282 research outputs found
Stretching Homopolymers
Force induced stretching of polymers is important in a variety of contexts.
We have used theory and simulations to describe the response of homopolymers,
with monomers, to force () in good and poor solvents. In good solvents
and for {{sufficiently large}} we show, in accord with scaling predictions,
that the mean extension along the axis for small , and
(the Pincus regime) for intermediate values of . The
theoretical predictions for \la Z\ra as a function of are in excellent
agreement with simulations for N=100 and 1600. However, even with N=1600, the
expected Pincus regime is not observed due to the the breakdown of the
assumptions in the blob picture for finite . {{We predict the Pincus scaling
in a good solvent will be observed for }}. The force-dependent
structure factors for a polymer in a poor solvent show that there are a
hierarchy of structures, depending on the nature of the solvent. For a weakly
hydrophobic polymer, various structures (ideal conformations, self-avoiding
chains, globules, and rods) emerge on distinct length scales as is varied.
A strongly hydrophobic polymer remains globular as long as is less than a
critical value . Above , an abrupt first order transition to a
rod-like structure occurs. Our predictions can be tested using single molecule
experiments.Comment: 24 pages, 7 figure
Depletion effects and loop formation in self-avoiding polymers
Langevin dynamics is employed to study the looping kinetics of self-avoiding
polymers both in ideal and crowded solutions. A rich kinetics results from the
competition of two crowding-induced effects: the depletion attraction and the
enhanced viscous friction. For short chains, the enhanced friction slows down
looping, while, for longer chains, the depletion attraction renders it more
frequent and persistent. We discuss the possible relevance of the findings for
chromatin looping in living cells.Comment: 4 pages, 3 figure
Generalizing DP-SGD with Shuffling and Batch Clipping
Classical differential private DP-SGD implements individual clipping with
random subsampling, which forces a mini-batch SGD approach. We provide a
general differential private algorithmic framework that goes beyond DP-SGD and
allows any possible first order optimizers (e.g., classical SGD and momentum
based SGD approaches) in combination with batch clipping, which clips an
aggregate of computed gradients rather than summing clipped gradients (as is
done in individual clipping). The framework also admits sampling techniques
beyond random subsampling such as shuffling. Our DP analysis follows the -DP
approach and introduces a new proof technique which allows us to derive simple
closed form expressions and to also analyse group privacy. In particular, for
epochs work and groups of size , we show a DP dependency
for batch clipping with shuffling.Comment: Update disclaimer
Batch Clipping and Adaptive Layerwise Clipping for Differential Private Stochastic Gradient Descent
Each round in Differential Private Stochastic Gradient Descent (DPSGD)
transmits a sum of clipped gradients obfuscated with Gaussian noise to a
central server which uses this to update a global model which often represents
a deep neural network. Since the clipped gradients are computed separately,
which we call Individual Clipping (IC), deep neural networks like resnet-18
cannot use Batch Normalization Layers (BNL) which is a crucial component in
deep neural networks for achieving a high accuracy. To utilize BNL, we
introduce Batch Clipping (BC) where, instead of clipping single gradients as in
the orginal DPSGD, we average and clip batches of gradients. Moreover, the
model entries of different layers have different sensitivities to the added
Gaussian noise. Therefore, Adaptive Layerwise Clipping methods (ALC), where
each layer has its own adaptively finetuned clipping constant, have been
introduced and studied, but so far without rigorous DP proofs. In this paper,
we propose {\em a new ALC and provide rigorous DP proofs for both BC and ALC}.
Experiments show that our modified DPSGD with BC and ALC for CIFAR- with
resnet- converges while DPSGD with IC and ALC does not.Comment: 20 pages, 18 Figure
Hogwild! over Distributed Local Data Sets with Linearly Increasing Mini-Batch Sizes
Hogwild! implements asynchronous Stochastic Gradient Descent (SGD) where
multiple threads in parallel access a common repository containing training
data, perform SGD iterations and update shared state that represents a jointly
learned (global) model. We consider big data analysis where training data is
distributed among local data sets in a heterogeneous way -- and we wish to move
SGD computations to local compute nodes where local data resides. The results
of these local SGD computations are aggregated by a central "aggregator" which
mimics Hogwild!. We show how local compute nodes can start choosing small
mini-batch sizes which increase to larger ones in order to reduce communication
cost (round interaction with the aggregator). We improve state-of-the-art
literature and show ) communication rounds for heterogeneous data
for strongly convex problems, where is the total number of gradient
computations across all local compute nodes. For our scheme, we prove a
\textit{tight} and novel non-trivial convergence analysis for strongly convex
problems for {\em heterogeneous} data which does not use the bounded gradient
assumption as seen in many existing publications. The tightness is a
consequence of our proofs for lower and upper bounds of the convergence rate,
which show a constant factor difference. We show experimental results for plain
convex and non-convex problems for biased (i.e., heterogeneous) and unbiased
local data sets.Comment: arXiv admin note: substantial text overlap with arXiv:2007.09208
AISTATS 202
Theory of biopolymer stretching at high forces
We provide a unified theory for the high force elasticity of biopolymers
solely in terms of the persistence length, , and the monomer spacing,
. When the force f>\fh \sim k_BT\xi_p/a^2 the biopolymers behave as Freely
Jointed Chains (FJCs) while in the range \fl \sim k_BT/\xi_p < f < \fh the
Worm-like Chain (WLC) is a better model. We show that can be estimated
from the force extension curve (FEC) at the extension
(normalized by the contour length of the biopolymer). After validating the
theory using simulations, we provide a quantitative analysis of the FECs for a
diverse set of biopolymers (dsDNA, ssRNA, ssDNA, polysaccharides, and
unstructured PEVK domain of titin) for . The success of a specific
polymer model (FJC or WLC) to describe the FEC of a given biopolymer is
naturally explained by the theory. Only by probing the response of biopolymers
over a wide range of forces can the -dependent elasticity be fully
described.Comment: 20 pages, 4 figure
Roots of the derivative of the Riemann zeta function and of characteristic polynomials
We investigate the horizontal distribution of zeros of the derivative of the
Riemann zeta function and compare this to the radial distribution of zeros of
the derivative of the characteristic polynomial of a random unitary matrix.
Both cases show a surprising bimodal distribution which has yet to be
explained. We show by example that the bimodality is a general phenomenon. For
the unitary matrix case we prove a conjecture of Mezzadri concerning the
leading order behavior, and we show that the same follows from the random
matrix conjectures for the zeros of the zeta function.Comment: 24 pages, 6 figure
Generalizing DP-SGD with shuffling and batch clipping
Classical differential private DP-SGD implements individual clipping with random subsampling,
which forces a mini-batch SGD approach. We provide a general differential private algorithmic
framework that goes beyond DP-SGD and allows any possible first order optimizers (e.g., classical
SGD and momentum based SGD approaches) in combination with batch clipping, which clips an
aggregate of computed gradients rather than summing clipped gradients (as is done in individual
clipping). The framework also admits sampling techniques beyond random subsampling such as
shuffling. Our DP analysis follows the f -DP approach and introduces a new proof technique based
on a slightly stronger adversarial model which allows us to derive simple closed form expressions
and to also analyse group privacy. In particular, for E epochs work and groups of size g, we show a√gE DP dependency for batch clipping with shuffling
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