14,337 research outputs found
Non-convex Optimization for Machine Learning
A vast majority of machine learning algorithms train their models and perform
inference by solving optimization problems. In order to capture the learning
and prediction problems accurately, structural constraints such as sparsity or
low rank are frequently imposed or else the objective itself is designed to be
a non-convex function. This is especially true of algorithms that operate in
high-dimensional spaces or that train non-linear models such as tensor models
and deep networks.
The freedom to express the learning problem as a non-convex optimization
problem gives immense modeling power to the algorithm designer, but often such
problems are NP-hard to solve. A popular workaround to this has been to relax
non-convex problems to convex ones and use traditional methods to solve the
(convex) relaxed optimization problems. However this approach may be lossy and
nevertheless presents significant challenges for large scale optimization.
On the other hand, direct approaches to non-convex optimization have met with
resounding success in several domains and remain the methods of choice for the
practitioner, as they frequently outperform relaxation-based techniques -
popular heuristics include projected gradient descent and alternating
minimization. However, these are often poorly understood in terms of their
convergence and other properties.
This monograph presents a selection of recent advances that bridge a
long-standing gap in our understanding of these heuristics. The monograph will
lead the reader through several widely used non-convex optimization techniques,
as well as applications thereof. The goal of this monograph is to both,
introduce the rich literature in this area, as well as equip the reader with
the tools and techniques needed to analyze these simple procedures for
non-convex problems.Comment: The official publication is available from now publishers via
http://dx.doi.org/10.1561/220000005
Bayesian Matrix Completion via Adaptive Relaxed Spectral Regularization
Bayesian matrix completion has been studied based on a low-rank matrix
factorization formulation with promising results. However, little work has been
done on Bayesian matrix completion based on the more direct spectral
regularization formulation. We fill this gap by presenting a novel Bayesian
matrix completion method based on spectral regularization. In order to
circumvent the difficulties of dealing with the orthonormality constraints of
singular vectors, we derive a new equivalent form with relaxed constraints,
which then leads us to design an adaptive version of spectral regularization
feasible for Bayesian inference. Our Bayesian method requires no parameter
tuning and can infer the number of latent factors automatically. Experiments on
synthetic and real datasets demonstrate encouraging results on rank recovery
and collaborative filtering, with notably good results for very sparse
matrices.Comment: Accepted to AAAI 201
Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
A general framework for solving image inverse problems is introduced in this
paper. The approach is based on Gaussian mixture models, estimated via a
computationally efficient MAP-EM algorithm. A dual mathematical interpretation
of the proposed framework with structured sparse estimation is described, which
shows that the resulting piecewise linear estimate stabilizes the estimation
when compared to traditional sparse inverse problem techniques. This
interpretation also suggests an effective dictionary motivated initialization
for the MAP-EM algorithm. We demonstrate that in a number of image inverse
problems, including inpainting, zooming, and deblurring, the same algorithm
produces either equal, often significantly better, or very small margin worse
results than the best published ones, at a lower computational cost.Comment: 30 page
Arriving on time: estimating travel time distributions on large-scale road networks
Most optimal routing problems focus on minimizing travel time or distance
traveled. Oftentimes, a more useful objective is to maximize the probability of
on-time arrival, which requires statistical distributions of travel times,
rather than just mean values. We propose a method to estimate travel time
distributions on large-scale road networks, using probe vehicle data collected
from GPS. We present a framework that works with large input of data, and
scales linearly with the size of the network. Leveraging the planar topology of
the graph, the method computes efficiently the time correlations between
neighboring streets. First, raw probe vehicle traces are compressed into pairs
of travel times and number of stops for each traversed road segment using a
`stop-and-go' algorithm developed for this work. The compressed data is then
used as input for training a path travel time model, which couples a Markov
model along with a Gaussian Markov random field. Finally, scalable inference
algorithms are developed for obtaining path travel time distributions from the
composite MM-GMRF model. We illustrate the accuracy and scalability of our
model on a 505,000 road link network spanning the San Francisco Bay Area
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