198 research outputs found
Swarm-Based Optimization with Random Descent
We extend our study of the swarm-based gradient descent method for non-convex
optimization, [Lu, Tadmor & Zenginoglu, arXiv:2211.17157], to allow random
descent directions. We recall that the swarm-based approach consists of a swarm
of agents, each identified with a position, , and mass, . The
key is the transfer of mass from high ground to low(-est) ground. The mass of
an agent dictates its step size: lighter agents take larger steps. In this
paper, the essential new feature is the choice of direction: rather than
restricting the swarm to march in the steepest gradient descent, we let agents
proceed in randomly chosen directions centered around -- but otherwise
different from -- the gradient direction. The random search secures the descent
property while at the same time, enabling greater exploration of ambient space.
Convergence analysis and benchmark optimizations demonstrate the effectiveness
of the swarm-based random descent method as a multi-dimensional global
optimizer
Linearly convergent adjoint free solution of least squares problems by random descent
We consider the problem of solving linear least squares problems in a
framework where only evaluations of the linear map are possible. We derive
randomized methods that do not need any other matrix operations than forward
evaluations, especially no evaluation of the adjoint map is needed. Our method
is motivated by the simple observation that one can get an unbiased estimate of
the application of the adjoint. We show convergence of the method and then
derive a more efficient method that uses an exact linesearch. This method,
called random descent, resembles known methods in other context and has the
randomized coordinate descent method as special case. We provide convergence
analysis of the random descent method emphasizing the dependence on the
underlying distribution of the random vectors. Furthermore we investigate the
applicability of the method in the context of ill-posed inverse problems and
show that the method can have beneficial properties when the unknown solution
is rough. We illustrate the theoretical findings in numerical examples. One
particular result is that the random descent method actually outperforms
established transposed-free methods (TFQMR and CGS) in examples
Heuristic pattern search for bound constrained minimax problems
This paper presents a pattern search algorithm and its hybridization
with a random descent search for solving bound constrained minimax problems.
The herein proposed heuristic pattern search method combines the Hooke and
Jeeves (HJ) pattern and exploratory moves with a randomly generated approxi-
mate descent direction. Two versions of the heuristic algorithm have been applied
to several benchmark minimax problems and compared with the original HJ pat-
tern search algorithm
That rare random descent : an approach to the cantos of sylvia plath
The intent of the following thesis is to establish an approach to the body of Sylvia Plath\u27s poetry, to provide some insight into an artistic career which lasted less than ten years. [...] The approach I establish, then, takes the poems in their chronological order. I have attempted an understanding of her mautre work, thus, on its own terms, terms Plath herself established, subsequently broke from and re-styled, so that I might dispel the stultifying effect her suicide has had on our critical interpretation
Compressed sensing of data with a known distribution
Compressed sensing is a technique for recovering an unknown sparse signal
from a small number of linear measurements. When the measurement matrix is
random, the number of measurements required for perfect recovery exhibits a
phase transition: there is a threshold on the number of measurements after
which the probability of exact recovery quickly goes from very small to very
large. In this work we are able to reduce this threshold by incorporating
statistical information about the data we wish to recover. Our algorithm works
by minimizing a suitably weighted -norm, where the weights are chosen
so that the expected statistical dimension of the corresponding descent cone is
minimized. We also provide new discrete-geometry-based Monte Carlo algorithms
for computing intrinsic volumes of such descent cones, allowing us to bound the
failure probability of our methods.Comment: 22 pages, 7 figures. New colorblind safe figures. Sections 3 and 4
completely rewritten. Minor typos fixe
Perceptron learning with random coordinate descent
A perceptron is a linear threshold classifier that separates examples with a hyperplane. It is perhaps the simplest learning model that is used standalone. In this paper, we propose a family of random coordinate descent algorithms for perceptron learning on binary classification problems. Unlike most perceptron learning algorithms which require smooth cost functions, our algorithms directly minimize the training error, and usually achieve the lowest training error compared with other algorithms. The algorithms are also computational efficient. Such advantages make them favorable for both standalone use and ensemble learning, on problems that are not linearly separable. Experiments show that our algorithms work very well with AdaBoost, and achieve the lowest test errors for half of the datasets
A Note on Shortest Developments
De Vrijer has presented a proof of the finite developments theorem which, in
addition to showing that all developments are finite, gives an effective
reduction strategy computing longest developments as well as a simple formula
computing their length.
We show that by applying a rather simple and intuitive principle of duality
to de Vrijer's approach one arrives at a proof that some developments are
finite which in addition yields an effective reduction strategy computing
shortest developments as well as a simple formula computing their length. The
duality fails for general beta-reduction.
Our results simplify previous work by Khasidashvili
Probabilistic Rewriting: On Normalization, Termination, and Unique Normal Forms
While a mature body of work supports the study of rewriting systems, even
infinitary ones, abstract tools for Probabilistic Rewriting are still limited.
Here, we investigate questions such as uniqueness of the result (unique limit
distribution) and we develop a set of proof techniques to analyze and compare
reduction strategies. The goal is to have tools to support the operational
analysis of probabilistic calculi (such as probabilistic lambda-calculi) whose
evaluation is also non-deterministic, in the sense that different reductions
are possible.
In particular, we investigate how the behavior of different rewrite sequences
starting from the same term compare w.r.t. normal forms, and propose a robust
analogue of the notion of "unique normal form". Our approach is that of
Abstract Rewrite Systems, i.e. we search for general properties of
probabilistic rewriting, which hold independently of the specific structure of
the objects.Comment: Extended version of the paper in FSCD 2019, International Conference
on Formal Structures for Computation and Deductio
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