#  Effects of exact kinematics and the Sudakov
form factor on the dipole amplitude

T. Goda,1, K. Kutak1 and S. Sapeta1

1 The H. Niewodniczański Institute of Nuclear Physics PAN,
Radzikowskiego 152, 31-342 Kraków, Poland

###### Abstract

We investigate effects of exact gluon kinematics on the parameters of the
Golec-Biernat–Wüsthoff, and Bartels–Golec-Biernat–Kowalski saturation models.
The resulting fits show some differences, particularly, in the normalization
of the dipole cross section $\sigma_{0}$. The refitted models are used for the
dijet production process in DIS to investigate effects of the Sudakov form
factor at Electron Ion Collider energies.

IFJPAN-IV-2023-2

##  1 Introduction

$q$$x$, $\kappa$$x/z$, $k$

Figure 1: Kinematic variables of the inclusive DIS.

Factorization of scales plays central role in Quantum Chromodynamics (QCD). In
particular, within collinear factorization approach, long-distance effects can
be isolated into objects called collinear parton distribution functions (PDFs)
[1, 2]. The collinear PDFs are not fully perturbatively calculable but they
obey perturbative evolution equations and are process-independent [2]. That is
to say that one needs to determine initial condition by fit to data, and the
obtained PDFs can be used universally.

One of the main type of processes that is suited particularly well to the
studies of proton structure is the deep inelastic scattering (DIS) [1, 2]. The
data from HERA has enabled us to answer many questions in that domain and
largely improved our picture of interior of a proton. It is therefore very
exiting that the next generation of DIS machines, the Election Ion Collider
(EIC) [3] is making its way and will soon allow us to uncover more details
about hadron structure.

At large center of mass energy and fixed value of the photon virtuality, $Q$,
one probes the region of small Bjorken scaling variable, $x$. In that region,
which is dominated by gluons [1, 2, 4], the approach of High Energy
Factorization, also called $k_{T}$-factorization [5, 6, 7, 8, 9, 10], has
proven to be suited particularly well. In this framework, the interaction of
the photon with gluons happens through boson-gluon fusion or quark impact
factor, as depicted in Fig. 1. An object, which is central to the description
of such process is called the dipole gluon density,
$\mathcal{F}^{\mathrm{dipole}}(x,k_{T}^{2})$, and it is a type of transverse
momentum dependent (TMD) [11, 12, 13, 14] PDF. Due to its clear picture, one
often uses a position-space counterpart of the gluon density, called dipole
cross section. It appears in the $k_{T}$-factorization formula [6, 12, 13,
14], which we shall discuss in the next section, and plays a role similar to
that of the collinear parton distribution function in the collinear
factorization.

Much effort has been made to study this object, notably the work of Balitsky,
Fadin, Kuraev and Lipatov (BFKL) [15, 16] has predicted sharp rise of cross
sections with $1/x$. However, such rise violates unitarity and the Froissart
bound [1, 2, 4, 17]. In Ref. [18] it has been recognized that gluon
recombination can tame the growth of gluons. The interplay of linear and
nonlinear terms leads to a phenomenon called gluon saturation and
corresponding evolution equations are known as Balitsky-Kovchegov (BK) [19,
20] equation or JIMWLK[21, 22, 23, 24, 25, 26] equation.

Despite the success of the aforementioned evolution equations, there exist a
number of phenomenological models of the dipole cross section, which are
popular due to to their simplicity. A notable example is the model proposed by
Golec-Biernat and Wus̈thoff (GBW) [27] and its extension by Bartels, Golec-
Biernat and Kowalski (BGK) [28]. We will use these two specific models in our
study. We note, however, that other saturation models have been discussed in
the literature [29, 30, 31, 32].

In the present study, we fit the GBW and BGK models to HERA data [33] in the
$k_{T}$-factorization formula for $F_{2}$, instead of the original dipole
factorization [34] formula. As discussed below, this allows us to relatively
easy investigate the role of the exact gluon kinematics. Since inclusive
observables do not reveal the $k_{T}$ dependence well, in order to demonstrate
the effects of exact kinematics we consider a less inclusive observable which
is more sensitive to the $k_{T}$ dependence of the gluon distribution. The
resulting models are used for predictions of dijet production in DIS at
Electron Ion Collider (EIC) [35], including effects of the Sudakov form
factor, which resums logarithms of the small transverse momentum. This process
is known to be sensitive to another type of gluon density, called Weitzsäcker-
Williams (WW) [12, 13]. For that process, we study, in particular, the angular
correlation of jets, and that of the scattered electron and the jets. For
other papers addressing dijet production at the EIC we refer the Reader to
[36, 37, 38, 39, 40].

The paper is organized as follows. In the next section, we present theoretical
framework which we use and outline differences between the
$k_{T}$-factorization and the dipole factorization. In Sec. 3, results of the
new fits to HERA data in the $k_{T}$-factorization formula are presented. In
Sec. 4, we apply the results from the previous section to compute
distributions for dijet production in DIS at the EIC and make comparisons to
our earlier study of Ref. [41].

##  2 The framework

The DIS cross section (structure function) factorises and can be written in a
form

$d\sigma=\sum_{a}\phi_{a/h}\otimes H_{\gamma a\to X},$ (1)

where the hard function, $H_{\gamma a\to X}$, involving a parton $a$ in the
initial state, is calculable perturbatively, and $\phi_{a/h}$ is the parton
distribution function of $a$ in a hadron $h$. The symbol $\otimes$ denotes
appropriate convolution. All nonperturbative effects are absorbed in
$\phi_{a/h}$, while $H_{\gamma a\to X}$ can be computed order by order in
$\alpha_{s}$.

We will study factorization in two versions: defined in momentum and position
space, respectively. As mentioned earlier, in the fit, we use the GBW and BGK
models. The models were originally formulated in the position-space version of
the $k_{T}$-factorization formula. In the GBW model, the dipole cross section
has the form [27]

$\displaystyle\sigma_{\mathrm{GBW}}(x,r)$
$\displaystyle=\sigma_{0}\left(1-e^{-r^{2}/R^{2}_{0}}\right)$
$\displaystyle\mathrm{where}$ $\displaystyle R^{-2}_{0}$
$\displaystyle=\frac{Q_{0}^{2}}{4}\left(\frac{x_{0}}{x}\right)^{\lambda}.$ (2)

The dipole cross section is related to the dipole gluon density by the Fourier
transform

$\alpha_{s}\mathcal{F}^{\mathrm{dipole}}(x,k_{T}^{2})=\frac{N_{c}}{4\pi}\int\frac{d^{2}\mathbf{r}}{(2\pi)^{2}}e^{i\mathbf{k}_{t}\cdot\mathbf{r}}\nabla_{\mathbf{r}}^{2}\sigma_{\mathrm{dipole}}(x,r).$
(3)

The essence of the GBW model is encoded in the $x$-dependent saturation scale
$Q_{s}^{2}(x)=Q^{2}_{0}(x_{0}/x)^{-\lambda}$, which separates the saturation
region and the scaling region. An extension of the above model was proposed by
Bartels, Golec-Biernat and Kowalski [28] who incorporated the DGLAP evolution
in the GBW dipole cross section (2) by modifying the exponent to

$R_{0}^{-2}=\frac{\pi^{2}\alpha(\mu^{2})xg(x,\mu^{2})}{3\sigma_{0}},$ (4)

where

$\displaystyle\mu^{2}$ $\displaystyle=\frac{C}{r^{2}}+\mu_{0}^{2}$
$\displaystyle\mathrm{and}$ $\displaystyle xg(x,Q^{2}_{0})$
$\displaystyle=A_{g}x^{-\lambda_{g}}(1-x)^{5.6},$ (5)

thus improving the description of data at higher $Q^{2}$.

While these models enjoyed much success in the phenomenology of DIS, including
the diffractive and photo-production processes [27, 42], it is worth
mentioning that they both use certain kinematic approximation which is
specific to LO dipole factorization and are not there in the
$k_{T}$-factorization. This was recognized in [43]. In the following, we will
investigate effects of these approximations.

Firstly, let us outline the factorization formulas. With $q^{\prime}\equiv
q+xp$, one decomposes $k$ and $\kappa$, defined in Fig. 1, as

$\displaystyle\kappa$ $\displaystyle=\alpha p-\beta q^{\prime}+\kappa_{t}$
$\displaystyle\mathrm{and}$ $\displaystyle k$ $\displaystyle=ap-
bq^{\prime}+k_{T}.$ (6)

For the contribution depicted in Fig. 1, the structure function $F_{2}$
factorizes with a change of variable
${\boldsymbol{\kappa}^{\prime}}_{t}\equiv{\boldsymbol{\kappa}_{t}}-(1-\beta)\mathbf{k}_{t}$,
to the form [44, 45]

$F_{2}(x,Q^{2})=\sum_{f}e_{f}^{2}\frac{Q^{2}}{2\pi}\int\frac{dk^{2}_{t}}{k_{T}^{2}}\int^{1}_{0}d\beta\int
d{\kappa^{\prime}}_{t}\alpha_{s}(\mu^{2})\mathcal{F}^{\mathrm{dipole}}(x/z,k_{T}^{2})\Theta(1-x/z)\\\
\left[\left(\beta^{2}+(1-\beta)^{2}\right)\left(\frac{I_{1}}{2\pi}-\frac{I_{2}}{2\pi}\right)+\left(m_{f}^{2}+4Q^{2}\beta^{2}(1-\beta)^{2}\right)\left(\frac{I_{3}}{2\pi}-\frac{I_{4}}{2\pi}\right)\right],$
(7)

where

$\frac{1}{z}=1+\frac{{\kappa^{\prime}}^{2}_{t}+m_{f}^{2}}{\beta(1-\beta)Q^{2}}+\frac{k^{2}_{t}}{Q^{2}}\,.$
(8)

One should note that the gluons are not probed directly, thus the argument of
the gluon density is $x/z$ rather than $x$. If one, instead, uses
$1/z=1+4m_{f}^{2}/Q^{2}$ and assumes that $\mu$ is independent of
$\kappa^{\prime}$ and $k_{T}$, the above formula can be written in the impact
parameter space [27, 46] (i.e. as a dipole factorization formula)

$F_{2}\left(x,Q^{2}\right)=\sum_{f}e_{f}^{2}\frac{Q^{2}}{4\pi^{2}}\int^{1}_{0}d\beta\int\frac{d^{2}\mathbf{r}}{(2\pi)^{2}}\left|\Psi\left(\tilde{x},\beta,Q^{2}\right)\right|^{2}\sigma_{\mathrm{dipole}}\left(\tilde{x},r\right),$
(9)

where the photon wave function,
$\left|\Psi\left(x,\beta,Q^{2}\right)\right|^{2}$, describes splitting of the
incoming photon into a $q\overline{q}$ pair with light-cone momenta fractions
$\beta$ and $1-\beta$ respectively, and the dipole cross section,
$\sigma_{\mathrm{dipole}}\left(\tilde{x},r\right)$, describes the interaction
of the colour dipole of size $r$ with the proton. In Ref. [27],
$\tilde{x}=x(1+4m_{f}/Q^{2})$ with nonzero value of $m_{f}$ was used even for
light flavours, which is necessary to partially substitute $k_{T}$ and
${\kappa^{\prime}}_{T}$ in Eq. (8). In the present study, we use Eq. (7) to
fit the models, where the gluon density is obtained by evaluating Eq. (3).

Considering the small-$r$ limit of the GBW model

$\left.\sigma_{\mathrm{GBW}}\left(x,r\right)\right|_{r\ll Q_{s}}\approx
r^{2}Q_{s}^{2}/4\,,$ (10)

comparison to

$\left.\sigma_{\mathrm{dipole}}\left(x,r\right)\right|_{r\ll
Q_{s}}\approx\frac{r^{2}\pi^{2}\alpha_{s}(\mu^{2})xg(x,\mu^{2})}{3}\,,$ (11)

from BGK, suggests that the GBW model is an approximation in which
$\alpha_{s}(\mu^{2})$ and $xg(x,\mu^{2})$ are independent of $r$ [28]. For
this reason, in one version of the model discussed below, we shall account for
the running coupling in Eq. (7) by assuming that $\alpha_{s}$ in Eq. (3) is
constant for the GBW model, and thus explicitly multiply it by the running
coupling

$\alpha_{s}(\mu^{2})\mathcal{F}^{\mathrm{dipole}}(x,k_{T}^{2})\rightarrow\alpha_{s}(\mu)\frac{\alpha_{s}\mathcal{F}^{\mathrm{dipole}}(x,k_{T}^{2})}{0.2},$
(12)

where

$\alpha_{s}(\mu^{2})=\frac{1}{\frac{11C_{A}-2n_{f}}{12\pi}\log\left(\frac{\mu^{2}}{\Lambda_{\mathrm{QCD}}^{2}}\right)},$
(13)

and $\Lambda_{\mathrm{QCD}}^{2}=0.09\;\mathrm{GeV}^{2}$.

The factor 0.2 is an arbitrary normalization, whose effect is absorbed by
$\sigma_{0}$, and hence bears no importance. (For a more detailed analysis of
the dipole gluon density from the BGK model see Ref. [47]111In Ref. [47], the
coupling constant was treated differently and the large $k_{T}$-region was
matched to the derivative of $xg(x)$, while we have used series transformation
to accelerate the high-$k_{T}$ integration. Their treatment is also different
from the original BGK paper [28]. As a consequence, their gluon density is
positive in the large $k_{T}$-region and the overall large-$k_{T}$ behaviour
is significantly different.) While there is some ambiguity on what the
argument of $\alpha_{s}(\mu^{2})$ should be, we follow Ref. [45] and use

$\mu^{2}=k_{T}^{2}+{\kappa^{\prime}}_{t}^{2}+m_{f}^{2}+\mu^{2}_{0},$ (14)

where we add $\mu^{2}_{0}=4\;\mathrm{GeV}^{2}$ in order to freeze the coupling
at low scales.

##  3 Fits to $F_{2}$ data

We fitted the GBW and BGK models to the $F_{2}$ data from HERA [33]222For
description of $F_{2}$ at NLO accuracy within dipole factorisation see Ref.
[48] . A numerical program was written with help of CERNLIB (DPSIPG) [49], GSL
[50], CUBA [51] and ROOT [52] libraries to evaluate $F_{2}$. The fitting was
performed using MnMigrad and MnSimplex of ROOT::Minuit2 [53].

The data were selected to be in the range $0.045\leq Q^{2}\leq
650\;\mathrm{GeV^{2}}$, $x<0.01$. As with the previous fit [54], the $c$ and
$b$ flavours were taken into account with the mass $1.3$ and
$4.6\;\mathrm{GeV}$, respectively. As discussed earlier, we take the light
quarks to be massless.

We studied the following cases

  * •

GBW model with the fixed coupling in $k_{T}$-factorization ($k_{T}$-GBW),

  * •

GBW model with the running coupling in $k_{T}$-factorization (rc-$k_{T}$-GBW),

  * •

BGK model in $k_{T}$-factorization ($k_{T}$-BGK),

and, as a reference, we provide the following results from Ref. [54]

  * •

GBW model with massless light quarks in the dipole factorization ($r$-GBW),

  * •

GBW model with massive light quarks in the dipole factorization ($r$-GBW-
massive),

  * •

BGK model in the dipole factorization ($r$-BGK).

- |  $\sigma_{0}$ [mb] | $x_{0}\left(10^{-4}\right)$ | $\lambda$ | $\chi^{2}/\mathrm{dof}$
---|---|---|---|---
$r$-GBW | 1.907e+01 | 2.582e+00 | 3.219e-01 | 4.438e+00
$r$-GBW-massive | 2.384e+01 | 1.117e+00 | 3.082e-01 | 5.274e+00
$k_{t}$-GBW | 3.344e+01 | 1.333e+00 | 3.258e-01 | 4.396e+00
rc-$k_{t}$-GBW | 1.520e+01 | 2.648e+00 | 3.211e-01 | 2.447e+00

- |  $\sigma_{0}$ [mb] | $A_{g}$ | $\lambda_{g}$ | $C$ | $\mu_{0}^{2}\left[\mathrm{GeV^{2}}\right]$ | $\chi^{2}/\mathrm{dof}$
---|---|---|---|---|---|---
$r$-BGK | 2.326e+01 | 1.181e+00 | 8.317e-02 | 3.294e-01 | 1.873e+00 | 1.556e+00
$k_{t}$-BGK | 3.470e+01 | 1.048e+00 | 2.205e-01 | 2.391e-01 | 9.954e-01 | 1.527e+00

Table 1: Fit parameters of respective models. The parameters of the dipole
factorization cases are from Ref. [54].

The results of the fits are summarized in Tab. 1. The fit quality of
$k_{T}$-GBW is almost unchanged w.r.t. $r$-GBW, while rc-$k_{T}$-GBW shows
remarkable improvement, almost halving the $\chi^{2}$ value. This is in line
with the observation made in Refs. [55, 56, 57] that in the BK evolution, the
running coupling corrections have considerable effect.

Another notable point is that, except for the normalization $\sigma_{0}$, the
parameters are very similar, particularly those of rc-$k_{T}$-GBW are almost
identical to those of $r$-GBW. While the GBW model remains almost unaffected,
the BGK model seems to show slightly more change. The difference in
$\sigma_{0}$ is similar to that of GBW and other parameters changed
moderately.

In Fig. 2, the plots of the dipole cross section, the dipole gluon density and
the saturation scale for the GBW and BGK models are shown. The dipole cross
section and the gluon density are both normalized by $\sigma_{0}$ in order to
show the effects of other parameters better, and the saturation scale is
defined as a ridge of the dipole gluon density in the $(x,k_{T}^{2})$ plane.

In the plots on the left hand side, one can see the effects of changes in the
parameters of the GBW model. The difference between the rc-$k_{T}$-GBW and
$r$-GBW is negligible, while the $k_{T}$-GBW is slightly shifted, compared to
others, by the change in $x_{0}$.

On th right hand side of Fig. 2, the same plots are shown for the BGK model.
Unlike in the GBW case, the connection between the differences shown in the
plots and the differences in the parameters is less clear. Nevertheless, the
differences are more prominent in the small-$x$ region.

Comparisons of the results with the $F_{2}$ data are shown in Figs. 3 and 4.
In Fig. 3, the differences between $r$-GBW and $k_{T}$-GBW are not visible,
while rc-$k_{T}$-GBW shows sizable difference from the others, particularly in
the large-$Q^{2}$ region. Recalling that the parameters of rc-$k_{T}$-GBW and
$r$-GBW are very similar, the difference in $F_{2}$ is almost entirely due to
the coupling constant. The improvement in the fit quality is depicted as a
histogram of the $\chi^{2}$-vale per number of points at the bottom of Fig. 3.
Here, the improvement in the large-$Q^{2}$ region is very clear. In Fig. 4,
the differences between $r$-BGK and $k_{T}$-BGK are hardly visible. In the
histogram at the bottom, one can see some differences, but they cancel out
mostly, making little improvement overall.

(a) GBW

(b) BGK

Figure 2: The dipole cross section, the dipole gluon density at
$x=10^{-2},\;10^{-6}$, and the saturation scale for the GBW (left) and the BGK
(right) models. Note that the dipole cross section and the gluon density are
normalized with $\sigma_{0}^{-1}$.

Figure 3: Comparison of $F_{2}$ fro GBW with HERA data. The histogram shows
the $\chi^{2}$ value per data point in each frame. Improvement by the running
coupling (rc-$k_{T}$) is clearly visible in the high-$Q^{2}$region, while the
new fit ($k_{T}$) shows only marginal improvement.

Figure 4: Comparison of $F_{2}$ from BGK with HERA data. The histogram shows
the $\chi^{2}$ value per data point in each frame. The overall fit quality
remain similar but the quality in each frame changes noticeably. In
particularly, the $k_{T}$-formula ($k_{T}$) shows better quality at small
$Q^{2}$.

Let us now take a closer look at the parameter $\sigma_{0}$. Recall that the
difference between the $k_{T}$-factorization formula and the dipole
factorization formula is in $x/z$, and this enters in the GBW formalism as
$\tilde{x}$. It is easy to see that, as $x$ grows, the dipole cross section
gets suppressed. (Keeping in mind the suppression by the photon wave function
in the large-$r$ region.) Such effect was discussed previously in Ref. [58] in
the context of the BK equation. In fact, this suppression is the motivation
given in Ref. [27] for such modification of $x$, so that, in the small-$Q^{2}$
limit, the total cross section remains finite. Since

$\left(1+\frac{4m_{f}^{2}}{Q^{2}}\right)\leq\left(1+\frac{k_{T}^{2}}{Q^{2}}+\frac{{\kappa^{\prime}}_{t}^{2}+m_{f}^{2}}{\beta(1-\beta)Q^{2}}\right),$
(15)

the $k_{T}$-factorization case receives more suppression. Consequently, the
normalization factor $\sigma_{0}$ rises to compensate the suppression.
Therefore, one can understand the change in $\sigma_{0}$ as a direct
consequence of the key difference between the $k_{T}$-factorization and the
dipole factorization.

##  4 Dijet production at EIC

Figure 5:  Weizsäcker-Williams gluon density at $x=10^{-3}$. Top row:
comparison of the dipole factorization fit and $k_{T}$-factorization fit
results. Bottom row: comparison of the respective models with and without the
Sudakov factor, at $\mu=17,\;67\ \mathrm{GeV}$.The green dotted line is the KS
gluon [41], and the green dashed line is the rcBK gluon [59].

Figure 6:  Azimuthal correlation of the jets and the scattered electron in the
Breit frame. Top: Comparison of dipole factorization fit and
$k_{T}$-factorization fit. Middle & Bottom: Effect of the Sudakov form factor.
The green dotted line is the KS gluon [41], and the green dashed line is the
rcBK gluon [59].

Figure 7: Azimuthal correlation of the jets and the scattered electron in the
lab frame. Top: Comparison of dipole factorization fit and
$k_{T}$-factorization fit. Middle & Bottom: Effect of the Sudakov form factor.
The green dotted line is the KS gluon [41], and the green dashed line is the
rcBK gluon [59].

Figure 8: Azimuthal correlation of the jets in the Breit frame. Top:
Comparison of dipole factorization fit and $k_{T}$-factorization fit. Middle &
Bottom: Effect of the Sudakov form factor. The green dotted line is the KS
gluon [41], and the green dashed line is the rcBK gluon [59].

The $F_{2}$ structure function is an inclusive object and it is weakly
sensitive to the shape of the gluon density. To probe that shape better, we
shall now apply the dipole cross sections obtained in the previous section to
the jet correlations at the EIC, following closely the method of Ref. [41].

We consider dijet production in DIS

$e+p\rightarrow e+J_{1}+J_{2}+X\,.$ (16)

At the leading order, in the small-$x$ limit, this process is dominated by
$q\overline{q}$ jets [13]. It is therefore closely related to the dipole
picture we discussed earlier. In the Breit frame, where the photon momentum is
given by $q=(0,0,0,Q)$, at the leading order, the jets momentum imbalance
$p_{T}\equiv\left|p_{1T}+p_{2T}\right|$ equals the gluon transverse momentum
$k_{T}$, where $p_{1T}$ and $p_{2T}$ are transverse momenta of the jets. This
makes dijets an interesting process. For the region where $p_{T}\ll P_{T}\sim
p_{1T},\,p_{2T}$, one may use power counting to take leading order in
$p_{T}/P_{T}$, which leads to the transverse-momentum-dependent (TMD)
factorization.

In the large-$N_{c}$ limit of the TMD factorization, there are two types of
gluon densities, namely the dipole gluon density and the Weizsäcker-Williams
(WW) gluon density [12, 13, 60, 61]. It was shown in Ref. [13], that the dijet
process in DIS can directly probe the WW gluon,
$\mathcal{F}^{\mathrm{WW}}(x,k^{2})$, where the differential cross section
factorizes as

$\frac{d\sigma^{\gamma^{*}p\rightarrow
q\overline{q}X}}{d\mathrm{P.S.}}=\mathcal{F}^{\mathrm{WW}}(x,k_{T}^{2})H_{\gamma^{*}g^{*}\rightarrow
q\overline{q}},$ (17)

with the hard function $H_{\gamma^{*}g^{*}\rightarrow q\overline{q}}$
describing interactions of an off-shell photon with an off-shell gluon
producing a $q\overline{q}$ pair. $\mathcal{F}^{\mathrm{WW}}(x,k^{2})$, has an
interpretation as a number density of gluons inside a proton, while
$\mathcal{F}^{\mathrm{dipole}}(x,k_{T}^{2})$ does not have such an
interpretation [12, 13, 61].

We carry out our study in the framework of the Improved Transverse-Momentum-
Dependent (ITMD) factorization [62, 60]333We limit ourselves to unpolarized
contribution as it is the lading one. For the polarized one, see Ref. [38].
This is implemented in the program KaTie [63], which we use to compute the
cross sections. The ITMD factorization is a generalization of the TMD
factorization, where the momentum imbalance in TMD is restricted to be small
[62, 60]. That is to say, ITMD resums $(Q_{s}/k_{T})^{n}$ and
$(k_{T}/P_{T})^{n}$ [62, 60], thus extends the region of applicability up to
$k_{T}\sim P_{T}$. The difference of Eq. (17) from the regular TMD is that the
hard function has an off-shell gluon, $g^{*}$, thus rendering the $k_{T}$
dependence in the hard function as well [62].

Under the Gaussian approximation and assuming $\theta$-like profile of the
proton, one can write [60, 61, 12, 13]

$\mathcal{F}^{\mathrm{WW}}(x,k^{2})=\frac{C_{F}}{2\pi^{3}\alpha_{s}}\int^{\infty}_{0}\frac{dr}{r}J_{0}(rk)\sigma_{\mathrm{dipole\,Adj.}}(x,r),$
(18)

with the adjoint dipole cross section

$\sigma_{\mathrm{dipole\,Adj.}}(x,r)=\sigma_{0}\left(1-\left(1-\frac{\sigma_{\mathrm{dipole}}(x,r)}{\sigma_{0}}\right)^{C_{A}/C_{F}}\right).$
(19)

For the region where the TMD factorization is applicable, $Q_{s}\sim k_{T}\ll
P_{T}\sim Q$, one needs to resum the large Sudakov logarithms $\log(k_{T}/Q)$,
as well as $\log(1/x)$ [13]. It was shown in Refs. [64, 65, 14] that
consistent resummation of such logarithms is possible owing to the separation
of corresponding regions (see also Ref. [66]). Resummation of the Sudakov
logarithms is achieved by the formula

$\mathcal{F}^{\mathrm{WW}}(x,k^{2},\mu^{2})=\frac{C_{F}}{2\pi^{3}\alpha_{s}}\int^{\infty}_{0}\frac{dr}{r}J_{0}(rk)e^{-S(r,\mu^{2})}\sigma_{\mathrm{dipole\,Adj.}}(x,r),$
(20)

where, we use the Sudakov form factor [65, 14],

$S(r,\mu^{2})=\frac{\alpha_{s}N_{c}}{4\pi}\ln^{2}\left(\frac{\mu^{2}r^{2}}{4e^{-2\gamma_{E}}}\right),$
(21)

in which $\gamma_{E}$ is the Euler-Mascheroni constant, and we set
$\alpha_{s}=0.2$. Following Ref. [41], we study the azimuthal correlations of
jets and the final state electron in DIS , where it was argued that this
observable is sensitive to the soft emissions and the saturation effects. In
this study we focus only on the proton case. The kinematical cuts suggested in
Ref. [41] are

$\displaystyle E_{e}$ $\displaystyle=15\mathrm{GeV}$ $\displaystyle E_{p}$
$\displaystyle=135\mathrm{GeV}$ $\displaystyle Q^{2}$
$\displaystyle>1\mathrm{GeV}^{2}$ $\displaystyle 0.1<$ $\displaystyle\nu<0.85$
$\displaystyle\Delta R_{\mathrm{Breit}}$ $\displaystyle<1$ $\displaystyle
p^{\mathrm{Breit}}_{1,2\,T}$ $\displaystyle>3\mathrm{GeV}$ $\displaystyle-4<$
$\displaystyle y_{1,2\,\mathrm{lab}}<-1.$

Grids of the Weizsäcker-Williams gluon density were produced by evaluating
Eqs. (19) and (20). The gluon density at $x=10^{-3}$ is plotted in Fig. 5 with
the hard-scale-independent Kutak-Sapeta (KS) gluon [67, 41, 68] and the
running-coupling BK (rcBK) gluon density [69, 57, 59]. Both of these gluon
densities are solutions of evolution equations and treat better perturbative
tail at large $k_{T}$. Furthermore, the KS gluon takes into account resummed
corrections of higher orders, i.e. kinematical constraint and nonsingular (at
low $z$) elements of DGLAP splitting functions [45].

Clearly, as shown in Fig. 5, the GBW and BGK models fall much more quickly
than the KS and rcBK gluon densities. In general, expected behaviour in the
large-$k_{T}$ region is $\sim k_{T}^{-2}$ [12, 13], while
$\sigma_{\mathrm{GBW}}$ behaves like $\sim e^{-k_{T}^{2}}$. As in Ref. [41],
the Sudakov factor enhances in the small-$k_{T}$ region and suppresses in the
large-$k_{T}$ region. In other words, it broadens the distribution. In
comparison to the result of Ref. [41], the effect of broadening by the Sudakov
factor is significantly more pronounced in the case of the GBW and BGK models.
The hard-scale-dependent GBW and BGK models, as a consequence, become closer
to the KS and rcBK gluons, cf. Fig. 2 of Ref. [41].

Figs. 6 and 7 show electron-jets azimuthal correlation in the Breit and in the
Lab frame, respectively. In the top row of Fig. 6, we see, for both the GBW
and BGK models, better agreements of results with the KS and rcBK for the new
$k_{T}$-factorization fits. However the overall normalization of the gluon
density depends on the coupling $\alpha_{s}$, which we assumed to be 0.2.
Nevertheless, it shows clearly the effect of the parameter $\sigma_{0}$. In
the middle and the bottom row, it shows the effects of the Sudakov form
factor, which qualitatively agrees with that of Ref. [41], by lowering the
cross section.

Fig. 7 shows the electron-jets correlation in the Lab frame. Here, the
difference between KS and GBW and BGK is more prominent, while rcBK shows
similar pattern to the GBW and BGK models and therefore we can attribute the
effects to importance of higher order corrections, as are accounted for in KS
gluon. The effects of the Sudakov factor are similar to those in Ref. [41] at
relatively high $\Delta\phi$, while at smaller $\Delta\phi$, the effect is
reversed (i.e. the cross sections were slightly lowered in Ref. [41], while
here, they are significantly increased).

Finally, Fig. 8 shows the jet-jet correlations in the Breit frame. Again, the
GBW and BGK models exhibit considerable deviation from the KS gluon. The
difference from the previous plot is the disagreement of the GBW/BGK models
and the rcBK in the small-$\Delta\phi$ region. Similarly to the previous
plots, the Sudakov factor affects the models somewhat differently from the KS
gluon in Ref. [41]. The effect enhances the cross section considerably in the
small-$\Delta\phi$ region, making it closer to KS gluon result.

The results shown in Figs. 7 and 8 are natural, as back-to-back configuration
in the respective observable corresponds to the small-$k_{T}$ region of gluon
densities and, as it can be seen clearly in Fig. 5, the GBW and BGK gluons do
not fare well in the large-$k_{T}$ region. That is to say that the enhancement
in the small-$\Delta\phi$ region is a direct consequence of the broadening by
the Sudakov factor.

##  5 Summary

We have fitted the GBW and BGK saturation models to HERA data [33] using the
$k_{T}$-factorization for the structure function $F_{2}$. The main difference
between the $k_{T}$\- and the dipole factorizations is an argument of the
gluon, $x/z$, appearing in the former and being replaced by
$x\,(1+4m_{f}^{2}/Q^{2})$ in the latter. In fact, the massive light quarks
used in Ref. [27] partially simulate the factor $1/z$, and our fit result
indicates such effect, as expected.

We found that the dipole factorization can reproduce the result of the
$k_{T}$-factorization formula quite well. The only major difference is in the
normalization parameter $\sigma_{0}$, which increases significantly for the
$k_{T}$-factorization case. We argued that this change is a direct
consequences of the kinematic approximation used in the dipole factorization
formula. We have also observed that the explicit inclusion of the running
coupling in the GBW model has significant effect on fit quality, particularly
in the large-$Q^{2}$ region, where the GBW model performs poorly.

We have applied the new results from our fits for predictions of the dijet
process in DIS at EIC. Additionally, effects of the Sudakov form factor were
investigated for that process. Results of the electron-jets correlation in the
Breit frame agree qualitatively with those of Ref. [41]. Other results, namely
the electron-jets in the Lab frame and the jet-jet correlation in the Breit
frame, show considerable effects of the Sudakov form factor, which broadens
the gluon density.

## Acknowledgment

We are grateful to Krzysztof Golec-Biernat, Piotr Kotko and Andreas van
Hameren for useful discussions. The project is partially supported by the
European Union’s Horizon 2020 research and innovation program under grant
agreement No. 824093\. TG and SS are partially supported by the Polish
National Science Centre grant no. 2017/27/B/ST2/02004.

##  Appendix A $I_{i}$ in Eq. (7)

The functions $I_{1},\,I_{2},\,I_{3},\,I_{4}$ used in Eq. (7) are defined as
[44]

$\begin{split}\frac{I_{1}}{2\pi}=\frac{N_{1}N_{2}+N_{3}^{2}}{\left(N^{2}_{1}+2N_{1}N_{2}+N_{3}^{2}\right)^{3/2}},&\hskip
28.45274pt\frac{I_{2}}{2\pi}=\frac{N_{3}-(1-2\beta)N_{1}}{(N_{1}+N_{4})\sqrt{N^{2}_{1}+2N_{1}N_{2}+N_{3}^{2}}},\\\
\frac{I_{3}}{2\pi}=\frac{N_{1}+N_{2}}{\left(N^{2}_{1}+2N_{1}N_{2}+N_{3}^{2}\right)^{3/2}},&\hskip
28.45274pt\frac{I_{4}}{2\pi}=\frac{2(1-\beta)}{(N_{1}+N_{4})\sqrt{N^{2}_{1}+2N_{1}N_{2}+N_{3}^{2}}},\end{split}$
(22)

for

$\begin{split}N_{1}\equiv\beta(1-\beta)Q^{2}+m_{f}^{2},&\hskip
56.9055ptN_{2}\equiv{\kappa^{\prime}}_{t}^{2}+(1-\beta)^{2}k_{T}^{2},\\\
N_{3}\equiv{\kappa^{\prime}}_{t}^{2}-(1-\beta)^{2}k_{T}^{2},&\hskip
56.9055ptN_{4}\equiv{\kappa^{\prime}}_{t}^{2}+\beta(1-\beta)k_{T}^{2}.\end{split}$
(23)

## References

  * [1] R. G. Roberts.  The Structure of the proton: Deep inelastic scattering.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2 1994.
  * [2] John Collins.  Foundations of perturbative QCD, volume 32.  Cambridge University Press, 11 2013.
  * [3] An Assessment of U.S.-Based Electron-Ion Collider Science.  The National Academies Press, Washington, DC, 2018.
  * [4] Yuri V. Kovchegov and Eugene Levin.  Quantum Chromodynamics at High Energy, volume 33.  Oxford University Press, 2013.
  * [5] S. Catani, M. Ciafaloni, and F. Hautmann.  GLUON CONTRIBUTIONS TO SMALL x HEAVY FLAVOR PRODUCTION.  Phys. Lett. B, 242:97–102, 1990.
  * [6] S. Catani, M. Ciafaloni, and F. Hautmann.  High-energy factorization and small x heavy flavor production.  Nucl. Phys. B, 366:135–188, 1991.
  * [7] John C. Collins and R. Keith Ellis.  Heavy quark production in very high-energy hadron collisions.  Nucl. Phys. B, 360:3–30, 1991.
  * [8] S. Catani, M. Ciafaloni, and F. Hautmann.  High-energy factorization in QCD and minimal subtraction scheme.  Phys. Lett. B, 307:147–153, 1993.
  * [9] S. Catani and F. Hautmann.  Quark anomalous dimensions at small x.  Phys. Lett. B, 315:157–163, 1993.
  * [10] S. Catani and F. Hautmann.  High-energy factorization and small x deep inelastic scattering beyond leading order.  Nucl. Phys. B, 427:475–524, 1994.
  * [11] C. J. Bomhof, P. J. Mulders, and F. Pijlman.  The Construction of gauge-links in arbitrary hard processes.  Eur. Phys. J. C, 47:147–162, 2006.
  * [12] Fabio Dominguez, Bo-Wen Xiao, and Feng Yuan.  $k_{t}$-factorization for Hard Processes in Nuclei.  Phys. Rev. Lett., 106:022301, 2011.
  * [13] Fabio Dominguez, Cyrille Marquet, Bo-Wen Xiao, and Feng Yuan.  Universality of Unintegrated Gluon Distributions at small x.  Phys. Rev. D, 83:105005, 2011.
  * [14] Bo-Wen Xiao, Feng Yuan, and Jian Zhou.  Transverse Momentum Dependent Parton Distributions at Small-x.  Nucl. Phys. B, 921:104–126, 2017.
  * [15] I. I. Balitsky and L. N. Lipatov.  The Pomeranchuk Singularity in Quantum Chromodynamics.  Sov. J. Nucl. Phys., 28:822–829, 1978.
  * [16] E. A. Kuraev, L. N. Lipatov, and Victor S. Fadin.  The Pomeranchuk Singularity in Nonabelian Gauge Theories.  Sov. Phys. JETP, 45:199–204, 1977.
  * [17] V. Barone, M. Genovese, Nikolai N. Nikolaev, E. Predazzi, and B. G. Zakharov.  Unitarization of structure functions at large 1/x.  Phys. Lett. B, 326:161–167, 1994.
  * [18] L. V. Gribov, E. M. Levin, and M. G. Ryskin.  Semihard Processes in QCD.  Phys. Rept., 100:1–150, 1983.
  * [19] I. Balitsky.  Operator expansion for high-energy scattering.  Nucl. Phys. B, 463:99–160, 1996.
  * [20] Yuri V. Kovchegov.  Small x F(2) structure function of a nucleus including multiple pomeron exchanges.  Phys. Rev. D, 60:034008, 1999.
  * [21] Alex Kovner and J. Guilherme Milhano.  Vector potential versus color charge density in low x evolution.  Phys. Rev. D, 61:014012, 2000.
  * [22] Alex Kovner, J. Guilherme Milhano, and Heribert Weigert.  Relating different approaches to nonlinear QCD evolution at finite gluon density.  Phys. Rev. D, 62:114005, 2000.
  * [23] Edmond Iancu, Andrei Leonidov, and Larry D. McLerran.  Nonlinear gluon evolution in the color glass condensate. 1.  Nucl. Phys. A, 692:583–645, 2001.
  * [24] Jamal Jalilian-Marian, Alex Kovner, and Heribert Weigert.  The Wilson renormalization group for low x physics: Gluon evolution at finite parton density.  Phys. Rev. D, 59:014015, 1998.
  * [25] Jamal Jalilian-Marian, Alex Kovner, Andrei Leonidov, and Heribert Weigert.  The Wilson renormalization group for low x physics: Towards the high density regime.  Phys. Rev. D, 59:014014, 1998.
  * [26] Jamal Jalilian-Marian, Alex Kovner, Andrei Leonidov, and Heribert Weigert.  The BFKL equation from the Wilson renormalization group.  Nucl. Phys. B, 504:415–431, 1997.
  * [27] Krzysztof J. Golec-Biernat and M. Wusthoff.  Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction.  Phys. Rev. D, 59:014017, 1998.
  * [28] J. Bartels, Krzysztof J. Golec-Biernat, and H. Kowalski.  A modification of the saturation model: DGLAP evolution.  Phys. Rev. D, 66:014001, 2002.
  * [29] Henri Kowalski and Derek Teaney.  An Impact parameter dipole saturation model.  Phys. Rev. D, 68:114005, 2003.
  * [30] Larry D. McLerran and Raju Venugopalan.  Computing quark and gluon distribution functions for very large nuclei.  Phys. Rev. D, 49:2233–2241, 1994.
  * [31] Jeffrey R. Forshaw, G. Kerley, and Graham Shaw.  Extracting the dipole cross-section from photoproduction and electroproduction total cross-section data.  Phys. Rev. D, 60:074012, 1999.
  * [32] E. Iancu, K. Itakura, and S. Munier.  Saturation and BFKL dynamics in the HERA data at small x.  Phys. Lett. B, 590:199–208, 2004.
  * [33] I. Abt, A. M. Cooper-Sarkar, B. Foster, V. Myronenko, K. Wichmann, and M. Wing.  Investigation into the limits of perturbation theory at low $Q^{2}$ using HERA deep inelastic scattering data.  Phys. Rev. D, 96(1):014001, 2017.
  * [34] Alfred H. Mueller and Bimal Patel.  Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes.  Nucl. Phys. B, 425:471–488, 1994.
  * [35] R. Abdul Khalek et al.  Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report.  Nucl. Phys. A, 1026:122447, 2022.
  * [36] Paul Caucal, Farid Salazar, Björn Schenke, Tomasz Stebel, and Raju Venugopalan.  Back-to-back inclusive dijets in DIS at small $x$: Gluon Weizsäcker-Williams distribution at NLO.  4 2023.
  * [37] Paul Caucal, Farid Salazar, and Raju Venugopalan.  Dijet impact factor in DIS at next-to-leading order in the Color Glass Condensate.  JHEP, 11:222, 2021.
  * [38] Renaud Boussarie, Heikki Mäntysaari, Farid Salazar, and Björn Schenke.  The importance of kinematic twists and genuine saturation effects in dijet production at the Electron-Ion Collider.  JHEP, 09:178, 2021.
  * [39] Pieter Taels, Tolga Altinoluk, Guillaume Beuf, and Cyrille Marquet.  Dijet photoproduction at low x at next-to-leading order and its back-to-back limit.  JHEP, 10:184, 2022.
  * [40] Tolga Altinoluk, Renaud Boussarie, Cyrille Marquet, and Pieter Taels.  TMD factorization for dijets + photon production from the dilute-dense CGC framework.  JHEP, 07:079, 2019.
  * [41] Andreas van Hameren, Piotr Kotko, Krzysztof Kutak, Sebastian Sapeta, and Elżbieta Żarów.  Probing gluon number density with electron-dijet correlations at EIC.  Eur. Phys. J. C, 81(8):741, 2021.
  * [42] Krzysztof J. Golec-Biernat and M. Wusthoff.  Saturation in diffractive deep inelastic scattering.  Phys. Rev. D, 60:114023, 1999.
  * [43] A. Bialas, H. Navelet, and Robert B. Peschanski.  QCD dipole model and K(t) factorization.  Nucl. Phys. B, 593:438–450, 2001.
  * [44] M. Kimber.  Unintegrated parton distributions.  PhD thesis, Durham U., 2001.
  * [45] J. Kwiecinski, Alan D. Martin, and A. M. Stasto.  A Unified BFKL and GLAP description of F2 data.  Phys. Rev. D, 56:3991–4006, 1997.
  * [46] Nikolai N. Nikolaev and B. G. Zakharov.  Color transparency and scaling properties of nuclear shadowing in deep inelastic scattering.  Z. Phys. C, 49:607–618, 1991.
  * [47] Agnieszka Łuszczak, Marta Łuszczak, and Wolfgang Schäfer.  Unintegrated gluon distributions from the color dipole cross section in the BGK saturation model.  Phys. Lett. B, 835:137582, 2022.
  * [48] G. Beuf, H. Hänninen, T. Lappi, and H. Mäntysaari.  Color Glass Condensate at next-to-leading order meets HERA data.  Phys. Rev. D, 102:074028, 2020.
  * [49] K. S. Kolbig.  PROGRAMS FOR COMPUTING THE LOGARITHM OF THE GAMMA FUNCTION AND THE DIGAMMA FUNCTION FOR COMPLEX ARGUMENT.  4 1972.
  * [50] M. Galassi, J. Davies, J. Theiler, B. Gough, and G. Jungman.  GNU Scientific Library - Reference Manual, Third Edition, for GSL Version 1.12 (3. ed.).  2009\.
  * [51] T. Hahn.  CUBA: A Library for multidimensional numerical integration.  Comput. Phys. Commun., 168:78–95, 2005.
  * [52] R. Brun and F. Rademakers.  ROOT: An object oriented data analysis framework.  Nucl. Instrum. Meth. A, 389:81–86, 1997.
  * [53] Fred James and Matthias Winkler.  MINUIT User’s Guide.  6 2004.
  * [54] Tomoki Goda, Krzysztof Kutak, and Sebastian Sapeta.  Sudakov effects and the dipole amplitude.  Nucl. Phys. B, 990:116155, 2023.
  * [55] J. L. Albacete, N. Armesto, J. G. Milhano, C. A. Salgado, and U. A. Wiedemann.  Numerical analysis of the Balitsky-Kovchegov equation with running coupling: Dependence of the saturation scale on nuclear size and rapidity.  Phys. Rev. D, 71:014003, 2005.
  * [56] Javier L. Albacete and Yuri V. Kovchegov.  Solving high energy evolution equation including running coupling corrections.  Phys. Rev. D, 75:125021, 2007.
  * [57] Javier L. Albacete, Nestor Armesto, Jose Guilherme Milhano, Paloma Quiroga-Arias, and Carlos A. Salgado.  AAMQS: A non-linear QCD analysis of new HERA data at small-x including heavy quarks.  Eur. Phys. J. C, 71:1705, 2011.
  * [58] K. Kutak and A. M. Stasto.  Unintegrated gluon distribution from modified BK equation.  Eur. Phys. J. C, 41:343–351, 2005.
  * [59] Martin Hentschinski, Krzysztof Kutak, and Robert Straka.  Maximally entangled proton and charged hadron multiplicity in Deep Inelastic Scattering.  Eur. Phys. J. C, 82(12):1147, 2022.
  * [60] A. van Hameren, P. Kotko, K. Kutak, C. Marquet, E. Petreska, and S. Sapeta.  Forward di-jet production in p+Pb collisions in the small-x improved TMD factorization framework.  JHEP, 12:034, 2016.  [Erratum: JHEP 02, 158 (2019)].
  * [61] Bo-Wen Xiao.  Low-$x$ Physics in $pA$ Collisions and at the EIC.  Nucl. Phys. A, 967:257–264, 2017.
  * [62] P. Kotko, K. Kutak, C. Marquet, E. Petreska, S. Sapeta, and A. van Hameren.  Improved TMD factorization for forward dijet production in dilute-dense hadronic collisions.  JHEP, 09:106, 2015.
  * [63] A. van Hameren.  KaTie : For parton-level event generation with $k_{T}$-dependent initial states.  Comput. Phys. Commun., 224:371–380, 2018.
  * [64] A. H. Mueller, Bo-Wen Xiao, and Feng Yuan.  Sudakov Resummation in Small-$x$ Saturation Formalism.  Phys. Rev. Lett., 110(8):082301, 2013.
  * [65] A. H. Mueller, Bo-Wen Xiao, and Feng Yuan.  Sudakov double logarithms resummation in hard processes in the small-x saturation formalism.  Phys. Rev. D, 88(11):114010, 2013.
  * [66] Maxim Nefedov.  Sudakov resummation from the BFKL evolution.  Phys. Rev. D, 104(5):054039, 2021.
  * [67] Krzysztof Kutak and Sebastian Sapeta.  Gluon saturation in dijet production in p-Pb collisions at Large Hadron Collider.  Phys. Rev. D, 86:094043, 2012.
  * [68] N. A. Abdulov et al.  TMDlib2 and TMDplotter: a platform for 3D hadron structure studies.  Eur. Phys. J. C, 81(8):752, 2021.
  * [69] Ian Balitsky.  Quark contribution to the small-x evolution of color dipole.  Phys. Rev. D, 75:014001, 2007.


10 1

Математические основы и численные методы моделирования Mathematical modeling
and numerical simulation

??.12.2021. ??.12.2021.

519.856 Stochastic optimization in digital pre-distortion of the signal
††thanks: Работа Д.А. Пасечнюка и А.М. Райгородского поддержана грантом
Российского научного фонда (проект №21-71-30005). The research of D.
Pasechnyuk and A. Raigorodskii was supported by Russian Science Foundation
grant (project No. 21-71-30005).

Артём Вадимович Алпатов A. V. Alpatov Artem V. Alpatov ‘‘Samara Lyceum of
information technologies (Basic school of the RAS)’’, Samara, 443096, Russia

Егор Александрович Петерс E. A. Peters Egor A. Peters 2]ЦДНИТТ ‘‘УникУм’’ при
КузГТУ,
Россия, 650000, г. Кемерово, ул. Красноармейская, д. 117 CCSETC at KuzSTU
‘‘UnicUm’’, Kemerovo, 650000, Russia

Дмитрий Аркадьевич Пасечнюк D. A. Pasechnyuk Dmitry A. Pasechnyuk 3]Московский
физико-технический институт,
Россия, 141701, г. Долгопрудный, Институтский пер., д. 9 Moscow Institute of
Physics and Technology, Moscow, 141701, Russia

4]Исследовательский центр доверенного искусственного интеллекта ИСП РАН,
Россия, 109004, г. Москва, ул. А. Солженицына, д. 25 [4]ISP RAS Research
Center for Trusted Artificial Intelligence, Moscow, 109004, Russia

Андрей Михайлович Райгородский A. M. Raigorodskii Andrei A. Raigorodskii

# Стохастическая оптимизация в задаче цифрового предыскажения сигнала

А. В. Алпатов<EMAIL_ADDRESS>ГАОУ СО ‘‘Самарский лицей
информационных технологий (Базовая школа РАН)’’,
Россия, 443096, г. Самара, ул. Больничная, д. 14а  Е. А. Петерс
<EMAIL_ADDRESS>[  Д. А. Пасечнюк<EMAIL_ADDRESS>[  [  А. М.
Райгородский<EMAIL_ADDRESS>
###### Abstract

В данной статье осуществляется сравнение эффективности некоторых современных
методов и практик стохастической оптимизации применительно к задаче цифрового
предыскажения сигнала (DPD), которое является важной составляющей процесса
обработки сигнала на базовых станциях, обеспечивающих беспроводную связь. В
частности, рассматривается два круга вопросов о возможностях применения
стохастических методов для обучения моделей класса Винера–Гаммерштейна в
рамках подхода минимизации эмпирического риска: касательно улучшения глубины и
скорости сходимости данного метода оптимизации, и относительно близости самой
постановки задачи (выбранной модели симуляции) к наблюдаемому в
действительности поведению устройства. Так, в первой части этого исследования,
внимание будет сосредоточено на вопросе о нахождении наиболее эффективного
метода оптимизации и дополнительных к нему модификаций. Во второй части
предлагается новая квази-онлайн постановка задачи и соответственно среда для
тестирования эффективности методов, благодаря которым результаты численного
моделирования удаётся привести в соответствие с поведением реального прототипа
устройства DPD. В рамках этой новой постановки далее осуществляется повторное
тестирование некоторых избранных практик, более подробно рассмотренных в
первой части исследования, и также обнаруживаются и подчёркиваются
преимущества нового лидирующего метода оптимизации, оказывающегося теперь
также наиболее эффективным и в практических тестах. Для конкретной
рассмотренной модели максимально достигнутое улучшение глубины сходимости
составило 7% в стандартном режиме и 5% в онлайн постановке (притом что метрика
сама по себе имеет логарифмическую шкалу). Также благодаря дополнительным
техникам оказывается возможным сократить время обучения модели DPD вдвое,
сохранив улучшение глубины сходимости на 3% и 6% для стандартного и онлайн-
режимов соответственно. Все сравнения производятся с методом оптимизации Adam,
который был отмечен как лучший стохастический метод для задачи DPD из
рассматриваемых в предшествующей работе [Pasechnyuk et al., 2021], и с методом
оптимизации Adamax, который оказывается наиболее эффективным в предлагаемом
онлайн-режиме.

###### keywords:

цифровое предыскажение

###### keywords:

обработка сигнала

###### keywords:

стохастическая оптимизация

###### keywords:

онлайн обучение

In this paper, we test the performance of some modern stochastic optimization
methods and practices in application to digital pre-distortion problem, that
is a valuable part of processing signal on base stations providing wireless
communication. In first part of our study, we focus on search of the best
performing method and its proper modifications. In the second part, we
proposed the new, quasi-online, testing framework that allows us to fit our
modelling results with the behaviour of real-life DPD prototype, retested some
selected of practices considered in previous section and approved the
advantages of the method occured to be the best in real-life conditions. For
the used model, maximum achieved improvement in depth was 7% in standard
regime and 5% in online one (metric itself is of logarithmic scale). We also
achieved a halving of the working time preserving 3% and 6% improvement in
depth for the standard and online regime, correspondingly. All comparisons are
made to the Adam method, which was highlighted as the best stochastic method
for DPD problem in paper [Pasechnyuk et al., 2021], and to the Adamax method,
that is the best in the proposed online regime.

digital pre-distortion signal processing stochastic optimization online
learning

## Introduction

Consider a base station providing some form of wireless communication. Let it
be required to transmit with its help some given signal. The signal is
represented as a sampled array $x\in\mathbb{C}^{m}$. One of the
transformations that must be applied to the signal before transmission is
amplification. To carry out this transformation, the PA device is most often
used, whose action on the signal, ideally, has the form
$\mathcal{PA}(x)=a\cdot x$, where $a\gg 1$. However, hardware imperfections
lead to a deviation from this law; in particular, the $\mathcal{PA}(x)$ signal
has out-of-band spurious harmonics, resulting in interference. This is
illustrated in Fig. 1.

Figure 1:  Power spectral density plot of original signal (Input signal, blue
dotted line) and power amplified signal (Output signal, red line). Picture
from Wisell D. H. Exploring the sampling rate requirements for behavioural
amplifier modelling // XVIII IMEKO World Congress. — 2006. — p. 1-4

To prevent noise in the transmitted signal, there is digital pre-distortion
technology. Specifically, there is a DPD device acting on the signal just
before the PA action, so that the result is the signal
$\mathcal{PA}(\mathcal{DPD}(x))$. If we omit the factor $a$, which is easy to
accomplish by scaling the signal, then we can see that for an ideal action it
must hold that $\mathcal{DPD}=\mathcal{PA}^{-1}$ (Fig. 2). Thus, the task is
reduced to the inversion of $\mathcal{PA}$ function. However, the
$\mathcal{PA}$ function actually depends on the characteristics of the
environment, the signal x itself, and on time, so it can only be inverted
numerically. To do this, we choose a certain parametric family of functions
$\\{\mathcal{DPD}_{\theta}\\}_{\theta\in\Theta}$ to describe DPD and choose
such a parametrization $\theta$ in which
$\mathcal{PA}\circ\mathcal{DPD}_{\theta}$ is closest to the identical
transformation.

Figure 2:  The intuitive principle of digital pre-distortion: DPD acts as an
inverse of power amplifiers function, so that the resulting transformation is
near linear (identical) Picture from [Pasechnyuk et al., 2021]

In this study, to model the behaviour of DPD we use the Wiener–Hammerstein
cascade models family [Ghannouchi, Hammi, Helaoui, 2015]. The structure of
these models is very similar to the structure of multilayer neural network,
but instead of the neurons it consists of cells specific for signal
processing. Fig. 3 shows the principle scheme of such a cell with polynomial
non-linearity within.

Figure 3:  The structure of the Wiener–Hammerstein model cell with polynomial
non-linearity. Picture from [Ghannouchi, Hammi, Helaoui, 2015]

Now, when we defined the parametrization for DPD, we should provide a more
optimization-friendly formulation to the problem of inverting the
$\mathcal{PA}$ function. The point is that to do this we need to have the
perfect pre-distortion $\overline{y}$ for some signal $x$, to train the model
to repeat this predistortion, i.e. exactly invert the $\mathcal{PA}$ function.
Using the standard empirical risk minimization framework, we then obtain some
optimization problem with the sum-like function. The following formulation,
proposed and described in details in [Pasechnyuk et al., 2021], is the central
one for this paper (we denote the $k$-th element in array $x$ by $[x]_{k}$ and
$x_{k}$).

$\frac{1}{m}\sum_{k=1}^{m}|[\mathcal{DPD}_{\theta}(x)]_{k}-\bar{y}_{k}|^{2}\,\rightarrow\,\min_{\theta}.$
(1)

So, the both two research questions of this study correspond to this problem
formulation and concern how to solve it properly, taking into account its
original motivation.

  * $\boldsymbol{RQ_{1}}$

Which stochastic optimization method is the most suitable to solve the stated
problem with such a specific data and model behind?

In broad strokes, this question was considered in course of study [Pasechnyuk
et al., 2021] for the different groups of numerical methods. Here we focus on
the stochastic methods (indeed, (1) can be easily randomized by terms), and
consider several recently proposed approaches in this field.

  * $\boldsymbol{RQ_{2}}$

How one should organize the training procedure and evaluation of the model to
agree with the practical behaviour of DPD device?

The operation of the full DPD technology stack is more complicated than it is
representated in (1), and the main difference is its online nature. It is even
more online than it is implied, for example, in the expectation minimization
problems, because the source of signal (and hence a joint distribution of
$(x,\overline{y})$, continuing the analogy) also changes in time. We propose a
way to simulate this specificity, and eliminate contradictions between our
experimental results and some results of laboratory tests with DPD device.

## Standard learning framework

In this section we consider the standard setup for the problem stated above,
i.e. train the model on some known data (it is important for signal processing
task, that this subset of the whole data is not random chosen but is the
prefix of the ordered signal array) and then test the model on the whole
signal (NB: not the remaining part, as usual). In our experiments, length of
training signal prefix is 75% of the whole signal’s length (albeit it is
deliberately more than needed, see [Pasechnyuk et al., 2021] for details, to
avoid any approximation issues that are out of the scope this time). We use
the segment with $m=2\cdot 10^{5}$ ticks from the real-life 80 MHz signal as a
dataset. As we shall see, this way to train/test model is slightly naïve to
represent the behaviour of the real-live DPD device (being standard and widely
used nonetheless), but thanks to it we can easily test many possible
techniques in the simplest (and computationally inexpensive) conditions to
select the few best to work with further.

To assess the results of evaluating model on test split of dataset we use the
following logarithmic scaling analogue of Mean Squared Error metric with
normalization, that is specific to the signal processing tasks:

$\text{NMSE}(y,\overline{y}):=10\cdot\log_{10}\frac{\sum_{k=1}^{m}|y_{k}-\overline{y}_{k}|^{2}}{\sum_{k=1}^{m}|x_{k}|^{2}}\text{
dB}.$

Source code of the model used to obtain the following experiments results is
available at https://github.com/dmivilensky/Sirius-2021-DPD-optimization
(Python 3 + PyTorch). All the computations were performed on the standard
personal computer without GPU acceleration.

### Comparison of algorithms

Figure 4:  The convergence curves of different optimization methods in
standard regime at initial epochs

Let us start with the testing of different optimization in application to this
standardly formulated learning problem. The convergence of some most known
ones was examined in [Pasechnyuk et al., 2021] recently, so there is no need
for us to present the same results for our model: in comparison with table,
that is shown in Fig. 5, they are just transposed a couple of decibels higher
(in our research, we use the Wiener–Hammerstein model with significantly less
parameters than for the model considered in [Pasechnyuk et al., 2021], so our
score will be a little worse — it is just a specific of particular used
model).

Figure 5:  Summary of the result of training the DPD model with different
optimization methods in standard setting. Table from [Pasechnyuk et al., 2021]

Besides, in this section we focus on the starter methods, i.e. that ones the
convergence of which is the most efficient among all tested methods at the
beginning of the training process. That’s why we show the convergence of all
the methods on the first 20 or 40 epochs. The first reason to this is that
long term convergence in standard training regime is already considered in
[Pasechnyuk et al., 2021], so we should clarify the short term one, that was
not yet considered (it is important for the multi-method optimization
schemes). The second is that the convergence on the late epochs and the depth
of finally obtained minimum is more the question of late convergence rate of
method, not convergence depth, and we will see in online learning framework,
that methods presented as the best here are not well adopted to converge with
a good rate at late epoch, that is important in practice, because the optimal
point we seek for is changing together with data segment. So, let us avoid all
this complicated questions in this section, and focus on the starting
convergence, i.e. the behaviour of methods far from local minimum.

So, in Fig. 4, Fig. 6 and Fig. 7 we present the convergence curves of
different tested optimization methods. Further, we describe our observations
on it.

  1. 1.

Starting convergence of Adamax is not very well, despite the fact that it is
one of the most efficient methods in long term perspective. Some of the
methods, like RMSprop, are just unstable for the model and signal of used
size.

  2. 2.

The best among methods from Fig. 4 are Adam, as expected, and Lookahead(Adam)
(we also denote it by LaAdam), i.e. the Adam enveloped with Lookahead
algorithm from [Zhang et al., 2019]. The latter demonstrates the faster
convergence.

  3. 3.

The only method that outperforms Adam is Shampoo [Gupta, Koren, Singer, 2014],
it is the best starter method for DPD training for now.

  4. 4.

Lookahead gives very notable results. It significantly improves and stabilizes
the convergence of Adam, that is shown in Fig. 6. Unfortunately, there is no
effect of Lookahead on Shampoo. Nevertheless, convergence of Shampoo is
phenomenally stable without Lookahead too.

Figure 6:  The comparison of Adam and Shampoo [Gupta, Koren, Singer, 2014]
optimization methods, and the effect of Lookahead envelope on training the DPD
model

Another novel optimization method we tested is Accelerated mini-batch SGD
version from [Woodworth, Srebro, 2021] (we name it AccMbSGD). This method is
specialized for the overparametrized optimization problems. The point is that
in our case the number of parameters in model is significantly bigger than
number of terms in sum in empirical risk, so it is reasonable to assume that
value of NMSE in minimum is near the zero (and it is). This is the
characteristic property of overparametrization, and mentioned method is
theoretically optimal for this case.

  1. 5.

Unfortunately, the convergence of AccMbSGD is not so well as expected.
Moreover, this method is complicated in hyperparameter tuning. It seems that
it is not adopted for DPD training. The corresponding curves are presented in
Fig. 7.

Figure 7:  The comparison of Adam and AccMbSGD [Woodworth, Srebro, 2021]
optimization methods in application to training the DPD model

### Dynamic batch size

It is well-known practice to use mini-batching in stochastic optimization
methods to improve the convergence (due to the less variance of oracle) and
speed up the calculations (it is typical for training neural networks).
Usually, the size of the batch is just a hyperparameter of the method and is
the same at the every iteration. Nevertheless, it is reasonable to strive to
increase of it, if the latter is computationally efficient. On the other hand,
this heuristic is quite legitimate: indeed, at the late steps of the method
current point is already close enough to minimum and the step direction
efficiency becomes more sensitive to the inaccuracies of the stochastic
gradient approximation (which just can be reduced by increasing the size of
the batch).

Some theoretical justifications on this technique are provided in [Zhao, Xie,
Li, 2020]. The practical efficiency of batch size adaptation is also shown in
[Devarakonda, Naumov, Garland, 2017] for some deep learning problems. So,
further we check if the changing of batch size affects the convergence of Adam
optimization method in our case. We considered both increasing and decreasing
changing, as well as the different changing rates.

  1. 1.

In our experiments, Adam with reducing batch size demonstrated less stable and
less deep convergence than the Adam with fixed batch size (as expected).

  2. 2.

The growth of batch size following the exponential law lead to the improvement
of convergence, but its effect is less than for linear law.

  3. 3.

The best formula for changing the batch size in our case is
$b=200+120\cdot\text{epoch}$. Using it, we obtained a 3% improvement in NMSE
(Fig. 8a) and reduced training time by 2 times (Fig. 8b). The results are
summarized in Table 1.

  4. 4.

The positive effect in training time increases with an increase in the
coefficient in the latter linear formula (it is $120$ there). But here we
faced with the trade-off, because the increase in the coefficient can also
worsen the limit depth of convergence (value of NMSE). In signal processing
task the learning time is minor metric, so it is reasonable to stick around
$120$.

(a) The dependency of NMSE value (dB) on the number of training epochs

(b) The dependency of working time (s) on the number of training epoch

Figure 8: The effect of batch size changing on training the DPD model. It leads to the more deep convergence of Adam (left plot) and slower growth in working time (right plot) | NMSE (dB) | time (s)
---|---|---
fixed | -34.90 | 660
changing | -35.92 | 320
Table 1: Summary of the result of training the DPD model with fixed and with
changing batch size

In the experiments above we used the Adam optimization method to demonstrate
the efficiency of dynamic batch size approach. The Adam was chosen as the
leader among the methods for standard learning framework [Pasechnyuk et al.,
2021]. Nevertheless, these results are similar for all the tested optimization
methods (they are listed in Table 4).

### Dynamic learning rate

Another important hyperparameter of the optimization methods is its learning
rate, i.e. some starting value of the step size. Further, we describe the
results of our experiments on scheduling (changing) the value of learning rate
during the training. It is known that for some specific models it may be
essential to use a proper learning rate scheduler to obtain the satisfactory
accuracy [Xiong et al., 2020]. So, we tested some of the widely used
schedulers in our case to check if it affects convergence significantly.

The first group of schedulers consists of the simple monotonically-decaying
ones, and here we found the most suitable option for the growth rate and
configuration of scheduler. We tested both the increasing and decreasing
changing of learning rate, and obtain the following results:

  1. 1.

The most proper formula to change the learning rate is a linear one (bounded
below): $\text{lr}\leavevmode\nobreak\ =\leavevmode\nobreak\
\max\\{10^{-2}-10^{-4}\cdot\text{epoch},\leavevmode\nobreak\ 6\cdot
10^{-3}\\}$. The effect of scheduling is nearly the same independent of law it
follows, if the lower and upper bounds for learning rate are that ($6\cdot
10^{-3}$ and $10^{-2}$, correspondingly).

  2. 2.

The learning rate increasing strategy leads to a destabilization of
convergence (as expected).

  3. 3.

The reference score, achieved by the Adam without learning rate scheduling, is
$\text{NMSE}=-34.29$ dB. For the properly scheduled learning rate this score
becomes
$\text{NMSE}=\hbox{\pagecolor{ForestGreen!40}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>dB. Thus, the improvement is 7%. The convergence curves of some tested options
are presented in Fig. 9

(a) The dependency of NMSE value (dB) on the number of training epochs (b) The
dependency of NMSE value (dB) on the training time (s)

Figure 9: The effect of learning rate changing on training the DPD model. When
decreasing, it leads to the significantly more deep convergence of Adam

The results summarized above are related to only the monotone schedulers. On
the other hand, it becomes more popular to use some cyclic strategies in some
cases to improve the properties of obtained solution. For instance, in
[Izmailov et al., 2008] it was proposed the scheduler called SWA (Stochastic
Weight Averaging), changing learning rate in cyclic manner and averaging the
points after the smaller steps. In many practical deep learning problems it
allows to achieve around 1% improvement in accuracy even without a precise
tuning of scheduler. The behaviour of cyclic learning rate scheduling without
averaging is also considered in [Smith, 2017]. We tested both SWA and simple
cycling strategies to compare their efficiency on our problem.

  1. 4.

The SWA scheduler( with one epoch of fixed learning rate, period of learning
rate decreasing equal to one epoch, and same lower and upper bounds on
learning rate as in previous experiment) worsens convergence in comparison
with reference score obtained by Adam without any scheduling.

  2. 5.

The same cyclic scheduling without averaging the points, in opposite, allows
to obtain significant improvement.

  3. 6.

For the Adam optimization method, the improvement in depth given by cyclic
learning rate is 5%. For the Adamax, it is 7%. The results of experiment are
summarized in Table 2.

NMSE (dB) | Adam | Adamax
---|---|---
fixed | -33.30 | -29.66
cyclic | -35.00 | -31.86
Table 2: Summary of the result of training the DPD model with fixed and with
cyclically changing learning rate

  4. 7.

We suppose that cyclic scheduling for our problem indeed behaves like it
described in [Garipov et al., 2008] (one of the main prerequisite papers for
SWA). But averaging itself fails and it is not asymptotically equivalent to
FGE algorithm from [Garipov et al., 2008] in fact. It may be because the
structure of local minima in considered problem is not so well as in common
deep learning tasks, i.e. instead of situation like in Fig. 10a we have
something like Fig. 10b, so we see the loss in score after averaging.

(a) The behaviour of SWA near the local minimum in many deep learning problems
cases. Picture is similar to that from [Izmailov et al., 2008]

(b) The probable behaviour of SWA in case of more complicated structured local
minimum.

Figure 10: The two probable cases of SWA [Izmailov et al., 2008] operation. We
denote the result of averaging intermediate points $w_{1}$, $w_{2}$, $w_{3}$
by $w$. The left figure demonstrates the simple task for SWA, the right one
maybe explained the losing of SWA we observed.

Finally, it is reasonable to assess the learning rate scheduling techniques we
considered in compound with the previously described batch size changing, and
check if we can combine them to obtain the better scores. We considered to
test only a paired technique of increasing the batch size and monotone
decreasing of learning rate, since these two are efficient and simple enough
to be combined without additional synchronizations.

  1. 8.

Increasing batch size with decreasing learning rate is efficient, but the
effect of learning rate here is weaker then without batch size changing: it is
only a 0.5%, see Fig. 11. This is near the scale of score oscillations from
epoch to epoch, so this improvement may be unstable.

  2. 9.

For this compound regime the resulting score is
$\text{NMSE}=\hbox{\pagecolor{Gray!40}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>dB. The reference value for this experiment is when batch size increases but
learning rate is fixed, it is $\text{NMSE}=-35.27$ dB.

  3. 10.

Increasing learning rate still leads to a divergence, but this effect become
more pronounced when batch size increases.

(a) The dependency of NMSE value (dB) on the number of training epochs (b) The
dependency of NMSE value (dB) on the training time (s)

Figure 11: The effect of both learning rate changing and batch size increasing
on training the DPD model. When learning rate decreasing, it leads to a little
more deep convergence of Adam, while if increasing, it leads to a significant
destabilization

These results are somewhat unexpected. One can think that if batch size
becomes bigger, the stability of method becomes better, and at least with
decreasing learning rate (that is very careful heuristic) it should have
manifested itself in the improvement of scores, but it did not. Anyway, we can
say, that combining of batch size changing with learning rate scheduling is
not very efficient, so in any particular case it will be necessary to make
choice between big improvement in time and good improvement in depth. It
mentioned, that working time is minor metric for signal processing tasks.
Nevertheless, it would be interesting to know that in online regime changing
batch size is just more efficient with respect to the depth of convergence
than both of learning rate scheduling policies.

### Regularization

The last of non-optimizational modifications considered in the course of our
study is a regularization. It is very common to add the regularization term in
the problem to control the properties of the solution, but the regularization
may also be a good heuristic to improve the convergence, due to its impact on
(local) conditionality of the problem. This practical property of
regularization seems to be underexamined and is bypassed in many new research
papers on application of optimization methods to learning problems.

In out experiment, we considered only some widely-known regularization terms,
namely $\ell_{1}$ (lasso), $\ell_{2}$ (ridge) and $\ell_{1}+\ell_{2}$ (elastic
net). To check their impact on convergence, we tested the efficiency of
different optimization methods, listed in Table 4, on the corresponding
regularized problems. And there are the results it gave:

  1. 1.

Unlike the most of results presented in previous sections, the impact of
regularization is significantly dependent on the chosen optimization method.
So, Adam and Adamax are sensitive to it, while RAdam [Liu et al., 2019] does
not totally (see Table 3).

  2. 2.

On the other hand, one can conclude from the left part of Table 3 that the
standard regularization (one like $\lambda\|\theta\|^{2}_{2}$, we name it
zero-centered; we set $\lambda=10^{-4}$) does not affect the convergence. We
present the minimum-aggregated values of NMSE, so we hide some probable
destabilization effects here — they can appear in RAdam, but not in Adam or
Adamax (see Fig. 12).

The point is that it more efficient here is to use more complicated
regularization with moving center (one like
$\lambda\|\theta-\overline{\theta}\|^{2}_{2}$, we name it prox-centered). We
tested the version with a period of updates equal to one epoch and
$\overline{\theta}$ set to point obtained after the last iteration of method.
This configuration resembles the scheme of inexact proximal gradient method a
little.

  1. 3.

So, this approach is efficient and gives 2% improvement to the Adam
optimization method. The results are summarized in Table 3.

  2. 4.

It is unusual that the best result is obtained with $\ell_{1}$-regularization
(and also with elastic net, but the effect is smaller, so we can conclude that
$\ell_{2}$ part of it only suppresses the effect of $\ell_{1}$ one).

method | reference | min NMSE, zero-centered | min NMSE, prox-centered
---|---|---|---
${\ell_{1}}$ | ${\ell_{2}}$ | ${\ell_{1}+\ell_{2}}$ | ${\ell_{1}}$ | ${\ell_{2}}$ | ${\ell_{1}+\ell_{2}}$
Adam | -33.30 | -33.30 | -33.30 | -33.30 | -33.99 | -33.31 | -33.81
RAdam | -31.88 | -31.88 | -31.88 | -31.88 | -31.88 | -31.88 | -31.88
Table 3: Summary of the result of training the DPD model with different
regularization terms

(a) The dependency of NMSE value (dB) on the number of training epochs for
Adam (b) The dependency of NMSE value (dB) on the number of training epochs
for RAdam

Figure 12: The effect of prox-centered regularization on training the DPD
model

## Quasi-online learning framework

It was said before that the standard learning framework is not suitable for
learning the DPD models due to some disagreements with the practical devices
behaviour. The point is that if we use some optimization methods, for example,
Adam and Adamax, to fit the parameters of the real technical prototype, and
then test it on some real life signal, we will get that Adamax (significantly)
outperforms Adam, while our experiments in previous section predict the
opposite. The reason is that the operation of technical prototype is different
from the simulating scheme used before, due to the specifics of signal
processing task. In fact, the DPD model retrains periodically to be able to
work with the new incoming signal. Unlike machine learning, the data in this
case is updated with a big frequency (80 MHz or more) and, moreover, change
its properties, for example, due to the change of the interlocutor on the
telephone line. A very simplified circuit diagram of DPD operation is
presented in Fig. 13, and represents the looped process. So, if our results
disagree with the experiment, we firstly should take into account this
cyclicality as the most notable difference with the standard machine learning
framework.

Figure 13: Schematic diagram of the operation and learning of the DPD device
during signal processing

Further, we propose another framework to train/test DPD models, that includes
the retraining procedures and more correct evaluation, and test some of the
techniques proposed above in new, more close to real, conditions.

### New framework and comparison of algorithms

As it was noted in the preamble of section, the provided comparison of methods
and testing any practices in the previously postulated conditions have little
in common with the behaviour of these objects observed in reality. At least,
it is difficult to find real analogies for training and test datasets, and yet
the NMSE value we use is a function not only of the applied numerical method,
but of these two parameters too. Of course, we are not able to avoid this
functional dependency. We propose to replace these two variables with the most
natural one — the whole given signal segment.

Let us turn to the operation of DPD device. Following the scheme from Fig. 13:
DPD collects information about some small segment of signal, then it trains on
it and is predicting distortion for some new small segment, while collecting
information about it, then it repeats this procedure over and over again. The
only thing we cannot wave away here is ‘‘over and over again’’, because we are
limited with the given dataset, but we can exactly simulate all the other
steps. Namely, let us split our dataset into some number of segments with the
same length and train and test model on the two consequent segments
correspondingly at the every training era. This is shown in Fig. 14.

Figure 14: Description of an online learning framework: a diagram of data
segments used at different moments in signal processing by the model

There are two options to assess model in this framework. The most obvious one
is the value of NMSE from the last era, i.e. evaluated on the last segment of
dataset. But this metric has two flaws: 1) it is evaluated only on the last
segment of data, that may lead to an approximation issues (like overestimation
of error), if this segment is small; 2) it does not represent the performance
of the method the whole signal, and even if all segments except the last one
were predicted absolutely incorrect, score can be good enough. To deal with
these problems, we also use the mean-aggregated version of NMSE:

$\text{mean
NMSE}(T,y,\overline{y}):=\frac{1}{T}\sum_{i=1}^{T}NMSE(y_{i},\overline{y}_{i})\text{
dB}.$

It is in fact a very proper choice for assessment the performance of online
optimization methods, because it is usually used in their theoretical analysis
[Hazan, Kale, 2014] (and is connected with so called regret metric).

So, let us firstly reassess the optimization methods in new setting without
any additional modifications. In Table 4 we present the results for some of
the methods with corresponding values of NMSE and mean NMSE. And here we see
the key difference of new approach with the previous one: Adamax algorithm now
is the best performing method and by a large margin — it is 3% better than
Adam in NMSE, and also have better rate of convergence at the late eras (see
Fig. 15a). The latter is the most important property of the optimization
method in our task, because it determines the possible depth of solution given
by this method. We see, that despite the starting lag of Adamax it managed to
overtake methods like Shampoo and Adam, that seems to be stick in local minima
or just have decline in convergence rate.

Method | NMSE (dB) | mean NMSE (dB)
---|---|---
Adamax | -33.04 | -30.21
Adam | -31.94 | -30.17
Shampoo | -31.06 | -29.99
DiffGrad | -30.41 | -28.76
RMSprop | -30.18 | -27.96
LaAdam | -29.12 | -27.90
RAdam | -29.35 | -27.24
AccMbSGD | -27.18 | -24.65
Yogi | -26.41 | -24.29
Table 4: Summary of the result of training the DPD model with different
optimization methods in new quasi-online setting

For the other methods we see mostly the same results as that for standard
learning framework. Shampoo is still on of the best starter methods, and
AccMbSGD demonstrates not well convergence rate. The Fig. 15b shows the mean-
aggregated version of the same plots, and we can make the similar conclusions
from it, except that leadership of Adamax here is not so pronounced, but it is
just an effect of averaging.

(a) The dependency of NMSE value (dB) on the working time (s)

(b) The dependency of mean time NMSE value (dB) on the working time (s)

Figure 15: The convergence curves of different optimization methods in quasi-
online regime with evaluation after every era

### Dynamic batch size

Now, we should revisit the practices we proposed for the standard learning
framework in the view of new one. Let us start with the batch size changing.
Further we consider only the most efficient policy we identified above (linear
increasing of batch size), but at the same time, we test two modifications of
this scheme, that are given by the more complicated structure of testing
framework itself.

Namely, the first option is to change batch size linearly in dependence on
current era (number of learning segment), and the second option is to bind
batch size with the number of epoch within every of the segments every era.
The choice of policy here is in fact the trade-off between depth of
convergence in ‘‘expected’’ score and depth of convergence for an every
segment correspondingly.

We tested the both policies, and the result is that era changing (first
option) and epoch changing (second option) are competitive depending of what
the metric we use. In the standard NMSE, era changing performs better, but
after averaging epoch changing becomes a better option. It means, that it is
more efficient to achieve a little improvement in convergence at the every era
(for the every small piece of signal) instead of improving the outer loop
convergence. It is quite unexpected result.

The results of this experiment are summarized in Table 5. The use of the epoch
changing batch size allows to obtain 6% improvement in mean NMSE metric for
the considered setting. Fig. 16 shows the convergence of Adamax with the
different batch size changing policies in more details for two metrics.

| NMSE (dB) | mean NMSE (dB)
---|---|---
fixed | -33.15 | -28.96
era changing | -33.63 | -29.85
epoch changing | -33.49 | -30.66
Table 5: Summary of the result of training the DPD model with fixed and with
changing batch size in quasi-online regime

(a) The dependency of NMSE value (dB) on the working time (s) (b) The
dependency of mean time NMSE value (dB) on the working time (s)

Figure 16: The effect of batch size changing on training the DPD model

### Regularization

In the case of regularization, we do not need to take into account the
difference between eras ans epochs, so to test the performance of this
modifications we just carried out one more experiment of learning DPD model
with Adamax optimization method (it is the best one in online setting) with
$\ell_{1}$-regularization (that was the best and the most stable option in
standard learning framework experiments). The result approves the positive
impact of $\ell_{1}$-regularization: comparing to the reference value of
$\text{mean NMSE}=-30.72$ dB, given by the model trained without any
regularization, $\ell_{1}$-regularization allows to achieve $\text{mean
NMSE}=\hbox{\pagecolor{Gray!40}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>dB, that is 0.3% better. Unfortunately, this is near the scale of oscillations
in convergence on the late iterations of the method, so we conclude that
regularization gives no improvement in online regime (but also does not impair
the convergence depth).

We adhere to the explanation of this that the oscillations in online regime
are itself really perceptible, and the effect of our modification is just
suppressed by multiple switching between different data segments. Another
reason is that after the last change of segment, we again start to seek new
optimal point (corresponding to this segment) from some arbitrary point, and
due to the small value of regularization coefficient, we have the situation
similar to that from standard framework experiment, but within the small
subset of data. In other words, regularization impacts convergence only on the
last segment, and impact is small because of smallness of this segment.

### Cyclic learning rate

At last, we reproduced the experiments with cyclic learning rate scheduling in
the new online framework. Now we applied this modification to the new best
method, Adamax. We tested both options for the period of cycling: once in
epoch or once in era, and chose the epoch — there is no qualitative difference
between the results in two these configurations, but increasing the period
leads to a decrease of this positive effect itself. So, we used learning rate
cycling scheduling with the period equal to one epoch together with Adamax,
for which the improvement was 1% in mean NMSE, and with Adam, with the similar
resulting 1% improvement in mean NMSE. The decrease in effect in comparison
with that one for standard learning framework is explained by the same reasons
as presented in previous paragraph.

## Conclusion

In the introduction, we posed two central questions of this research: which
optimization method is the most suitable for training the DPD model, and what
the framework we need to use to simulate the DPD operation in more close to
real conditions and to agree our simulation with the results observed in
practice. In the course of our study, we carried out a big series of
clarifying and detailing experiments to come to the two desired answers. In
the first section, dedicated to the standard learning framework, we summarize
those that relate to the first question, presenting the comparison of many
modern optimization methods and revealing the gist of proposed modifications
designed to improve their convergence. In the second section, we propose the
new quasi-online learning framework and demonstrate that the ranking of the
methods in our new experiments is mostly in line with that observed in close
to real conditions, and also reassess some of proposed modifications to
approve their stability.

To express the achieved improvements in numbers: in standard learning
framework, the depth of convergence (value of NMSE metric, that is logarithmic
scaled itself) was improved in 7% (thanks to steadily decreasing learning rate
changing for Adam or cyclic scheduling for Adamax), while the maximum achieved
improvement in online regime is 6% in mean NMSE (and given by the increasing
of batch size at every epoch). In terms of working time, we have achieved a
twofold improvement (with changing of batch size), that preserves 3% and 6%
improvement in NMSE and mean NMSE for the standard and online regime,
correspondingly.

Interim results arising in the course of the study confirm that the considered
DPD model is really specific and requires a separate study of the
effectiveness of optimization methods as applied to its training: many
approaches, common for example in the field of training neural networks, work
here unexpectedly or do not work at all. The author’s team hopes that the
proposed results will be useful for other researchers of DPD models, and that
the questions asked will stimulate a deeper study of problems in the vicinity
of this work.

## Acknowledgments

The work on this research was supported and organized in cooperation with the
Educational Сenter ‘‘Sirius’’ and Educational Foundation ‘‘Talent and
Success’’, Russia, during the Project science and technology program ‘‘Big
Challenges’’. The assistance provided by the mentor of program, Anton Gusev,
was greatly appreciated. The topic of research is a part of joint project of
Moscow Institute of Physics and Technology and Huawei Russian Research
Institute. We would like to thank Andrey Vorobyev, Eugeniy Yanitskiy and Anna
Nikolaeva for openness to cooperation and interest in this study. We wish to
extend out special thanks to prof. Alexander Gasnikov for all the helpful
discussions and his unwavering favor for this project. We also thank Liliana
Baislanova and Mansur Zainullin for their participation in setting up some
numerical experiments.

## References

  * [Haykin, 2008] Haykin S. S. Adaptive filter theory. — Pearson Education India, 2008.
  * [Ghannouchi, Hammi, Helaoui, 2015] Ghannouchi F. M., Hammi O., Helaoui M. Behavioral modeling and predistortion of wideband wireless transmitters. — John Wiley & Sons, 2015.
  * [Schreurs et al., 2008] Schreurs D. et al. RF power amplifier behavioral modeling. — New York, USA: Cambridge university press, 2008.
  * [Pasechnyuk et al., 2021] Pasechnyuk D. et al. Non-convex optimization in digital pre-distortion of the signal // arXiv preprint arXiv:2103.10552. — 2021.
  * [Kingma, Ba, 2014] Kingma D. P., Ba J. Adam: A method for stochastic optimization // arXiv preprint arXiv:1412.6980. — 2014.
  * [Woodworth, Srebro, 2021] Woodworth B., Srebro N. An Even More Optimal Stochastic Optimization Algorithm: Minibatching and Interpolation Learning // arXiv preprint arXiv:2106.02720. — 2021.
  * [Gupta, Koren, Singer, 2014] Gupta V., Koren T., Singer Y. Shampoo: Preconditioned stochastic tensor optimization // International Conference on Machine Learning. — PMLR, 2018. — p. 1842-1850.
  * [Liu et al., 2019] Liu L. et al. On the variance of the adaptive learning rate and beyond // arXiv preprint arXiv:1908.03265. — 2019.
  * [Zhang et al., 2019] Zhang M. R. et al. Lookahead optimizer: k steps forward, 1 step back // arXiv preprint arXiv:1907.08610. — 2019.
  * [Izmailov et al., 2008] Izmailov P. et al. Averaging weights leads to wider optima and better generalization // arXiv preprint arXiv:1803.05407. — 2018.
  * [Garipov et al., 2008] Garipov T. et al. Loss surfaces, mode connectivity, and fast ensembling of dnns // Proceedings of the 32nd International Conference on Neural Information Processing Systems. — 2018. — p. 8803-8812
  * [Smith, 2017] Smith L. N. Cyclical learning rates for training neural networks // 2017 IEEE winter conference on applications of computer vision (WACV). — IEEE, 2017. — p. 464-472.
  * [Zhao, Xie, Li, 2020] Zhao S. Y., Xie Y. P., Li W. J. Stagewise Enlargement of Batch Size for SGD-based Learning // arXiv preprint arXiv:2002.11601. — 2020.
  * [Devarakonda, Naumov, Garland, 2017] Devarakonda A., Naumov M., Garland M. Adabatch: Adaptive batch sizes for training deep neural networks // arXiv preprint arXiv:1712.02029. — 2017.
  * [Xiong et al., 2020] Xiong R. et al. On layer normalization in the transformer architecture // International Conference on Machine Learning. — PMLR, 2020. — p. 10524-10533.
  * [Hazan, Kale, 2014] Hazan E., Kale S. Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization // The Journal of Machine Learning Research. — 2014. — V. 15. — No. 1. — p. 2489-2512.