11institutetext: Fraunhofer IAIS, Sankt Augustin, Germany
11email<EMAIL_ADDRESS>22institutetext: Saarland
University, 22email<EMAIL_ADDRESS>
# Communication-Efficient Distributed Online Learning with Kernels

Michael Kamp ✉ 11 Sebastian Bothe 11 Mario Boley 22 Michael Mock 11

###### Abstract

We propose an efficient distributed online learning protocol for low-latency
real-time services. It extends a previously presented protocol to kernelized
online learners that represent their models by a support vector expansion.
While such learners often achieve higher predictive performance than their
linear counterparts, communicating the support vector expansions becomes
inefficient for large numbers of support vectors. The proposed extension
allows for a larger class of online learning algorithms—including those
alleviating the problem above through model compression. In addition, we
characterize the quality of the proposed protocol by introducing a novel
criterion that requires the communication to be bounded by the loss suffered.

##  1 Introduction

We consider the problem of distributed online learning for low-latency real-
time services [4, 10]. In this scenario, a learning system of $m\in\mathbb{N}$
connected local learners provides a real-time prediction service on multiple
dynamic data streams. In particular, we are interested in generic distributed
online learning protocols that treat concrete learning algorithms as a black-
box. The goal of such a protocol is to provide, in a communication efficient
way, a service quality similar to a serial setting in which all examples are
processed at a central location. While such an optimal predictive performance
can be trivially achieved by centralizing all data, the required continuous
communication usually exceeds practical limits (e.g., bandwidth constraints
[1], latency [21, 8], or battery power [16, 5]). Similarly, communication
limits can be satisfied trivially by letting all local learners work in
isolation. However, this usually comes with a loss of service quality that
increases with the number of local learners.

In previous work, we presented a protocol that effectively reduces
communication while providing strict loss bounds for a class of algorithms
that perform loss-proportional convex updates of linear models [10]. That is,
algorithms that update linear models in the direction of a convex set with a
magnitude proportional to the instantaneous loss (e.g., Stochastic Gradient
Descent [2], or Passive Aggressive [3]). The protocol is able to cease
communication as soon as no loss is suffered anymore. However, for most
realistic problems this cannot be achieved by linear models. Thus, a more
complex hypothesis class is desirable that enables the learners to achieve
zero loss and thus reach quiescence.

Kernelized online learning algorithms can provide such an extended hypothesis
class, but practical versions of these algorithms do not perform loss-
proportional convex updates (e.g., [12, 15, 20]). Therefore, in this paper we
extend the class of algorithms to _approximately_ loss-proportional convex
updates (Sec. 2). This relaxation is particularly crucial for kernelized
online learners for streams that represent the model by its support vector
expansion. These learners use this relaxation in order to reduce the number of
support vectors, since otherwise a monotonically increasing model size would
render them infeasible in streaming settings.

Also, for the first time we characterize the quality of the proposed protocol
by introducing a novel criterion for efficient protocols that requires a
strict loss bound and ties the loss to the allowed amount of communication. In
particular, the criterion implies that the communication vanishes whenever the
loss approaches zero. We bound the loss and communication of the proposed
protocol and show for which class of learning algorithms it fulfills the
efficiency criterion (Sec. 3).

Figure 1:  1 Trade-off between cumulative error and cumulative communication,
and 1 cumulative communication over time of a distributed learning system
using the proposed protocol. The learning task is classifying instances from
the UCI SUSY dataset with $4$ learners, each processing $1000$ instances.
Parameters of the learners are optimized on a separate set of $200$ instances
per learner.

While the strict loss bound required in our criterion can be achieved by
periodically communicating protocols [14, 4], their communication never
vanishes, independent of their loss, which is also required for efficiency. By
communicating only when it significantly improves the service quality, our
protocol achieves similar service quality as any periodically communicating
protocol while communicating less by a factor depending on its in-place loss.

To further amplify this advantage, we apply methods from serial kernelized in-
stream learning approaches. These approaches reduce the number of support
vectors, e.g., by truncating individual support vectors with small weights
[12], or by projecting a single support vector on the span of the remaining
ones [15, 20].

We illustrate the impact of the choice of the hypothesis class on the
predictive performance and communication as well as the impact of model
compression on an example dataset in Fig. 1. In this example, we predicted the
class of instances drawn from the SUSY dataset from the UCI machine learning
repository [13]. The learning systems using linear models continuously suffer
loss resulting in a large cumulative error, but since the linear models are
small compared to support vector expansions, the cumulative communication is
small. A continuously synchronizing protocol using support vector expansions
has a significantly smaller loss at the cost of very high communication, since
each synchronization requires to send models with a growing number of support
vectors. Using the proposed dynamic protocol, this amount of communication can
be reduced without losing in prediction quality. In addition, when using model
compression the communication can be further reduced to an amount similar to
the linear model, but at the cost of prediction quality.

We further discuss the behavior of our protocol with respect to the trade-off
between predictive performance and communication, and point out the strengths
and weaknesses of the protocol in Sec. 4.

##  2 Distributed Online Learning with Kernels

In this section, we provide preliminaries and describe the protocol, extend it
from linear function spaces to kernel Hilbert spaces, and provide an
effectiveness criterion for distributed online learning. For that, we consider
distributed online learning protocols $\Pi=(\mathcal{A},\sigma)$ that run an
online learning algorithm $\mathcal{A}$ on a distributed system of
$m\in\mathbb{N}$ local learners and exchange information between these
learners using a synchronization operator $\sigma$.

#### Preliminaries:

The online learning algorithm $\mathcal{A}=(\mathcal{H},\varphi,\ell)$ run at
each local learner ${i\in[m]}$ maintains a local model $f^{i}\in\mathcal{H}$
from a function space $\mathcal{H}$ using an update rule $\varphi$ and a loss
function $\ell$. That is, at each time point $t\in\mathbb{N}$, each learner
$i$ observes an individual input $\left(x_{t}^{i},y_{t}^{i}\right)$ drawn
independently from a time-variant distribution ${P_{t}\\!:X\times Y\to[0,1]}$
over an input space $X\times Y$ . Based on this input and the local model, the
local learner provides a service whose quality is measured by the loss
function ${\ell\\!:\mathcal{H}\times X\times Y\to\mathbb{R}_{+}}$. After
providing the service, the local learner updates its local model using the
update rule ${\varphi\\!:\mathcal{H}\times X\times Y\to\mathcal{H}}$ in order
to minimize the cumulative loss. The synchronization operator
$\sigma\\!:\mathcal{H}^{m}\to\mathcal{H}^{m}$ transfers the current model
configuration ${\mathbf{f}=\left(f^{1},\dots,f^{m}\right)}$ of $m$ local
models to the synchronized configuration $\sigma(\mathbf{f})$. In the
following, we recapitulate the dynamic protocol presented in [10] as well as
two baseline protocols, i.e., a continuously and a periodic protocol.

Given an online learning algorithm $\mathcal{A}$, the periodic protocol
$\mathcal{P}=(\mathcal{A},\sigma_{b})$ synchronizes every $b\in\mathbb{N}$
time steps the current model configuration ${\mathbf{f}}$ by replacing all
local models by their joint average
$\overline{\mathbf{f}}=\nicefrac{{1}}{{m}}\sum_{i=1}^{m}f^{i}$. That is, the
synchronization operator is given by

$\sigma_{b}(\mathbf{f}_{t})=\begin{cases}\left(\overline{\mathbf{f}}_{t},\dots,\overline{\mathbf{f}}_{t}\right),&\text{
if }b\mid t\\\ \enspace\mathbf{f}_{t}=(f_{t}^{1},\dots,f_{t}^{m}),&\text{
otherwise}\\\ \end{cases}\enspace.$

A special case of this is the continuous protocol
$\mathcal{C}=(\mathcal{A},\sigma_{1})$ that continuously synchronizes every
round, i.e.,
$\sigma_{1}\left(\mathbf{f}\right)=\left(\overline{\mathbf{f}},\dots,\overline{\mathbf{f}}\right)$.

The dynamic protocol $\mathcal{D}=(\mathcal{A},\sigma_{\Delta})$ synchronizes
the local learners using a dynamic operator $\sigma_{\Delta}$ [10]. This
operator only communicates when the model divergence

$\delta(\mathbf{f})=\frac{1}{m}\sum_{i=1}^{m}\left\|f^{i}-\overline{\mathbf{f}}\right\|^{2}$
(1)

exceeds a divergence threshold $\Delta$. That is, the dynamic averaging
operator is defined as

$\sigma_{\Delta}(\mathbf{f}_{t})=\begin{cases}(\overline{\mathbf{f}}_{t},\dots,\overline{\mathbf{f}}_{t}),&\text{
if }\delta(\mathbf{f}_{t})>\Delta\\\ \mathbf{f}_{t},&\text{ otherwise}\\\
\end{cases}\enspace.$

In order to decide when to communicate, each local learner $i\in[m]$ monitors
the local condition $\|f_{t}^{i}-r_{t}\|^{2}\leq\Delta$ for a reference model
$r_{t}\in\mathcal{H}$ that is common among all learners (see [11, 19, 6, 7]
for a more general description of this method). The local conditions guarantee
that if none of them is violated, the divergence does not exceed the threshold
$\Delta$. The closer the reference model is to the true average of local
models, the tighter are the local conditions. Generally, the first choice for
the reference model is the average model from the last synchronization step.
Note, however, that there are several refinements of this choice that can be
used in practice to further reduce communication.

#### Efficiency Criterion:

In the following, we introduce performance measures in order to analyze the
dynamic protocol and compare it to the continuous and periodic protocols. We
measure the predictive performance of a distributed online learning system
until time $T\in\mathbb{N}$ by its cumulative loss

$L(T,m)=\sum_{t=1}^{T}\sum_{i=1}^{m}\ell(f_{t}^{i},\left(x_{t}^{i},y_{t}^{i}\right))\enspace.$

Performance guarantees are typically given by a loss bound $\textbf{L}(T,m)$,
i.e., for all possible input sequences it holds that
$L(T,m)\leq\textbf{L}(T,m)$. These bounds can be defined with respect to a
sequence of reference models, in which case they are referred to as (shifting)
regret bounds.

We measure its performance in terms of communication by its cumulative
communication

$C(T,m)=\sum_{t=1}^{T}c(\mathbf{f}_{t})\enspace,$

where $c\\!:\mathcal{H}^{m}\to\mathbb{N}$ measures the number of bytes
required by the learning protocol to synchronize models
$\mathbf{f}_{t}=\left(f^{1}_{t},\dots,f^{m}_{t}\right)$ at time $t$.

There is a natural trade-off between communication and loss of a distributed
online learning system. On the one hand, a loss similar to a serial setting
can be trivially achieved by continuous synchronization. On the other hand,
communication can be entirely omitted. The trade-off for these two extreme
protocols can be easily determined: if the cumulative loss of an online
learning algorithm $\mathcal{A}$ is bounded by $\textbf{L}_{\mathcal{A}}(T)$,
the loss of a permanently centralizing system with $m$ local learners running
$\mathcal{A}$ is bounded by
$\textbf{L}_{\mathcal{C}}(T,m)=\textbf{L}_{\mathcal{A}}(mT)$, i.e., the loss
bound of a serial online learning algorithm processing $mT$ inputs. The
protocol transmits $\mathcal{O}\left(m\right)$ messages of size up to
$\mathcal{O}\left(T\right)$ in every of the $T$ points in time. At the same
time, the loss of a distributed system without any synchronization is bounded
by $\textbf{L}(T,m)=m\textbf{L}_{\mathcal{A}}(T)$, whereas the communication
is $C(T)=0$.

The communication bound of an adaptive protocol should only depend on
$\textbf{L}_{\mathcal{A}}(T)$ and not on $T$, while at the same time retaining
the loss bound of the serial setting. In the following definition we formalize
this in order to provide a strong criterion for effectiveness of distributed
online learning protocols.

###### Definition 1

A distributed online learning protocol $\Pi=(\mathcal{A},\sigma)$ processing
$mT$ inputs is consistent if it retains the loss bound of the serial online
learning algorithm $\mathcal{A}$, i.e.,

$\textbf{L}_{\Pi}(T,m)\in\mathcal{O}\left(\textbf{L}_{\mathcal{A}}(mT)\right)\enspace.$

The protocol is adaptive if its communication bound is linear in the number of
local learners $m$ and the loss bound $\textbf{L}_{\mathcal{A}}(mT)$ of the
serial online learning algorithm, i.e.,

$C_{\Pi}(T,m)\in\mathcal{O}\left(m\textbf{L}_{\mathcal{A}}(mT)\right)\enspace.$

An efficient protocol is adaptive and consistent at the same time. In the
following section we theoretically analyze the performance of the dynamic
protocol with respect to this efficiency criterion.

#### Extension to Kernel Methods:

The protocols presented above are defined for models from a Euclidean vector
space. In this paper, we generalize $\mathcal{H}$ to be a reproducing kernel
Hilbert space
${\mathcal{H}=\\{f\\!:X\to\mathbb{R}|f(\cdot)=\sum_{j=1}^{\dim{F}}w_{j}\Phi_{j}(\cdot)\\}}$
with kernel function $k\\!:X\times X\to\mathbb{R}$, feature space $F$, and a
mapping ${\Phi\\!:X\to F}$ into the feature space [18]. The kernel function
corresponds to an inner product of input points mapped into feature space,
i.e.,
$k(x,x^{\prime})=\sum_{j=1}^{\dim{F}}\xi_{j}\Phi_{j}(x)\Phi_{j}(x^{\prime})$
for constants $\xi_{1},\xi_{2},\dots\in\mathbb{R}$. Thus, we can express the
model in its support vector expansion, or dual representation, i.e.,
$f(\cdot)=\sum_{x\in S}\alpha_{x}k(x,\cdot)$ with a set of support vectors
${S=\\{x_{1},\dots,x_{\left|S\right|}\\}\subset X}$ and corresponding
coefficients $\alpha_{x}\in\mathbb{R}$ for all $x\in S$. This implies that the
linear weights $w=(w_{1},w_{2},\dots)\in F$ defining $f$ are given implicitly
by $w_{i}=\sum_{x\in S}\xi_{i}\alpha_{x}\Phi_{i}(x)$. In order to apply the
previously defined synchronization protocols to models from a reproducing
kernel Hilbert space, we determine how to calculate the average of a model
configuration and its divergence. For that, let
${\mathbf{f}=\left(f^{1},\dots,f^{m}\right)}\subset\mathcal{H}$ be a model
configuration with corresponding weight vectors
$\left(w^{1},\dots,w^{m}\right)\subset F$, where each model $i\in[m]$ has
support vectors $S^{i}=\\{x^{i}_{1},\dots,x^{i}_{|S^{i}|}\\}\subset X$ and
coefficients $\alpha^{i}_{x}$ for all $x\in S^{i}$. The average is given by

$\overline{\mathbf{f}}(\cdot)=\frac{1}{m}\sum_{i=1}^{m}f^{i}(\cdot)=\frac{1}{m}\sum_{i=1}^{m}\sum_{j=1}^{\dim{F}}w^{i}_{j}\Phi_{j}(\cdot)=\frac{1}{m}\sum_{i=1}^{m}\sum_{j=1}^{\dim{F}}\sum_{x\in
S^{i}}\xi_{j}\alpha^{i}_{x}\Phi_{j}(x)\Phi_{j}(\cdot)\enspace.$

We can simplify the above equation to
$\overline{\mathbf{f}}(\cdot)=\frac{1}{m}\sum_{i=1}^{m}\sum_{x\in
S^{i}}\alpha^{i}_{x}k(x,\cdot)$. By defining the union of support vectors
$\overline{S}=\bigcup_{i\in[m]}S^{i}=\\{s_{1},\dots,s_{|\overline{S}|}\\}$ and
augmented coefficients $\overline{\alpha}^{i}_{s}\in\mathbb{R}$, which are
given by

$\overline{\alpha}^{i}_{s}=\begin{cases}\alpha_{x}^{i},&\text{ if }x=s\\\
0,&\text{ otherwise}\end{cases}\enspace,$

the dual representation of the average directly follows.

###### Proposition 2

For a model configuration
${\mathbf{f}=\left(f^{1},\dots,f^{m}\right)}\subset\mathcal{H}$, where each
model $i\in[m]$ has augmented coefficients $\overline{\alpha}^{i}_{s}$ for
$s\in\overline{S}$, the average $\overline{\mathbf{f}}\in\mathcal{H}$ is given
by

$\overline{\mathbf{f}}(\cdot)=\sum_{s\in\overline{S}}\left(\frac{1}{m}\sum_{i=1}^{m}\overline{\alpha}^{i}_{s}\right)k(s,\cdot)\enspace,$

with support vectors $\overline{S}$ and coefficients
$\overline{\alpha}_{s}=\nicefrac{{1}}{{m}}\sum_{i=1}^{m}\overline{\alpha}^{i}_{s}$
for all $s\in\overline{S}$.

Using this definition of the average, we now define the distance between
models in $\mathcal{H}$ and the divergence $\delta$ of a model configuration
$\mathbf{f}\subset\mathcal{H}$. For an individual model ${f^{i}}$ and the
average ${\overline{\mathbf{f}}}$, the distance induced by the inner product
of $\mathcal{H}$ is defined by
${\left\|f^{i}-\overline{\mathbf{f}}\right\|=\langle
f^{i},f^{i}\rangle+\langle\overline{\mathbf{f}},\overline{\mathbf{f}}\rangle-2\langle
f^{i},\overline{\mathbf{f}}\rangle}$, i.e.,

$\begin{split}\left\|f^{i}-\overline{\mathbf{f}}\right\|=\sum_{x\in
S^{i}}\left(\alpha_{x}^{i}\right)^{2}k(x,x)+\sum_{s\in\overline{S}}\left(\overline{\alpha}_{s}\right)^{2}k(s,s)-2\sum_{x\in
S^{i}}\sum_{s\in\overline{S}}\alpha_{x}^{i}\overline{\alpha}_{s}k(x,s)\enspace.\end{split}$

Using this distance, we can compute the divergence (Eq. 1) for models from a
reproducing kernel Hilbert space.

##  3 Performance Guarantees

In order to determine the performance of the dynamic protocol, we start by
extending the definition of loss-proportional convex update rules. This allows
us to bound the loss for kernelized online learning algorithms that reduce
their model size using a compression step.

Let $\varphi\\!:\mathcal{H}\times X\times Y\to\mathcal{H}$ be a loss-
proportional convex update rule, then $\widetilde{\varphi}$ is an
approximately loss-proportional convex update rule if for all
$f\in\mathcal{H}$, $x\in X$, and $y\in Y$ it holds that
$\|\widetilde{\varphi}(f,x,y)-\varphi(f,x,y)\|\leq\epsilon$. With this, we can
bound the distance between two models after the approximate update step.

###### Lemma 3

For two models $f,g\in\mathcal{H}$ and an approximately loss-proportional
convex update rule $\widetilde{\varphi}$, with
$\|\widetilde{\varphi}(f,x,y)-\varphi(f,x,y)\|\leq\epsilon$ for the
corresponding loss-proportional convex update rule $\varphi$, it holds that

$\begin{split}\|\widetilde{\varphi}(f,x,y)-\widetilde{\varphi}(g,x,y)\|^{2}\leq\|f-g\|^{2}-\gamma^{2}\left(\ell(f,x,y)-\ell(g,x,y)\right)^{2}+2\epsilon^{2}\enspace.\end{split}$

###### Proof

We abbreviate $\varphi(f,x,y)$ as $\varphi(f)$. Then
$\|\widetilde{\varphi}(f)-\varphi(f)\|\leq\epsilon$ implies for
$f,g\in\mathcal{H}$ that
${\|\widetilde{\varphi}(f)-\widetilde{\varphi}(g)\|^{2}\leq\|\varphi(f)-\varphi(g)\|^{2}+2\epsilon^{2}}$.
Together with the result from Lm. 4 in [10], i.e.,
${\|\varphi(f)-\varphi(g)\|^{2}\leq\|f-g\|^{2}-\gamma^{2}\left(\ell(f)-\ell(g)\right)^{2}}$,
follows the result. ∎

Using Lm. 3, we can bound the loss of our protocol.

###### Theorem 4

Let $\mathcal{A}$ be an online learning algorithm with $\gamma$-loss-
proportional convex update rule $\varphi$. Let
$\mathbf{d}_{1},\dots\mathbf{d}_{T}$ and $\mathbf{p}_{1},\dots,\mathbf{p}_{T}$
be two sequences of model configurations such that
$\mathbf{d}_{1}=\mathbf{p}_{1}$ and the first sequence is maintained by the
dynamic protocol $\mathcal{D}=(\mathcal{A},\sigma_{\Delta})$ and the second by
the periodic protocol $\mathcal{P}=(\mathcal{A},\sigma_{b})$. That is, for
$t=1,\dots,T$ the sequence is defined by
$\mathbf{d}_{t+1}=\sigma_{\Delta}\left(\varphi(\mathbf{d}_{t})\right)$, and
$\mathbf{p}_{t+1}=\sigma_{b}\left(\varphi(\mathbf{p}_{t})\right)$
respectively. Then it holds that

$L_{\mathcal{D}}(T,m)\leq
L_{\mathcal{P}}(T,m)+\frac{T}{\gamma^{2}}(\Delta+2\epsilon^{2})\enspace.$

###### Proof

First note that for simplicity we abbreviate $\ell(f_{t},x_{t},y_{t})$ by
$\ell(f_{t})$. We combine our Lm. 3 with Lm. 3 from [10] which states that

$\frac{1}{m}\sum_{i=1}^{m}\|\sigma_{\Delta}(\mathbf{d})^{i}-\sigma_{b}(\mathbf{p})^{i}\|^{2}\leq\frac{1}{m}\sum_{i=1}^{m}\|d^{i}-p^{i}\|^{2}+\Delta\enspace.$

This yields for all $t\in[T]$ that

$\sum_{i=1}^{m}\left\|d_{t+1}^{i}-p_{t+1}^{i}\right\|^{2}\leq\sum_{i=1}^{m}\left\|d_{t}^{i}-p_{t}^{i}\right\|^{2}-\gamma^{2}\sum_{i=1}^{m}\left(\ell(d_{t}^{i})-\ell(p_{t}^{i})\right)^{2}+\Delta+2\epsilon^{2}\enspace.$

By applying this inequality recursively for $t=1,\dots,T$ it follows that

$\begin{split}\sum_{i=1}^{m}\left\|d_{t+1}^{i}-p_{t+1}^{i}\right\|^{2}\leq&\sum_{i=1}^{m}\left\|d_{1}^{i}-p_{1}^{i}\right\|^{2}+T(\Delta+2\epsilon^{2})-\gamma^{2}\sum_{t=1}^{T}\sum_{i=1}^{m}\left(\ell(d_{t}^{i})-\ell(p_{t}^{i})\right)^{2}.\end{split}$

Using $\mathbf{d}_{1}=\mathbf{p}_{1}$, we conclude that

$\begin{split}\sum_{t=1}^{T}\sum_{i=1}^{m}\left(\ell(d_{t}^{i})-\ell(p_{t}^{i})\right)^{2}\leq&\frac{1}{\gamma^{2}}\left(T(\Delta+2\epsilon^{2})-\sum_{i=1}^{m}\left\|d_{t+1}^{i}-p_{t+1}^{i}\right\|^{2}\right)\leq\frac{1}{\gamma^{2}}T\Delta\\\
\Leftrightarrow
L_{\mathcal{D}}(T)^{m}-L_{\mathcal{P}}(T)^{m}\leq&\frac{1}{\gamma^{2}}T(\Delta+2\epsilon^{2})\end{split}$

∎

By setting the communication period $b=1$, this result also holds for the
continuous protocol $\mathcal{C}$.

The result of Thm. 4 is similar to the original loss bound of the dynamic
protocol but also accounts for the inaccuracy of the update rule, e.g.,
because of model compression. We can apply the original consistency result: if
the continuous protocol is consistent, then the dynamic protocol is consistent
as well. For Stochastic Gradient Descent it has been shown that the dynamic
protocol is consistent for linear models [10]. From Thm. 4 follows that the
dynamic protocol remains consistent for approximately loss-proportional update
rules. Note that for static target distributions, consistency can be achieved
by a decreasing divergence threshold and compression error, i.e.,
$\Delta_{t}=t^{\nicefrac{{-1}}{{2}}}$ and $\epsilon=t^{\nicefrac{{-1}}{{4}}}$.

We now provide communication bounds for the dynamic protocol. For that, assume
that the $m$ learners maintain models in their support vector expansion. Let
$S_{t}^{i}\subset\mathbb{R}^{d}$ denote the set of support vectors of learner
$i\in[m]$ at time $t$ and $\alpha^{i}_{t}$ the corresponding coefficients. Let
$B_{x}\in\mathcal{O}\left(d\right)$ be the number of bytes required to
transmit one support vector and $B_{\alpha}\in\mathcal{O}\left(1\right)$ be
the number of bytes required for the corresponding weight. Furthermore, let
$I:\mathbb{N}\times[m]\rightarrow\\{0,1\\}$ be an indicator function that is
$1$ if for learner $i$ at time $t$ a new support vector has been added during
the update.

We assume that a designated coordinator node performs the synchronizations,
i.e., all local learners transmit their models to the coordinator which in
turn sends the synchronized model back to each learner. Furthermore, we assume
that all protocols apply the following trivial communication reduction
strategy. Let $t^{\prime}$ be the time of last synchronization. Assume the
coordinator stored the support vectors of the last average model
$\overline{S}_{t^{\prime}}$. Whenever a learner $i$ has to send its model to
the coordinator, it sends all support vector coefficients $\alpha$ but only
the new support vectors, i.e., only $S_{t}^{i}\setminus S_{t^{\prime}}^{i}$.
This avoids redundant communication at the cost of higher memory usage at the
coordinator side. In turn, after averaging the models, the coordinator sends
to learner $i$ all support vector coefficients, but only the support vectors
$\overline{S}_{t}\setminus S_{t}^{i}$.

We start by bounding the communication of a continuous protocol $\mathcal{C}$,
i.e., one that transmits all models from each learner in each round. The
trivial communication reduction technique discussed above implies that in each
round, a learner transmits its full set of support vector coefficients and
potentially one support vector—depending on whether a new support vector was
added in this round. Thus, at time $t$ learner $i$ submits

$|S_{t}^{i}|B_{\alpha}+I(t,i)B_{x}$ (2)

bytes to the coordinator. The coordinator transmits to learner $i\in m$ all
support vector coefficients of the average model and all its support vectors,
except the support vectors $S_{t}^{i}$ of the local model at learner $i$.
Thus, it transmits the following amount of bytes.

$\left|\overline{S}_{t}\right|B_{\alpha}+\left|\overline{S}_{t}\setminus
S^{i}_{t}\right|B_{x}=\left|\bigcup_{j=1}^{m}S_{t}^{j}\right|B_{\alpha}+\left|\bigcup_{j=1}^{m}S_{t}^{j}\setminus
S_{t}^{i}\right|B_{x}\enspace.$ (3)

With this we can derive the following communication bound.

###### Proposition 5

The communication of the continuous protocol $\mathcal{C}$ on $m\in\mathbb{N}$
learners until time $T\in\mathbb{N}$ is bound by

$\begin{split}C_{\mathcal{C}}(T,m)\leq
Tm2|\overline{S}_{T}|B_{\alpha}+m|\overline{S}_{T}|B_{x}\leq
m^{2}T^{2}B_{\alpha}+m^{2}TB_{x}\in\mathcal{O}\left(m^{2}T^{2}\right)\enspace.\end{split}$

###### Proof

The constantly synchronizing protocol transmits at each time step from each
learner a set of support vector coefficients and potentially one support
vector to the coordinator. The amount of bytes is given in Eq. 2. The
coordinator transmits the averaged model back to each learner with an amount
of bytes as given in Eq. 3. Summing up the communication over $T\in\mathbb{N}$
time points and $m$ learners yields

$\begin{split}C_{\mathcal{C}}(T,m)&=\sum_{t=1}^{T}\sum_{i=1}^{m}\left(|S_{t}^{i}|B_{\alpha}+I(t,i)B_{x}+\left|\bigcup_{j=1}^{m}S_{t}^{j}\right|B_{\alpha}+\left|\bigcup_{j=1}^{m}S_{t}^{j}\setminus
S_{t}^{i}\right|B_{x}\right)\\\
&=\sum_{t=1}^{T}\sum_{i=1}^{m}\left(|S_{t}^{i}|B_{\alpha}+\left|\overline{S}_{t}\right|B_{\alpha}+I(t,i)B_{x}+\left|\overline{S}_{t}\setminus
S_{t}^{i}\right|B_{x}\right)\enspace.\end{split}$

We analyze this sum separately in terms of bytes required for sending the
support vectors and bytes for sending the coefficients. The amount of bytes
for sending the support vectors is bounded by $m|S_{T}^{i}|B_{x}$, as we show
in the following.

$\begin{split}&\sum_{t=1}^{T}\sum_{i=1}^{m}I(t,i)B_{x}+\left|\overline{S}_{t}\setminus
S_{t}^{i}\right|B_{x}=\underbrace{\sum_{t=1}^{T}\sum_{i=1}^{m}I(t,i)B_{x}}_{=|\overline{S}_{T}|B_{x}}+\sum_{t=1}^{T}\sum_{i=1}^{m}\left|\overline{S}_{t}\setminus
S_{t}^{i}\right|B_{x}\\\
=&|\overline{S}_{T}|B_{x}+\sum_{t=1}^{T}\sum_{i=1}^{m}\left|\left(\bigcup_{j=1}^{m}S_{t}^{j}\setminus\bigcup_{j=1}^{m}S_{t-1}^{j}\right)\setminus\left(S_{t}^{i}\setminus\overline{S}_{t-1}\right)\right|B_{x}\\\
\leq&|\overline{S}_{T}|B_{x}+\sum_{t=1}^{T}\sum_{i=1}^{m}\sum_{\stackrel{{\scriptstyle
j=1}}{{j\neq
i}}}^{m}I(t,i)B_{x}\leq|\overline{S}_{T}|B_{x}+\sum_{t=1}^{T}\sum_{i=1}^{m}(m-1)I(t,i)B_{x}\\\
\leq&|\overline{S}_{T}|B_{x}+(m-1)|\overline{S}_{T}|B_{x}=m|\overline{S}_{T}|B_{x}\enspace.\end{split}$

We now bound the amount of bytes required for sending the support vector
coefficients.

$\begin{split}\sum_{t=1}^{T}\sum_{i=1}^{m}\underbrace{|S_{t}^{i}|}_{\leq|\overline{S}_{T}|}B_{\alpha}+\underbrace{|\overline{S}_{t}|}_{\leq|\overline{S}_{T}|}B_{\alpha}\leq\sum_{t=1}^{T}\sum_{i=1}^{m}2|\overline{S}_{T}|B_{\alpha}=Tm2|\overline{S}_{T}|B_{\alpha}\enspace.\end{split}$

From $\left|\overline{S}_{T}\right|\leq mT$ and the fact that we regard
$B_{\alpha}\in\mathcal{O}\left(1\right)$ and
$B_{x}\in\mathcal{O}\left(d\right)$ as constants we can follow that

$\begin{split}C_{\mathcal{C}}(T,m)\leq
2Tm|\overline{S}_{T}|B_{\alpha}+m|\overline{S}_{T}|B_{x}\leq
m^{2}T^{2}B_{\alpha}+m^{2}TB_{x}\in\mathcal{O}\left(m^{2}T^{2}\right)\enspace.\end{split}$

∎

Note that this communication bound implies that—unlike for linear
models—synchronizing models in their support vector expansion requires even
more communication than centralizing the input data. However, in real-time
prediction applications, the latency induced by central computation can exceed
the time constraints, rendering continuous synchronization a viable approach
nonetheless.

Similarly, the communication of a periodic protocol $\mathcal{P}$ that
communicates every $b\in\mathbb{N}$ steps ($b$ is often referred to as mini-
batch size) can be bounded by

$C_{\mathcal{P}}(T,m)\leq\frac{T}{b}2m|\overline{S}_{T}|B_{\alpha}+m|\overline{S}_{T}|B_{x}\leq\frac{T}{b}m^{2}TB_{\alpha}+m^{2}TB_{x}\in\mathcal{O}\left(\frac{1}{b}m^{2}T^{2}\right)\enspace.$

We now for the first time provide a communication bound for the dynamic
protocol $\mathcal{D}$. For that, we first bound the number of synchronization
steps and then analyze the amount of communication per synchronization.

###### Proposition 6

Let $\mathcal{A}=(\mathcal{H},\widetilde{\varphi},\ell)$ be an online learning
algorithm with an approximately loss-proportional convex update rule
$\widetilde{\varphi}$ for which holds
that${\|f-\widetilde{\varphi}(f,x,y)\|\leq\eta\ell(f,x,y)}$. The number of
synchronizations $V_{\mathcal{D}}(T)$ of the dynamic protocol $\mathcal{D}$
running $\mathcal{A}$ in parallel on $m$ nodes until time $T\in\mathbb{N}$
with divergence threshold $\Delta$ is bounded by

$V_{\mathcal{D}}(T)\leq\frac{\eta}{\sqrt{\Delta}}L_{\mathcal{D}}(T,m)\enspace.$

where $L_{\mathcal{D}}(T,m)$ denotes the cumulative loss of $\mathcal{D}$.

###### Proof

For this proof, we abbreviate $\ell(f_{t}^{i},x_{t}^{i},y_{t}^{i})$ as
$\ell(f_{t}^{i})$ and $\widetilde{\varphi}(f_{t}^{i},x_{t}^{i},y_{t}^{i})$ as
$\widetilde{\varphi}(f_{t}^{i})$. The dynamic protocol synchronizes if a local
condition $\|f_{t}^{i}-r_{t}\|^{2}\leq\Delta$ is violated. Now assume that at
$t=1$ all models are initialized with $f_{1}^{1}=\dots=f_{1}^{m}$ and
$r_{1}=\overline{\mathbf{f}}_{1}$, i.e., for all local learners $i$ it holds
that $\|f_{1}^{i}-r_{1}\|=0$. A violation, i.e.,
$\|f_{t}^{i}-r_{t}\|>\sqrt{\Delta}$, occurs if one local model drifts away
from $r_{t}$ by more than $\sqrt{\Delta}$. After a violation, a
synchronization is performed and $r_{t}=\overline{\mathbf{f}}_{t}$, hence
$\|f_{t}^{i}-r_{t}\|=0$ and the situation is again similar to the initial
setup for $t=1$. In the worst case, a local learner drifts continuously in one
direction until a violation occurs. Hence, we can bound the number of
violations $V_{i}(T)$ at a single learner $i$ by the sum of its drifts divided
by $\sqrt{\Delta}$:

$\begin{split}V_{l}(T)\leq\frac{1}{\sqrt{\Delta}}\sum_{t=1}^{T}\|f_{t}^{i}-f_{t+1}^{i}\|=&\frac{1}{\sqrt{\Delta}}\sum_{t=1}^{T}\underbrace{\|f_{t}^{i}-\widetilde{\varphi}(f_{t}^{i})\|}_{\leq\eta\ell(f_{t}^{i})}\leq\frac{1}{\sqrt{\Delta}}\sum_{t=1}^{T}\eta\ell(f_{t}^{i})\enspace.\end{split}$

With this, we can bound the amount of points in time $t\in[T]$ where at least
one learner $l$ has a violation, i.e., $V(T)$. In the worst case, all
violations at all local learners occur at different time points, so that we
can upper bound $V(T)$ by the sum of local violations $V_{i}(T)$ which is
again upper bounded by the cumulative sum of drifts of all local models:

$V(T)\leq\sum_{i=1}^{m}V_{i}(T)\leq\frac{1}{\sqrt{\Delta}}\sum_{t=1}^{T}\sum_{i=1}^{m}\eta\ell(f_{t}^{i})=\frac{\eta}{\sqrt{\Delta}}L_{\mathcal{D}}(T,m)\enspace.$

∎

In the following theorem we bound the overall communication by combining this
bound on the number of synchronizations with an analysis of the amount of
bytes transfered per synchronization.

###### Theorem 7

Let $\mathcal{A}=(\mathcal{H},\widetilde{\varphi},\ell)$ be an online learning
algorithm with approximately loss-proportional update rule
$\widetilde{\varphi}$ and
$\|f-\widetilde{\varphi}(f,x,y)\|\leq\eta\ell(f,x,y)$. The amount of
communication $C_{\mathcal{D}}(T,m)$ of the dynamic protocol $\mathcal{D}$
running $\mathcal{A}$ in parallel on $m$ nodes until time $T\in\mathbb{N}$
with divergence threshold $\Delta$ is bounded by

$\begin{split}C_{\mathcal{D}}(T,m)\leq\frac{\eta}{\sqrt{\Delta}}L_{\mathcal{D}}(T,m)\left(2m\left|\overline{S}_{T}\right|B_{\alpha}\right)+m\left|\overline{S}_{T}\right|B_{x}\end{split}$

###### Proof

Assume that at time $T$, the dynamic protocol performs a synchronization.
Then, similar to the argument for the continuous protocol, the support vector
set at time $T$ is similar for all learners and independent of the number of
synchronization steps before. In particular, it is the same if a
synchronization was performed in every time step. Thus, again the amount of
bytes required for sending the support vectors is bounded by
$m\left|\overline{S}_{T}\right|B_{x}$. Let
${\theta\\!:\mathbb{N}\to\\{0,1\\}}$ be an indicator function such that
$\theta(t)=1$ if at time $t$ the dynamic protocol performed a synchronization
and $\theta(t)=0$ otherwise. Then, the amount of bytes required to send all
the support vector coefficients until time $T$ is

$\begin{split}\sum_{t=1}^{T}\theta(t)\sum_{i=1}^{m}\left(\left|S_{t}^{i}\right|+\left|\overline{S}_{t}\right|\right)B_{\alpha}\leq&\underbrace{\sum_{t=1}^{T}\theta(t)}_{=V_{\mathcal{D}}(T)}\sum_{l=1}^{m}2|\overline{S}_{T}|B_{\alpha}\leq\underbrace{\frac{\eta}{\sqrt{\Delta}}L_{\mathcal{D}}(T,m)}_{\text{Prop.~{}\ref{prop:numberViolations}}}\left(2m|\overline{S}_{T}|B_{\alpha}\right)\\\
\end{split}$

Together with the amount of bytes required for exchanging all support vectors
this yields
$C_{\mathcal{D}}(T,m)\leq\frac{\eta}{\sqrt{\Delta}}L_{\mathcal{D}}(T,m)\left(2m|\overline{S}_{T}|B_{\alpha}\right)+m\left|\overline{S}_{T}\right|B_{x}$.
∎

Note that the loss bounds for online learning algorithms are typically sub-
linear in $T$, e.g., optimal regret bounds for static target distributions are
in $\mathcal{O}(\sqrt{T})$. In these cases, the dynamic protocol has an amount
of communication in $\mathcal{O}(m^{2}T\sqrt{T})$ which is smaller than
$\mathcal{O}(m^{2}T^{2})$ of the continuously and periodic protocols by a
factor of $\sqrt{T}$.

In the original case of linear models instead, the dynamic protocol only
transmits $m$ weight vectors of fixed size per synchronization. In this case
the amount of communication per synchronization is bounded by a constant. If
for an online learning algorithm $\mathcal{A}$ and the periodic protocol
$\mathcal{P}$ it holds that
$\textbf{L}_{\mathcal{P}}(T,m)\leq\textbf{L}_{\mathcal{A}}(mT)$, then by Thm.
4 it also holds that
$\textbf{L}_{\mathcal{D}}(T,m)\leq\textbf{L}_{\mathcal{A}}(mT)$. This implies
that the dynamic protocol is adaptive. In the following corollary, we show
that for linear models, the dynamic protocol is adaptive when using the
Stochastic Gradient Descent algorithm.

###### Corollary 8

The dynamic protocol $\mathcal{D}=(\text{SGD},\sigma_{\Delta})$ using
Stochastic Gradient Descent SGD with linear models is adaptive, i.e.,

$C_{\mathcal{D}}(T,m)\in\mathcal{O}\left(m\textbf{L}_{\text{SGD}}(mT)\right)$

###### Proof

The amount of synchronizations of the dynamic protocol is bounded by $V(T)$
(see Prop. 6). In each synchronization, each learner transmits one linear
model, i.e., one weight vector of fixed size to the coordinator. The
coordinator submits one averaged weight vector back to each learner. Thus, the
amount of communication per synchronization is bounded by
$\mathbf{c}_{m}\in\mathbb{N}$, where
$\mathbf{c}_{m}\in\mathcal{O}\left(m\right)$. Then, the total communication is
bounded by

$C_{\mathcal{D}}(T,m)\leq\mathbf{c}_{m}\frac{\eta}{\sqrt{\Delta}}L_{\mathcal{D}}(T,m)\in\mathcal{O}\left(mL_{\mathcal{D}}(T,m)\right)\enspace.$

The dynamic protocol retains the loss bound of Stochastic Gradient Descent
[10], i.e., $L_{\mathcal{D}}(T,m)\leq\textbf{L}_{\text{SGD}}(mT)$ ∎

Unfortunately, from Thm. 4 also follows that the dynamic protocol applied to
kernelized online learning algorithms that do not bound the size of their
models does not comply to the strict notion of adaptivity as given in Def. 1.
That is, because the model size and thus the size of each message to and from
the coordinator can grow with $T$. Nonetheless, the theorem guarantees that if
the learners do not suffer loss anymore, the dynamic protocol reaches
quiescence.

In order to make the dynamic protocol adaptive in the strict sense of Def. 1,
the model size has to be bounded. For kernelized online learning in streams,
several model compression techniques have been proposed [12, 15, 20]. These
techniques typically guarantee that the compression error is bounded, i.e.,
for the compressed model $\widetilde{f}$ it holds that
$\left\|f-\widetilde{f}\right\|\leq\epsilon$. From this directly follows that
if the base algorithm uses a loss-proportional convex update rule $\varphi$,
the compressed version is an approximately loss-proportional convex update
rule $\widetilde{\varphi}$.

One approach to compressing the support vector expansion is to project a new
support vector on the span of the remaining ones and thus avoid adding it to
the support set. Another one is to truncate support vectors with small
coefficients. For the projection approach (e.q., described in [15]) the error
bound is independent of the learning algorithm. However, there is no bound on
the number of support vectors. Thus, even though the model size is reduced in
practice, there is no formal bound on the model size. For the truncation
approach, however, [12] have shown that an error bound as well as a bound on
the number of support vectors can be achieved when using Stochastic Gradient
Descent. Specifically, for a fixed model size of $\tau$ support vectors, they
have shown that the compression error is bound by
$\left\|f-\widetilde{f}\right\|\leq\epsilon\in\mathcal{O}\left(\frac{1}{\lambda}(1-\lambda)^{\tau}\right)$,
where $\lambda\in\mathbb{R}$ is the learning rate of the Stochastic Gradient
Descent algorithm (SGD). Therefore, we can follow that the dynamic protocol
with SGD using kernel models compressed by truncation is adaptive.
Specifically for SGD, [4] have shown that periodic synchronizations retain the
serial loss bound of SGD. It is consistent in this setting, because the
dynamic protocol in turn retains the loss bounds of any periodic protocol.
Since it is both consistent and adaptive, the dynamic protocol is efficient.

##  4 Discussion

Figure 2:  2 Trade-off between cumulative error and cumulative communication
and 2 cumulative communication over time of the dynamic protocol versus a
periodic protocol. $32$ learners perform a stock price prediction task using
SGD (learning rate $\eta$ and regularization parameter $\lambda$ optimized
over $200$ instances, with $\eta=10^{-10}$, $\lambda=1.0$ for the periodic
protocol, and $\eta=1.0$, $\lambda=0.01$ for the dynamic protocol) updates,
either with linear models or with non-linear models (Gaussian kernel with
number of support vectors limited to $50$ using the truncation approach of
[12]).

The dynamic protocol, extended to kernel methods, yields for the first time a
theoretically efficient tool to learn non-linear models for distributed real-
time services, in settings where communication is a major bottleneck. For
that, it can employ online kernel methods together with model compression
techniques, which reduce, or bound the number of support vectors. The
efficiency of the protocol is characterized by a novel criterion that ties a
tight loss bound to the required amount of communication—a criterion which is
not satisfied by the state of the art of periodically communicating protocols.

While we provided a theoretical analysis, the advantage of the dynamic
protocol in combination with kernel methods can also be shown in practice:
Fig. 2 shows the results of an experiment on financial data [9], where $32$
learners predicted the stock price of a target stock. We can see that for this
difficult learning task linear models perform poorly compared to non-linear
models using a Gaussian kernel function. Simultaneously, the communication
required to periodically synchronize these non-linear models is larger than
for linear models by more than two orders of magnitude. Using the dynamic
protocol with kernel models we could reduce the error by an order of magnitude
compared to using linear models (a reduction by a factor of $18$). At the same
time, the communication is reduced by more than three orders of magnitude
compared to the static protocol (by a factor of $2433$), which is yet an order
of magnitude smaller than the communication when using linear models (by a
factor of $10$). Moreover, within less than $2000$ rounds, the dynamic
protocol reaches quiescence, as it is implied by the efficiency criterion.

A limit of the employed notion of efficiency is that it only takes into
account the sum of messages but not the peak communication. In large data
centers, where the distributed learning system is run next to other processes,
the main bottleneck is the overall amount of transmitted bytes and a high peak
in communication can often be handled by the communication infrastructure or
evened out by a load balancer. In smaller systems, however, high peak
communication can become a serious problem for the infrastructure and it
remains an open problem how it can be reduced. Note that the frequency of
synchronizations in a short time interval can actually be bounded by a trivial
modification of the dynamic protocol: local conditions are only checked after
a mini-batch of examples have been observed. Thus, the peak communication is
upper bounded in the same way as with a periodic protocol, while still
dynamically reducing the overall amount of communication.

When analyzing the reason for practical efficiency, model compression has
proven to be a crucial factor, since storing and evaluating models with large
numbers of support vectors can become infeasible—even in serial settings. In a
distributed setting, transmitting large models furthermore induce high
communication costs, which is aggravated by averaging local models, because
the synchronized model consists of the union of all local support vectors. For
the model truncation approach of [12], we have shown that the efficiency
criterion is satisfied, but other model compression approaches might be
favorable in certain scenarios. Thus, an interesting direction for future
research is to study the relationship between loss and model size of those
model compression techniques in order to extend the results on efficiency.

Also, alternative approaches to ensuring constant model size could be
investigated, e.g., a finite dimensional approximation of the feature map
$\Phi\\!:X\to\mathcal{H}$ of a reproducing kernel Hilbert space $\mathcal{H}$,
such as Random Fourier Features [17]. It remains an open problem how tight
loss bounds combined with communication bounds can be derived in these
settings.

Finding the right divergence threshold for the dynamic protocol, i.e., one
that suits the desired trade-off between service quality and communication, is
in practice a neither intuitive nor trivial task. The threshold can be
selected using a small data sample, but the communication for a given
threshold can vary over time and is also influenced by other parameters of the
learner. Thus, another direction for future research is to investigate an
adaptive divergence threshold. This could allow for a more direct selection of
the desired trade-off between service quality (i.e., predictive performance)
and communication.

### Acknowledgments

This research has been supported by the EU FP7-ICT-2013-11 under grant 619491
(FERARI).

## References

  * [1] Barroso, L.A., Clidaras, J., Hölzle, U.: The datacenter as a computer: An introduction to the design of warehouse-scale machines. Synthesis lectures on computer architecture 8(3), 1–154 (2013)

  * [2] Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge university press (2004)
  * [3] Crammer, K., Dekel, O., Keshet, J., Shalev-Shwartz, S., Singer, Y.: Online passive-aggressive algorithms. The Journal of Machine Learning Research 7, 551–585 (2006)
  * [4] Dekel, O., Gilad-Bachrach, R., Shamir, O., Xiao, L.: Optimal distributed online prediction using mini-batches. The Journal of Machine Learning Research 13, 165–202 (2012)
  * [5] Deligiannakis, A., Kotidis, Y., Roussopoulos, N.: Bandwidth-constrained queries in sensor networks. The VLDB Journal 17(3), 443–467 (2008)
  * [6] Gabel, M., Keren, D., Schuster, A.: Communication-efficient distributed variance monitoring and outlier detection for multivariate time series. In: Proceedings of the 28th International Parallel and Distributed Processing Symposium (IPDPS). pp. 37–47. IEEE (2014)
  * [7] Giatrakos, N., Deligiannakis, A., Garofalakis, M., Sharfman, I., Schuster, A.: Prediction-based geometric monitoring over distributed data streams. In: Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data. pp. 265–276 (2012)
  * [8] Heires, K.: Budgeting for latency: If i shave a microsecond, will i see a 10x profit. Securities Industry News 22(1) (2010)
  * [9] Kamp, M., Boley, M., Gärtner, T.: Beating human analysts in nowcasting corporate earnings by using publicly available stock price and correlation features. In: 2013 IEEE 13th International Conference on Data Mining Workshops. pp. 384–390 (2013)
  * [10] Kamp, M., Boley, M., Keren, D., Schuster, A., Sharfman, I.: Communication-efficient distributed online prediction by dynamic model synchronization. In: Machine Learning and Knowledge Discovery in Databases, pp. 623–639. Springer (2014)
  * [11] Keren, D., Sharfman, I., Schuster, A., Livne, A.: Shape sensitive geometric monitoring. IEEE Transactions on Knowledge and Data Engineering 24(8), 1520–1535 (2012)
  * [12] Kivinen, J., Smola, A.J., Williamson, R.C.: Online learning with kernels. IEEE Transactions on Signal Processing 52(8), 2165–2176 (2004)
  * [13] Lichman, M.: UCI machine learning repository (2013), http://archive.ics.uci.edu/ml
  * [14] Mcdonald, R., Mohri, M., Silberman, N., Walker, D., Mann, G.S.: Efficient large-scale distributed training of conditional maximum entropy models. In: Advances in Neural Information Processing Systems. pp. 1231–1239 (2009)
  * [15] Orabona, F., Keshet, J., Caputo, B.: Bounded kernel-based online learning. The Journal of Machine Learning Research 10, 2643–2666 (2009)
  * [16] Predd, J.B., Kulkarni, S.R., Poor, H.V.: Distributed learning in wireless sensor networks. John Wiley & Sons: Chichester, UK (2007)
  * [17] Rahimi, A., Recht, B.: Random features for large-scale kernel machines. In: Advances in Neural Information Processing Systems (NIPS). pp. 1177–1184 (2007)
  * [18] Schölkopf, B., Smola, A.J.: Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT press (2001)
  * [19] Sharfman, I., Schuster, A., Keren, D.: A geometric approach to monitoring threshold functions over distributed data streams. Transactions on Database Systems (TODS) 32(4) (2007)
  * [20] Wang, Z., Vucetic, S.: Online passive-aggressive algorithms on a budget. In: International Conference on Artificial Intelligence and Statistics. pp. 908–915 (2010)
  * [21] Yuan, S., Wang, J., Zhao, X.: Real-time bidding for online advertising: measurement and analysis. In: Proceedings of the Seventh International Workshop on Data Mining for Online Advertising. p. 3 (2013)


# Beware of CaBER: filament thinning rheometry doesn’t give ‘the’ relaxation
time of polymer solutions

A. Gaillard1<EMAIL_ADDRESS>M. A. Herrada2  A. Deblais1  J.
Eggers3  D. Bonn1  1 Van der Waals-Zeeman Institute, University of Amsterdam,
Science Park 904, Amsterdam, Netherlands 2 Depto. de Mecánica de Fluidos e
Ingeniería Aeroespacial, Universidad de Sevilla, Sevilla E-41092, Spain 3
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1
TW, United Kingdom

###### Abstract

Capillary Breakup Extensional Rheometry (CaBER) is commonly used to measure
‘the’ viscoelastic relaxation time $\tau$ of polymer solutions. We show
experimentally that this method in fact only provides an apparent relaxation
time that increases significantly when increasing the plate diameter and
droplet volume. Similar results are obtained with a Dripping-onto-Substrate
(DoS) method. This dependence on the flow history before the formation of the
viscoelastic filament is in contradiction with polymer models such as
Oldroyd-B that predict a filament thinning rate $1/3\tau$ which is a material
property independent of geometrical factors. We show that this is not due to
artefacts such as solvent evaporation or polymer degradation and that it
cannot always be explained by the finite extensibility of polymer chains.

###### keywords:

Viscoelasticity; Polymers; Capillary flows

##  1 Introduction

When polymers are added to a low-viscosity solvent such as water, the
extensional rheology of the resulting solution is usually measured by indirect
techniques where the (extensional) strain and strain rate are not controlled,
unlike for high-viscosity polymer solutions or melts for which reliable
extensional rheometers are available, e.g. Meissner’s RME (Rheometric Melt
Elongation rheometer) and FiSER (Filament Stretching Extensional Rheometer).
Most indirect techniques for low-viscosity polymer solutions aim at forming a
liquid filament undergoing capillary-driven thinning. Historically, this was
first achieved by placing a drop of liquid between two horizontal plates which
are then separated beyond the range of stable liquid bridges (Bazilevsky et
al., 1997; Anna & McKinley, 2001; Stelter et al., 2000), a technique now known
as CaBER (Capillary Breakup Extensional Rheometry). Alternative techniques,
also based on the Rayleigh-Plateau instability, were proposed to avoid
inertio-capillary oscillations of the end-drops in the original CaBER step-
strain (rapid) plate separation protocol which prohibits measurement of very
short relaxation times (Rodd et al., 2004). The general idea is to generate a
viscoelastic filament slowly, e.g. by separating the plates at a constant low
velocity (Slow Retraction Method or SRM) (Campo-Deano & Clasen, 2010), by
dripping a droplet in air from a nozzle at a low flow rate (Tirtaatmadja et
al., 2006) or, in a more recent technique, by slowly bringing a solid
substrate in contact with a drop hanging steadily from a nozzle (Dripping-
onto-Substrate or DoS) (Dinic et al., 2017).

In all these techniques, after an initial inertial and/or viscous regime, an
elastic regime emerges where the elastic stresses arising from the stretching
of polymer chains dominate and give rise to a cylindrical filament that thins
exponentially in time for a wide range of dilute and semi-dilute polymer
solutions. This is consistent with the Oldroyd-B model that predicts
(Bazilevsky et al., 1990; Entov & Hinch, 1997):

$h=h_{1}\exp{\left(-\frac{t-t_{1}}{3\tau}\right)}$ (1)

where $h$ is the minimum filament radius, $\tau$ the viscoelastic relaxation
time of the polymer solution, $t_{1}$ the time marking the onset of the
elastic regime, and $h_{1}=h(t_{1})$ the filament radius at that time. For a
step-strain CaBER protocol, in which polymer molecules do not relax during the
fast plate separation, the model predicts $h_{1}=(Gh_{i}^{4}/2\gamma)^{1/3}$
where $G=\eta_{p}/\tau$ is the elastic modulus, $\eta_{p}$ the polymer
contribution to the shear-viscosity, $\gamma$ the surface tension and $h_{i}$
the radius of the initial liquid column (Clasen et al., 2006a). It is also
generally accepted that for a polymer solution with a spectrum of relaxation
times, the longest one rapidly dominates (Anna & McKinley, 2001).

The general consensus is that geometrical parameters, in particular the size
of the system, can only influence $h_{1}$ and not the thinning rate
$|\dot{h}/h|=1/3\tau$ of the filament since $\tau$ is a material property. In
particular, Bazilevsky et al. (1997) and Miller et al. (2009) checked that the
filament thinning rate was independent of the sample volume and of the plate
separation speed and, in a step-strain plate separation protocol, on the final
plate separation distance. This suggests that it is independent of the history
of the polymer deformation prior to the elastic (exponential) regime. However,
Rajesh et al. (2022) recently tested polymer solutions of different solvent
viscosities with a dripping method and reported a larger thinning rate for a
smaller nozzle radius.

In this communication, we investigate the role of the drop size on the
thinning dynamics of CaBER filaments with a slow plate retraction protocol and
show that CaBER does not measure the relaxation time properly; rather it
yields an apparent relaxation time that depends on the way the viscoelastic
filament is formed, and hence on the specific geometry used for the
experiments.

##  2 Materials and methods

We use three different liquids: two solutions of poly(ethylene oxide) (PEO) of
molecular weight $M_{w}=4\times 10^{6}$ g/mol, one in water with concentration
$500$ ppm, referred to as PEOaq, and one in a more viscous solvent with
concentration $25$ ppm, referred to as PEOvisc, and a $1000$ ppm solution of
poly(acrylamide/sodium acrylate) (HPAM) [70:30] of molecular weight
$M_{w}=18\times 10^{6}$ g/mol in water with $1$ wt% NaCl to screen
electrostatic interactions and make the chain flexible instead of semi-rigid.
Both polymers were provided by Polysciences (ref. 04030-500 for PEO and
18522-100 for HPAM). For the PEOvisc solution, the solvent is an aqueous
Newtonian $30$ wt% $20,000$ g/mol PEG solution. The different concentrations
were chosen to ensure that all three liquids have comparable filament thinning
rates. After slowly injecting the polymer powder to a vortex generated by a
magnetic stirrer, solutions were homogenised using a mechanical stirrer at low
rotation speed for about $16$ hours.

Name00 | $\rho$ | $\gamma$ | $\eta_{s}$ | $c$ | $c/c^{*}$ | $\eta_{0}$ | $\eta_{p}$ | $n$ | $1/\dot{\gamma}_{c}$ | $\tau_{m}$
---|---|---|---|---|---|---|---|---|---|---
|  (kg/m3) | (mN/m) |  (mPa0s) | (ppm) |  |  (mPa0s) |  (mPa0s) |  | (ms) | (ms)
PEOaq0 | 998 | 62.5 | 0.92 | 500 | 1.86 | 3.0 | 2.08 | 0.93 | 120 | 240
PEOvisc | 1048 | 56.0 | 245 | 25 | – | 248 | 3.3 | 1 | – | 110
HPAM0 | 998 | 72.0 | 0.92 | 1000 | – | 15 | 14 | 0.78 | 410 | 100
Table 1: Properties of the three polymer solutions. $\rho$ is the density,
$\gamma$ the surface tension, $\eta_{s}$ the solvent viscosity, $c$ the
polymer concentration, $c^{*}$ the critical overlap concentration, $\eta_{0}$,
$n$ and $\dot{\gamma}_{c}$ the Carreau fitting parameters of the shear
viscosity, $\eta_{p}=\eta_{0}-\eta_{s}$, and $\tau_{m}$ the maximum CaBER
relaxation time measured for the largest plates.

The shear viscosity $\eta$ of these solutions was measured at the temperature
of filament thinning experiments with a MRC-302 rheometer from Anton Paar
equipped with a cone plate geometry (diameter $50$ mm, angle $1^{\circ}$ and
truncation gap $53$ $\mu$m). The PEOvisc solution is a Boger fluid with a
constant shear viscosity while the two others are shear-thinning and are well
described by the Carreau law
$\eta(\dot{\gamma})=\eta_{0}(1+(\dot{\gamma}/\dot{\gamma}_{c})^{2})^{(n-1)/2}$
where $\eta_{0}$ is the zero-shear viscosity, $n$ is the shear-thinning
exponent and $\dot{\gamma}_{c}$ is the shear rate marking the onset of shear
thinning. These values, along with the solvent viscosity $\eta_{s}$, the
density $\rho$ and the surface tension $\gamma$ measured with a pendent drop
method, are reported in table 1. We also report the value of $c/c^{*}$ for the
PEOaq solution for which viscosity measurements for different PEO
concentrations gave an intrinsic viscosity $[\eta]=2.87$ m3/kg and hence a
critical overlap concentration $c^{*}=0.77/[\eta]=0.268$ kg/m3.

In our home-made CaBER setup, a droplet of volume $V$ is placed between two
horizontal plates of radius $R_{0}$ and the top plate, which is controlled by
a motor, is moved down until it is fully wetted by the liquid. As the top
plate is moved up, a stable capillary bridge is formed, as shown in the left
inset of figure 1(a). Beyond a certain critical value of the plate separation
distance $L$, the bridge becomes unstable and collapses under the action of
surface tension, transiently leading to the formation of a nearly cylindrical
filament which is the signature of viscoelastic pinch-off, as shown in the
right inset of figure 1(a). Instead of moving the top plate at a constant low
velocity close to the bridge instability threshold, i.e. like in SRM (Campo-
Deano & Clasen, 2010), we move it by 10 $\mu$m steps, waiting about one second
between each step (longer than the solution’s relaxation time) to allow
polymers to relax back to equilibrium before the next step. We stop moving the
top plate once capillary-driven thinning starts.

Figure 1: (a) Time evolution of the minimum bridge / filament radius $h$ in
the step-by-step plate separation protocol for the PEOaq solution for plate
diameters $2R_{0}=3.5$ mm and a droplet volume $V^{*}=V/R_{0}^{3}\approx 2.4$.
Inertio-capillary oscillations are visible after each step. Inset images
correspond to a stable liquid bridge (left) and to a thinning filament of the
same liquid with $2R_{0}=7$ mm and $V^{*}=2.4$. (b) $h(t)$ in log-lin for the
PEOaq solution tested with plate diameters, $2R_{0}=2$, $3.5$, $5$ and $7$ mm,
with $V^{*}\approx 2.4$. Inset images correspond to three times labelled 1 to
3 indicated on the $h(t)$ curve plus a fourth later time where $h$ is below
our spatial resolution for $2R_{0}=7$ mm. The time reference $t_{1}$ marks the
onset of the elastic regime.

The process is recorded by a high-magnification objective mounted on a high-
speed camera (Phantom TMX 7510) and images are analysed by a python code. A
typical time-evolution of $h$ is shown in figure 1(a). The purpose of this
step-by-step plate separation protocol is to extract the value of the last
stable bridge radius $h_{0}$ which, since steps are small, can be considered
as the initial bridge radius at the onset of capillary thinning.

##  3 Results

Figure 1(b) compares the time-evolution of the minimum bridge / filament
radius $h$ for the PEOaq solution tested with plate diameters $2R_{0}$ between
$2$ and $7$ mm with a fixed non-dimensional droplet volume
$V^{*}=V/R_{0}^{3}\approx 2.4$. Although all filaments thin exponentially in
time in the elastic regime, as suggested by the fairly straight curves, they
thin faster for smaller plates. This is in apparent contradiction with the
Oldroyd-B model which predicts a rate of exponential thinning
$|\dot{h}/h|=1/3\tau$ (see equation 1) which should be the same for all
filaments, provided that the liquid doesn’t change so that the (longest)
relaxation time $\tau$ of the polymer solution is the same. As we present in
Appendix A, similar results are found in DoS (Dripping-onto-Substrate) with
filament thinning rates increasing with decreasing nozzle size.

To quantify these differences, we introduce an apparent (or effective)
relaxation time $\tau_{e}$ such that $|\dot{h}/h|=1/3\tau_{e}$ in the elastic
regime. It is plotted in figure 2(a) as a function of the initial bridge
radius $h_{0}$ for all polymer solutions, plate diameters and droplet volumes.
We observe that $\tau_{e}$ increases with both $R_{0}$ and $V^{*}$ and that
all data points for a given solution collapse on a single curve when plotted
against $h_{0}$, which is itself an increasing function of both $R_{0}$ and
$V^{*}$. This suggests that $h_{0}$ is in fact the only relevant geometrical
parameter of the problem. This is in agreement with the accepted idea that
polymer deformations during capillary thinning are only influenced by the
local extensional flow in the bridge / filament of maximum extension rate
$\dot{\epsilon}=-2\dot{h}/h$ at its thinnest point, where the dot means
$\mathrm{d}/\mathrm{d}t$, while the top and bottom end droplets act as passive
liquid reservoirs, their size not directly influencing the pinch-off dynamics.

The apparent relaxation time varies significantly, up to a factor $4$ within
the typical range of plate diameters used for CaBER experiments, see figure
2(a). However, the fairly linear increase of $\tau_{e}$ with $h_{0}$ in figure
2(a) cannot continue indefinitely. In order to observe the expected saturation
of $\tau_{e}$ for larger values $h_{0}$, we had to move to much larger plate
diameters, up to $2R_{0}=25$ mm. For plate diameters $2R_{0}\geq 10$ mm, the
top end-drop doesn’t cover the top plate fully because of gravity, as shown in
the inset of figure 2(b). In fact, there is always a thin liquid film covering
the top plate since we always clean the plates using plasma treatment, which
makes the aluminium plates more hydrophilic and prevents dewetting. Note also
that the top end-drop is not at the centre of the the top plate since the two
plates are not perfectly parallel.

Figure 2: Apparent relaxation time $\tau_{e}$ against $h_{0}$ for all three
solutions for plate diameters between $2$ and $7$ mm (a) and between $2$ and
$25$ mm (b). In (a), data points of the same colour correspond to the same
$R_{0}$ with different droplet volumes $V^{*}=V/R_{0}^{3}\approx 1.3$, $2.4$
and $3.2$. In (b), a single volume is used for each plate diameter
($V^{*}\approx 0.3$ for the smallest plates and $0.11$ for the largest
plates). The inset in (b) shows the shape of a stable liquid bridge for
$2R_{0}=20$ mm.

In spite of this lack of full coverage, we find that the critical minimum
bridge radius $h_{0}$ marking the onset of the Rayleigh-Plateau instability
increases with $R_{0}$, allowing us to explore a wider range $h_{0}$ values,
as shown in figure 2(b) where the apparent relaxation time $\tau_{e}$ seems to
saturate to a maximum value $\tau_{m}$, reported in table 1, at large $h_{0}$.
Since no clear plateau is observed, especially for the PEOaq solution, the
value of $\tau_{m}$ is only an estimation. In figure 2(b), we only show one
data point for each plate diameter, with $V^{*}$ between $0.3$ for the
smallest plates and $0.11$ for the largest plates.

Note that, the PEO solutions used in figures 2(b) are not the same as the ones
used in figure 2(a) and and have apparent relaxation times about $30$% larger
for $2R_{0}=7$ mm, while the shear viscosity was only up to $10$ % larger,
meaning that the shear rheology parameters in table 1 are representative of
both solutions. These differences are due to slightly different preparation
protocols, e.g. agitation times, for a given recipe.

##  4 Interpretations

The apparent disagreement between experiments and equation 1 implies that
either the liquid changes, becoming less elastic for lower values of $h_{0}$,
or that the Oldroyd-B model, from which equation 1 is derived, misses some
important features of polymer dynamics in extensional flows.

First of all, although the relaxation time measured in filament thinning is
known to increase with polymer concentration (Tirtaatmadja et al., 2006;
Clasen et al., 2006b), solvent evaporation can’t explain the observed increase
of the apparent relaxation time with increasing droplet size since the bulk
polymer concentration would increase quicker for smaller droplets due to their
larger surface to volume ratio, leading to larger concentrations for smaller
droplets. Besides, all experiments are carried out at high relative humidity
($>80$%) and repeating an experiment several times over the course of $10$
minutes doesn’t lead to a monotonic increase or decrease of $\tau_{e}$ over
time, beyond small variations of less than $5$%. The latter observation also
argues against polymer degradation as a possible explanation. Moreover,
$\tau_{e}$ is observed to increase with $h_{0}$ for both PEO and HPAM
solutions, even though HPAM is less fragile than PEO.

Therefore, if the liquid is in fact the same for each experiment, the
Oldroyd-B model fails to describe the full polymer dynamics in the bridge /
filament. In particular, differences in the history of polymer deformations
for different drop sizes could lead to different “initial” states of polymers
at the onset of the elastic regime, which could result in different filament
thinning rates. We now discuss whether finite extensibility of polymer chains,
as described by the FENE-P model, can account for such differences.

Figure 3: (a) $h(t)$ for the PEOaq solution tested with different plate
diameters. The elastic regime is fitted by equation 3 (see text) with
$\tau=\tau_{m}$, using $h_{1}$ and $A_{1}/L^{2}$ as fitting parameters. (b)
Time-evolution $h$ from experiments and of $A_{zz}$ calculated from equation 2
with $\dot{\epsilon}=-2\dot{h}/h$ using experimental values of $h$, with
$\tau=\tau_{m}$ and various values $L^{2}$ for the PEOaq solution tested with
a plate diameter $2R_{0}=10$ mm, using $A_{zz}=1$ at $h=h_{0}$ as the initial
condition.

Following Wagner et al. (2015), for a uniaxial extensional flow, the polymeric
part of the normal stress is $\sigma_{p,zz}=G(fA_{zz}-1)$ in the flow
direction $z$ where $G$ is the elastic modulus and $A_{zz}$ is the normal part
of the conformation tensor ${\bf A}$ which follows

$\dot{A}_{zz}-2\dot{\epsilon}A_{zz}=-\frac{fA_{zz}-1}{\tau},$ (2)

where $\dot{\epsilon}$ is the extension rate, $\tau$ the relaxation time and
$f=(1-\mathrm{tr}({\bf A})/L^{2})^{-1}$ where $L$ is the ratio of the fully
unravelled chain size to its equilibrium size . In this model the stress
diverges as $A_{zz}$ approaches its limit value $L^{2}$. In the elastic
regime, we assume that polymers are far from equilibrium and that the axial
stress dominates over the radial stress, i.e. $A_{zz}\gg 1>A_{rr}$. Assuming
negligible inertia and solvent viscosity, the force balance equation is
$(2X-1)\gamma/h=\sigma_{p,zz}$ with $X=3/2$ in the Oldroyd-B limit (Eggers et
al., 2020). Assuming a small correction due to finite extensibility, combining
the force balance with equation 2 with $\dot{\epsilon}-2\dot{h}/h$ leads to
the ordinary differential equation $(3+A_{zz}/L^{2})\dot{A}_{zz}=A_{zz}/\tau$
which has an implicit solution

$\frac{t-t_{1}}{\tau}=\frac{A_{zz}-A_{1}}{L^{2}}+3\ln{\left(\frac{A_{zz}}{A_{1}}\right)},$
(3)

where $A_{1}=A_{zz}(t_{1})$ quantifies the amount of stretching of polymers in
the $z$-direction at the onset of the elastic regime at time $t_{1}$. In the
Oldroyd-B limit $L^{2}\to\infty$, we recover the expected exponential growth
$A_{zz}=A_{1}\exp{((t-t_{1})/3\tau)}$. For a finite extensibility, the stress
diverges and $h\to 0$ in finite time when $A_{zz}$ saturates to $L^{2}$, which
occurs sooner as $A_{1}/L^{2}$ is closer to one. Consequently, as $A_{1}$
approaches $L^{2}$, increasingly larger filament thinning rates will be
observed. This is shown in figure 3(a) where the elastic regime of the PEOaq
solution, tested with different plate diameters, is compared with the
predictions of equation 3 where we have chosen the maximum relaxation time
$\tau_{m}$ measured at large $h_{0}$ as the relaxation time of the FENE-P
model. We use the fact that, according to the force balance equation, the
filament radius is $h/h_{1}=f_{1}A_{1}/fA_{zz}$ where
$f_{1}=(1-A_{1}/L^{2})^{-1}$ and $h_{1}=h(t_{1})$. We use $A_{1}/L^{2}$ and
$h_{1}$ as fitting parameters to obtain a good agreement between model and
experiments. Most importantly, we have to impose that $A_{1}/L^{2}$ gets
closer to one as $h_{0}$ decreases to capture the observed thinning rates, all
larger than $1/3\tau_{m}$. For $2R_{0}=2$ mm for example, we need
$A_{1}/L^{2}=0.93$, meaning that polymers are almost fully extended at the
onset of the elastic regime. Equally good fits can be obtained for the PEOvisc
and HPAM solutions and the corresponding values of the fitting parameters
$A_{1}/L^{2}$ are plotted against $h_{0}$ in figure 4(a) (light purple).

Although promising, this explanation is only valid if the FENE-P model indeed
predicts that polymers are more stretched at the onset of the elastic regime
as $h_{0}$ decreases. To test this, we use equation 2 to calculate
$A_{zz}(t)$, using the experimental values of $h(t)$ for
$\dot{\epsilon}(t)=-2\dot{h}/h$, although this expression is only valid at the
thinnest bridge radius. In other words, we calculate the predictions of the
model for the experimental history of extension rates. Importantly, we do
_not_ assume large polymer deformations ($A_{zz}\gg 1$), since polymers are at
rest at the onset of capillary thinning, i.e. $A_{zz}=1$ when $h=h_{0}$. We
use $f=(1-A_{zz}/L^{2})^{-1}$ since when $f\neq 1$, axial stresses dominate
over radial stresses. In order to circumvent the issue of calculating
$\dot{h}$ from experimental values of $h$, we introduce a function $y(t)$ such
that $A_{zz}=y/h^{4}$, which gives
$\dot{A}_{zz}+4(\dot{h}/h)A_{zz}=\dot{y}/h^{4}$, so that equation 2 becomes
$\tau\dot{y}=h^{4}-y/(1-y/(h^{4}L^{2}))$ which doesn’t involve $\dot{h}$
anymore. To solve this equation, we use a standard ODE solver, using spline
interpolation to create a $t\to h(t)$ function based on experimental values of
$h$. This equation can be integrated analytically in the Oldroyd-B limit, as
shown by Bazilevsky et al. (2001).

Figure 4: (a) Values of $A_{1}/L^{2}$ used as fitting parameters (light
purple, see e.g. figure 3(a)), and values calculated from the FENE-P model
(dark blue, see e.g. figure 3(b)) for $\tau=\tau_{m}$ and values of $L^{2}$
discussed in the text, against $h_{0}$ for all liquids and plate diameters.
(b) Same values of $A_{1}$ calculated from the FENE-P model (dark blue),
compared with $(h_{0}/h_{1})^{4}$ (light orange). All values of $A_{1}$
calculated from FENE-P are the same as in Oldroyd-B except for the PEOvisc
solution where Oldroyd-B values are shown with empty blue symbols.

The results are shown in figure 3(b) for the PEOaq solution tested with a
plate diameter $2R_{0}=10$ mm, with $\tau=\tau_{m}$ for the relaxation time of
the FENE-P model, along with various values of $L^{2}$, including the
Oldroyd-B limit $L^{2}\to+\infty$. As expected, values of $A_{zz}$ calculated
from FENE-P coincide with Oldroyd-B until it saturates when reaching $L^{2}$.
In particular, the value of $A_{1}$ becomes independent of $L^{2}$ when
$L^{2}$ is sufficiently large and becomes indistinguishable from the values
predicted by the Oldroyd-B model.

The values of $A_{1}$ calculated from the FENE-P model with $\tau=\tau_{m}$
are plotted in figure 4(a) against $h_{0}$ for all three solutions (dark blue)
as $A_{1}/L^{2}$ where we chose $L^{2}=49000$, $20000$ and $6200$ for the
PEOaq, PEOvisc and HPAM solutions, respectively. For each liquid, the values
of $L^{2}$ (a priory unknown) is chosen so that the calculated value of
$A_{1}/L^{2}$ at the largest $h_{0}$ coincides exactly with the fitting value
(light purple) used to fit the elastic regime with equation 3. We find that,
while the fitting values of $A_{1}$ increase towards $L^{2}$ as $h_{0}$
decreases, the calculated values of $A_{1}$ only do so for the PEOvisc
solution, for which a good agreement is found with the fitting values, and do
not for the PEOaq and HPAM solutions for which $A_{1}$ is fairly constant. For
these last two, no other value of $L^{2}$ can lead to a better agreement since
decreasing $L^{2}$ would just shift all calculated values towards the limit
$A_{1}/L^{2}=1$.

In order to better understand this, we compare these calculated values of
$A_{1}$ (dark blue) with their upper limit $(h_{0}/h_{1})^{4}$ (light orange)
in figure 4(b). This upper limit corresponds to a relaxation time $\tau$ so
large that polymer relaxation (right hand side of equation 2) is always
negligible in the Newtonian regime ($t<t_{1}$), a case where equation 2 can be
integrated as $A_{zz}h^{4}=h_{0}^{4}$. Differences in values of
$(h_{0}/h_{1})^{4}$ among the three polymer solutions are due to differences
in $h_{1}$ which is an increasing function of $\eta_{p}$, as will be discussed
in a separate paper. We find that $A_{1}$ is very close to the
$(h_{0}/h_{1})^{4}$ limit for the PEOaq and HPAM solutions at low $h_{0}$,
with increasingly lower values as $h_{0}$ increases. This is consistent with
the fact that the Deborah number $\mbox{{De}}=\tau_{m}/\tau_{R}$ decreases
from $500$ ($\gg 1$) to $5$ as $h_{0}$ increases where $\tau_{R}=\sqrt{\rho
h_{0}^{3}/\gamma}$ is the Rayleigh time scale relevant for the thinning
dynamics of such low-viscosity liquids with Ohnesorge numbers
$\mbox{{Oh}}=\eta_{s}/\sqrt{\rho\gamma h_{0}}$ up to $7\times 10^{-3}\ll 1$.
The greater difference between $A_{1}$ and $(h_{0}/h_{1})^{4}$ for the PEOvisc
solution in figure 4(b) is due to the slower thinning dynamics in the
Newtonian regime, caused by a larger solvent viscosity with Ohnesorge numbers
$270$ times larger than for the two other solutions. As $h_{0}$ increases, the
Newtonian thinning dynamics gets slower since the inertio- and visco-capillary
time scales $\tau_{R}$ and $\tau_{\mathrm{visc}}=\eta_{s}h_{0}/\gamma$
increase, which is the reason why $A_{1}$ decreases with $h_{0}$ for the
PEOvisc solution.

Figure 5: (a) $h(t)$ from experiments and simulations for the PEOvisc solution
for different plate diameters. Simulations are performed for $2R_{0}=2$, $3.5$
and $5$ mm only, using the FENE-P model with $\tau=\tau_{m}$ and
$L^{2}=20000$. The elastic regime is fitted by equation 3 where, like in
figure 3(a), $A_{1}/L^{2}$ and $h_{1}$ are used as fitting parameters. (b)
$h(t)$ from DoS experiments with different nozzle diameters for the PEOaq
solution. We report the values of the maximum diameter $2R_{0}^{*}$ of the top
end-drop which is between the inner and outer nozzle diameter. Top images
correspond to the steadily hanging drop (left) and to four times labelled 1 to
4 indicated on the $h(t)$ curve for $2R_{0}^{*}=3.27$ mm. Inset: $\tau_{e}$
vs. $h_{1}$ from CaBER and DoS for the PEOaq solution.

The FENE-P model can hence only explain the increase of the apparent
relaxation time $\tau_{e}$ with $h_{0}$ for the (dilute) PEOvisc solution,
which has the highest solvent viscosity out of our three solutions. This is
checked further by numerical simulations of the axisymmetric problem performed
for fixed material properties $\eta_{s}$ and $\eta_{p}$ from shear rheology,
$\tau=\tau_{m}$ and the value of $L^{2}=20000$ used in figure 4. The equations
and numerical methods are details in Rubio et al. (2020). The results are
shown in figure 5(a) for the three smallest plates and the corresponding
experimental droplet volumes in terms of the time evolution of the minimum
bridge / filament radius. Simulations are able to capture the Newtonian regime
quite well and provide a reasonable agreement with experiments in the elastic
regime. In particular, the filament thinning rate varies with the plate
diameter, consistent with experiments, while the Oldroyd-B model would give a
constant thinning rate $1/3\tau_{m}$. Simulations could not be continued far
enough to compare with the full experimental time window. The figure also
features the analytic solution 3 for the elastic regime where, like in figure
3(a), $A_{1}/L^{2}$ and $h_{1}$ are used as fitting parameters.

##  5 Conclusions and discussion

We have shown experimentally that the thinning rate of filaments of various
polymer solutions is not just a material property but depends on the size of
the system in both CaBER (with slow plate separation) and DoS experiments.
Although all filaments are observed to thin exponentially, as predicted by the
Oldroyd-B model (see equation 1), we show that the inferred apparent
relaxation time $\tau_{e}$ measured in CaBER increases as the minimum bridge
radius $h_{0}$ marking the onset of capillary thinning increases, which is a
function of both the plate diameter and droplet volume, and seems to saturate
at large $h_{0}$. These observations hence suggest that CaBER relaxation times
reported in the literature are not universal since testing a given polymer
solution with different plate diameters and droplet volumes yields different
results, at least for slow-plate-retraction CaBER protocols. We demonstrate
that the variation of $\tau_{e}$ with $h_{0}$ is not caused by solvent
evaporation or polymer degradation and can’t be universally explained by
finite extensibility effects described by the FENE-P model. These observations
suggest that the single-mode Oldroyd-B and FENE-P models miss some important
features of polymer dynamics in extensional flows.

A physical interpretation is still lacking and a better constitutive equation,
capturing the deformation-history-dependent filament thinning rate, is needed.
Models including a conformation-dependent drag accounting for hydrodynamic
interactions between unravelling polymer chains in extensional flows could be
the key.

Acknowledgements. We thank Louison Laruelle and Carmen van Poelgeest for
preliminary experimental work.

Declaration of Interests. The authors report no conflict of interest.

##  Appendix A Dripping-onto-substrate (DoS)

In DoS experiments, a horizontal substrate (here, a plasma-treated aluminium
plate) is moved slowly upward until being in contact with a liquid droplet
hanging steadily from a nozzle. As shown in the image sequence in figure 5(b),
a fast spreading of the liquid on the plate leads to the pinch-off of the
bridge connecting the substrate to the nozzle. This transiently leads to the
formation of an exponentially thinning filament, as shown by the time-
evolution of the minimum bridge / filament radius $h$ in figure 5(b), where
the PEOaq solution is tested with four different nozzle diameters. As in CaBER
experiments, the apparent relaxation time extracted from the filament thinning
rate increases with the droplet size, here quantified by the nozzle diameter.
This apparent relaxation time $\tau_{e}$ is plotted in the inset of figure
5(b) for both CaBER and DoS experiments against the filament radius $h_{1}$
marking the onset of the elastic regime which, unlike $h_{0}$ in CaBER, is
easily definable in both methods. The relatively good collapse of the data
points on a single curve suggests a universal physical mechanism for the
dependence of the apparent relaxation time on the size of the system,
independent of the exact method used.

## References

  * Anna & McKinley (2001) Anna, S. L. & McKinley, G. H. 2001 Elasto-capillary thinning and breakup of model elastic liquids. Journal of Rheology 45 (1), 115–138.
  * Bazilevsky et al. (1997) Bazilevsky, AV, Entov, VM, Lerner, MM & Rozhkov, AN 1997 Failure of polymer solution filaments. Polymer Science Series ac/c of Vysokomolekuliarnye Soedineniia 39, 316–324.
  * Bazilevsky et al. (1990) Bazilevsky, AV, Entov, VM & Rozhkov, AN 1990 Liquid filament microrheometer and some of its applications. In Third European Rheology Conference and Golden Jubilee Meeting of the British Society of Rheology, pp. 41–43. Springer.
  * Bazilevsky et al. (2001) Bazilevsky, AV, Entov, VM & Rozhkov, AN 2001 Breakup of an oldroyd liquid bridge as a method for testing the rheological properties of polymer solutions. Polymer Science Series AC/C of Vysokomolekuliarnye Soedineniia 43 (7), 716–726.
  * Campo-Deano & Clasen (2010) Campo-Deano, L. & Clasen, C. 2010 The slow retraction method (srm) for the determination of ultra-short relaxation times in capillary breakup extensional rheometry experiments. Journal of Non-Newtonian Fluid Mechanics 165 (23-24), 1688–1699.
  * Clasen et al. (2006a) Clasen, Christian, Eggers, Jens, Fontelos, Marco A, Li, JIE & McKinley, Gareth H 2006a The beads-on-string structure of viscoelastic threads. Journal of Fluid Mechanics 556, 283–308.
  * Clasen et al. (2006b) Clasen, C., Plog, J. P., Kulicke, W.-M, Owens, M., Macosko, C., Scriven, L. E., Verani, M. & McKinley, G. H. 2006b How dilute are dilute solutions in extensional flows? Journal of Rheology 50 (6), 849–881.
  * Dinic et al. (2017) Dinic, Jelena, Jimenez, Leidy Nallely & Sharma, Vivek 2017 Pinch-off dynamics and dripping-onto-substrate (DoS) rheometry of complex fluids. Lab on a Chip 17 (3), 460–473.
  * Eggers et al. (2020) Eggers, Jens, Herrada, Miguel Angel & Snoeijer, JH 2020 Self-similar breakup of polymeric threads as described by the oldroyd-b model. Journal of fluid mechanics 887, A19.
  * Entov & Hinch (1997) Entov, V. M. & Hinch, E. J. 1997 Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid. Journal of Non-Newtonian Fluid Mechanics 72 (1), 31–53.
  * Miller et al. (2009) Miller, Erik, Clasen, Christian & Rothstein, Jonathan P 2009 The effect of step-stretch parameters on capillary breakup extensional rheology (caber) measurements. Rheologica acta 48, 625–639.
  * Rajesh et al. (2022) Rajesh, Sreeram, Thiévenaz, Virgile & Sauret, Alban 2022 Transition to the viscoelastic regime in the thinning of polymer solutions. Soft Matter 18 (16), 3147–3156.
  * Rodd et al. (2004) Rodd, L. E., Scott, T. P., Cooper-White, J. J. & McKinley, G. H. 2004 Capillary break-up rheometry of low-viscosity elastic fluids .
  * Rubio et al. (2020) Rubio, Manuel, Vega, Emilio J, Herrada, Miguel A, Montanero, José M & Galindo-Rosales, Francisco J 2020 Breakup of an electrified viscoelastic liquid bridge. Physical Review E 102 (3), 033103.
  * Stelter et al. (2000) Stelter, M, Brenn, Günter, Yarin, AL, Singh, RP & Durst, Franz 2000 Validation and application of a novel elongational device for polymer solutions. Journal of Rheology 44 (3), 595–616.
  * Tirtaatmadja et al. (2006) Tirtaatmadja, V., McKinley, G. H. & Cooper-White, J. J. 2006 Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration. Physics of fluids 18 (4), 043101.
  * Wagner et al. (2015) Wagner, Caroline, Bourouiba, Lydia & McKinley, Gareth H 2015 An analytic solution for capillary thinning and breakup of FENE-P fluids. Journal of Non-Newtonian Fluid Mechanics 218, 53–61.