, such as the Lasso, group Lasso or empirical risk minimization with
convex constraints.
In this work, we propose and analyze ProxASAGA, a fully asynchronous sparse
method inspired by SAGA, a variance reduced incremental gradient algorithm. The
proposed method is easy to implement and significantly outperforms the state of
the art on several nonsmooth, large-scale problems. We prove that our method
achieves a theoretical linear speedup with respect to the sequential version
under assumptions on the sparsity of gradients and block-separability of the
proximal term. Empirical benchmarks on a multi-core architecture illustrate
practical speedups of up to 12x on a 20-core machine.Comment: Appears in Advances in Neural Information Processing Systems 30 (NIPS
2017), 28 page
The Case for Full-Matrix Adaptive Regularization
Adaptive regularization methods come in diagonal and full-matrix variants.
However, only the former have enjoyed widespread adoption in training
large-scale deep models. This is due to the computational overhead of
manipulating a full matrix in high dimension. In this paper, we show how to
make full-matrix adaptive regularization practical and useful. We present GGT,
a truly scalable full-matrix adaptive optimizer. At the heart of our algorithm
is an efficient method for computing the inverse square root of a low-rank
matrix. We show that GGT converges to first-order local minima, providing the
first rigorous theoretical analysis of adaptive regularization in non-convex
optimization. In preliminary experiments, GGT trains faster across a variety of
synthetic tasks and standard deep learning benchmarks
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