2 research outputs found
Weighted SGD for Regression with Randomized Preconditioning
In recent years, stochastic gradient descent (SGD) methods and randomized
linear algebra (RLA) algorithms have been applied to many large-scale problems
in machine learning and data analysis. We aim to bridge the gap between these
two methods in solving constrained overdetermined linear regression
problems---e.g., and regression problems. We propose a hybrid
algorithm named pwSGD that uses RLA techniques for preconditioning and
constructing an importance sampling distribution, and then performs an SGD-like
iterative process with weighted sampling on the preconditioned system. We prove
that pwSGD inherits faster convergence rates that only depend on the lower
dimension of the linear system, while maintaining low computation complexity.
Particularly, when solving regression with size by , pwSGD
returns an approximate solution with relative error in the objective
value in
time. This complexity is uniformly better than that of RLA methods in terms of
both and when the problem is unconstrained. For
regression, pwSGD returns an approximate solution with relative
error in the objective value and the solution vector measured in prediction
norm in time. We also provide lower bounds on the coreset
complexity for more general regression problems, indicating that still new
ideas will be needed to extend similar RLA preconditioning ideas to weighted
SGD algorithms for more general regression problems. Finally, the effectiveness
of such algorithms is illustrated numerically on both synthetic and real
datasets.Comment: A conference version of this paper appears under the same title in
Proceedings of ACM-SIAM Symposium on Discrete Algorithms, Arlington, VA, 201
Aligning Points to Lines: Provable Approximations
We suggest a new optimization technique for minimizing the sum of non-convex real functions that satisfy a property that we call
piecewise log-Lipschitz. This is by forging links between techniques in
computational geometry, combinatorics and convex optimization. As an example
application, we provide the first constant-factor approximation algorithms
whose running-time is polynomial in for the fundamental problem of
\emph{Points-to-Lines alignment}: Given points and
lines on the plane and , compute the matching
and alignment (rotation matrix and a translation vector
) that minimize the sum of Euclidean distances between each point to its corresponding
line.
This problem is non-trivial even if and the matching is given. If
is given, the running time of our algorithms is , and even
near-linear in using core-sets that support: streaming, dynamic, and
distributed parallel computations in poly-logarithmic update time.
Generalizations for handling e.g. outliers or pseudo-distances such as
-estimators for the problem are also provided.
Experimental results and open source code show that our provable algorithms
improve existing heuristics also in practice. A companion demonstration video
in the context of Augmented Reality shows how such algorithms may be used in
real-time systems