# Arrhenius Crossover Temperature of Glass-Forming Liquids Predicted by an
Artificial Neural Network

B.N. Galimzyanov<EMAIL_ADDRESS>M.A. Doronina
<EMAIL_ADDRESS>A.V. Mokshin<EMAIL_ADDRESS>Kazan
Federal University, 420008 Kazan, Russia  Udmurt Federal Research Center of
the Ural Branch of the Russian Academy of Sciences, 426067 Izhevsk, Russia

###### Abstract

The Arrhenius crossover temperature, $T_{A}$, corresponds to a thermodynamic
state wherein the atomistic dynamics of a liquid becomes heterogeneous and
cooperative; and the activation barrier of diffusion dynamics becomes
temperature-dependent at temperatures below $T_{A}$. The theoretical
estimation of this temperature is difficult for some types of materials,
especially silicates and borates. In these materials, self-diffusion as a
function of the temperature $T$ is reproduced by the Arrhenius law, where the
activation barrier practically independent on the temperature $T$. The purpose
of the present work was to establish the relationship between the Arrhenius
crossover temperature $T_{A}$ and the physical properties of liquids directly
related to their glass-forming ability. Using a machine learning model, the
crossover temperature $T_{A}$ was calculated for silicates, borates, organic
compounds and metal melts of various compositions. The empirical values of the
glass transition temperature $T_{g}$, the melting temperature $T_{m}$, the
ratio of these temperatures $T_{g}/T_{m}$ and the fragility index $m$ were
applied as input parameters. It has been established that the temperatures
$T_{g}$ and $T_{m}$ are significant parameters, whereas their ratio
$T_{g}/T_{m}$ and the fragility index $m$ do not correlate much with the
temperature $T_{A}$. An important result of the present work is the analytical
equation relating the temperatures $T_{g}$, $T_{m}$ and $T_{A}$, and that,
from the algebraic point of view, is the equation for a second-order curved
surface. It was shown that this equation allows one to correctly estimate the
temperature $T_{A}$ for a large class of materials, regardless of their
compositions and glass-forming abilities.

###### keywords:

machine learning; physical properties; organic compounds; metallic alloys;
silicates; borates

##  1 Introduction

In the last decade, interest in the study of phase transformations in glass-
forming liquids has increased significantly [1, 2, 3]. There is increasing
evidence that such transformations can be related to the ability of a liquid
to form a glassy state [4, 5, 6]. The results of recent studies show that the
glass-forming ability of a liquid depends on the specifics of changes in its
atomistic structure and collective dynamics near the melting temperature
$T_{m}$ [7, 8, 9]. The beginning of such the changes in the dynamics of a
liquid corresponds to the Arrhenius crossover temperature $T_{A}$ [10, 11, 12,
13]. It is generally accepted that the atoms of a liquid do not form any bound
structures above $T_{A}$. In this case, the dependence of the logarithm of
viscosity on the reverse temperature obeys a linear law (so-called high-
temperature Arrhenius behavior). Below $T_{A}$, individual groups of atoms
become less mobile, which manifests in the deviation of viscosity from the
Arrhenius behavior, which is typical for equilibrium liquids [14, 15, 16].

The existing empirical and theoretical methods for estimating $T_{A}$ are
mainly based on analysis of the temperature-dependence of liquid viscosity (or
the structural relaxation time) and on determining the high-temperature linear
regime in this relationship [17, 18, 19, 20]. As a rule, linear approximation
methods most accurately characterize this linear regime. Such approximations
are applicable only if the viscosity of the liquid is determined for a wide
temperature range covering temperatures above and below the melting
temperature ($T_{m}$). For organic (molecular) compounds and polymers
belonging to the class of the so-called fragile glass formers, viscosity
increases rapidly with decreasing temperature, which makes it possible to
determine the deviation from the high-temperature Arrhenius behavior. For the
so-called strong glass formers, including most metal melts, silicates and
borates, the Arrhenius behavior practically does not change even when passing
through the melting temperature and entering the region of supercooled melt.
This is displayed in blurring the region of transition from the high-
temperature Arrhenius behavior to the low-temperature nonlinear regime.
Therefore, the accuracy of the temperature estimation can be low, and the
estimated values of $T_{A}$ practically do not correlate with the other
physical characteristics of liquids. For example, an expression was proposed
by A. Jaiswal et al. which relates the fragility index $m$ with the $T_{A}$
values of various glass-forming liquids. This expression takes into account
the temperature dependence of the transport properties (mainly self-diffusion)
and the dynamics of atoms near the glass transition [8]. This expression gives
a correct correspondence between $m$ and $T_{A}$ in the case of molecular
glasses, though the results of calculations can differ greatly from empirical
data in the case of metallic and optical glasses. Further, the analytical
expression was proposed by T. Wen at al., according to which the glass-forming
ability of liquid is related to the reverse temperature $1/T_{A}$: i.e., the
higher the $T_{A}$, the worse the liquid forms a stable glassy state [21].
However, this rule is valid only for a narrow class of glass formers that are
similar in composition (mainly for metallic glasses). Therefore, obtaining an
analytical expression that allows one to determine $T_{A}$ based on the known
key physical characteristics of glass-forming liquids remains an unsolved
task. It is obvious that the correct solution of this task is possible using
machine learning methods, which will allow us to reveal hidden relationships
between physical characteristics and determine the most significant factors in
estimating $T_{A}$ [22, 23, 24, 25, 26].

The purpose of the present study was to determine how physical characteristics
associated with the overall kinetics of supercooled liquids correlate with
each other. These characteristics are primarily

  1. (i)

the glass transition temperature ($T_{g}$), at which liquid becomes amorphous
upon rapid cooling,

  2. (ii)

the melting temperature ($T_{m}$),

  3. (iii)

the Arrhenius crossover temperature ($T_{A}$),

  4. (iv)

the Kauzmann temperature,

  5. (v)

the high-temperature limit ($T_{\infty}$), at which the viscosity tends to
zero,

  6. (vi)

the temperature ($T_{0}$) associated with the transition to a non-ergodic
phase (for example, in the mode-coupling theory),

  7. (vii)

the temperature ratio of $T_{g}/T_{m}$, which is considered as one of the
criteria for the glass-forming ability of liquids and

  8. (viii)

the fragility index $m$, which determines the rate of change in viscosity with
temperature.

Some of these characteristics come to the fore for several reasons. First of
all, these characteristics are available for experimental measurements. In
addition, they are presented in various models that reproduce the kinetics and
transport properties of supercooled melts. Model equations for the shear
viscosity—such as the equations of the Vogel–Fulcher–Tammann–Hesse [12], Mauro
et al. [27], Avramov–Milchev [28] and the equation obtained in the framework
of the mode-coupling theory [29]—contain three or even more parameters to
reproduce the viscosity over a range of the temperatures. This indicates that
it is necessary to consider some temperature pairs associated with the
supercooled melt phase. It is important to note that these temperature pairs
occur in arbitrary combinations, which indirectly indicates the presence of
correlations between ”critical” temperatures in some way related to the glass
transition. Moreover, this fact is directly supported by previous results
relating to the description of the temperature dependence of the viscosity and
crystallization rate characteristics of supercooled melts by the scale
relations [9, 3, 30], where only the melting and glass transition
temperatures, $T_{m}$ and $T_{g}$, appear as input parameters. Thus, the
determination of specific correlation relationships between the ”critical”
temperatures of the kinetics of viscous melts is an important task, the
solution of which will contribute to a deeper understanding of the
solidification processes (glass transition and crystallization).

In the present work, the Arrhenius crossover temperature $T_{A}$ is predicted
for various types of glass-forming liquids, including silicates, borates,
metal melts and organic compounds using the machine learning method. The most
significant factors among the physical characteristics of these glass-forming
liquids are determined. Taking into account these factors, an analytical
equation is obtained that allows one to accurately relate the temperature
$T_{A}$ to the physical properties of glass-forming liquids.

##  2 Data Set and Machine Learning Model

Using an appropriate set of physical properties as the neural network input
parameters is a crucial for correct predicting the Arrhenius crossover
temperature. These physical properties must uniquely characterize the nature
of the material and must be determined with high accuracy by experimental or
simulation methods. Here, it is quite reasonable to choose the fragility index
($m$), the melting temperature ($T_{m}$), the glass transition temperature
($T_{g}$) and the so-called reduced glass transition temperature
($T_{g}/T_{m}$), whose values are known for almost all types of glass-forming
liquids and can be found in the scientific literature. Moreover, for some
organic and metallic glass formers, the phenomenological relation between
$T_{g}$ and $T_{A}$ is known [5, 15]. For most silicates and borates, there is
no known correlation between these two temperatures. At the same time, there
can be hidden relationships, which are usually revealed using machine learning
methods.

The initial data set for machine learning included experimental and calculated
data as well as information from databases (e.g., ITPhyMS-Information
technologies in physical materials science, Materials Project) [8, 12, 31].
For our purpose, different glass-forming materials were selected, among which
were silicates, borates, organic compounds and metallic alloys (Cu, Zr, Ti,
Ni, Pd-based) (see Table S1 in Supplementary Materials). We chose systems for
which the melting temperature, the glass transition temperature and the
fragility index are known. This data set was divided into the sets
corresponding to training, validation and test regimes. The training and
validation sets included all organic compounds and metallic alloys, along with
several silicates and borates, for which $T_{A}$ is known. The machine
learning model was created on the basis of the training data set. The accuracy
of the neural network was checked using the validation data set. The test set
included only silicates and borates, for which $T_{A}$ was predicted. Note
that to create an artificial neural network, we used instances for which all
parameters are known. Predictions were made only for those systems for which
the temperature $T_{A}$ is unknown. The reliability of the obtained results is
quite expected, since the formation of the neural network was performed using
the data for systems of all categories, including those for which further
predictions were made.

In the present work, the machine learning model was a feedforward artificial
neural network (see. Figure 1). This model has one input layer with four
neurons, for which the values of the melting temperature, the glass transition
temperature, the ratio $T_{g}/T_{m}$ and the fragility index were taken from
the data set. The values of this physical characteristics were renormalized
and presented in the range [$0$, $1$]. Next two layers of the neural network
were hidden and consisted of $20$ neurons. In the output layer we had only one
neuron, which determined $T_{A}$. For the initiation of the neural network,
the values of all neurons and their weight coefficients were assigned randomly
from the range [$0$, $1$]. Subsequently, calculation of the values of all
neurons was carried out as follows [32]:

$n_{i}^{(k)}=f\left(\sum_{j=1}^{N_{k-1}}w_{ij}^{(k-1)}n_{j}^{(k-1)}+b_{i}^{(k)}\right).$
(1)

Here, $n_{i}^{(k)}$ is the value of the $i$th neuron in the $k$th layer
($k=2,\,3,\,4$); $w_{ij}^{(k-1)}$ is the value of the $(k-1)$th layer weight
going from a neuron with index $j$ to a neuron with index $i$ from the $k$th
layer; $b_{i}^{(k)}$ is the bias weight acting on a neuron with index $i$ from
the $k$th layer; $N_{k-1}$ is the number of neurons in the $(k-1)$th layer;
function $f(...)$ is the sigmoid-type logistic function:

$f(x)=\frac{1}{1-\exp(-x)}.$ (2)

The neural network was trained using the backpropagation algorithm, according
to which the value of the weight coefficient was adjusted as follows [33]:

$w_{ij}^{(k),\,new}=w_{ij}^{(k)}-\gamma\frac{\partial\chi(s)}{\partial
w_{ij}^{(k)}}.$ (3)

$\gamma$ is the training rate, the value of which is usually chosen in the
range [$0$, $1$]. In the present work, we took the rate of $\gamma=0.3$ as
optimal for the considered machine learning model. The value of the loss
function $\chi(s)$ is determined as

$\chi(s)=\frac{1}{2}\left[n_{1}^{(4)}(s,l)-n(l)\right]^{2},$ (4)

where $s$ is the training iteration number (i.e., epoch number);
$n_{1}^{(4)}(s,l)$ is the value of the output neuron at the $s$th epoch for
the $l$th element from the training data set; $n(l)$ is the required value of
the output neuron for the $l$th element. To train the machine learning model,
$2400$ epochs were used. The gradient of the loss function with respect to
each weight was computed by the chain rule, according to which Equation (3)
can be represented in the following form:

$w_{ij}^{(k),\,new}=w_{ij}^{(k)}-\gamma\delta_{i}n_{i}^{(k)}\frac{e^{-W_{i}^{(k)}}}{\left[1+e^{-W_{i}^{(k)}}\right]^{2}},$
(5)

where

$\delta_{i}=\begin{cases}n_{1}^{(4)}(l)-n(l)&\text{if $i$ is the output layer
neuron}\\\ \sum_{j}w_{ij}\delta_{j}&\text{if $i$ is a neuron of the hidden
layers}\end{cases},$
$W_{i}^{(k)}=\sum_{j=1}^{N_{k-1}}w_{ij}^{(k-1)}n_{j}^{(k-1)}.$ (6)

This backpropagation algorithm allows one to control the training procedure.
The criterion for finishing this procedure is the minimal error between the
results of the output neuron and the required values from the validation data
set.

Figure 1: Scheme of the machine learning model based on the feedforward
artificial neural network.

##  3 Identification of Significant Physical Properties

To identify the physical characteristics that are most significant for
estimating the temperature $T_{A}$, calculations were carried out for various
combinations of the neural network’s input parameters. As shown in Figure 2a,
retraining of the machine learning model was performed for various
combinations of $T_{m}$, $T_{g}$, $T_{g}/T_{m}$ and $m$ using the training and
validation data sets. For each considered combination, the root mean square
error was calculated:

$\xi=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(T_{A}^{(pred)}-T_{A}^{(emp)}\right)}.$
(7)

Obviously, the smaller the value of $\xi$, the more accurately $T_{A}$ is
determined. In Equation (7), $T_{A}^{(pred)}$ is the Arrhenius crossover
temperature predicted by machine learning model; $T_{A}^{(emp)}$ is the
empirical Arrhenius crossover temperature; $N$ is the number of elements in
the data set. The obtained results reveal that the most significant physical
quantities correlating with $T_{A}$ are the glass transition temperature
$T_{g}$ and the melting temperature $T_{m}$. This is confirmed by the
relatively small value of the mean absolute error, which does not exceed
$\xi=11.4$ K. The quantities $T_{g}/T_{m}$ and $m$ are less significant in the
estimation of the temperature $T_{A}$, which is clearly manifested in the
relatively large $\xi$ with values of up to $25.8$ K. The smallest error
$\xi\approx 10.5$ K is obtained by taking into account all four physical
quantities at which the best agreement between the empirical values of $T_{A}$
and the result of the machine learning model is observed.

Figure 2b shows that the empirical and predicted temperatures $T_{A}$
correlate well with each other. Regarding organic compounds, an insignificant
variation between the empirical and predicted $T_{A}$ can be observed for
saccharides (for example, fructose, trehalose). This was mainly due to
insufficient of data in the training set for this class of materials. For
metal melts, the variation in the values of $T_{A}$ can be observed for alloys
based on rare earth elements (for example, Pr60Ni10Cu20Al10). The empirical
and predicted values of $T_{A}$ have a minimum variation for silicates and
borates. This result indicates that artificial neural networks have good
trainability with respect to these materials. The reason for this could be
that the change in viscosity of a silicate and borate melt occurs in a similar
way in a wide temperature range, including near the melting temperature [12].
Such universality in the temperature dependencies of viscosity is kept when
the composition of the melts changes, for example, by adding alkali oxides
(Li2O, Na2O, K2O, etc.) or metal oxides (Al2O3, MgO, PbO, etc.).

Figure 2: (a) Diagram of the root mean square error $\xi$ of estimation of the
Arrhenius crossover temperature $T_{A}$ calculated for various combinations of
the quantities $T_{m}$, $T_{g}$, $T_{g}/T_{m}$ and $m$, which were the inputs
of the machine learning model. Inset: $T_{A}^{(pred)}$ and $T_{A}^{(emp)}$ are
the predicted and empirical Arrhenius crossover temperatures, respectively.
(b) Correspondence between the empirical $T_{A}$ and the $T_{A}$ predicted by
the machine learning model using the validation data set.

##  4 Regression Model for Arrhenius Crossover Temperature

Figure 3a shows the correspondence between the glass transition temperature
$T_{g}$ and the predicted temperature $T_{A}$ for various glass-forming
liquids. For organic compounds, the correspondence between $T_{A}$ and $T_{g}$
is reproduced according to the linear law $T_{A}\simeq k\cdot T_{g}$ with
$k=1.4$. It is noteworthy that this correspondence between $T_{A}$ and $T_{g}$
was predicted earlier (for example, see Refs. [8, 34]). For metallic glass
formers, there is a relationship between $T_{A}$ and $T_{g}$ of the form
$T_{A}=k\cdot T_{g}$, where $k=2.0\pm 0.2$. As a rule, such a relationship
between temperatures $T_{A}$ and $T_{g}$ is universal for metal alloys
containing two to five different components [8]. For silicates and borates,
there is no clear correlation between $T_{A}$ and $T_{g}$: the known laws do
not reproduce the correspondence between $T_{A}$ and $T_{g}$. The results
given in Figure 3(b) reveal the obvious correlation between $T_{A}$ and
$T_{m}$ for silicates and borates, whereas variation in values of these
temperatures is more pronounced for organic and metallic glass formers.
Despite this, the correspondence between $T_{A}$ and $T_{m}$ is reproduced by
the linear law

$T_{A}=k\cdot T_{m}\,\,\,(\text{where}\,\,k=1.1\pm 0.15)$ (8)

regardless of the type of glass-forming liquid. It is noteworthy that this
result agrees with the results of Refs. [35, 36].

Relationship (8) is an empirical result that has no theoretical explanation
and is only an approximation. The error of this relationship depends both on
the specific type of material and on the category to which this material
belongs (i.e., organic, metallic, silicate). As shown in Figure 3b,
relationship (8) only qualitatively reproduces the empirical data for a large
data set. At the same time, one can be convinced that for certain categories
of materials, this relationship yields very accurate results. Thus, for
example, the available data for organic materials and metallic systems are
more correctly reproduced by the quadratic polynomials than by the linear
relationship (see Figure 3b). On the other hand, the results for silicates and
borates reveal a general trend of increasing $T_{A}$ with $T_{m}$, which can
be described by the linear relationship $T_{A}=aT_{m}+b$, where the parameters
$a$ and $b$ take different values for materials from different categories. In
this regard, it is quite natural to expect that the overall correlation
between $T_{A}$ and $T_{m}$ is not as so simple as prescribed by relationship
(8), and it requires taking into account other physical characteristics.

For implicit (”hidden”) correlations between different parameters, it is quite
natural and often feasible that the parameters do not appear in the resulting
correlation relation as single additive terms, but in the form of combinations
(products or ratios). For example, in contrast to the methodology of
artificial neural networks, this is most clearly manifested in the method of
joint accounting for arguments using the Kolmogorov–Gabor polynomial [37, 38,
39]:

$y(x_{1},...,x_{n})=a_{0}+\sum_{i=1}^{n}a_{i}x_{i}+\sum_{i=1}^{n}\sum_{j=i}^{n}a_{ij}x_{i}x_{j}+\sum_{i=1}^{n}\sum_{j=i}^{n}\sum_{k=j=1}^{n}a_{ijk}x_{i}x_{j}x_{k}+\ldots,$
(9)

which determines the relationship of a parameter $y$ with the parameters
$x_{1}$, $x_{2}$, … $x_{i}$,… In the obtained model of the artificial neural
network, the appearance of the parameter $T_{g}/T_{m}$, together with the
individual parameters $T_{g}$ and $T_{m}$, directly indicates that the
Arrhenius crossover temperature $T_{A}$ correlates not only with the absolute
values of the melting and glass transition temperatures for different systems,
but also with their ratio. This result is fully consistent with the
theoretical description of crystallization rate characteristics of supercooled
melts within the reduced temperature scale $\widetilde{T}_{MG}$ and universal
scaled relations [30, 9]. This point is discussed in detail in Ref. [30] (see
text on page 104502-2).

Figure 3: (a) Correspondence between the glass transition temperature $T_{g}$
and the predicted value of the Arrhenius crossover temperature $T_{A}$ for
different types of glass formers. (b) Correlation between the melting
temperature $T_{m}$ and the predicted temperature $T_{A}$. The dashed and
dotted lines show the interpolation by the quadratic polynomials:
$T_{A}=409-1.23T_{m}+0.003T_{m}^{2}$ in the case of organic materials and
$T_{A}=-2161+4.6T_{m}-0.0014T_{m}^{2}$ for metallic systems. The dot-dash
lines show the linear fit by equations $T_{A}=318+0.9T_{m}$ (for silicates)
and $T_{A}=465+0.71T_{m}$ (for borates).

To obtain a general expression relating the temperatures $T_{g}$, $T_{m}$ and
$T_{A}$, the reproducibility of these temperatures was tested in the framework
of the nonlinear regression model:

$T_{A}(T_{g},T_{m})=\sum_{i=1}^{k}\left(a_{i}T_{g}^{i}+b_{i}T_{m}^{i}+c_{i}T_{g}^{i}T_{m}^{i}\right).$
(10)

The temperatures $T_{g}$ and $T_{m}$ are input parameters determined by
experimental values; the temperature $T_{A}$ is the resulting factor; $k$ is
an integer value chosen during the regression analysis; $a_{i}$, $b_{i}$ and
$c_{i}$ are the fitting coefficients, whose values are determined by
enumeration to obtain the best agreement between the empirical temperature
$T_{A}$ and the result of Equation (10).

The values of the fitting coefficients were determined by regression analysis:
$a_{1}=b_{1}=0.7016$, $a_{2}=-7.52\times 10^{-4}$ K-1, $c_{1}=4.43\times
10^{-4}$ K-1. As was found, all other fitting coefficients equal zero. Thus,
with these values of the fitting coefficients, we obtained the minimum error
between the empirical $T_{A}$ and the result of Equation (10) for the
considered glass-forming liquids. Thus, the temperatures $T_{g}$, $T_{m}$ and
$T_{A}$ can be related by the nonlinear regression model:

$T_{A}(T_{g},T_{m})=a_{1}T_{g}+a_{2}T_{g}^{2}+b_{1}T_{m}+c_{1}T_{g}T_{m}.$
(11)

In algebra, an equation of this type is known as the equation of a second-
order curved surface. Figure 4 shows that Equation (11) correctly determines
the correspondence between the temperatures $T_{g}$, $T_{m}$ and $T_{A}$ for
all considered glass formers. The average error between the empirical data and
the result of Equation (11) is $\sim$10%. The plane surface corresponds to the
data for organic compounds and metal melts. The deviation from this surface
and its transformation into a curved surface occurs due to taking into account
the data for silicates and borates (see Figure 4b). Therefore, Equation (11)
can be applied to determine $T_{A}$ for various types of materials, regardless
of composition.

Figure 4: (a) Correspondence between the Arrhenius crossover temperature
($T_{A}$), the melting temperature ($T_{m}$) and the glass transition
temperature ($T_{g}$). Circle and square markers denote predicted and
empirical data, respectively. These data are compared with the results of
Equation (11), which are presented as a curved surface. (b) This figure is
from an another foreshortening, which allows one to consider the curved
surface.

Note that Equation (11) is an empirical result, the rigorous physical meaning
of which has not yet been established. This is also true for relationship (8),
which also has no a clear physical meaning. On the other hand, Equation (11)
shows that the three key temperatures associated with a change in kinetic
regime (as in the case of $T_{A}$) and with a change in thermodynamic phase
(as for $T_{m}$ and $T_{g}$) correlate in some universal way with each other
for melts that are different in physical nature. The necessity of the
quadratic contribution in Equation (11) to reproduce the empirical data
becomes obvious if these data are represented in the space of three
parameters—temperatures $T_{A}$, $T_{m}$ and $T_{g}$—as shown in Figure 4b. As
can be clearly seen in this representation, the empirical data form the
second-order curved surface, for the analytical reproduction of which the
presence of the quadratic contributions $T_{g}^{2}$ and $T_{g}T_{m}$, are
necessary. Moreover, since the curvature of this surface is expressed
significantly, its projection onto the coordinate plane ($T_{A}$, $T_{m}$)
will give a certain curve that can be reproduced by a straight line only
approximately (for example, as prescribed by relationship (8)). It should be
noted that such representation of the empirical data in ($T_{A}$, $T_{m}$,
$T_{g}$)-space was not expected and originally carried out; and Equation (11)
is a direct result of the regression analysis.

Figure 5: Root mean square relative error between the empirical values of
$T_{A}$ and actual $T_{A}$, which is computed by different methods for
silicates, borates, metallic systems and organic materials.

To determine the error in estimating $T_{A}$ for materials of various
categories, the root mean square relative error (RMSRE) was calculated:

$\text{RMSRE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(\frac{T_{A}^{(emp)}-T_{A}^{(calc)}}{T_{A}^{(emp)}}\right)^{2}},$
(12)

where $T_{A}^{(emp)}$ is the empirical value of $T_{A}$; $T_{A}^{(calc)}$ is
the $T_{A}$ computed by various methods—a machine learning model, by
relationship (8) and by Equation (11). Figure 5 shows that the accuracy at
estimation of $T_{A}$ depends on the applied method and the category of
material. Thus, for silicates and borates, the error of machine learning
prediction is lower than that of other methods. In this case, Equation (11) is
more accurate than relationship (8). For metallic systems, the errors of all
methods are comparable, although Equation (11) produces the smallest error.
For organic materials, the machine learning prediction is more accurate than
other methods. In this case, the error of Equation (11) is higher than the
error of relationship (8). This is due to the fact that for materials with
complex structures, such as organic materials, the glass transition
temperature is determined ambiguously. Namely, for this category of materials,
the temperature $T_{g}$ in relation to the melting temperature $T_{m}$ can
vary widely compared to silicate, borate and metallic systems. For example,
for organic materials, the variation in $T_{g}/T_{m}$ exceeds $0.5$, whereas
for borate, silicate and metallic systems this variation is usually less than
$0.4$.

##  5 Conclusions

The physical characteristics of various type glass-forming liquids were
determined using a machine learning model—that is, those which are most
significant to correct prediction/estimation of the Arrhenius crossover
temperature. Such significant factors are the glass transition temperature and
the melting temperature. It has been established that the fragility index and
the reduced glass transition temperature ($T_{g}/T_{m}$), which is directly
related to the glass-forming ability of a liquid, are insignificant factors.
These factors do not affect the accuracy of $T_{A}$ estimation. The
correctness of the obtained results was confirmed by the presence of a good
correlation between the empirical values of $T_{A}$ and the $T_{A}$ predicted
by a machine learning model. Moreover, the result of the machine learning
model gives the correct relationships between the temperatures $T_{A}$,
$T_{g}$ and $T_{m}$, which agree with the previously established empirical
rules $T_{A}\simeq 1.1T_{m}$ (for all types of liquids), $T_{A}\simeq
1.4T_{g}$ (for organic compounds) and $T_{A}\simeq 2.0T_{g}$ (for metallic
systems). Based on the results of nonlinear regression analysis, an equation
was obtained that allows one to determine the temperature $T_{A}$ by using
known temperatures $T_{g}$ and $T_{m}$. It was shown that this equation gives
the correct values of $T_{A}$ for various types of liquids, including
silicates and borates, for which direct estimation of $T_{A}$ can be
difficult.

## Acknowledgment

This research was funded by the Russian Science Foundation (project no.
19-12-00022).

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# One Size Doesn’t Fit All:
Personalised Affordance Design for Social Robots

Guanyu Huang  University of SheffieldSheffieldUnited Kingdom
<EMAIL_ADDRESS>0000-0002-0266-6987 and  Roger K. Moore  University
of SheffieldSheffieldUnited Kingdom<EMAIL_ADDRESS>0000-0003-0065-3311

(2018; 2023)

###### Abstract.

In human-robot interaction, personalisation is essential to achieve more
acceptable and effective results. Placing users in the central role, many
studies have focused on enhancing social robots’ abilities to perceive and
understand users. However, little is known about improving users’ perceptions
and interpretation of a social robot in spoken interactions. The work
described in the paper aims to find out what affects the personalisation of a
social robot’s affordance, namely appearance, voice and language behaviours.
The experimental data presented here is based on an ongoing project. It
demonstrates the many and varied ways in which people’s preferences for a
social robot’s affordance differ under different circumstances. It also
examines the relationship between such preferences and people’s expectations
of a social robot’s characteristics like competence and warmth. It also shows
that individuals have different perceptions of the same robot’s language
behaviours. These results demonstrate that one-sized personalisation does not
fit all. Personalisation should be considered a comprehensive approach,
including appropriate affordance design, to suit the users’ expectations of
social roles.

affordance design, personalisation, user needs, perception, stereotype content
model(SCM)

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; June 03–05, 2018; Woodstock,
NY††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: ; March 13, 2023; Stockholm, SE††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Human-centered computing Empirical
studies in HCI††ccs: Human-centered computing User studies††ccs: Human-
centered computing Empirical studies in interaction design††ccs: Human-
centered computing Scenario-based design††ccs: Human-centered computing
Laboratory experiments††ccs: General and reference Surveys and overviews

##  1\. Introduction

Personalisation is a type of adaptation. Personalised human-robot interaction
aims to improve the effectiveness and acceptability of a robot by adapting a
robot’s behaviours to interact with a specific individual or group. Such
adjustment is based on physical and social factors in the given environment
where the interaction takes place, as well as users’ preferences, behaviours,
and characteristics (Hellou et al., 2021). Along with the development of
robots in social domains, research for personalised HRI has focused on making
robots learn and adapt to users. These adaptations can be a long-term process
(Portugal et al., 2015; Gockley et al., 2005; Lee et al., 2012) and sometimes
culturally sensitive (Tuyen et al., 2018; Kanda et al., 2010). For short-term
individual interaction, most studies have paid attention to enhancing robots’
abilities to perceive and interpret users’ attention (McColl and Nejat, 2013),
personality (Aly and Tapus, 2013), behaviour (Gross et al., 2009) and
abilities (Torrey et al., 2006).

Whilst it is important to develop a robot’s abilities to adapt its behaviours,
it is also worth paying attention to a social robot’s static features which do
not change over the duration of interactions, such as its appearance and
voice. Such features help form the first impression, a fundamental factor in
human-human encounters with regard to people’s social cognition (e.g., trust
and rapport) (Ambady and, eds.). Also, they are related to a robot’s
affordance, namely its perceptual action possibilities. According to the
Affordance Theory(Gibson, 1977; Matei, 2020), affordance shapes people’s
perception of an object and affects people’s behaviours. Thus, it is important
to design appropriate affordances for robots to develop a satisfactory first
impression and optimise users’ experiences in the HRI.

Two dimensions widely used to measure people’s social perceptions are warmth
and competence (Cuddy et al., 2008; He et al., 2019). The former captures the
perceived friendliness and good intention. The latter captures the perceived
ability to deliver on those intentions. Given that social robots are used in
social domains to play social roles, it makes sense to investigate what people
expect robots to be like in different roles. A well-developed model to measure
warmth and competence in social roles is the stereotype content model (SCM),
which has been proven reliable in many experimental tests across different
cultural contexts (Cuddy et al., 2008; Russell and Fiske, 2008; Cuddy et al.,
2009). It has also been used in studies of how people perceive virtual agents
(e.g., (Bergmann et al., 2012)).

This paper adapts this model within a programme of empirical experimental
work. The experiment firstly aims to find out what people’s expectations of
social robots’ warmth and competence are in given social scenarios and how
such expectations affect people’s preferences for social robots’ affordances.
Secondly, it aims to identify personalised affordance design principles by
studying people’s preferences and perceptions of a social robot’s affordance.

##  2\. Research Hypothesis and Questions

The main hypothesis is that a social robot’s affordance is related to the
social scenarios in which it is used and to people’s expectations of a
specific social role. The research questions (RQs) are as follows.

  * •

RQ1. What preferences do people have for a social robot’s affordance in
general?

  * •

RQ2. Do such preferences vary when people know what the robot may be used for?

  * •

RQ3. How do people’s expected social perceptions of a social robot affect
their preference for a social robot’s affordance?

    * –

RQ3a. What levels of warmth and competence do people expect from a social
robot in different roles?

    * –

RQ3b. How do the expected warmth and competence level affect people’s
preferences for robots’ looks?

    * –

RQ3c. How do the expected warmth and competence level affect people’s
preferences for robots’ voices?

##  3\. Research Methods

To answer the above questions, an experiment is conducted by means of a
questionnaire. It asks about participants’ demographic information and
experience with speech-enabled technologies. It also asks for participants’
expectations for a speaking social robot’s look and voice under different
circumstances (1) without any information about what the robot is used for;
(2) with the information provided about the robot’s role in six social
scenarios, which includes three public and three private scenarios. The
robot’s role in each scenario is also provided. They are a receptionist robot
at a museum, a leisure robot at home, a private health-care robot, a private
language teacher robot, a waiter robot at a restaurant and a finance adviser
robot at a bank. The full description of the scenarios is attached in the
appendix.

A 5-point Likert scale is adopted, on a scale of 1-5, with ‘1’ as ‘the least’
and ‘5’ as ‘the most’. Participants are asked to indicate how much they agree
with the statements (e.g., ‘A robot’s look should be as human-like as
possible.’) when they do not know what the robots are used for. Later, when
provided with scenario information, participants are asked to indicate their
preferred level (e.g., ‘The robot needs to be competent.’, ‘The robot needs to
have a human-like look.’).

Given that there are four variables to measure in six scenarios, and each
participant goes through each condition (within groups), the statistical
software G*Power is used to conduct a prior test to determine the required
sample size. By selecting the ANOVA test (repeated measures), with the desired
large effect size of 0.5, alpha of 0.05, power of 0.8, a moderate correlation
of 0.5, and the type of effect size measure being ‘Partial Eta Squared’, a
total sample size of at least 42 participants is required.

The study is a part of the empirical human-robot interaction experiment, which
has received ethical approval from the Department of Computer Science at the
University of Sheffield.

##  4\. Current Results

Up to the revision date (28 February 2023), 68 participants completed the
survey. The experiment is expected to end in early March. All survey results
so far are valid to be processed. According to the survey results, 35.3% of
the participants are aged between 18-24 years old, 39.7% of them are 25-34
years old, 13.2% of them are 35-44 years old and 11.8% of them are above 45
years old. The gender distribution is 57.4% female, 35.3% male and 7.4% non-
binary. Over 60% of the participants are British. Nearly 20% of participants
report that their English accents ‘occasionally cause problems in daily
communication’. Most participants claim they do not know much about speech
science (72%) or conversational artificial intelligence (55%).

For RQ1, when participants do not know what the social robot would be used
for, they generally prefer it to have a less human-like look (mean 1.9), a
more human-like voice (mean 3.1) and less human-like behaviour (mean 2.3).
After performing a one-way analysis of variance (ANOVA), it shows these
preferences have a statistically significant difference (p-value = 0.000). As
shown in Figure 1, the violin plot shape shows the probability density of each
set, and the box plot shows the range and means of the agreement level.

Figure 1. The graph shows people’s preferences for a generic social robot’s
affordance, with the highest preference for a human-like voice, followed by
human-like behaviours and a human-like look.

People’s preference for a social robot’s look, voice and behaviour is shown in
a combination of box and violin plots.

For RQ2, it is found that, in comparison with general preferences,
participants’ preference for a social robot’s affordance differs among the six
social roles provided. As shown in Table 1, participants still prefer a more
human-like voice than a human-like look, no matter which situation a robot is
used in. Also, participants’ preference for a human-like look increases
slightly when they know what social robots are used for. Still, people hold
different opinions about how human-like it needs to be according to the
robot’s roles in given social scenarios. After performing a one-way analysis
of variance (ANOVA), it shows (1) there is no significant difference between
the sets of human-like looks (p=0.354, ¿0.05); (2) there is a significant
difference between the sets of human-like voices (p-value=0.000).

Table 1. Situational Affordance Preference Robot roles | Human-like Look | Human-like Voice
---|---|---
| mean | std. | mean | std.
general use | 1.9 | 1.0 | 3.1 | 1.2
receptionist | 2.1 | 1.1 | 3.1 | 1.2
leisure companion | 2.2 | 1.1 | 3.5 | 1.4
home-carer | 2.2 | 1.2 | 2.9 | 1.3
language teacher | 2.1 | 1.1 | 4.3 | 1.1
waiter | 2.3 | 1.1 | 3.0 | 1.2
financial adviser | 2.3 | 1.2 | 3.1 | 1.2

To answer RQ3, further analysis is conducted to investigate what roles social
perception (warmth and competence) plays in people’s preferences of social
robots’ affordance. For RQ3a, as shown in Table 2, participants expect a
medium-high warmth of all social roles. Compared to that, participants hold
higher expectations of a social robot’s competence. As for RQ3b and RQ3c,
which is about potential correlative relations between expected warmth,
competence and human-like look and voice, a correlative matrix is produced
(Figure 2). It shows that the absolute correlation coefficients between social
dimensions (warmth and competence) and a robot’s affordance (look and voice)
are mostly less than 0.5. The degree of a human-like look is negatively
connected with competence and positively associated with warmth at a small
level. The degree of human-like voice is positively related to competence at a
small level and negatively correlated with warmth at a moderate level.

Table 2. Expected Social Perception of Situational Robot Roles robot roles | warmth | competence
---|---|---
| mean | std. | mean | std.
receptionist | 3.8 | 0.9 | 4.7 | 0.7
leisure companion | 4.3 | 1.0 | 4.4 | 0.8
home-carer | 3.7 | 1.3 | 4.7 | 0.8
language teacher | 3.7 | 1.2 | 4.8 | 0.7
waiter | 3.8 | 1.2 | 4.6 | 0.7
financial adviser | 3.5 | 1.3 | 4.8 | 0.7

Figure 2. The heatmap shows relations between people’s expectation of a robot
and their preference for its affordance.

Most of the heatmap is the cold colours or pale yellow colour.

##  5\. Discussion and Conclusion

The human-likeness of social robots is a controversial topic. The current
experiment shows that human-like affordances are not necessarily favoured.
People hold a medium-low preference for the human-like affordance in a social
robot, with a higher preference for a human-like voice. These general
preferences stay the same in situational conditions. The results of this study
are based on people’s subjective preferences, not their perception of real
social robots. Why do people not hold a strongly positive view of human-like
affordance? This study does not provide an answer to that. It could be caused
by people’s concerns about the potential ‘Uncanny Valley Effect’, which states
the more human-like an object is, the more affinity it gains until a point
where the eerie sensation occurs (Mori et al., 2012). It could be rooted in
people’s ethical concern about mixing boundaries between humans and artificial
agents or their preference for a transparent and helpful affordance to reduce
potential uncertainties when they interact with the social robot. Further
studies are needed to develop a better understanding of the causes.

Another aspect to discuss is that a social robot’s look and voice are not
preferred to be 100% aligned. In other words, a less human-like look and a
more human-like voice may work. The questions are, what the safe boundary of
such difference is and how we can measure it. If the difference between a
social robot’s voice and look is too big, it may generate mismatched
perceptual cues, which can trigger an uncanny feeling (Moore, 2012; Meah and
Moore, 2014; Kätsyri et al., 2015; MacDorman and Chattopadhyay, 2016). Thus,
whilst it is sensible to consider individual components of a social robot’s
affordance, its overall affordance should also be considered.

Furthermore, the degree of human-likeness differs dependent on the scenarios.
Take a robot’s voice for example. Whilst a moderate human-like voice (3.0 +/-
0.1) is acceptable under most circumstances, people’s opinions on a robot’s
voice could differ a lot in some scenarios: an almost human-like voice is
preferred for a language teacher role. This may be related to the professional
requirements of the role. Thus, it is important to gain further understandings
of people’s many different needs in scenarios.

As for the role of social perception, the results show some correlation
between people’s expectations of how warm and competent a social robot should
be and how it should look and sound. Such correlations are the opposite. For
example, if people would like a more competent social robot in a given
situation, they may like a slightly more human-like voice and a slightly less
human-like look. Given that the warmth and competence levels are measured by
one stated preference only and people may have different understandings of
warmth and competence, it is worth investigating further to enhance the
reliability of the result. Also, it will be interesting when the project
proceeds to see whether the expected social perceptions match real-life social
perceptions.

Furthermore, one essential element in HRI is a social robot’s behaviour. From
the current survey result, people do not expect a highly human-like behaviour
pattern in a social robot. There is one question to follow up on. That is, for
the robot used in the same social situation and having the same behaviour
patterns, how do affordance designs affect people’s perception of the role and
their interaction experience? To answer this question, an extended in-lab
session of the above experiment is conducted. In the session, the same group
of participants is invited to have a face-to-face interaction with a speaking
social robot in the human-robot-interaction lab. The social robot ‘Furhat’ is
used to interact with participants. It plays three roles in three
interactions: a quiz host, a joke maker and a scientist (within the subject).
Each role comes with three affordance settings: robot-like, human-like (adult)
and human-like (child) (between subjects). Roles’ order and affordance are
randomised to eliminate the sequence bias. Participants need to complete
rating tasks before and after their interactions and elaborate on their
thoughts during post-interaction interviews. Up to the submission date, 45
participants completed the in-lab session. It is too early to report the
effect of different looks and voices on the same robot roles. However, post-
interaction interviews show that individual users have different perceptions
of the same robot’s language behaviour. Taking the word ‘sweet’ (as a response
when users agree to have another joke) as an example, some participants find
it too childish and not sincere, whilst others find it light-hearted.
Participants also pointed out that the sociolinguistic cues contained in the
voice could affect their willingness to interact with certain robotic
characters.

Yet to be completed, the empirical data from the experiment presented in this
paper shows that personalised affordance design is critical in human-robot
interaction and should be considered part of a comprehensive approach to
optimising human-robot interaction. It also shows that personalised affordance
design takes place within a multi-dimensional space. It requires some design
principles that take social factors into consideration. This report only
covers one aspect of the experiment conducted so far. It is anticipated to be
a starting point for further discussions, such as which features make a
robot’s look and voice more likeable. Added to this is how participants’
previous experience and expectations affect their preference and perception of
a social robot’s affordance.

###### Acknowledgements.

This work was supported by the Centre for Doctoral Training in Speech and
Language Technologies (SLT) and their Applications, funded by UK Research and
Innovation [grant number EP/S023062/1].

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##  Appendix A Six Social Scenarios

  1. (1)

You are visiting a museum. There is a receptionist robot that can answer your
questions about items in the exhibition and the museum’s history.

  2. (2)

You are living alone at home. There is a leisure robot that can read your
stories, sing songs and play video clips with your photos.

  3. (3)

You are ill. There is a service robot that can bring you medicine and do
housework for you.

  4. (4)

You are learning a new language. There is a multi-lingual robot that can
demonstrate how to speak a word and show you what goes wrong with your
pronunciation.

  5. (5)

You are visiting a newly open restaurant, and there is a waiter robot. It can
take your order and bring you food.

  6. (6)

You are visiting a bank to sort out your mortgage. There is a banker robot
which can evaluate your situation and give you advice.


$\mathbf{x}_{\textbf{AN}}$ as $\mathbf{x}_{t+1}$. However we make clear that
Mb-SVRN uses
$\mathbf{x}_{t+1}=\mathbf{x}_{t}-\eta\hat{\mathbf{H}}^{-1}\bar{\mathbf{g}}_{t}$.
In this proof we denote
$\mathbf{x}_{t+1}=\mathbf{x}_{\textbf{AN}}=\mathbf{x}_{t}-\eta\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}$,
$\Delta_{t+1}=\mathbf{x}_{\textbf{AN}}-\mathbf{x}^{*}$, and
$\Delta_{t}=\mathbf{x}_{t}-\mathbf{x}^{*}$.

$\displaystyle\Delta_{t+1}$
$\displaystyle=\Delta_{t}-\eta\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}$
$\displaystyle=\Delta_{t}-\eta\hat{\mathbf{H}}^{-1}(\mathbf{g}_{t}-\mathbf{g}(\mathbf{x}^{*}))$
$\displaystyle=\Delta_{t}-\eta\hat{\mathbf{H}}^{-1}\int_{0}^{1}{\nabla^{2}{f}(\mathbf{x}^{*}+\theta\Delta_{t})\Delta_{t}d\theta}$
$\displaystyle=\Delta_{t}-\eta\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}\Delta_{t}$
$\displaystyle\Rightarrow\left\lVert\Delta_{t+1}\right\rVert_{\bar{\mathbf{H}}}$
$\displaystyle=\left\lVert(\mathbf{I}-\eta\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}})\Delta_{t}\right\rVert_{\bar{\mathbf{H}}},$

where
$\bar{\mathbf{H}}=\int_{0}^{1}{\nabla^{2}{f}(\mathbf{x}^{*}+\theta\Delta_{t})d\theta}$.
We upper bound $\left\lVert\Delta_{t+1}\right\rVert_{\bar{\mathbf{H}}}$ as,

$\displaystyle\left\lVert\Delta_{t+1}\right\rVert_{\bar{\mathbf{H}}}$
$\displaystyle=\left\lVert(\mathbf{I}-\eta\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}})\Delta_{t}\right\rVert_{\bar{\mathbf{H}}}$
$\displaystyle=\left\lVert\bar{\mathbf{H}}^{1/2}(\mathbf{I}-\eta\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}})\Delta_{t}\right\rVert$
$\displaystyle=\left\lVert(\mathbf{I}-\eta\bar{\mathbf{H}}^{1/2}\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}^{1/2})\bar{\mathbf{H}}^{1/2}\Delta_{t}\right\rVert$
$\displaystyle\leq\left\lVert\mathbf{I}-\eta\bar{\mathbf{H}}^{1/2}\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}^{1/2}\right\rVert\cdot\left\lVert\Delta_{t}\right\rVert_{\bar{\mathbf{H}}}.$

As $\mathbf{x}_{t}\in\mathcal{U}_{f}(c\epsilon_{0}\eta)$, for any
$0\leq\theta\leq 1$, we have
$\mathbf{x}^{*}+\theta\Delta_{t}\in\mathcal{U}_{f}(c\epsilon_{0}\eta)$, and
therefore
$\frac{1}{1+c\epsilon_{0}\eta}\mathbf{H}\preceq\bar{\mathbf{H}}\preceq(1+c\epsilon_{0}\eta)\mathbf{H}$.
Also we have
$\frac{1}{\sqrt{\alpha}}\tilde{\mathbf{H}}\preceq\hat{\mathbf{H}}\preceq\sqrt{\alpha}\tilde{\mathbf{H}}$
and
$\frac{1}{1+\epsilon_{0}\eta}\mathbf{H}\preceq\tilde{\mathbf{H}}\preceq(1+\epsilon_{0}\eta)\mathbf{H}$.
Combining these positive semidefinite orderings for $\tilde{\mathbf{H}}$ and
$\hat{\mathbf{H}}$ along with $1+\epsilon_{0}\eta<2$, we have
$\frac{1}{2\sqrt{\alpha}}\mathbf{H}\preceq\hat{\mathbf{H}}\preceq
2\sqrt{\alpha}\mathbf{H}$. We get,

$\displaystyle\frac{1}{2\sqrt{\alpha}(1+c\epsilon_{0}\eta)}\bar{\mathbf{H}}^{-1}\preceq\hat{\mathbf{H}}^{-1}\preceq
2\sqrt{\alpha}(1+c\epsilon_{0}\eta)\bar{\mathbf{H}}^{-1},$
$\displaystyle\frac{\eta}{2\sqrt{\alpha}(1+c\epsilon_{0}\eta)}\mathbf{I}\preceq\eta\bar{\mathbf{H}}^{1/2}\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}^{1/2}\preceq
2\eta\sqrt{\alpha}(1+c\epsilon_{0}\eta)\mathbf{I},$

implying that,

$\displaystyle\left\lVert\mathbf{I}-\eta\bar{\mathbf{H}}^{1/2}\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}^{1/2}\right\rVert\leq\max\\{1-\frac{\eta}{2\sqrt{\alpha}(1+c\epsilon_{0}\eta)},2\eta\sqrt{\alpha}(1+c\epsilon_{0}\eta)-1\\}.$

Since $\eta<\frac{1}{4\sqrt{\alpha}}$, and
$\epsilon_{0}<\frac{1}{8c\sqrt{\alpha}}$, maximum value would be
$1-\frac{\eta}{2\sqrt{\alpha}(1+c\epsilon_{0}\eta)}$. We get,

$\displaystyle\left\lVert\mathbf{I}-\eta\bar{\mathbf{H}}^{1/2}\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}^{1/2}\right\rVert\leq
1-\frac{\eta}{4\sqrt{\alpha}},$

and hence,

$\displaystyle\left\lVert\Delta_{t+1}\right\rVert_{\bar{\mathbf{H}}}\leq\left(1-\frac{\eta}{4\sqrt{\alpha}}\right)\left\lVert\Delta_{t}\right\rVert_{\bar{\mathbf{H}}}.$

Changing $\left\lVert\cdot\right\rVert_{\bar{\mathbf{H}}}$ to
$\left\lVert\cdot\right\rVert_{\mathbf{H}}$ we get,

$\displaystyle\left\lVert\Delta_{t+1}\right\rVert_{\mathbf{H}}$
$\displaystyle\leq\sqrt{1+c\epsilon_{0}\eta}\left\lVert\Delta_{t+1}\right\rVert_{\bar{\mathbf{H}}}\leq\sqrt{1+c\epsilon_{0}\eta}\cdot\left(1-\frac{\eta}{4\sqrt{\alpha}}\right)\left\lVert\Delta_{t}\right\rVert_{\bar{\mathbf{H}}}\leq(1+c\epsilon_{0}\eta)\cdot\left(1-\frac{\eta}{4\sqrt{\alpha}}\right)\left\lVert\Delta_{t}\right\rVert_{\mathbf{H}}$
$\displaystyle\leq\left(1-\eta\left(\frac{1}{4\sqrt{\alpha}}-c\epsilon_{0}\right)-\frac{c\epsilon_{0}\eta^{2}}{4\sqrt{\alpha}}\right)\left\lVert\Delta_{t}\right\rVert_{\mathbf{H}}.$

Using $\epsilon_{0}<\frac{1}{8c\sqrt{\alpha}}$, we conclude,

$\displaystyle\left\lVert\Delta_{t+1}\right\rVert_{\mathbf{H}}\leq\left(1-\frac{\eta}{8\sqrt{\alpha}}\right)\left\lVert\Delta_{t}\right\rVert_{\mathbf{H}}.$

###  A.3 Proof of Lemma 3.

In the proof, we will use a result from [19], stated as the following,

######  Lemma 12 (Matrix Bernstein: Corollary 4.1 from [19])

Let $\mathbf{Y}_{1},\mathbf{Y}_{2},..\mathbf{Y}_{m}\in\mathbb{C}^{d}$ be a
sequence of random vectors such that $\mathbb{E}\mathbf{Y}_{i}=0$,
$\left\lVert\mathbf{Y}_{i}\right\rVert<U$ almost surely $\forall 1\leq i\leq
m$. Denote
$\sigma^{2}:=\sum_{i=1}^{m}{\mathbb{E}\left\lVert\mathbf{Y}_{i}\right\rVert^{2}}$.
Then $\forall t^{2}>\sigma^{2}+\frac{tU}{3}$,

$\displaystyle\Pr\left(\left\lVert\sum_{i=1}^{m}\mathbf{Y}_{i}\right\rVert_{2}>t\right)\leq
28\text{exp}\left[-\frac{t^{2}/2}{\sigma^{2}+tU/3}\right].$

We now return to the proof of Lemma 3.

Proof  Define random vectors $\mathbf{v}_{i}$, as
$\mathbf{v}_{i}=\nabla\psi_{i}(\mathbf{x}_{t})-\nabla\psi_{i}(\tilde{\mathbf{x}})+\mathbf{g}(\tilde{\mathbf{x}})-\mathbf{g}(\mathbf{x}_{t})$.
Then $\mathbb{E}[\mathbf{v}_{i}]=0$. Also,

$\displaystyle\|\mathbf{v}_{i}\|^{2}$ $\displaystyle\leq
2\|\nabla\psi_{i}(\mathbf{x}_{t})-\nabla\psi_{i}(\tilde{\mathbf{x}})\|^{2}+2\|\mathbf{g}(\mathbf{x}_{t})-\mathbf{g}(\tilde{\mathbf{x}})\|^{2}$
$\displaystyle\leq
4\lambda\left(\psi_{i}(\tilde{\mathbf{x}})-\psi_{i}(\mathbf{x}_{t})-\langle\nabla\psi_{i}(\mathbf{x}_{t}),\tilde{\mathbf{x}}-\mathbf{x}_{t}\rangle\right)+4\lambda\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}_{t})-\langle\nabla{f}(\mathbf{x}_{t}),\tilde{\mathbf{x}}-\mathbf{x}_{t}\rangle\right)$
$\displaystyle\leq
2\lambda^{2}\|\tilde{\mathbf{x}}-\mathbf{x}_{t}\|^{2}+2\lambda^{2}\|\tilde{\mathbf{x}}-\mathbf{x}_{t}\|^{2}$
$\displaystyle=4\lambda^{2}\|\tilde{\mathbf{x}}-\mathbf{x}_{t}\|^{2}.$

We also have the following upper bound on the variance of $\mathbf{v}_{i}$:

$\displaystyle\mathbb{E}[\|\nabla{\psi}_{i}(\mathbf{x}_{t})-\nabla{\psi}_{i}(\tilde{\mathbf{x}})+\mathbf{g}(\tilde{\mathbf{x}})-\mathbf{g}(\mathbf{x}_{t})\|^{2}]$
$\displaystyle=\mathbb{E}[\|\nabla{\psi}_{i}(\mathbf{x}_{t})-\nabla{\psi}_{i}(\tilde{\mathbf{x}})-(\mathbf{g}(\mathbf{x}_{t})-\mathbf{g}(\tilde{\mathbf{x}}))\|^{2}]$
$\displaystyle\leq\mathbb{E}[\|\nabla{\psi}_{i}(\mathbf{x}_{t})-\nabla{\psi}_{i}(\tilde{\mathbf{x}})\|^{2}]$
$\displaystyle\leq
2\lambda\cdot\mathbb{E}[(\psi_{i}(\tilde{\mathbf{x}})-\psi_{i}(\mathbf{x}_{t})-\langle\nabla{\psi}_{i}(\mathbf{x}_{t}),\tilde{\mathbf{x}}-\mathbf{x}_{t}\rangle)]$
$\displaystyle=2\lambda\cdot(f(\tilde{\mathbf{x}})-f(\mathbf{x}_{t})-\langle\mathbf{g}(\mathbf{x}_{t}),\tilde{\mathbf{x}}-\mathbf{x}_{t}\rangle).$

Since $f$ is twice continuously differentiable, there exists a
$\theta\in[0,1]$ such that,

$\displaystyle
f(\tilde{\mathbf{x}})-f(\mathbf{x}_{t})-\langle\mathbf{g}(\mathbf{x}_{t}),(\tilde{\mathbf{x}}-\mathbf{x}_{t})\rangle=\frac{1}{2}(\tilde{\mathbf{x}}-\mathbf{x}_{t})^{\scriptscriptstyle{\top}}\nabla^{2}{f}(\mathbf{x}_{t}+\theta(\tilde{\mathbf{x}}-\mathbf{x}_{t}))^{\scriptscriptstyle{\top}}(\tilde{\mathbf{x}}-\mathbf{x}_{t}).$

Using the assumption that
$\mathbf{x}_{t},\tilde{\mathbf{x}}\in\mathcal{U}_{f}(c\epsilon_{0}\eta)$, with
$\mathbf{z}=\mathbf{x}_{t}+\theta(\tilde{\mathbf{x}}-\mathbf{x}_{t})$ we have
$\mathbf{z}_{t}\in\mathcal{U}_{f}(c\epsilon_{0}\eta)$. With
$\epsilon_{0}<\frac{1}{8c\sqrt{\alpha}}$, and $\eta<\frac{1}{4\sqrt{\alpha}}$,

$\displaystyle\frac{1}{2}(\tilde{\mathbf{x}}-\mathbf{x}_{t})^{\scriptscriptstyle{\top}}\nabla^{2}{f}(\mathbf{x}_{t}+\theta(\tilde{\mathbf{x}}-\mathbf{x}_{t}))^{\scriptscriptstyle{\top}}(\tilde{\mathbf{x}}-\mathbf{x}_{t})$
$\displaystyle=\frac{1}{2}\|\mathbf{x}_{t}-\tilde{\mathbf{x}}\|^{2}_{\nabla^{2}{f}(\mathbf{z})}$
$\displaystyle\leq\frac{1}{2}(1+c\epsilon_{0}\eta)\left\lVert\tilde{\mathbf{x}}-\mathbf{x}_{t}\right\rVert_{\mathbf{H}}^{2}$
$\displaystyle\leq\left\lVert\tilde{\mathbf{x}}-\mathbf{x}_{t}\right\rVert_{\mathbf{H}}^{2}.$

We get,

$\displaystyle\mathbb{E}\|\mathbf{v}_{i}\|^{2}\leq
2\lambda\left\lVert\tilde{\mathbf{x}}-\mathbf{x}_{t}\right\rVert_{\mathbf{H}}^{2}.$

So we take
$U=2\lambda\left\lVert\tilde{\mathbf{x}}-\mathbf{x}_{t}\right\rVert$ and
$\sigma^{2}=2b\lambda\left\lVert\tilde{\mathbf{x}}-\mathbf{x}_{t}\right\rVert_{\mathbf{H}}^{2}$.
Also note that, $U\leq
2\frac{\lambda}{\sqrt{\mu}}\left\lVert\tilde{\mathbf{x}}-\mathbf{x}_{t}\right\rVert_{\mathbf{H}}$.
Look for $t$ such that,

$\displaystyle\exp\left(-\frac{t^{2}/2}{\sigma^{2}+tU/3}\right)<\delta\frac{b^{2}}{n^{2}}$
$\displaystyle\iff$
$\displaystyle\frac{t^{2}/2}{\sigma^{2}+tU/3}>2\ln(n/b\delta)$
$\displaystyle\iff$ $\displaystyle
t^{2}>4\sigma^{2}\ln(n/b\delta)+\frac{4tU}{3}\ln(n/b\delta)$
$\displaystyle\iff$
$\displaystyle\frac{t^{2}}{2}+\frac{t^{2}}{2}>4\sigma^{2}\ln(n/b\delta)+\frac{4tU}{3}\ln(n/b\delta).$

Now in the regime of $b<\frac{8}{9}\kappa$ we have
$2\sigma\ln(n/b\delta)<\frac{4}{3}U\ln(n/b\delta)$. Consider
$\mathbf{Y}_{i}=\mathbf{v}_{i}$,
$t=\frac{8}{3}U\ln(n/b\delta)=\frac{4}{3}.\frac{2\lambda}{\sqrt{\mu}}\left\lVert\tilde{\mathbf{x}}-\mathbf{x}_{t}\right\rVert_{\mathbf{H}}\ln(\frac{n}{b\delta})$.
For this value of $t$, we get with probability $1-\delta\frac{b^{2}}{n^{2}}$,

$\displaystyle\left\lVert\sum_{i=1}^{b}{\mathbf{Y}_{i}}\right\rVert\leq\frac{8}{3}.\frac{2\lambda}{\sqrt{\mu}}\left\lVert\tilde{\mathbf{x}}-\mathbf{x}_{t}\right\rVert_{\mathbf{H}}\ln(\frac{n}{b\delta}).$

This means with probability $1-\delta\frac{b^{2}}{n^{2}}$,

$\displaystyle\left\lVert\mathbf{g}(\mathbf{x}_{t})-\bar{\mathbf{g}}(\mathbf{x}_{t})\right\rVert\leq\frac{16\lambda}{3b\sqrt{\mu}}\ln(n/b\delta)\left\lVert\mathbf{x}_{t}-\tilde{\mathbf{x}}\right\rVert_{\mathbf{H}}.$

Moreover, in the regime of $b\geq\frac{8}{9}\kappa$, we have
$2\sigma\ln(n/b\delta)\geq\frac{4}{3}U\ln(n/b\delta)$. In this case set
$t=2\sqrt{2}\sigma\ln(n/b\delta)=4\sqrt{b\lambda}\left\lVert\mathbf{x}_{t}-\tilde{\mathbf{x}}\right\rVert_{\mathbf{H}}\ln(n/b\delta)$.
We conclude,

$\displaystyle\left\lVert\mathbf{g}(\mathbf{x}_{t})-\bar{\mathbf{g}}(\mathbf{x}_{t})\right\rVert\leq\frac{4\sqrt{\lambda}}{\sqrt{b}}\ln(n/b\delta)\left\lVert\mathbf{x}_{t}-\tilde{\mathbf{x}}\right\rVert_{\mathbf{H}}.$

###  A.4 Proof of Lemma 4.

Proof  Since $\mathbf{g}^{*}=0$, we have
$\left\lVert\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}\right\rVert_{\mathbf{H}}=\left\lVert\hat{\mathbf{H}}^{-1}(\mathbf{g}_{t}-\mathbf{g}^{*})\right\rVert_{\mathbf{H}}$.
Consider the following,

$\displaystyle\left\lVert\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}\right\rVert_{\mathbf{H}}=\left\lVert\hat{\mathbf{H}}^{-1}(\mathbf{g}_{t}-\mathbf{g}^{*})\right\rVert_{\mathbf{H}}$
$\displaystyle=\left\lVert\hat{\mathbf{H}}^{-1}\int_{0}^{1}{\nabla^{2}{f}(\mathbf{x}^{*}+\theta(\mathbf{x}_{t}-\mathbf{x}^{*}))\Delta_{t}\cdot
d\theta}\right\rVert_{\mathbf{H}}.$

Denoting
$\bar{\mathbf{H}}=\int_{0}^{1}{\nabla^{2}{f}(\mathbf{x}^{*}+\theta(\mathbf{x}_{t}-\mathbf{x}^{*}))\cdot
d\theta}$, we get,

$\displaystyle\left\lVert\hat{\mathbf{H}}^{-1}(\mathbf{g}_{t}-\mathbf{g}^{*})\right\rVert_{\mathbf{H}}$
$\displaystyle=\left\lVert\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}\Delta_{t}\right\rVert_{\mathbf{H}}$
$\displaystyle=\left\lVert\mathbf{H}^{1/2}\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}\Delta_{t}\right\rVert$
$\displaystyle=\left\lVert\mathbf{H}^{1/2}\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}\mathbf{H}^{-1/2}\mathbf{H}^{1/2}\Delta_{t}\right\rVert$
$\displaystyle\leq\left\lVert\mathbf{H}^{1/2}\hat{\mathbf{H}}^{-1}\bar{\mathbf{H}}\mathbf{H}^{-1/2}\right\rVert\left\lVert\Delta_{t}\right\rVert_{\mathbf{H}}$
$\displaystyle=\left\lVert\underbrace{\mathbf{H}^{1/2}\hat{\mathbf{H}}^{-1}\mathbf{H}^{1/2}}\underbrace{\mathbf{H}^{-1/2}\bar{\mathbf{H}}\mathbf{H}^{-1/2}}\right\rVert\left\lVert\Delta_{t}\right\rVert_{\mathbf{H}}$
$\displaystyle\leq\left\lVert\mathbf{H}^{1/2}\hat{\mathbf{H}}^{-1}\mathbf{H}^{1/2}\right\rVert\cdot\left\lVert\mathbf{H}^{-1/2}\bar{\mathbf{H}}\mathbf{H}^{-1/2}\right\rVert\cdot\left\lVert\Delta_{t}\right\rVert_{\mathbf{H}}.$

Since we have $\hat{\mathbf{H}}\approx_{\sqrt{\alpha}}\tilde{\mathbf{H}}$ and
$\tilde{\mathbf{x}}\in\mathcal{U}_{f}(\epsilon_{0}\eta)$, we have
$\hat{\mathbf{H}}\approx_{\sqrt{\alpha}(1+\epsilon_{0}\eta)}\mathbf{H}$.
Furthermore for $\mathbf{x}_{t}\in\mathcal{U}_{f}(c\epsilon_{0}\eta)$, we have
$\bar{\mathbf{H}}\approx_{1+c\epsilon_{0}\eta}\mathbf{H}$, because for all
$\theta,\mathbf{x}^{*}+\theta(\mathbf{x}_{t}-\mathbf{x}^{*})\in\mathcal{U}_{f}(c\epsilon_{0}\eta)$.
So we use the results
$\left\lVert\mathbf{H}^{1/2}\hat{\mathbf{H}}^{-1}\mathbf{H}^{1/2}\right\rVert\leq\sqrt{\alpha}(1+\epsilon_{0}\eta)$
and
$\left\lVert\mathbf{H}^{-1/2}\bar{\mathbf{H}}\mathbf{H}^{-1/2}\right\rVert\leq(1+c\epsilon_{0}\eta)$
and get,

$\displaystyle\left\lVert\hat{\mathbf{H}}^{-1}(\mathbf{g}_{t}-\mathbf{g}^{*})\right\rVert_{\mathbf{H}}$
$\displaystyle\leq\sqrt{\alpha}(1+\epsilon_{0}\eta)(1+c\epsilon_{0}\eta)\left\lVert\Delta_{t}\right\rVert_{\mathbf{H}}$
$\displaystyle<2\sqrt{\alpha}\left\lVert\Delta_{t}\right\rVert_{\mathbf{H}}.$

###  A.5 Proof of Theorem 3.

In our proof of Theorem 3, we use a Master tail bound for adapted sequenced
from [32]. Let $\mathcal{F}_{k}$ be a filtration and random process
$(Y_{k})_{k\geq 0}$ be $\mathcal{F}_{k}$ measurable. Also let another random
process $V_{k}$ such that $V_{k}$ is $\mathcal{F}_{k-1}$ measurable. Consider
the difference sequence for $k\geq 1$,

$\displaystyle X_{k}=Y_{k}-Y_{k-1}.$

Also, assume the following relation holds for a function
$g:(0,\infty)\rightarrow[0,\infty]$:

$\displaystyle\mathbb{E}_{k-1}e^{\theta\mathbf{X}_{k}}\leq
e^{g(\theta)V_{k}}.$

Then we have,

######  Theorem 7 (Master tail bound for adapted sequences [32])

For all $\lambda,w\in\mathbb{R}$, we have,

$\displaystyle\Pr\left(\exists k\geq 0:Y_{k}\geq Y_{0}+\lambda\ \ and\ \
\sum_{i=1}^{k}{V_{i}}\leq\sigma^{2}\right)\leq\inf_{\theta>0}e^{-\lambda\theta+g(\theta)\sigma^{2}}.$

In our proof, we also use the following Lemma from [32]. The original version
of the result assumes that $\mathbb{E}[X]=0$, however, we show below that the
proof also holds for the case when $\mathbb{E}[X]<0$.

######  Lemma 13 (Freedman MGF, Lemma 6.7 from [33])

Let $X$ be a random variable such that $\mathbb{E}X\leq 0$ and $X\leq R$
almost surely. Then for any $\theta>0$ and $h(R)=\frac{e^{\theta R}-\theta
R-1}{R^{2}}$,

$\displaystyle\mathbb{E}e^{\theta X}\leq e^{h(R)\mathbb{E}[X^{2}]}.$

Proof  Consider the Taylor series expansion of $e^{\theta x}$,

$\displaystyle e^{\theta x}$ $\displaystyle=1+x+\frac{x^{2}}{2!}+\cdots$
$\displaystyle=1+x+x^{2}h(x).$

Replace $x$ with the random variable $X$ and take expectation on both sides,
we get,

$\displaystyle\mathbb{E}e^{\theta X}\leq 1+\mathbb{E}X+\mathbb{E}[X^{2}\cdot
h(X)].$

On the second term use $\mathbb{E}X\leq 0$ and on the third term use $X\leq R$
to get $h(X)\leq h(R)$ almost surely, we get,

$\displaystyle\mathbb{E}e^{\theta X}\leq 1+h(R)\cdot\mathbb{E}[X^{2}]\leq
e^{h(R)\cdot\mathbb{E}[X^{2}]}.$

We now return to the proof of Theorem 3.

Proof  If $Y_{k}$ is a submartingale we have,

$\displaystyle\mathbb{E}_{k-1}X_{k}\leq 0,$

and also we know that $X_{k}\leq R$. For $k\geq 1$, consider
$V_{k}=\mathbb{E}_{k-1}(X_{k}^{2})$. Clearly, $V_{k}$ is $\mathcal{F}_{k-1}$
measurable. We now establish the relation that
$\mathbb{E}_{k-1}e^{\theta\mathbf{X}_{k}}\leq e^{g(\theta)V_{k}}$. For any
$\theta>0$, consider a function
$h(x):[0,\infty]\rightarrow[\frac{\theta^{2}}{2},\infty)$ defined as follows,

$\displaystyle h(x)=\frac{e^{\theta x}-\theta x-1}{x^{2}}\ ,and\ \
h(0)=\frac{\theta^{2}}{2}.$

It is easy to show that $h(x)$ is an increasing function of $x$. Using Lemma
13 we get $\mathbb{E}_{k-1}e^{\theta\mathbf{X}_{k}}\leq e^{g(\theta)V_{k}}$
for $g(\theta)=\frac{e^{\theta R}-\theta R-1}{R^{2}}$. Now we can use the
Master tail bound Theorem 7. The only thing remaining to analyze is,

$\inf_{\theta>0}e^{\theta\lambda+g(\theta)\cdot\sigma^{2}}.$

Doing little calculus shows that, $\theta=\frac{1}{R}\ln\left(1+\frac{\lambda
R}{\sigma^{2}}\right)$ minimizes $e^{\theta\lambda+g(\theta)\cdot\sigma^{2}}$
and the minimum value is
$e^{-\frac{1}{2}\left(\frac{\lambda}{\sigma^{2}}+\frac{\lambda}{R}\right)}$.
Finally observe that
$e^{-\frac{1}{2}\left(\frac{\lambda}{\sigma^{2}}+\frac{\lambda}{R}\right)}\leq
e^{-\frac{1}{4}\min\left(\frac{\lambda}{\sigma^{2}},\frac{\lambda}{R}\right)}$.
This completes the proof.

###  A.6 Proof of Theorem 6.

Proof  Using the $\lambda$-smoothness of $f$, we know that

$\displaystyle f(\mathbf{x}_{t+1})\leq
f(\mathbf{x}_{t})+\mathbf{g}_{t}^{\scriptscriptstyle{\top}}(\mathbf{x}_{t+1}-\mathbf{x}_{t})+\frac{\lambda}{2}\|\mathbf{x}_{t+1}-\mathbf{x}_{t}\|^{2}.$

Substitute
$\mathbf{x}_{t+1}=\mathbf{x}_{t}-\eta\hat{\mathbf{H}}^{-1}\bar{\mathbf{g}}_{t}$,
we get,

$\displaystyle f(\mathbf{x}_{t+1})$ $\displaystyle\leq
f(\mathbf{x}_{t})-\eta\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\bar{\mathbf{g}}_{t}+\eta^{2}\frac{\lambda}{2}\|\hat{\mathbf{H}}^{-1}\bar{\mathbf{g}}_{t}\|^{2}$
$\displaystyle=f(\mathbf{x}_{t})-\eta\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\bar{\mathbf{g}}_{t}+\eta^{2}\frac{\lambda}{2}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}+\mathbf{g}_{t}\right)\|^{2}.$

Take total expectation $\mathbb{E}$ on both sides, by which we mean
expectation conditioned only on known $\tilde{\mathbf{x}}$. We get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{t+1})$
$\displaystyle\leq\mathbb{E}\left(f(\mathbf{x}_{t})-\eta\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\bar{\mathbf{g}}_{t}+\eta^{2}\frac{\lambda}{2}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}+\mathbf{g}_{t}\right)\|^{2}\right)$
$\displaystyle=\mathbb{E}\left(\mathbb{E}_{t}\left(f(\mathbf{x}_{t})-\eta\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\bar{\mathbf{g}}_{t}+\eta^{2}\frac{\lambda}{2}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}+\mathbf{g}_{t}\right)\|^{2}\right)\right).$

Here in the last inequality, $\mathbb{E}_{t}$ means conditional expectation,
conditioned on known $\mathbf{x}_{t}$. Analyzing the inner expectation
$\mathbb{E}_{t}$, we get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{t+1})$
$\displaystyle\leq\mathbb{E}\left(f(\mathbf{x}_{t})-\eta\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}+\eta^{2}\frac{\lambda}{2}\mathbb{E}_{t}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}+\mathbf{g}_{t}\right)\|^{2}\right),$

where we obtained that second term due to the unbiasedness of stochastic
gradient i.e., $\mathbb{E}_{t}[\bar{\mathbf{g}_{t}}]=\mathbf{g}_{t}$.
Furthermore using unbiasedness we have
$\mathbb{E}_{t}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}+\mathbf{g}_{t}\right)\|^{2}=\mathbb{E}_{t}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}\right)\|^{2}+\|\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}\|^{2}$.
We get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{t+1})$
$\displaystyle\leq\mathbb{E}\left(f(\mathbf{x}_{t})-\eta\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}+\eta^{2}\frac{\lambda}{2}\|\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}\|^{2}+\eta^{2}\frac{\lambda}{2}\mathbb{E}_{t}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}\right)\|^{2}\right).$

Since
$\|\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}\|^{2}\leq\left\lVert\hat{\mathbf{H}}^{-1}\right\rVert\cdot\|\hat{\mathbf{H}}^{-1/2}\mathbf{g}_{t}\|^{2}\leq\frac{\sqrt{\alpha}}{\mu}\|\hat{\mathbf{H}}^{-1/2}\mathbf{g}_{t}\|^{2}=\frac{\sqrt{\alpha}}{\mu}\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}$,
where we used $\hat{\mathbf{H}}\approx_{\sqrt{\alpha}}\tilde{\mathbf{H}}$ to
write
$\left\lVert\hat{\mathbf{H}}^{-1}\right\rVert\leq\frac{\sqrt{\alpha}}{\mu}$.
Substituting we get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{t+1})$
$\displaystyle\leq\mathbb{E}\left(f(\mathbf{x}_{t})-\eta\left(1-\frac{\eta\kappa\sqrt{\alpha}}{2}\right)\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}+\eta^{2}\frac{\lambda}{2}\mathbb{E}_{t}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}\right)\|^{2}\right).$

Again using $\hat{\mathbf{H}}\approx_{\sqrt{\alpha}}\tilde{\mathbf{H}}$, we
get
$\mathbf{g}_{t}^{\scriptscriptstyle{\top}}\hat{\mathbf{H}}^{-1}\mathbf{g}_{t}\geq\frac{1}{\lambda\sqrt{\alpha}}\|\mathbf{g}_{t}\|^{2}$.
Also note that $\eta<\frac{2}{\kappa\sqrt{\alpha}}$ and hence
$\left(1-\frac{\sqrt{\alpha}\eta\kappa}{2}\right)>0$. We get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{t+1})$
$\displaystyle\leq\mathbb{E}\left(f(\mathbf{x}_{t})-\frac{\eta}{\lambda\sqrt{\alpha}}\left(1-\frac{\eta\kappa\sqrt{\alpha}}{2}\right)\|\mathbf{g}_{t}\|^{2}+\eta^{2}\frac{\lambda}{2}\mathbb{E}_{t}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}\right)\|^{2}\right).$

Now use
$\mathbb{E}_{t}\|\hat{\mathbf{H}}^{-1}\left(\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}\right)\|^{2}\leq\frac{\alpha}{\mu^{2}}\mathbb{E}_{t}\|\bar{\mathbf{g}}_{t}-\mathbf{g}_{t}\|^{2}\leq\frac{8\lambda\alpha}{b\mu^{2}}\left(f(\mathbf{x}_{t})-f(\mathbf{x}^{*})+f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right)$.
Also due to $\mu$-strong convexity of $f$ we have $\|\mathbf{g}_{t}\|^{2}\geq
2\mu\left(f(\mathbf{x}_{t})-f(\mathbf{x}^{*})\right)$. We get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{t+1})$
$\displaystyle\leq\mathbb{E}\left(f(\mathbf{x}_{t})-\frac{2\eta\mu}{\lambda\sqrt{\alpha}}\left(1-\frac{\eta\kappa\sqrt{\alpha}}{2}\right)\left(f(\mathbf{x}_{t})-f(\mathbf{x}^{*})\right)+\eta^{2}\frac{4\lambda^{2}\alpha}{b\mu^{2}}\left(f(\mathbf{x}_{t})-f(\mathbf{x}^{*})+f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right)\right)$
$\displaystyle=\mathbb{E}\left(f(\mathbf{x}_{t})-\frac{2\eta}{\kappa\sqrt{\alpha}}\left(1-\frac{\eta\kappa\sqrt{\alpha}}{2}\right)\left(f(\mathbf{x}_{t})-f(\mathbf{x}^{*})\right)+\eta^{2}\frac{4\kappa^{2}\alpha}{b}\left(f(\mathbf{x}_{t})-f(\mathbf{x}^{*})+f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right)\right).$

Subtract $f(\mathbf{x}^{*})$ from both sides we get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{t+1})-f(\mathbf{x}^{*})$
$\displaystyle\leq\mathbb{E}\left(\underbrace{\left[1-\frac{2\eta}{\kappa\sqrt{\alpha}}+\eta^{2}\left(1+\frac{4\kappa^{2}\alpha}{b}\right)\right]}_{\boldsymbol{\xi}}\cdot\left(f(\mathbf{x}_{t})-f(\mathbf{x}^{*})\right)+\eta^{2}\frac{4\kappa^{2}\alpha}{b}\cdot\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right)\right)$
$\displaystyle=\boldsymbol{\xi}\cdot\mathbb{E}\left(f(\mathbf{x}_{t})-f(\mathbf{x}^{*})\right)+\eta^{2}\frac{4\kappa^{2}\alpha}{b}\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right)$
(26)

We denote
$\boldsymbol{\xi}:=1-\frac{2\eta}{\kappa\sqrt{\alpha}}+\eta^{2}\left(1+\frac{4\kappa^{2}\alpha}{b}\right)$.
Since we perform $\frac{n}{b}$ inner iterations before updating
$\tilde{\mathbf{x}}$, we recursively unfold the relation (26) for
$\frac{n}{b}$ times. This provides the following,

$\displaystyle\mathbb{E}f(\mathbf{x}_{n/b})-f(\mathbf{x}^{*})$
$\displaystyle\leq\boldsymbol{\xi}^{\frac{n}{b}}\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right)+\eta^{2}\frac{4\kappa^{2}\alpha}{b}\cdot\left(1+\boldsymbol{\xi}+\boldsymbol{\xi}^{2}+\cdots+\boldsymbol{\xi}^{\frac{n}{b}-1}\right)\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right)$
$\displaystyle=\boldsymbol{\xi}^{\frac{n}{b}}\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right)+\eta^{2}\frac{4\kappa^{2}\alpha}{b}\cdot\left(\frac{1-\boldsymbol{\xi}^{\frac{n}{b}}}{1-\boldsymbol{\xi}}\right)\cdot\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right).$

For $\eta<\frac{b}{5\kappa^{3}\alpha^{3/2}}$, we have
$\boldsymbol{\xi}<1-\frac{\eta}{\kappa\sqrt{\alpha}}$. Substituting upper
bound for $\boldsymbol{\xi}$ we get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{n/b})-f(\mathbf{x}^{*})\leq\left[\left(1-\frac{\eta}{\kappa\sqrt{\alpha}}\right)^{n/b}+\frac{4\eta\kappa^{3}\alpha^{3/2}}{b}\cdot\left(1-\left(1-\frac{\eta}{\kappa\sqrt{\alpha}}\right)^{n/b}\right)\right]\cdot\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right).$

Let $\eta<\frac{b}{8\kappa^{3}\alpha^{3/2}}$, we get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{n/b})-f(\mathbf{x}^{*})\leq\left(\frac{1}{2}\left(1-\frac{\eta}{\kappa\sqrt{\alpha}}\right)^{\frac{n}{b}}+\frac{1}{2}\right)\cdot\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right).$

Now if
$\left(1-\frac{\eta}{\kappa\sqrt{\alpha}}\right)^{\frac{n}{b}}<\frac{1}{2}$ we
get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{n/b})-f(\mathbf{x}^{*})<\frac{3}{4}\cdot\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right),$

otherwise if
$\left(1-\frac{\eta}{\kappa\sqrt{\alpha}}\right)^{\frac{n}{b}}\geq\frac{1}{2}$,
then we consider two cases based on the value of $\eta$. If
$\eta=\frac{2}{\kappa\sqrt{\alpha}}$ then we get,

$\displaystyle\mathbb{E}f(\mathbf{x}_{n/b})-f(\mathbf{x}^{*})<\left(1-\frac{1}{\alpha\kappa^{2}}\right)\cdot\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right),$

and if $\eta=\frac{b}{8\kappa^{3}\alpha^{3/2}}$ then,

$\displaystyle\mathbb{E}f(\mathbf{x}_{n/b})-f(\mathbf{x}^{*})<\left(1-\frac{b}{16\kappa^{4}\alpha^{2}}\right)\cdot\left(f(\tilde{\mathbf{x}})-f(\mathbf{x}^{*})\right).$

This completes the proof.

##  Appendix B Additional Plots for Synthetic Data Experiments

For designing synthetic data, we considered two random orthonormal matrices
$\mathbf{U}\in\mathbb{R}^{n\times d}$ and $\mathbf{V}\in\mathbb{R}^{d\times
d}$ and constructed a matrix
$\mathbf{A}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{\scriptscriptstyle{\top}}$
where $\mathbf{\Sigma}\in\mathbb{R}^{d\times d}$ is a diagonal matrix with
large condition number $(\kappa\approx 6000)$. In $\mathbf{\Sigma}$, we set
$(n-1)$ diagonal entries to be $n$ and only one diagonal entry as
$n\cdot\kappa$, artificially enforcing a very large condition number. For
constructing labels, we set
$b_{i}=\text{sign}(\mathbf{a}_{i}^{\scriptscriptstyle{\top}}\mathbf{x})$ where
$\mathbf{a}_{i}$ is the $i^{th}$ row of $\mathbf{A}$ and
$\mathbf{x}\in\mathbb{R}^{d}$ is a random vector drawn from an isotropic
multivariate Gaussian distribution. Also, we fix the regularization parameter
$\mu$ to $10^{-2}$.

Our empirical results for the synthetic data are presented in Figure 6. In
plot 6a, we observe the sensitivity of the peformance of SVRG to the gradient
mini-batch size, and step size. Unlike SVRG, we empirically observe the
robustness of Mb-SVRN to gradient mini-batch size. This is demonstrated
clearly in our plots by an almost flat-line alignment of red dots for
$b=10k,40k,$ and $100k$. Note that the optimal step size for SVRG is close to
$2^{-6}$ for all gradient mini-batch sizes, and the slight deviation from this
optimal step size leads to rapid deterioration in the convergence rate. On the
other hand, in Figures 6b-6d, we see the convergence rate curves for different
gradient mini-batch sizes exhibit a flatter behavior near their respective
optimal step sizes, demonstrating the robustness of Mb-SVRN to the step size.
In addition to robustness, the plots also demonstrate the faster overall
convergence rate of Mb-SVRN.

|
---|---
|

(a) SVRG

(b) Mb-SVRN, with $h=200$

(c) Mb-SVRN, with $h=1k$

(d) Mb-SVRN, with $h=10k$

Figure 6: Experiments on the synthetic logistic regression task. The red dots
on every curve mark the respective optimal convergence rate attained at the
optimal step size. The plot a) demonstrate the performance of SVRG for
different gradient mini-batch sizes, whereas plots b) c) and d) demonstrate
the performance of Mb-SVRN with different gradient and Hessian mini-batch
sizes.