A simple criterion and experiments for onset of flocculation in kaolin clay
suspensions

N. Rommelfanger1, B. Vowinckel1,3, Z. Wang2, R. Dohrmann4, E. Meiburg1 & P.
Luzzatto-Fegiz1

1Department of Mechanical Engineering, University of California, Santa
Barbara, CA 93106 2Department of Physics, University of California, Santa
Barbara, CA 93106 3Leichtweiß-Institut für Wasserbau, Technische Universität
Braunschweig, 38106 Braunschweig, Germany 4Bundesanstalt für Geowissenschaften
und Rohstoffe (BGR), Geozentrum Hannover, Stilleweg 2, 30655 Hannover, Germany

###### Abstract

Cohesive effects between fine-grained sediment particles greatly influence
their effective settling rate and erodibility. Many studies have observed a
qualitative difference in settling dynamics between clays in freshwater, where
particles remain dispersed, and in saltwater, where aggregates form and settle
rapidly. The critical coagulation concentration (CCC) of salt that separates
the two regimes however remains under-investigated, even though knowledge of
the CCC is crucial to understanding aggregation in settings such as estuaries,
where large salt concentration gradients occur. Furthermore, no simple
criterion exists to predict the CCC for clay suspensions. In this study,
systematic experiments are performed to determine the CCC, by measuring
transmitted light intensity through clay suspensions. To investigate the
effect of ion valence, sodium chloride (NaCl) and calcium chloride (CaCl2) are
used. For kaolin clay, the results show a CCC of 0.6 mM NaCl ($\approx 0.04$
ppt NaCl $=$ 0.04 PSU), and of 0.04 mM CaCl2 ($\approx 0.004$ ppt CaCl2).
Because these salinities are lower than those commonly observed in nature,
these findings indicate that kaolin clay should flocculate in nearly all
natural aquatic environments. Furthermore, due to the fact that tap water
often has salinities higher than this threshold, these results imply that
great care is needed in experiments, especially in large facilities where
using distilled water is not feasible. In addition, a simple criterion to
estimate the CCC for a kaolin clay suspension is derived. This criterion
predicts that flocculation occurs at extremely low salt concentrations and is
approximately independent of clay concentration, in agreement with the
experimental observations and consistent with experimental evidence from the
literature.

##  1 Introduction

Cohesive sediments comprising particles smaller than $\sim 63\,\upmu\text{m}$
[1, 2] are ubiquitous in aquatic environments such as rivers [3], lakes and
estuaries [4], fisheries and coastal ecosystems, and benthic habitats.
Cohesive sediments can experience flocculation, which greatly accelerates
their settling rate and qualitatively alters settling dynamics. Closely
related to this, cohesive forces also significantly alter the erodibility of
the sediment grains. Owing to their plasticity and cohesive strength, clay
minerals of grain size smaller than $\sim\leavevmode\nobreak\
2\,\upmu\text{m}$ are a key constituent of cohesive sediments [1, 5, 2]. The
cohesive interparticle forces responsible for flocculation are highly
dependent on salt concentration, making it important to develop clay
flocculation criteria that are both physically grounded, as well as tractable
enough to be readily used in practical applications.

The stability of clay suspensions (and therefore the onset of flocculation) is
determined by the interplay between repulsive electrostatic forces and
attractive van der Waals forces [6]. Kaolin clay is composed mainly of the
mineral kaolinite together with other sedimentary minerals, such as quartz,
feldspar, iron oxides and additional minerals in trace concentrations.
Kaolinite is one of the most common clay minerals found in the environment and
is widely used in laboratory models [7, 8, 9], and is therefore the focus of
this study. Kaolinite particles are plate-like in shape (as shown by the
micrograph in Fig. 1), with negatively charged faces. An electric double layer
forms around each particle when submerged in water, which leads to
electrostatic repulsion between the like-charged particles. By adding positive
ions in the form of dissolved salts, one may compress the double layer,
decreasing the repulsive forces relative to van der Waals attraction and
allowing aggregation to occur. The aggregates, or flocs, formed by this
process settle faster than individual particles, sweeping up other flocs in
their path, such that the suspended clay concentration decreases rapidly after
the onset of flocculation [10, 11]. The salt concentration at which the
repulsive forces become smaller than the attractive forces is known in the
literature on colloids as the ‘critical coagulation concentration’ (CCC) [12].

Although several investigations in environmental science have focused on clay
suspensions, conflicting observations have been reported regarding the
salinity required to trigger flocculation, with some researchers finding a
significant influence of salinity on flocculation, and others measuring no
influence. [13] measured the effect of shear on the flocculation of fine
sediment (composed partially of kaolinite) from the Detroit River. They
observed similar mean floc diameters for suspensions in both tap water and in
well water with added salts. The authors concluded that small changes in
freshwater composition do not strongly affect flocculation. [14] observed
increasing floc size in the freshwater Garonne and Dordogne Rivers (0.01 PSU)
while moving downstream towards the Gironde Estuary. [15] reported no
correlation between major ion concentration and floc size in six rivers in
southeastern Canada. [16] measured the flocculation properties of suspended
sediment taken from the Rhone River, and observed only a slight increase in
mean floc size when the river water’s salinity exceeded 25 ppt NaCl. They
concluded that salt-induced aggregation may not enhance flocculation at the
river mouth. [17] measured flocculation in the San Francisco Bay-Delta Estuary
and observed similar floc sizes in the freshwater Sacramento River (0.1 PSU)
as in saltwater.

Although field measurements are in disagreement about the effect of salt
concentration, laboratory experiments have consistently reported that
switching from freshwater to saltwater induces flocculation. [18] measured the
effect of shear, pH, and salinity on the flocculation of laboratory-made or
treated mud (treated with hydrogen peroxide) composed partially of kaolinite.
They found that at a shear rate of 35 s-1 and pH $=8$, mud with added salt
formed larger flocs than mud without added salt. [10] performed experiments
with either freshwater or saltwater of at least 5 ppt NaCl, and reported that
kaolin clay suspended in saltwater settled more rapidly than kaolin clay
suspended in freshwater. Although these laboratory studies show that
flocculation increases upon adding salt, this increase in flocculation is not
commonly observed in situ upon transitioning from saltwater to freshwater (for
example, in estuaries). Indeed, flocculation is often observed in freshwater
environments [19]. [20] measured flocculation in the Po River Delta, and
noticed that flocs had already formed upstream in the freshwater Po River,
where kaolinite composes part of the sediment [21].

In summary, while laboratory experiments have observed a consistent difference
in settling between pure freshwater and saltwater, field measurements and
experiments with natural samples have provided contrasting results, with
numerous studies reporting flocculation already taking place in freshwater,
such that a negligible difference in flocculation was observed after the
introduction of additional salt. Furthermore, laboratory studies of
flocculation do not routinely report the salinity threshold for aggregation.

An expression for the CCC can be derived through Derjaguin Landau Verwey
Overbeek (DLVO) theory, which calculates the energy balance of electrostatic
repulsion and van der Waals attraction between plate-like particles [22, 23].
This derivation is, however, mathematically complex, and the final expression
involves parameters that are difficult to measure and often unfamiliar to
those outside of chemical engineering, such as the system’s Hamaker parameter,
the closest approach distance between particles, and the surface potential of
the clay particles. Furthermore, there does not seem to be agreement over the
definition of these input parameters and over how to measure them; as a matter
of fact, the values selected by different users often vary by an order of
magnitude [24]. Moreover, despite its complexity, DLVO theory commonly
requires additional modifications to completely describe the relevant physics;
an example is given by the elastohydrodynamic corrections introduced by [6].
While manageable theories based on dimensional analysis have been developed
for particle size distributions in clay suspensions [25], the authors are not
aware of similarly tractable models for the CCC. A simple approach that could
predict the value of the CCC (or even its order of magnitude) using easily
found properties of the clay and ambient fluid would be very valuable in both
laboratory and field applications. This would be especially valuable, for
example, for predicting the onset of flocculation in estuaries, where a
gradient of salt concentration from $\approx 0-35$ PSU occurs as rivers mix
with coastal oceans, as well as for industrial applications, where countless
processes involve ionic solutions of varying salinity [26, 27, 28].

In this work, experiments are performed to accurately measure the CCC of
kaolin clay dispersed in NaCl and CaCl2 solutions, with a focus on testing the
influence of very low salinities. Using scaling arguments, a tractable
criterion is derived to predict the stability of clay suspensions. This
criterion uses input parameters that are well-characterized for many types of
clay, and provides a simpler alternative to DLVO theory.

##  2 Materials and Methods

Figure 1: Sample of dry kaolin powder used in the present experiments, imaged
with an environmental scanning electron microscope.

###  2.1 Clay Properties

Kaolin clay was purchased as a dry powder from Sigma-Aldrich (product #18616,
typical density of 2.65 g/cm3 [29]). The quality of this specific kaolin clay
was confirmed by X-ray diffraction of powdered specimen as well as oriented
slides following treatments with air dried and ethylene-glycol-solvated
conditions. Kaolinite
($\text{Al}_{2}\text{Si}_{2}\text{O}_{5}(\text{OH})_{4}$) dominated the clay
as expected. Traces of other phyllosilicates such as mica and expandable clay
minerals (smectite) along with minor amounts of quartz, feldspar and possibly
traces of tridymite (SiO2 polymorph) were also present.

The cation exchange capacity (CEC) of this specific kaolin clay was
$3\,\text{mEq}/100\,\text{g}$, measured using the Cu-trien method as described
in [30]. This CEC is within the range of $3-15\,\text{mEq}/100\,\text{g}$ from
the classic study of [31], which provides comprehensive CEC values for a wide
variety of clays. The fact that the CEC value is at the lower end of the
typical CEC range can partly be explained by the presence of admixtures of
expandable clay minerals (smectitic), which implies that the kaolinite itself
has a very low CEC. A PANalytical Axios XRF spectrometer was also used to
determine the chemical composition of the kaolin clay, shown in Table 1. The
relatively large K2O content of the kaolin clay was explained by the presence
of K-feldspar.

Table 1: Chemical composition of kaolin clay used in the present experimentsa Constituent | $\text{SiO}_{2}$ | $\text{Al}_{2}\text{O}_{3}$ | $\text{Fe}_{2}\text{O}_{3}$ | CaO | MgO | $\text{K}_{2}\text{O}$ | $\text{Na}_{2}\text{O}$ | MnO | $\text{TiO}_{2}$ | $\text{P}_{2}\text{O}_{5}$ | $\text{SO}_{3}$
---|---|---|---|---|---|---|---|---|---|---|---
% mass | 55.7 | 39.6 | 1.0 | 0.1 | 0.3 | 2.7 | 0.1 | 0.0 | 0.1 | 0.2 | $<0.01$
aDetermined by X-ray fluorescence following the procedure described by [32].

The specific surface area was determined as 8.6 m2/g, by using N2 and
Brunauer-Emmett-Teller (BET) analysis. A characteristic image of the kaolin
clay powder from an environmental scanning electron microscope (ESEM, FEI
Company XL30) is shown in Fig. 1. Kaolin clay particles are arranged as
typical booklets consisting of regularly-shaped kaolinite plates, and as
dispersed particles thereof. The plate-like particles have two long axes and
one short axis, and are typically $2\,\upmu\text{m}$ or less in width, and
between 10-120 nm in thickness [33, 34]. The particle size distribution of the
kaolin clay was measured using X-ray granulometry (Micromeritics SediGraph
5100), showing that 62.4% of the clay consisted of particles smaller than
$2\,\upmu\text{m}$, and 93.7% consisted of particles smaller than
$6.3\,\upmu\text{m}$.

###  2.2 Fluid Properties

As this study aimed to trigger and analyze flocculation in a precisely
controlled manner, deionized (DI) water was used as the ambient fluid. The DI
water had an electrical resistivity of 18 M$\Upomega$-cm; this high
resistivity was chosen to minimize the presence of ions in the ambient fluid
that could contribute to flocculation. The salts consisted of NaCl and
$\text{CaCl}_{2}$ from Sigma-Aldrich (product numbers 746398 and 793639
respectively). As is well known, NaCl is the most common salt found in
seawater [35]. The choice of $\text{CaCl}_{2}$ enabled the observation of the
effects of a divalent cation on flocculation, while still using chloride as
the anion to limit the number of independent variables. Below, salinity is
reported in units of molarity (M), that is, moles of salt per liter of
solution. Molarity may be converted to parts per thousand (ppt) by multiplying
the molarity by the salt’s molar mass: this is 58.44 g/mol for NaCl, and
110.98 g/mol for CaCl2. For NaCl solutions, practical salinity units (PSU,
commonly used in oceanography and limnology) are equivalent to ppt.

###  2.3 Experimental Apparatus

Various methods have been used to measure colloid sedimentation or empirically
define the critical coagulation concentration of colloids, and methods
employing changes in optical properties are common for clay suspensions [36,
37, 38, 39, 40, 41, 42, 12, 43]. Higher suspended clay concentrations result
in greater attenuation of light passing through a cuvette. Although measuring
changes in transmitted light intensity is an indirect way to measure the CCC,
its simplicity allows one to rapidly test a wide parameter space. Furthermore,
determining the CCC by analyzing macroscopic changes in suspended clay
concentration, as opposed to microscopic changes in particle size, is also
useful for practical applications where the amount of clay in suspension is
important, such as mining tailings [26], water treatment [27], and paper and
ceramic production [28].

The present investigation has been developed in preparation for microgravity
experiments aboard the International Space Station (ISS). Due to operational
constraints, the ISS setup uses the Binary Colloidal Alloy Test (BCAT)
apparatus, which has been repeatedly employed to investigate the
crystallization of colloid mixtures in microgravity [44, 45, 46] (ISS results
involving cohesive sediment – labelled ‘BCAT-CS’ – are presently being
analysed). The laboratory setup employed for this study was therefore modeled
after the BCAT apparatus, and is shown in Fig. 2. It comprises a camera (Nikon
D2Xs with 60 mm lens, f/8, and 1/60 s exposure time) fixed on a rail 16 cm
from a rack that holds five cuvettes. The cuvettes hold a volume of 10 mm by
45 mm by 4 mm each, where the optical light path for illumination through the
interior of the cuvette is 4 mm, such that the field of view is 10 mm by 45 mm
in width and height, respectively. Due to the small size of these cuvettes,
one should not expect the settling velocities of initially dispersed clay
suspensions (procedure described in detail below) to be independent of the
container size [47]. Preliminary tests were however conducted with different
containers, which showed that the qualitative analysis on the triggering of
flocculation was independent of container size (width, height) and container
shape (cuboid, cylinder, cone). As described below, data was collected after
the initial dynamics in the cuvette had ceased, in order to minimize the
influence of reduced settling velocity. These small volumes enabled the rapid
investigation of a wide parameter range of salinity and clay concentrations to
determine critical thresholds for flocculation. Behind the cuvette rack, an
LED light panel (Fotodiox C-218AS set to 14% intensity and 4500 K temperature)
uniformly illuminated the five cuvettes. The uniform lighting provided by the
light panel eliminated the need for an additional diffuser. The entire setup
was covered by black construction paper so that only the light passing through
the clay suspensions reached the camera.

Figure 2: The experimental setup. A camera (left) photographs a rack of five
cuvettes (right-center), which are evenly illuminated from behind by a diffuse
LED light panel (right). Black construction paper covering the entire setup
prevents ambient light from reaching the camera’s lens.

###  2.4 Experimental Procedure

Results are reported for kaolin clay suspensions of three different
concentrations: 4 ppt, 20 ppt, and 40 ppt. For clay suspensions, ppt is
defined as (clay mass) / (total mass)$\times 10^{3}$ (where both masses were,
of course, measured using the same units). These clay suspensions allowed
testing the effect of varying the clay concentration across one order of
magnitude, while maintaining a sensible range of transmitted light intensity
throughout the flocculation process. For each of these clay concentrations,
several salt concentrations were tested, with additional measurements
performed covering the concentration range around the onset of flocculation.
For both NaCl and CaCl2, these solutions ranged from $1.71\times 10^{-3}$ mM
to $1.71\times 10^{1}$ mM, with a total of 8-12 different salt concentrations
for each clay concentration. Because the focus of this investigation was on
salt concentrations around the CCC, solutions with salinity higher than 17.1
mM were not tested (note that 17.1 mM NaCl = 1 PSU).

For each combination of clay concentration and salinity, a sample volume of
100 mL was prepared. To remove contaminants, all clay samples were rinsed
before testing by mixing the clay with DI water, allowing the clay to settle
for at least 60 hours, and then removing the supernatant. This washing
techniques is consistent with those used by other studies of clay suspensions
[48]. Furthermore, this washing procedure ensured that the clay had soaked in
fluid for multiple days before experiments were conducted, thereby more
closely matching the state of clay found in natural aquatic settings.
Nevertheless, it was found that washed clay flocculated at approximately the
same CCC as unwashed clay.

To test whether small particles lost during the rinsing process had an impact
on flocculation, an alternative washing procedure was also employed, whereby
selected samples were centrifuged at 8000 rpm for three minutes before
removing the supernatant; this ensured that even the smallest particles were
retained. The flocculation behavior of the centrifuged samples was the same as
for samples that were not centrifuged.

Shortly before testing, all suspensions were shaken vigorously by using a
mixing accessory attached to a reciprocating saw running at approximately 1500
rpm for 15 s. The saw had a stroke length of 1/2 inch, thus accelerating the
suspensions at approximately 130 m/s2.

Once the sample was mixed, a pipette was immediately used to transfer small
volumes of mixture into the glass cuvettes. Well-used glass cuvettes were
used, as new glassware is often slightly alkaline [49], which may affect the
flocculation process at low salt concentrations. The cuvettes were re-agitated
by hand once more just before placing them in the rack and beginning recording
images with the camera. The loading procedure took about 40 seconds. The
camera took a grayscale photograph every 20 seconds for ten minutes.

###  2.5 Image Analysis

In order to minimize the influence of solid boundaries on lighting and on the
recorded settling behavior, a ‘working area’ was delineated in the center part
of the cuvette, as shown by the blue dashed lines in Fig. 3.

Figure 3: Example photograph of settling progress at ten minutes for cuvettes
with 20 ppt kaolin clay suspensions. Salt concentration (CaCl2) in the
solutions increases from left to right. The areas marked by blue dashed lines
are determined to be the “working areas” in each cuvette. Light intensity was
horizontally-averaged across the solid red lines, at 23.2 mm above the bottom
of the cuvettes. This averaged light intensity was used as a measure of
suspended clay concentration.

As a metric of suspended clay concentration, the light intensity was
horizontally-averaged at a height of 23.2 mm above the bottom of the cuvette,
for images acquired ten minutes after the start of the experiment (this height
is marked by the red lines in Fig. 3). This height was selected because, once
a steady state was reached (as discussed below), this vertical location was
distinct from both the settled clay on the cuvette floor and from the
unevenly-lit region near the top of the cuvette.

Figure 4 shows a time series of the light intensity at the reference height,
for 40 ppt kaolin clay suspensions with various concentrations of CaCl2,
ranging from $1.71\times 10^{-3}$ mM to $1.71\times 10^{1}$ mM. By ten
minutes, the light intensities had settled into steady values. The formation
of a defined sediment-supernatant interface in the solutions with
concentrations of at least $1.71\times 10^{-1}$ mM CaCl2 drastically affected
the transmitted light intensity once the front descended below the measurement
height. The formation of this front at high salt concentrations is consistent
with observations in [50] and [10], among others.

These fronts descended below the reference height within the first three
minutes; the light intensity remained relatively constant in all cuvettes
afterwards. Thus, the suspensions were in a well-established steady-state
regime at ten minutes, whereby one could observe how much clay had been left
in suspension after the early sedimentation behavior.

Figure 4: Example of horizontally-averaged transmitted light intensity 23.2 mm
above the bottom of the cuvette vs. time for 40 ppt kaolin clay suspensions
with various CaCl2 concentrations. Lines are included to guide the eye. The
formation of a sediment-supernatant interface in the solutions with CaCl2
concentrations at or above $1.71\times 10^{-1}$ mM, leads to a sudden increase
in light intensity when the interface descends below the measurement height,
between one and three minutes.

The transmitted light intensity depended heavily on the clay concentration of
the suspension under test. Therefore, in order to compare results with
different concentrations on the same axes, a normalized light intensity was
calculated for each clay concentration, ranging from zero to one. To achieve
this, the minimum light intensity was subtracted from the values at the end of
the ten minute period (typically from the cuvette with the lowest salt
concentration) for all cuvettes under test. The light intensities were then
divided by the updated maximum light intensity at the end of the ten minute
period (typically from the cuvette with the highest salt concentration).

Motivated by a visual inspection of the results, the data was fit at the end
of the ten minute period in MATLAB with an error function of the form:

$I_{\text{norm}}=A\,\text{erf}\left(B\,\text{log}\left(\frac{s}{s_{\text{CCC}}}\right)\right)+D$
(1)

where $I_{\text{norm}}$ is the normalized light intensity, s is the salt
concentration, and $A$, $B$, $s_{\text{CCC}}$, and $D$ are fit parameters.
With this fit, the CCC is defined as the concentration at the mid-line of the
error function curve, which marks the maximum rate of change in flocculation
with increasing NaCl or CaCl2 concentration. The CCC is often defined
arbitrarily [38, 12], as flocculation changes over a finite range of salt
concentrations from complete dispersion to complete aggregation. This
definition using a fit parameter is robust, and is therefore appropriate for
these CCC measurements.

##  3 Experimental results

Figure 5 shows the results for NaCl solutions. We tested 4 ppt, 20 ppt, and 40
ppt kaolin clay suspensions with NaCl concentrations ranging from $1.71\times
10^{-3}$ mM to $1.71\times 10^{1}$ mM. The CCC for each case is marked by a
star below the curves (for clarity, the stars are placed at an arbitrary
vertical coordinate); dashed lines are used to emphasize the mid-point of the
transition in the curve. Horizontal error bars around the stars denote 95%
confidence intervals for the CCC as calculated by the curve fit. The CCCs for
these kaolin clay suspensions lie around 0.6 mM NaCl ($\approx 0.04$ ppt NaCl
$=$ 0.04 PSU).

Figure 5: Normalized light intensity versus NaCl concentration for 4 ppt, 20
ppt, and 40 ppt kaolin clay solutions. The data is fit by an error function,
and the CCC is defined as the concentration at the mid-line of the curve. The
CCC of each curve is marked by a star of the same color below the curves, and
95% confidence intervals for the CCC are marked by horizontal error bars.

Although the CCCs of the kaolin clay suspensions with NaCl appear to be
somewhat lower than typically reported in the clay literature, this
discrepancy is consistent with differences in methods of defining the CCC; for
example, [51] based their CCC estimates on the value of concentration beyond
which no significant changes occur in the flocculation dynamics. By this
definition, [51] find a CCC of 3 mM NaCl, which is consistent with the data in
Fig. 5, as all curves indeed level off around 3 mM.

Fig. 6 shows results for tests with kaolin clay and CaCl2 solutions. Once
more, tests were performed for 4 ppt, 20 ppt, and 40 ppt kaolin clay
suspensions with CaCl2 concentrations ranging from $1.71\times 10^{-3}$ mM to
$1.71\times 10^{1}$ mM. Here, the CCCs were found to range from 0.034 mM CaCl2
($\approx 0.0038$ ppt CaCl2) for 4 ppt kaolin clay to 0.052 mM CaCl2 ($\approx
0.0058$ ppt CaCl2) for 40 ppt kaolin clay. Although the CCC in NaCl solutions
did not appear to depend on clay concentration, a slight increase of the CCC
with increasing clay concentration may be visible in the results for CaCl2
solutions.

Figure 6: Normalized light intensity versus CaCl2 concentration for 4 ppt, 20
ppt, and 40 ppt kaolin clay suspensions. The CCC of each curve is marked by a
star of the same color below the curves, and 95% confidence intervals for the
CCC are marked by horizontal error bars. The CCCs for these solutions are over
an order of magnitude smaller than the CCCs for NaCl solutions, ranging from
0.034 mM for 4 ppt kaolin clay to 0.052 mM for 40 ppt kaolin clay.

Incidentally, the fact that the 67 distinct cuvette time-series used to
compile Figs. 5, 6 yield data that consistently collapses to the same curves
suggests a good degree of reproducibility for these experiments.

##  4 Theory

The large difference in CCC between NaCl and CaCl2 solutions demonstrates the
challenge of reliably predicting the onset of flocculation for a given clay
suspension. This section presents a simple derivation of a dimensionless
quantity that may be used to determine the effect of various clay and solution
parameters on flocculation. For this purpose, the number of negative charges
on suspended clay particles are compared to the number of positive charges
surrounding the particles, and these two quantities are expressed in terms of
parameters commonly known for a given clay.

To begin, the clay particles are approximately represented as plates with
major dimension $d$ and thickness $\tau$. For kaolinite particles, $d\approx
10\tau$ [47]. Thus, the edge surface area of a particle is nearly one order of
magnitude smaller than the face surface area; for this reason, the edge
surface area is neglected in the following analysis.

The surface charge on the edge of the particles is also neglected, which is in
accordance with early findings of [52], who observed that exchangeable cations
were mainly adsorbed on the hydroxyl-free basal planes (tetrahedral sheet) and
to a lower amount on the hydroxyl-rich basal planes (octahedral sheet) rather
than at the edges. This was confirmed by [53] using atomic force microscopy.
Not only is the edge surface area much smaller than the face surface area, but
the edge charge is pH dependent, and the point of zero charge (PZC) of
kaolinite particles is near the pH measured for the solutions, at pH $=5.8$
[54]. Thus, the edge charge is expected to be insignificant in this order-of-
magnitude derivation.

One wishes to obtain an expression relating the number of charges per surface
area of the clay to the CEC. Because the CEC has units of number of charges
per mass, it must be multiplied by the mass per area ratio of the particles to
obtain the number of charges per surface area.

The area of one of the hexagonal faces of a clay particle is
$A_{\text{face}}=\frac{\sqrt{3}}{2}d^{2}$, as illustrated in figure 7. Because
here the focus is on establishing a dimensionless group that characterizes the
CCC, order-one constants are omitted, such that $A_{\text{face}}\sim d^{2}$,
where the symbol ‘$\sim$’ denotes ‘order of magnitude’. Therefore the mass $m$
and surface area $A$ of a clay particle are given by:

$m=\rho_{\text{clay}}A_{\text{face}}\,\tau\sim\rho_{\text{clay}}\,d^{2}\,\tau,$
(2) $A=2\,A_{\text{face}}+\frac{6}{\sqrt{3}}d\tau\sim 2d^{2},$ (3)

where $\tau\ll d$ was used. Thus, one obtains:

$\left(\frac{\text{\\#
charges}}{\text{surface}}\right)_{\text{clay}}\sim\text{CEC}\;\frac{m}{A}=\text{CEC}\;\frac{\rho_{\text{clay}}\,\tau}{2}.$
(4)

One must now solve for the number of ions in solution, per area of the
particle they are surrounding. This requires defining a thickness for the
‘cloud’ of ions enveloping the clay particle. A diffuse double-layer model is
used, as illustrated in Fig. 7, where the characteristic thickness of the
counterion cloud surrounding the particle is the Debye length $\kappa^{-1}$
[55]. The Debye length is calculated as:

$\kappa^{-1}=\sqrt{\frac{\epsilon_{0}\epsilon_{r}k_{B}T}{2000N_{a}e^{2}I}}$
(5)

where $\epsilon_{0}$ is the permittivity of free space, $\epsilon_{r}$ is the
medium’s dielectric constant, $k_{B}$ is the Boltzmann constant, $T$ is the
temperature, $N_{a}$ is Avogadro’s number, $e$ is the charge of an electron,
and $I$ is the ionic strength of the salt solution (in mol/L), defined below.
While this expression is derived for the case of ions with equal valence, it
is also a good approximation for the case of two ions with different valence
[55]. Additionally, while slight differences in the CCC of clays have been
reported for different salts with the same valence number, these differences
are generally less than one order of magnitude and thus fall within the error
of this approximation [56, 57, 58].

Figure 7: Definitions of clay particle dimensions and of double layer
thickness used in the model. The Stern layer consists of the cations adhered
to the negatively-charged clay surface. The diffuse double layer comprises the
cloud of ions beyond the Stern layer, where the Debye length $\kappa^{-1}$ is
the characteristic thickness of the double layer, calculated with Eq. (5).

The ionic strength $I$ of the salt solution expresses the number of charges
per unit volume, and is given by:

$I=\frac{1}{2}\sum_{i}z_{i}^{2}s_{i}$ (6)

where $s$ is the ion concentration (in mol/L) and $z$ is the ion’s valence
number, summed over each of the $i$ cations and anions in solution [37]. To
obtain the number of cations per area around the particle, one must multiply
the ionic strength of the cations by the ‘cloud’ volume per area ratio, or
simply the thickness of the cloud. It is assumed that the cations relevant to
screening the negatively-charged particle faces are found within a distance of
order $\kappa^{-1}$ of the particle surfaces, and that the effect of anions is
minimal within this region. Consistently with the earlier assumption about the
edges of the clay particles, cations around the edges are also neglected, due
to the small volume of the edge ‘cloud’ relative to the face ‘cloud’. Thus,
one considers a volume surrounding the faces of a clay particle, given by:

$V\sim 2d^{2}\kappa^{-1}.$ (7)

One thus obtains:

$\displaystyle\left(\frac{\text{\\#
charges}}{\text{surface}}\right)_{\text{salt}}$ $\displaystyle\sim
z_{\text{cation}}^{2}s_{\text{cation}}\frac{V}{A}$ (8)
$\displaystyle=\kappa^{-1}z_{\text{cation}}^{2}s_{\text{cation}}.$

A dimensionless parameter $\Phi$ is now introduced, expressing the ratio of
the effective number of cations in solution (8) to the number of anions on the
clay surface (4):

$\displaystyle\Phi$ $\displaystyle=\left.\left(\frac{\text{\\#
charges}}{\text{surface}}\right)_{\text{salt}}\,\right/\,\left(\frac{\text{\\#
charges}}{\text{surface}}\right)_{\text{clay}}$ (9)
$\displaystyle=\frac{2\kappa^{-1}z_{\text{cation}}^{2}s_{\text{cation}}}{\text{CEC}\;\rho_{\text{clay}}\tau}.$

For NaCl, $z_{\text{cation}}=1$ and $s_{\text{cation}}=s_{\text{NaCl}}$,
giving:

$\Phi=\frac{2\kappa^{-1}s_{\text{NaCl}}}{\text{CEC}\,\rho_{\text{clay}}\tau}.$
(10)

For CaCl2, $z_{\text{cation}}=2$ and $s_{\text{cation}}=s_{\text{CaCl}_{2}}$,
giving:

$\Phi=\frac{8\kappa^{-1}s_{\text{CaCl}_{2}}}{\text{CEC}\,\rho_{\text{clay}}\tau}.$
(11)

When the dimensionless quantity $\Phi\sim 1$, the magnitude of the positive
charge within $\kappa^{-1}$ of the faces of suspended clay particles is
equivalent to the magnitude of the negative surface charge on the clay
particles. Thus, it is expected that the surface charge will be effectively
screened, and the repulsive force between clay particles will be approximately
zero. This should lead to unimpeded flocculation. It must be noted, however,
that as salt concentration increases from zero, flocculation initially occurs
when the attractive van der Waals force becomes greater than the repulsive
electrostatic force (incidentally, this is equivalent to removing the energy
barrier in DLVO theory [12]). This is expected to occur for a critical value
of $\Phi$ below 1, as when $\Phi=1$ the repulsive force is already effectively
zero. This behavior would be consistent with experience from other areas of
hydrodynamics, where transitions between regimes are observed when the ratios
between stabilizing and destabilizing effects – as expressed by relevant
dimensionless groups – are several orders of magnitude away from unity [59,
60].

Below, a procedure is described for empirically determining the critical value
$\Phi_{\text{CCC}}$ for the onset of flocculation. Once $\Phi_{\text{CCC}}$ is
known, it is expected that a similar $\Phi_{\text{CCC}}$ should induce
flocculation for other salts (or clays in the kaolin group). This will be
verified with the experimental results. For many problems in geoscience, the
simple dimensional analysis presented here can sidestep the need to employ
significantly more complicated DLVO calculations to predict the necessary
conditions for flocculation.

Figure 8: Comparing plots of normalized light intensity vs. salt concentration
(Figs. 8a-c) and vs. the dimensionless group $\Phi$ (Figs. 8d-f) for 4 ppt, 20
ppt, and 40 ppt kaolin clay. The data is the same as was presented in Figs. 5
and 6. While the curves for NaCl and CaCl2 are separated by over an order of
magnitude when plotted against salt concentration, the curves are brought
within a factor of 1.5-2 when plotted against $\Phi$. Thus, $\Phi$ can be used
as an order-of-magnitude predictor of the onset of flocculation for various
clay concentrations and salts of different valence. Here, $\Phi_{\text{CCC}}$
falls between $1.3\times 10^{-3}$ and $2.4\times 10^{-3}$, so it is estimated
that the onset of flocculation will occur in clay suspensions when
$\Phi=O(10^{-3})$.

##  5 Using the parameter $\Phi$ to predict the onset of flocculation

Fig. 8 shows a comparison of normalized light intensity plotted against salt
concentration (Fig. 8a-c) and against the dimensionless group $\Phi$ (Fig.
8d-f) for 4 ppt, 20 ppt, and 40 ppt kaolin clay suspensions. The data is the
same as was presented in Figs. 5 and 6, but now results for NaCl and CaCl2 are
shown on the same plot. The NaCl and CaCl2 curves for all kaolin
concentrations are separated by a factor of 12-18 when plotted against salt
concentration.When plotted against $\Phi$, however, the NaCl and CaCl2 curves
draw within a factor of 1.5-2. This demonstrates the ability of the
dimensionless group $\Phi$ to approximately collapse the curves of suspensions
made with salts of different valence.

The empirically determined $\Phi_{\text{CCC}}$ is expected to be comparable
for NaCl and CaCl2 solutions. As mentioned above, when $\Phi\sim 1$, the
repulsive force between clay particles will be effectively zero; however,
because flocculation only requires that van der Waals forces can overcome
electrostatic repulsion, flocculation should be triggered at some
$\Phi_{\text{CCC}}<1$. For the data sets shown in Fig. 8, $\Phi_{\text{CCC}}$
falls between $1.3\times 10^{-3}$ and $2.4\times 10^{-3}$. Thus, is is
estimated empirically that for a given clay suspension, flocculation will be
triggered when $\Phi=O(10^{-3})$. For $\Phi\gg O(10^{-3})$, flocculation will
occur, and suspended clay will rapidly clear from the supernatant. For
$\Phi\ll O(10^{-3})$, flocculation will not occur and the suspension will be
stable. The dimensionless group $\Phi$ should therefore be useful to those
outside of the colloid community who want to predict the basic behavior of
clay suspensions. Users may determine whether their suspension will be stable
or unstable by substituting the proper numerical values for the materials in
their suspension, all of which are well-characterized for common clays, with
no need to employ significantly more complicated DLVO calculations.

##  6 Discussion

###  6.1 Implications for laboratory and environmental measurements of
flocculation

The results show that for nearly all cases outside of the laboratory, kaolin
clay suspensions will naturally have a salinity greater than the CCC.

As a matter of fact, these extremely small quantities of salt necessary to
induce flocculation can often already be found in tap water. For example,
water for UC Santa Barbara is supplied by the Goleta Water District, whose
water quality testing throughout 2018 showed an average of 2.4 mM Na+ and 2.0
mM Ca2+ present in the surface water supply [61]. These concentrations are
sufficient to induce flocculation in the kaolin clay, as the authors verified
with experiments with tap water in the laboratory. This may explain why [13]
did not observe a difference in flocculation upon adding more salt to tap
water, as the salinity of the tap water (likely also from the Goleta Water
District 30 years earlier) was already above the sediment’s CCC.

We also expect that [14] observed an increase in floc size upon moving down
the Garonne and Dordogne Rivers towards the Gironde Estuary because the very
low salinity of the rivers (0.01 PSU $=0.17$ mM NaCl) is below the CCC we
measured for the kaolin clay solutions. The lack of correlation between
salinity and floc size measured by [15] and [16] may be explained if the
(unreported) salinities of their river samples were already above their
sediment’s CCC.

These measurements may also explain the similar floc sizes observed by [17] in
the Sacramento River and the San Francisco Bay-Delta estuary, as the river’s
salinity of 0.1 PSU ($=1.7$ mM NaCl) is above the CCC for the kaolin clay
solutions. Additionally, the low salinity reported in [19] is above the CCC we
measured, suggesting that salt would have induced flocculation whether or not
organic matter was present.

Lastly, the salt content of the Po River has been repeatedly measured to
contain calcium concentrations above the CCC of kaolin clay [62], so although
[20] concludes that the presence of both calcium and organic matter lead to
flocculation in the Po, the calcium’s presence alone was likely sufficient to
trigger flocculation.

Although many have observed a qualitative difference in settling between
freshwater and saltwater in the laboratory, the extremely small magnitude of
the CCC is often overlooked and unaccounted for in experimental design. For
example, two otherwise identical laboratory studies of turbidity currents may
find qualitatively different results depending on the cleanliness of the clay,
on the water source (including effects due to variations in local tapwater
properties), as well as on the presence of salt residues after past
experiments involving even small amounts of dissolved salts.

###  6.2 Effects of ion valence

The CCC that we measured for each kaolin clay concentration was over an order
of magnitude smaller in CaCl2 solutions than NaCl solutions: 18.9 times
smaller for 4 ppt kaolin clay suspensions, 13.7 times smaller for 20 ppt
kaolin clay suspensions, and 11.5 times smaller for 40 ppt kaolin clay
suspensions. This behavior is expected based on the Schulze-Hardy rule and
subsequent developments, where the ratio of CCC${}_{\text{CaCl}_{2}}$ /
CCC${}_{\text{NaCl}}$ is expected to range between 1/4 for low potentials and
1/64 for high potentials [63].

These results also align well with other experiments, where it is common to
observe at least one order of magnitude smaller CCC with CaCl2 solutions than
with NaCl solutions [64, 40, 65].

The CCCs for the NaCl solutions in Fig. 5 do not appear to depend on clay
concentration. This is consistent with previous results that have found that
the CCC is independent of clay concentration when salts are not adsorbed in
the Stern layer (e.g. [40]; the Stern layer is illustrated in Fig. 7).
Monovalent cations do not experience as strong of an attraction to the
negatively-charged clay surface as do multivalent cations, such as Ca2+ [66],
and it appears that there is little Na+ adsorption in the Stern layer in this
case.

The CCCs for the CaCl2 solutions in Fig. 6 increase slightly with increasing
kaolin concentration, which may be explained by adsorption of Ca2+ ions in the
Stern layer. This adsorption of Ca2+ leads to greater surface charge
neutralization than is observed for Na+, resulting in a very low CCC [67]. If
the CaCl2 concentration is fixed and the clay concentration is increased, the
effect of this charge neutralization becomes less noticeable in the presence
of more clay particles with fully-charged surfaces. Thus, the CCC may be
observed to increase with clay concentration in CaCl2 solutions.

##  7 Conclusions

The onset of particle aggregation is an important threshold for determining
the dynamics of environmental flows. Here a simple criterion was presented, to
predict the stability of a clay suspension using the dimensionless group
$\Phi$, along with experimental results to empirically determine
$\Phi_{\text{CCC}}=O(10^{-3})$. This newly defined dimensionless quantity
$\Phi$ is an accessible tool for understanding flocculation in clay, providing
a tractable alternative to the more laborious calculations of DLVO theory.

It is expected that the use of the parameter $\Phi$ will allow for simple
control of clay suspensions in aquatic systems ranging from the laboratory to
the field. The experimental results show that the salt concentration needed to
induce flocculation in kaolin clay is lower than what is typically observed
even in tap water, and thus kaolin clay suspensions will flocculate in most
environmental fluid mechanics applications.

## Acknowledgments

This work was supported by NSF ENG CBET Fluid Dynamics award #1638156. N.R.
was supported by a UCSB MRL RISE award. B.V. gratefully acknowledges the
Feodor-Lynen scholarship provided by the Alexander von Humboldt foundation
(Germany) and the German Research Foundation (DFG) grant VO2413/2-1. E.M.
furthermore gratefully acknowledges support by the Army Research Office under
grant W911NF-18-1-0379.

The authors thank Sage Davis at the UCSB Micro-Environmental Imaging and
Analysis Facility for his assistance in operating the ESEM, and Otger Campas
and Renwei Chen for access and assistance with the centrifuge. The authors
also thank Brandon Grunberg, Kevin Hill, Balaji Subramanian, Meital Carmi,
Omar Curiel, Rochishnu Chowdhury, Daniel Newton and Rena Yang for
contributions to the development of the experimental setup and procedures.

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# Improving Stability of Fine-Tuning Pretrained Language Models
via Component-Wise Gradient Norm Clipping

Chenghao Yang1, Xuezhe Ma2
1University of Chicago
2University of Southern California
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Fine-tuning over large pretrained language models (PLMs) has established many
state-of-the-art results. Despite its superior performance, such fine-tuning
can be unstable, resulting in significant variance in performance and
potential risks for practical applications. Previous works have attributed
such instability to the catastrophic forgetting problem in the top layers of
PLMs, which indicates iteratively fine-tuning layers in top-down manner is a
promising solution. In this paper, we first point out that this method does
not always work out due to different convergence speeds of different
layers/modules. Inspired by this observation, we propose a simple component-
wise gradient norm clipping method to adjust the convergence speed for
different components. Experiment results demonstrate that our method achieves
consistent improvements in terms of generalization performance, convergence
speed and training stability. The codebase can be found at
https://github.com/yangalan123/FineTuningStability.

##  1 Introduction

Fine-tuning over large pretrained language models (PLMs), which achieved
remarkable performance over various benchmarks, has become the de facto
paradigm for several current natural language processing (NLP) systems.
However, fine-tuning can be unstable in terms of significant variance in
metrics, resulting in even worse-than-random failed models (Devlin et al.,
2019; Lee et al., 2019; Dodge et al., 2020; Mosbach et al., 2020).

Catastrophic forgetting (Kirkpatrick et al., 2017) during fine-tuning of PLMs
is one common explanation for this instability (Lee et al., 2019), i.e., PLMs
may lose their rich domain-agnostic knowledge acquired by language model
pretraining in the process of fine-tuning. Through layer-replacement
experiments between pretrained models and fine-tuned models, Mosbach et al.
(2020) further connected the catastrophic forgetting problem to the
optimization problem on top layers.

These findings give rise to a straightforward way to enhance the fine-tuning
stability: how about fine-tuning the model from top to bottom to reduce the
parameter changes and hence mitigate the catastrophic forgetting problem? This
is reminiscent of the gradual unfreezing (Howard and Ruder, 2018), which does
layer-wise top-down fine-tuning and unfreezing new layers only when the layers
above have been fine-tuned. Therefore, the newly unfrozen layer would only be
tuned for a slightly easier optimization problem at each iteration, leading to
much fewer changes to the parameters.

Figure 1: Fine-tuning performance over time on RTE datasets. Here we fine-tune
over BERT-large-uncased model (Devlin et al., 2019). In each iteration, we
train the model for $3$ epochs and the errorbar is plotted based on $5$
different runs.

However, based on a comprehensive case study of the gradual unfreezing method,
we obtained empirical observations beyond our expectations (§2). Our analysis
further reveals a possible reason: _different components (e.g., feed-forward
networks at different layers, fully connected matrices and biases at output
layer) converge at varying speeds._ Thus, components in upper layers, which
have converged to local optima, cannot easily be fine-tuned with newly
unfrozen parameters.

(a)  Compared with original BERT model

(b) Compared with previous iteration

Figure 2: Parameter update at each iteration for GU. Updates that are too
small cannot be seen in this figure (e.g., for the $22$-th layer, or
“layer.21” in the figure update is too small to plot out in Fig. 2(b)). We
only plot the first $4$ iterations as we observe that the performance is
almost stable by the fourth iteration. As different layers can have components
with different dimensionalities, we show the component-wise maximum rooted
mean squared difference for each layer.

Based on this observation, we propose a simple component-wise gradient
clipping method to stabilize the fine-tuning process (§3). This method
achieves significant empirical improvements of fine-tuning stability in terms
of the variance and the failed run percentage over three tasks (§4). In
summary we make the following contributions:

  1. 1.

We find that component-wise convergence speed divergences is the key challenge
in fine-tuning stability, based on the case study of the gradual unfreezing
method.

  2. 2.

Based on our observation, we propose a new simple component-wise gradient
clipping method to help stabilize fine-tuning, which achieves empirical
improvements of fine-tuning stability over previous methods.

##  2 A Bitter Case Study: Layer-wise Gradual Unfreezing

Mosbach et al. (2020) attributed the instability problem in fine-tuning
process to the catastrophic forgetting in the top layers, through a layer
replacement experiment between pretrained and fine-tuned models. Following
this empirical observation, the instability problem might be mitigated if we
can minimize the edits to the pretrained model parameters, especially in top
layers. This inspires us to mitigate the instability problem via gradual
unfreezing (GU, Howard and Ruder (2018)).

Specifically, suppose we are working with a model $M$ with $L$ layers
parameterized by $\\{\theta^{(i)},1\leq i\leq L\\}$. GU tunes $M$ for $L$
iterations. At $k$-th ($k$ start from $0$) iteration, we only tune a subset of
parameters $R^{(k)}=\\{\theta^{(i)},L-k\leq i\leq L\\}$, where
$\theta^{(i)}(L-k+1\leq i\leq L)$ is also tuned in $k-1$-th iteration. In each
iteration, we tune the parameter for $E$ epochs, where $E$ is large enough for
convergence.111We select $E$ based on preliminary experiments. Detailed
algorithm is shown in Algorithm 1 at Appendix A.

(a)  Failed Run

(b) Success Run

Figure 3: Gradient norm across different parameters at layer 22 on RTE dataset
fine-tuning. Here the left figure (“Failed Run”) refers to the case when the
fine-tuned model cannot beat the majority classifier and the right figure
(“Success Run”) otherwise. Different colors represent different parameters.
Due to space limitation, legends are omitted and we cannot show results from
all layers. But in our observation, most layers have similar phenomenon.

### Failure of Gradual Unfreezing

From Fig. 1, the accuracy of gradual unfreezing is significantly worse than
full fine-tuning,222As pointed out by Mosbach et al. (2020), without bias
correction, the original results in Devlin et al. (2019) paper can be pretty
bad and unstable, so we add bias correction on top of Devlin et al. (2019) to
make a strong baseline. although it indeed achieves smaller update to
pretrained model parameters compared with full fine-tuning (Fig. 2(a)).

### Convergence Racing between Parameters

To investigate the reason behind this unsatisfying performance of GU, we plot
the component-wise maximum update (measured by component-wise maximum rooted
mean squared difference,333We also plot the parameter change measured by
cosine distance and put the figures in Appendix C as different layers can have
multiple components with different dimensions) at each layer in Fig. 2(b).
Clearly, from the very beginning, the parameter updates for both early-tuned
parameters and newly-unfrozen parameters have quickly diminished in GU so not
many updates happen later when new parameters join. Based on this observation,
we hypothesize that the failure of GU is because the early-tuned parameters
have already converged in early iterations and cannot be re-activated later to
adapt for newly-unfrozen parameters.

To verify this hypothesis, we simply modify GU to stop using early-tuned
parameters in the previous iteration and instead copy the weight from the
pretrained BERT model. For simplicity, we term this method as “GU (restart)”.
Note that full fine-tuning is just the last iteration of GU (restart) when all
layers are unfrozen. We plot GU (restart) performance in Fig. 1, and observe
that GU (restart) achieves consistently much better performance than GU.
However, GU (restart) is much more unstable than GU, and the best performance
is only reached when almost all layers have been tuned.

### Unbalanced Gradients across Parameters

To investigate the cause of the convergence racing problem, we analyze the
distribution of parameters’ gradients from different components, by plotting
the gradient norm in Fig. 3.444In Fig. 3(b) and Fig. 3(a), we do the fine-
tuning by following the recommended setting in Devlin et al. (2019). Here we
follow the terminology in previous work Dodge et al. (2020); Mosbach et al.
(2020) that, if a fine-tuned model checkpoint can’t beat the majority
classifier,555Here “majority classifier” means simply using the majority
labels in the training dataset as the predicted label. then it is a “Failed
Run” and “Success Run” otherwise. Comparing Fig. 3(b) and Fig. 3(a), we can
find that convergence racing still exists under normal full fine-tuning. The
success run wins by making all parameters update roughly following the same
trends and making gradient norms well-bounded.

### Discussion on Fine-Tuning Stability

From the case study over GU, we observe that there is a convergence racing
problem between parameters of model components, which would lead to
incompetent or unstable performance. Based on existing studies, we argue that
a robust PLM fine-tuning method should satisfy the following principles:

  1. 1.

Updating the full set of parameters, not just a subset of layers.

  2. 2.

No significant parameter updates to avoid catastrophic forgetting problem.

  3. 3.

Adjusting the convergence speed for parameters to mitigate the convergence
racing.

Mosbach et al. (2020) proposed to use small learning rate (e.g., $1e^{-5}$) to
mitigate catastrophic forgetting, together with longer fine-tuning process.
However, unbalanced gradients across different components is another factor
for catastrophic forgetting and convergence racing, which is not resolved by
only using small learning rates.

Approach | RTE | MRPC | COLA
---|---|---|---
Std | Mean | Max | Std | Mean | Max | Std | Mean | Max
Devlin et al. (2019) | 4.5 | 50.9 | 67.5 | 3.9 | 84.0 | 91.2 | 25.6 | 45.6 | 64.6
\+ bias correction | 4.0 | 70.6 | 75.5 | 2.1 | 89.2 | 92.2 | 11.1 | 59.2 | 64.1
Lee et al. (2019) | 7.9 | 65.3 | 74.4 | 3.8 | 87.8 | 91.8 | 20.9 | 51.9 | 64.0
Mosbach et al. (2020) | 2.7 | 67.3 | 71.1 | 0.8 | 90.3 | 91.7 | 1.8 | 62.1 | 65.3
CWGNC | 1.3 | 73.1 | 75.1 | 0.6 | 90.5 | 91.5 | 1.7 | 62.2 | 65.0
Table 1: Experiment results for fine-tuning benchmark. Boldfaced numbers are
best under each criterion.

##  3 The Component-Wise Gradient Norm Clipping Method

Bearing these principles in mind, we propose the component-wise gradient norm
clipping (CWGNC) method. The components are different parameters serving
different functionalities. For example, in a Transformer-based neural
architecture, components can be the weight matrices of Key-Query-Value in
Transformer architectures, the weight and bias terms in feed-forward layers,
etc. For those optimizers which maintain first-order and/or second-order bias
correction terms (e.g., Adam Kingma and Ba (2015) and AdamW Loshchilov and
Hutter (2018) as the popular optimizer in PLM literature), our proposed norm
clipping operation happens before the bias correction terms computation and
does not interfere with normal bias correction process.

By clipping the gradient norm of each component individually, we aim to
balance the distribution of gradients across different components to adjust
their convergence speed, hence mitigating the convergence racing problem.

##  4 Experiments

To evaluate our CWGNC method, we follow Mosbach et al. (2020) to run our CWGNC
method over $25$ different runs and report aggregated results on the
validation set of three different datasets: RTE (Acc), MRPC (F1) and COLA
(MCC). We report the standard deviation (Std), averaged performance (Mean) and
maximum performance (Max). Hyperparameters are tuned on held-out data. More
implementation details are in Appendix B.

### Baseline Setting

For baselines, we first consider the original BERT paper reported results
Devlin et al. (2019) and we run original BERT fine-tuning with bias correction
to make a stronger baseline following the observation in Mosbach et al.
(2020). We also compare to Mixout regularization methods Lee et al. (2019) and
“simple but hard-to-beat baseline” proposed by Mosbach et al. (2020).

### Experiment Results

Experiment results are shown in Table 1. Here we can see our CWGNC achieves
significantly better performance in terms of averaged performance and standard
deviation. Compared with previous state-of-the-art results from (Mosbach et
al., 2020), our method only needs to tune $5$ epochs and does not need to wait
for $20$ epochs even on these small datasets so our method indeed converges
faster.

Our method is also robust to a wide range of the selection of gradient norm
clipping thresholds. We show this in Table 2 on COLA dataset as we find that
fine-tuning methods are particularly unstable on COLA dataset. Here we see
that within a reasonable range of threshold ($<1$), the model performance
would be mostly maintained and the standard deviation is well controlled. If
we use a significantly larger threshold ($>=1$), less control is enforced by
CWGNC and it will degrade to normal full model fine-tuning if we further
increase the threshold.

Threshold | COLA
---|---
Std | Mean | Max
0.01 | 1.3 | 61.6 | 64.3
0.05 | 1.7 | 62.2 | 65.0
0.1 | 1.4 | 61.4 | 65.1
0.5 | 1.3 | 61.6 | 64.1
1 | 1.3 | 61.4 | 63.8
5 | 15.4 | 57.3 | 65.0
Table 2: CWGNC fine-tuning results on COLA under different gradient norm
clipping thresholds.

##  5 Conclusion

In this paper, we investigate the instability problem of fine-tuning large
pretrained language models. Inspired by previous works, we first experiment
with gradual unfreezing methods, which should help minimize the updates in top
layers and ease the catastrophic forgetting problem. However, further
experiment results do not support that and we find it is because there is a
convergence racing issue between different parameters, namely that early-
converged parameters can limit the search space for other parameters. Based on
this finding, we propose to do component-wise gradient norm clipping and
achieve significant improvement on averaged performance, smaller standard
deviation and quicker convergence speed. Our method is robust to the selection
of gradient norm clipping threshold. In the future, we will try to study
whether the component racing problem also exists in pretrained language models
with different sizes and different pretraining methods.

##  6 Limitations

In this paper we mainly work with one particularly popular large pretrained
language model BERT Devlin et al. (2019). While we believe our empirical
investigation and conclusion is widely applicable to a wide range of current
transformer-based large pretrained language models, more experiments and
theoretical explanations are needed for further research. Also, due to
computational resources limitation, we cannot investigate whether the most
recent large models like T0 Sanh et al. (2022) are stable under fine-tuning.
We follow the evaluation protocols in previous works Dodge et al. (2020); Lee
et al. (2019); Mosbach et al. (2020) to investigate the instability of fine-
tuning on small datasets including COLA, RTE and MRPC. But for real
challenging low-resource situation, we believe it can be more complicated and
more investigation is needed.

## Acknowledgements

This material is based on research sponsored by Air Force Research Laboratory
(AFRL) under agreement number FA8750-19-1-1000. The U.S. Government is
authorized to reproduce and distribute reprints for Government purposes
notwithstanding any copyright notation therein. The views and conclusions
contained herein are those of the authors and should not be interpreted as
necessarily representing the official policies or endorsements, either
expressed or implied, of Air Force Laboratory, DARPA or the U.S. Government.

## References

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##  Appendix A Gradual Unfreezing Algorithm

The gradual unfreezing algorithm we implemented following Howard and Ruder
(2018) is shown in Algorithm 1.

Algorithm 1 Gradual Unfreezing Fine-Tuning

A multi-layer Transformer PLM $M$ with its $L$-layer parameters
$\\{\theta^{(i)},1\leq i\leq L\\}$. The maximum number of iterations $T$, the
dataset $D$, the optimizer ${Opt}$, the number of epochs for each iteration
$E$.

$1\leq T\leq L$

$\hat{R}^{(0)}\leftarrow\phi$ $\triangleright$ $\hat{R}^{(l)}$ represents to-
be-tuned parameters at $l$-th iteration.

while $T\leq L$ do

$\hat{R}^{(T)}\leftarrow\hat{R}^{(T-1)}\cup\\{\theta^{(L-T+1)}\\}$

for $j=1\to E$ do

$\mathcal{L}\leftarrow$ Forward$(M,D)$

$G\leftarrow$ Backward$(\mathcal{L},L-T+1,L)$

$\triangleright$ Only needs to compute the gradient for top-$T$ layers

Update$(Opt,G,\hat{R}^{(T)})$

$\triangleright$ Apply $G$ over $\hat{R}^{(T)}$

Replace$(M,\hat{R}^{(T)},L-T+1,L)$

$\triangleright$ Replace the updated layers back to make sure the updated
parameters are involved in next epoch forward process

end for

$T\leftarrow T+1$

end while

##  Appendix B Implementation Details

Our codebase is based on Huggingface Transformers Wolf et al. (2020) example
fine-tuning scripts and will be released later. We tune models using our
method for $5$ epochs. For weight decay and warm-up steps, we follow the
settings original fine-tuning method as described in (Devlin et al., 2019). We
here report the result with clipping threshold $0.05$ as we empirically find
it works better on held-out dataset. We also show later that our method is
actually pretty robust to a wide range of threshold picking in Table 2.

##  Appendix C Gradient Update Measured by Cosine Similarities

In the main text we measures the update via root-mean-square difference. Here
we show that if measured by cosine-similarity, we would obtain similar
conclusion. Here, Fig. 4 is the cosine similarity version for Fig. 2(b). Fig.
5 is the cosine similarity version for Fig. 2(a). Note that because smaller
cosine similarity indicates more changes, in contrast to square-mean-root
difference, we use the component-wise minimum cosine similarity to represent
the update at each layer.

Figure 4: The scales of incremental parameter update at each iteration in GU
(compared with previous iteration, measured by cosine similarity). As
different layers can have different components with different
dimensionalities, we show the component-wise maximum rooted mean squared
difference for each layer. We only plot the first $4$ iterations as we observe
that the performance is almost stable by the $4$-th iteration. $22$-th layer
(“layer.21” in the figure) update is too small to plot out in this figure.
Figure 5: The scales of parameter update at each iteration in GU (compared
with original BERT model, measured by cosine similarity). Too small updates
cannot be seen in this figure. We only plot the first $4$ iterations as we
observe that the performance is almost stable in $4$-th iteration.