# Dual optical frequency combs with ultra-low relative phase jitters from 550
nm to 1020 nm for precision spectroscopy

Eunmi Chae  1,2 Kota Nakashima  3 and Kosuke Yoshioka1  1Photon Science
Center, School of Engineering, The University of Tokyo, Bunkyo-ku, Tokyo,
Japan
2Department of Physics, Korea University, Seongbuk-gu, Seoul, Republic of
Korea
3Department of Applied Physics,School of Engineering, The University of Tokyo,
Bunkyo-ku, Tokyo, Japan<EMAIL_ADDRESS>
###### Abstract

Here, ultra-low relative phase jitters over a wide optical spectrum were
achieved for dual Ti:Sapphire optical frequency combs. The two optical
frequency combs were independently phase-locked to a Sr optical lattice clock
laser delivered through a commercial optical fiber network. We confirmed that
the relative phase jitters between the two combs integrated from 8.3 mHz to
200 kHz were below 1 rad, corresponding to a relative linewidth of below 8.3
mHz, over the entire wavelength of the optical frequency combs ranging from
550 nm to 1020 nm. Our work paves the way for ultrahigh-precision dual-comb
spectroscopy covering a wide optical spectral range with a simple setup, and
provides an absolute optical frequency reference with great stability over a
wide range of wavelengths.

††journal: oe

##  1 Introduction

Ultra-precision spectroscopy plays a key role in modern science and
technology, such as tests of fundamental physics, establishment of an optical
frequency standard, and precision timing control for GPS and satellites [1, 2,
3, 4, 5, 6, 7, 8, 9]. A laser with a narrow linewidth and low phase noise is
essential for ultra-precision spectroscopy. Preparing an accurate optical
reference over a broad wavelength range is necessary to meet the wavelength
requirements of deep ultraviolet(UV) to far infrared(IR) applications. The
most advanced technique to reduce the phase noise of a laser is to lock the
laser to a high-finesse cavity. The state-of-the-art laser has a sub-10 mHz
linewidth when locked to a silicon Fabry-Pérot cavity at 124 K [10]. However,
high-finesse cavities are available only at a few selected wavelengths.
Moreover, the setup for such an ultrastable laser requires a considerable
amount of time and resources, including a temperature stabilized high-Q cavity
in a shielded environment. Therefore, it would be ideal to distribute the
phase coherence of an ultrastable optical reference over various locations and
optical frequencies.

Recent progress in optical fiber networks has enabled the distribution of the
stability of an ultrastable laser to remote locations with unprecedented
precision. In addition, by stabilizing an optical frequency comb (OFC) to the
received reference laser, all longitudinal modes of the OFC, over several
hundreds of THz, can acquire the same precision as the reference laser. In
previous research, we have shown that the stability of the optical reference
can be reproduced with no degradation by the Ti:Sapphire OFC under challenging
circumstances 1) in the presence of a 2.9 km-long fiber network and 2) without
the local phase-purification cavity [11]. A longitudinal mode of the OFC with
optical frequency nearing that of the optical reference was used to examine
the phase noise and stability of the OFC in the previous experiment. However,
the performance of other longitudinal modes, especially those with wavelengths
far from that of the optical reference, remains unclear. Every optical
frequency in a stabilized OFC is described as $f_{n}=f_{\mathrm{CEO}}+n\times
f_{\mathrm{rep}}$ ($f_{\mathrm{CEO}}$, $f_{\mathrm{rep}}$, and $n$ are the
carrier-envelope offset(CEO) frequency, the repetition frequency, and the mode
number, respectively). In principle, the phase noises of all longitudinal
modes of an OFC are tightly related to each other by this equation. In
practice, however, fluctuations of various physical origins can degrade the
phase coherence of the modes that are away from the reference optical
frequency. To validate the application of the OFC in ultra-precision
spectroscopy over a wide wavelength range, the phase noise has to be
characterized at various wavelengths. In a previous research on Ti:Sapphire
OFC, a sub-Hz residual linewidth was confirmed at the wavelength of 250 nm
away from the optical reference [12]. People have also demonstrated a fiber
OFC with sub-Hz linewidths over the wavelength range of 1200 nm to 1720 nm
[13].

Here, we studied the residual phase noise of the two OFCs by subjecting them
to optical heterodyne detection, within the 550 – 1020 nm wavelength range.
Two identical Ti:Sapphire OFCs were prepared and stabilized independently to a
remotely located Sr optical lattice clock laser through an optical fiber
network, as introduced in a previous study [11]. Heterodyne beatnotes were
detected at various wavelengths after overlapping the two pulse trains
precisely in the time domain. The phase noise on the detected heterodyne
beatnotes contains only the residual phase noise of the OFCs because the
common noise of the optical reference and the fiber network cancel out. The
residual linewidth of the OFCs over a wide range of wavelengths was analyzed
based on the detected phase noise.

##  2 Experimental Setup

Figure 1: Overall experimental setup. The optical standard at the remote lab
is delivered to the main lab through a telecom-wavelength fiber network (UT-
NET) with fiber noise cancellation. Two identical OFCs, with self-referenced
$f_{\mathrm{CEO}}$s, are phase-locked to the optical standard. The beat signal
of the two OFCs is measured at various optical wavelengths to evaluate the
residual phase noise of the OFCs. DM: dichroic mirror, BS: beam splitter, G:
grating, PD: photo detector, PCF: photonic crystal fiber.

The overall experimental setup consists of two Ti:Sapphire OFCs that are
phase-locked to a remotely placed ultrastable laser for Sr optical lattice
clock (optical standard) as described in ref. [11]. To measure the heterodyne
beat signal of the two OFCs, the timing of optical pulses from the two OFCs
are precisely synchronized. The residual linewidth of the OFC with respect to
the optical standard is evaluated by analyzing the phase noise of the
heterodyne beat signal.

###  2.1 Stabilization of the OFCs

The setup of this experiment comprises two areas (Fig.1). We constructed two
identical home-made Ti:Sapphire OFCs with output powers and repetition
frequencies ($f_{\textrm{rep}}$) of approximately 500 mW and 117 MHz,
respectively. After broadening the wavelength range using photonic crystal
fibers, the $f_{\mathrm{CEO}}$s of the OFCs were stabilized by a self-
referencing technique with an $f$–$2f$ interferometer. The optical standard,
an ultranarrow-linewidth laser at 698 nm for a Sr optical lattice clock[14],
was located at a laboratory (remote lab) about 1 km away from the location of
the two OFCs (main lab). The optical standard was delivered to the main lab
through a 2.9 km telecom-wavelength optical fiber network (UT-NET, University
of Tokyo), phase noise of which was actively canceled[11]. One of the
frequency modes of the OFCs was phase-locked to the delivered optical standard
with a frequency offset ($f_{\mathrm{off}}$). All RF signals and measurement
apparatus were referenced by a GPS-disciplined Rb standard. A detailed
description of the stabilization and its performance is given in ref. [11].

In the previous experiment, the out-of-loop measurement at the wavelength near
the optical standard detected the phase noise of 0.54 rad integrated from 8.3
mHz to 1.95 MHz with respect to the optical standard. However, this phase
noise included high-frequency phase noise from the repeater laser of the
optical standard. The high-frequency phase noise cannot be transfered to the
OFC due to the intrinsic narrow linewidth of the OFC. To correctly evaluate
the phase noise of the OFC itself, we constructed two identical OFCs and
evaluated the beat signal at various optical wavelengths.

###  2.2 Heterodyne beatnote measurement of the two OFCs

The optical paths of the two OFCs were carefully overlapped using a beam
splitter so that the two light waves would interfere with each other. The full
optical spectrum of the OFCs, 550 nm to 1100 nm, was used in the experiment.
To reduce the phase noise due to different optical paths, we minimized the
lengths of the out-of-loop path of the two OFCs. Once the paths were combined,
a grating was employed to separate the wavelengths of the OFCs. The phase
noise at selected wavelength ranges was measured by placing a slit between the
grating and a photo detector, as shown in Fig. 1. By rotating the grating, we
changed the wavelength range to $549$ nm, $650$ nm, $750$ nm, $850$ nm, $948$
nm, and $1020$ nm. The post-slit wavelength was measured using a spectrometer
(Thorlabs, CCS200/M).

The pulse width of the spectrally resolved OFCs was approximately 100 fs. For
the two OFCs to interfere, a prerequisite is to synchronize the arrival times
of the two OFC pulses within the pulse width. In other words, the two OFCs
should have the same $f_{\mathrm{rep}}$, and their repetition phases should
match. The frequency of the heterodyne beat ($f_{\mathrm{beat}}$) then appears
at $f_{\mathrm{CEO,1}}-f_{\mathrm{CEO,2}}$. To synchronize the timing of the
pulses, we first chose the $f_{\mathrm{CEO}}$s and $f_{\mathrm{off}}$s, and
match the $f_{\mathrm{rep}}$s of the two OFCs by properly chose the
longitudinal mode locked to the optical reference. We set
$f_{\mathrm{CEO,1}}=10.3$ MHz, $f_{\mathrm{off,1}}=-11.1$ MHz,
$f_{\mathrm{CEO,2}}=-11.1$ MHz, and $f_{\mathrm{off,2}}=10.3$ MHz, which
resulted in $f_{\mathrm{rep}}=117$ MHz and $f_{\mathrm{beat}}=21.4$ MHz.

Matching the repetition phases is more challenging because the path-length of
the two OFCs should be matched within a 30 $\mu$m - spatial spread of the
pulses. To achieve this, we intentionally changed the $f_{\mathrm{rep}}$ of
one OFC by a small amount ($<10$ mHz), thereby changing the relative position
between the two femtosecond pulses as a function of time. A heterodyne beat at
21.4 MHz appears when the two pulses overlap. At this moment, we immediately
set $f_{\mathrm{rep}}$ of the OFC back to its original value.

Figure 2: Heterodyne beat signal between the two OFCs at 650 nm. For
reference, the in-loop spectra of $f_{\mathrm{CEO}}$ and $f_{\mathrm{off}}$
are also shown. (a) Resolution bandwidth of 300 Hz, (b) resolution bandwidth
of 1 Hz.

##  3 Results and Discussions

The heterodyne beat signal at 650 nm, measured by a spectrum analyzer
(Anritsu, MS2720T), is shown in Fig. 2 (red lines). The signal was amplified
by 50 dB using a home-made amplifier. The center frequency of the beat was
21.4 MHz as expected. We observed a central coherent peak, the width of which
is limited by the instrumental resolution. The phase noise at low frequencies
matched very well the in-loop phase noise of $f_{\mathrm{off}}$. However, at
high frequencies, the heterodyne beat between the two OFCs had a significantly
smaller phase noise. This indicates that the high frequency noise of
$f_{\mathrm{off}}$ is from the repeater laser of the optical standard and is
rejected during the active stabilization process of the OFCs.

The phase noise of the heterodyne beat was also measured after subtracting the
carrier frequency to further examine the low frequency components of the phase
noise. We compared the phase of the heterodyne beat to a reference RF signal
generator using a home-made digital phase-frequency discriminator (DPFD). The
output of the DPFD was measured using an oscilloscope (Pico Technology,
PicoScope5244B) for 120 s with a sampling rate of 3. 9 MHz. The two OFCs were
phase-locked during the 120 s measurement without any phase jumps. However,
the output power fluctuation of the OFCs caused the phase signal of the
resulting beat to be discontinuous, at wavelength of 550 nm, 850 nm, 950 nm,
and 1020 nm. The discontinuities were especially significant for 550 nm and
1020 nm, corresponding to the edges of the OFC optical spectra. This occurred
when the input signal voltage was insufficient relative to the threshold
voltage of the DPFD. In order to concentrate only on the phase noise, we
selected time windows where there were no discontinuous phase signals for each
wavelength.

The power spectral density (PSD) of the phase noise obtained from the raw data
had a spectral range of 8.3 mHz to 1.95 MHz. We confirmed that there were
negligible amount of phase noise at frequencies higher than 20 kHz due to the
intrinsic narrow linewidth of our OFCs. Therefore, the PSD components upto 20
kHz were used for the following analysis.

The measured phase noise had a linear drift on the order of 0.003 rad/s, with
magnitude and sign varing over time. The drift is due to the different optical
paths between the OFCs. The drifts, as they are not essential for our
discussion, are excluded in the following analysis.

Figure 3: (a) PSD and (b) integrated phase jitters of the phase noise of the
heterodyne beat between the two OFCs at each wavelength.

The PSD of the phase noise of the beat between the two OFCs is shown in Fig. 3
(a). All PSDs display identical shapes over the entire wavelength range of
over an octave of the OFCs, indicating that no additional phase noise was
added to the OFCs. Below 1 Hz, the PSDs show the typical $1/f^{2}$ dependence
of white frequency modulation noise. Fig. 3 (b) shows the accumulated phase
noise of each OFC calculated from the PSDs. The accumulated phase is smaller
than 1 rad over the whole wavelength range. The residual linewidth of the OFC
is smaller than the resolution of our measurement, namely 8.3 mHz over the
whole wavelength range. The relative linewidth demonstrated here is two orders
of magnitudes narrower than the previous works [12, 13].

When the $f_{\textrm{CEO}}$ and $f_{\textrm{rep}}$ of an OFC are stabilized,
the phase accumulation should increase linearly as a function of the optical
frequency of the OFC [13]. As shown in Fig. 4, the accumulated phase jitters
at 1 Hz follow a linear function over most of the optical spectrum of the OFC.
The accumulated phase jitter at 298 THz ($=1020$ nm) is more pronounced than
expected because of the optical power fluctuation. The timing jitter of the
OFC pulses is estimated to be approximated 150 as from the accumulated phase
jitter.

Figure 4: Accumulated phase jitter at 1 Hz for various wavelengths. The black
dashed line indicates the linear guideline. The y intercept of this line is
zero. The frequency of the optical standard is represented by the gray
vertical dashed line.

The two tightly phase-locked OFCs with absolute frequency accuracy are
excellent tools for the precision dual-comb spectroscopy [15]. The optical
bandwidth and the frequency accuracy are the two features that describe the
performance of the dual comb spectroscopy. The optical bandwidth of the dual-
comb spectroscopy is determined by the ratio between the relative linewidth of
the two OFCs and $f_{\textrm{rep}}$. Therefore, it is important that the OFCs
maintain the narrow relative linewidth over a wide optical wavelength range to
achieve a large optical bandwidth. The frequency resolution and accuracy are
assured when the two OFCs have absolute frequency modes precisely described by
$f_{n}=f_{\mathrm{CEO}}+n\times f_{\mathrm{rep}}$ with narrow linewidth over
the whole spectral range. Our result with < 1 rad phase jitters over 120 s for
an octave-spanning wavelength range would allow dual comb spectroscopy with an
unprecedented precision.

Fig. 5 presents an example of a dual-comb spectroscopy signal using our OFCs.
We selected wavelengths in the 655 to 665 nm range (optical frequency with of
7.2 THz, approximately 60,000 frequency modes) using a band-pass filter for
this test. We set the $\Delta f_{\textrm{rep}}=26.45$ Hz in this measurement.
Fig. 5 (a) is the raw data taken by a photo detector with 150 MHz bandwidth
and a 16-bit digitizer with a 100 MHz sampling frequency. The interference
signals of the two OFCs appear every $1/26.45\sim 40$ ms. By zooming into Fig.
5 (b), the beat inside the pulse can be observed. The RF spectrum of the
signal is obtained by performing a Fourier transform of the data after
filtering out the noise during periods of no interference (Fig. 5 (c)). The
width of the RF spectrum is approximately 1 MHz, which is comparable to the
expected value of $26.45\times 60,000\sim 1.6$ MHz. Fig. 5 (d) shows the
zoomed in spectrum where one can clearly resolve each mode of the RF comb
separated by 26.45 Hz with the linewidth of 1 Hz limited by our measurement
time of 1 s. This confirms that our OFCs maintain their coherence over 1 s.

Figure 5: Demonstration of the dual-comb spectroscopy using our OFCs. Owing to
the long coherence time of the two OFCs, RF spectra with a 1 Hz linewidth
could be obtained. (a)(b): The raw interference data. (c)(d) The RF spectrum
obtained by Fourier transform. Every RF mode is separated by 26.45 Hz and the
linedwidth of 1 Hz is set by the measurement time.

##  4 Conclusion

We constructed two Ti:Sapphire OFCs that were tightly phase-locked to each
other over a wide wavelength range, 550 nm to 1020 nm. The two OFCs were
independently phase-locked to an optical lattice clock at 698 nm via an
optical fiber network. The phase noise at different wavelengths was measured
by taking heterodyne beatnotes of the two OFCs. The resulting phase jitters
displayed a linear increase as a function of the optical frequencies, which is
in agreement with the theoretical prediction. Phase jitters of less than 1 rad
integrated from 8.3 mHz to 20 kHz indicate that the two OFCs are tightly
phase-locked to each other over the entire wavelength range of 550 nm to 1020
nm. The phase jitter measured here corresponds to a relative linewidth of <
8.3 mHz and a timing jitter of 150 as. Our result shows that these OFCs
transfer the coherence of a remote optical clock laser with minimal phase
noise to arbitraty wavelengths in the range stated above. Therefore, the
referenced OFCs with the current setup can be utilized for state-of-the-art
precision spectroscopy over a wide wavelength range. Also, the ultrastable OFC
demonstrated here is an excellent tool for dual-comb spectroscopy covering a
wide optical bandwidth with a Hz-level resolution and a pump-probe measurement
with sub-fs timing resolution. It also provides an absolute optical frequency
reference with great stability over a wide range of wavelengths.

Funding This work is supported by Kakenhi (JP17H06205, JP20K05357), MEXT
Quantum Leap Flagship Program (MEXT QLEAP) Grant No. JPMXS0118067246, PRESTO,
JST (JPMJPR190B). E. C. is supported by NRF of Korea (2020R1A4A1018015,
2021R1C1C1009450) and a Korea University Grant (K2009521).

Acknowledgments The authors thank H. Katori for providing access to the clock
laser and other equipment in his laboratory. The authors appreciate the
technical support provided by T. Takano, N. Ohmae, T. Akatsuka and I.
Ushijima. The authors thank M. Kuwata-Gonokami for presenting the conceptual
idea in Advanced Photon Science Alliance (APSA) project that led to the
present work, and providing basic equipment at the OFC site.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

## References

  * [1] R. Altmann, L. Dreissen, E. Salumbides, W. Ubachs, and K. Eikema, “Deep-ultraviolet frequency metrology of H2 for tests of molecular quantum theory,” Physical Review Letters 120, 043204 (2018).
  * [2] J. Biesheuvel, J.-P. Karr, L. Hilico, K. S. E. Eikema, W. Ubachs, and J. C. J. Koelemeij, “Probing QED and fundamental constants through laser spectroscopy of vibrational transitions in HD+,” Nature Communications 7, 10385 (2016).
  * [3] J. K. Webb, J. A. King, M. T. Murphy, V. V. Flambaum, R. F. Carswell, and M. B. Bainbridge, “Indications of a spatial variation of the fine structure constant,” Physical Review Letters 107, 191101 (2011).
  * [4] N. Huntemann, B. Lipphardt, C. Tamm, V. Gerginov, S. Weyers, and E. Peik, “Improved limit on a temporal variation of mp/me from comparisons of Yb+ and Cs atomic clocks,” Physical Review Letters 113, 210802 (2014).
  * [5] R. Godun, P. Nisbet-Jones, J. Jones, S. King, L. Johnson, H. Margolis, K. Szymaniec, S. Lea, K. Bongs, and P. Gill, “Frequency ratio of two optical clock transitions in ${}^{1}71$Yb+ and constraints on the time variation of fundamental constants,” Physical Review Letters 113, 210801 (2014).
  * [6] C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland, “Optical clocks and relativity.” Science (New York, N.Y.) 329, 1630 (2010).
  * [7] C. Lisdat, G. Grosche, N. Quintin, C. Shi, S. Raupach, C. Grebing, D. Nicolodi, F. Stefani, A. Al-Masoudi, S. Dörscher, S. Häfner, J.-L. Robyr, N. Chiodo, S. Bilicki, E. Bookjans, A. Koczwara, S. Koke, A. Kuhl, F. Wiotte, F. Meynadier, E. Camisard, M. Abgrall, M. Lours, T. Legero, H. Schnatz, U. Sterr, H. Denker, C. Chardonnet, Y. Le Coq, G. Santarelli, A. Amy-Klein, R. Le Targat, J. Lodewyck, O. Lopez, and P.-E. Pottie, “A clock network for geodesy and fundamental science,” Nature Communications 7, 12443 (2016).
  * [8] A. Derevianko, “Atomic clocks and dark-matter signatures,” Journal of Physics: Conference Series 723, 012043 (2016).
  * [9] B. M. Roberts, G. Blewitt, C. Dailey, M. Murphy, M. Pospelov, A. Rollings, J. Sherman, W. Williams, and A. Derevianko, “Search for domain wall dark matter with atomic clocks on board global positioning system satellites,” Nature Communications 8, 1195 (2017).
  * [10] D. G. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. M. Robinson, J. Ye, F. Riehle, and U. Sterr, “1.5 $\mu$m lasers with sub-10 mHz linewidth,” Physical Review Letters 118, 263202 (2017).
  * [11] E. Chae, K. Sugiyama, K. Yoshioka, K. Nakashima, and T. Ikeda, “Direct phase-locking of a Ti:Sapphire optical frequency comb to a remote optical frequency standard,” Optics Express 27, 15649 (2019).
  * [12] A. Bartels, C. W. Oates, L. Hollberg, and S. A. Diddams, “Stabilization of femtosecond laser frequency combs with subhertz residual linewidths,” Optics Letters 29, 1081 (2004).
  * [13] W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock, and M. M. Fejer, “Fiber-laser frequency combs with subhertz relative linewidths,” Optics Letters 31, 3046 (2006).
  * [14] T. Takano, M. Takamoto, I. Ushijima, N. Ohmae, T. Akatsuka, A. Yamaguchi, Y. Kuroishi, H. Munekane, B. Miyahara, and H. Katori, “Geopotential measurements with synchronously linked optical lattice clocks,” Nature Photonics 10, 662 (2016).
  * [15] I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3, 414 (2016).


11institutetext: V. Novitskii 22institutetext: Moscow Institute of Physics and
Technology, Russia
22email<EMAIL_ADDRESS>33institutetext: A. Gasnikov
44institutetext: Moscow Institute of Physics and Technology, Russia
Institute for Information Transmission Problems RAS, Russia
Weierstrass Institute for Applied Analysis and Stochastics, Germany

# Improved Exploiting Higher Order Smoothness in Derivative-free Optimization
and Continuous Bandit††thanks: This work is based on results achieved by 63
Conference MIPT held in November 2020. The research of A. Gasnikov was
partially supported by the Ministry of Science and Higher Education of the
Russian Federation (Goszadaniye) 075-00337-20-03, project no. 0714-2020-0005.
The work of V. Novitskii was supported by Andrei M. Raigorodskii Scholarship
in Optimization.

Vasilii Novitskii  Alexander Gasnikov

(Received: date / Accepted: date)

###### Abstract

We consider $\beta$-smooth (satisfies the generalized Hölder condition with
parameter $\beta>2$) stochastic convex optimization problem with zero-order
one-point oracle. The best known result was akhavan2020exploiting :

$\mathbb{E}\left[f(\overline{x}_{N})-f(x^{*})\right]=\tilde{\mathcal{O}}\left(\dfrac{n^{2}}{\gamma
N^{\frac{\beta-1}{\beta}}}\right)$

in $\gamma$-strongly convex case, where $n$ is the dimension. In this paper we
improve this bound:

$\mathbb{E}\left[f(\overline{x}_{N})-f(x^{*})\right]=\tilde{\mathcal{O}}\left(\dfrac{n^{2-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{\beta}}}}{\gamma
N^{\frac{\beta-1}{\beta}}}\right).$

###### Keywords:

zeroth-order optimization convex problem stochastic optimization one-point
bandit smoothing kernel

##  1 Introduction

We study the problem of zero-order stochastic optimization in which the aim is
to minimize an unknown convex or strongly convex function where no gradient
realization is given but a function value is available at each iteration with
some additive noise $\xi$. We also study a closely related problem of
continuous stochastic bandits. These problems have received significant
attention in the literature (see akhavan2020exploiting ; bach2016highly ;
gasnikov2014stochastic ; gasnikov2015gradient ; gasnikov2017stochastic ;
zhang2020boosting ; bubeck2017kernel ; larson2019derivative-free ;
conn2009introduction ; spall2003introduction ) and are fundamental for many
application where the derivative of function is not available or it is hard to
calculate derivatives.

The goal of this paper is to exploit higher order smoothness of the function
to improve the performance of projected gradient-like algorithms. Our approach
is outlined in Algorithm 1, in which a sequential algorithm gets at each
iteration two function values under some noise. At each iteration the
algorithm gets function values at points $x_{k}+\delta_{k}$ and
$x_{k}-\delta_{k}$, where $\delta_{k}=\tau_{k}r_{k}e_{k}$. Here $r_{k}$ is
uniformly distributed random variable, $e_{k}$ is uniformly distributed on the
Euclidean sphere, $\tau_{k}$ – is tunable parameter of the algorithm, the
smaller $\tau_{k}$ is, the smaller approximation error of the gradient
$\|\widetilde{g_{k}}-\nabla f(x_{k})\|$ is (in this article we use only the
Euclidean norm) but the bigger variance of $\|\widetilde{g_{k}}\|$ is, so the
trade-off between these terms is needed. Our approach uses kernel smoothing
technique proposed by Polyak and Tsybakov in polyak1990optimal , this helps to
exploit higher order smoothness.

Algorithm 1 Zero-order Stochastic Projected Gradient

Requires:  Kernel $K:[-1,1]\rightarrow\mathbb{R}$, step size $\alpha_{k}>0$,
parameters $\tau_{k}$.

Initialization:  Generate scalars $r_{1},\dots,r_{N}$ uniformly on $[-1,1]$
and vectors $e_{1},\dots,e_{N}$ uniformly on the Euclidean unit sphere
$S_{n}=\\{e\in\mathbb{R}^{n}:\,\|e\|=1\\}$.

for $k=1,\dots,N$ do

  1. 1.

$y_{k}:=f(x_{k}+\tau_{k}r_{k}e_{k})+\xi_{k}$,
$y_{k}^{\prime}:=f(x_{k}-\tau_{k}r_{k}e_{k})+\xi_{k}^{\prime}$

  2. 2.

Define
$\widetilde{g_{k}}:=\frac{n}{2\tau_{k}}(y_{k}-y_{k}^{\prime})e_{k}K(r_{k})$

  3. 3.

Update $x_{k+1}:=\Pi_{Q}(x_{k}-\alpha_{k}\widetilde{g_{k}})$

end for

Output: $\left\\{x_{k}\right\\}_{k=1}^{N}$.

In algorithms like Algorithm 1 the two possibilities are usually considered.
The first one is to obtain a function value in one point with some noise
(”one-point” multi-armed bandit), the second is to observe function values in
two points with the same noise at each iteration (”two-point” multi-armed
bandit). The use of three and more points do not make dramatic difference to
the results for two points duchi2015optimal . Note that despite our algorithm
gets two function values for iteration, they are obtained with different noise
$\xi_{k}$ and $\xi^{\prime}_{k}$, so it is correct to regard Algorithm 1 one-
point and to compare it with one-point algorithms.

In this paper we study higher order smooth functions $f$ functions satisfying
the generalized Hölder condition with parameter $\beta>2$ (see inequality (1)
below).

We address the question: what is the performance of Algorithm 1, namely the
explicit dependency of the convergence rate on the main parameters $n$
(dimension), $N$, $\gamma$ (strong convexity parameter for strongly convex
functions), $\beta$. To handle this task we prove upper bound for Algorithm 1.

Contributions. Out main contributions can be summarized as follows:

  1. 1.

For strongly-convex case: under an adversarial noise assumption (see
Assumption 1) we establish for all $\beta>2$ the upper bound of order
$\mathcal{O}\left(\dfrac{n^{2-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{\beta}}}}{\gamma
N^{\frac{\beta-1}{\beta}}}\right)$ for the optimization error of Algorithm 1
for strongly convex case.

  2. 2.

For convex case: under an adversarial noise assumption (see Assumption 1) we
establish for all $\beta>2$ that after
$N(\varepsilon)=\mathcal{O}\left(\dfrac{n^{2+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{\beta-1}}}}{\varepsilon^{2+\frac{2}{\beta-1}}}\right)$
iterations of Algorithm 1 for the regularized function
$f_{\gamma}(x):=f(x)+\frac{\varepsilon}{2R^{2}}\|x-x_{0}\|^{2}$ we achieve the
optimization error less than or equal to $\varepsilon$.

For clarity we compare our results with state-of-the-art ones in Table 1
(dependence of optimization error $\varepsilon$ on the number of iteration
$N$, dimension $n$ and $\beta$, $\gamma$) and Table 2 (dependence of the
number of iteration $N$ on the optimization error $\varepsilon$, dimension $n$
and $\beta$, $\gamma$). To summarize the results we use
$\tilde{\mathcal{O}}()$ , where $\tilde{\mathcal{O}}()$ coincides with
$\mathcal{O}()$ up to the logarithmic factor.

Table 1: The dependence of optimization error ($\varepsilon$) on $N$ (number of iterations), $n$ (dimension), $\gamma$, $\beta$ | strongly convex | convex
---|---|---
lower bound akhavan2020exploiting  | $\mathcal{O}\left(\min\left(\dfrac{n}{\gamma N^{\frac{\beta-1}{\beta}}},\dfrac{n}{\sqrt{N}}\right)\right)$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{O}\left(\min\left(\dfrac{\sqrt{n}}{N^{\frac{\beta-1}{2\beta}}},\dfrac{n}{\sqrt{N}}\right)\right)}$
this work (2020) | $\tilde{\mathcal{O}}\left(\dfrac{n^{2-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{\beta}}}}{\gamma N^{\frac{\beta-1}{\beta}}}\right)$ | $\tilde{\mathcal{O}}\left(\dfrac{n^{1-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{2\beta}}}}{N^{\frac{\beta-1}{2\beta}}}\right)$
Akhavan, Pontil, Tsybakov (2020) akhavan2020exploiting  | $\tilde{\mathcal{O}}\left(\dfrac{n^{2}}{\gamma N^{\frac{\beta-1}{\beta}}}\right)$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\mathcal{O}}\left(\dfrac{n}{N^{\frac{\beta-1}{2\beta}}}\right)}$
Bach, Perchet (2016) bach2016highly  | $\mathcal{O}\left(\dfrac{n^{2-\frac{2}{\beta+1}}}{(\gamma N)^{\frac{\beta-1}{\beta+1}}}\right)$ | $\mathcal{O}\left(\dfrac{n^{1-\frac{1}{\beta+1}}}{N^{\frac{\beta-1}{2(\beta+1)}}}\right)$
Gasnikov and al. (2015), $\beta=2$, gasnikov2017stochastic  | $\tilde{\mathcal{O}}\left(\dfrac{n}{\sqrt{\gamma N}}\right)$ | $\tilde{\mathcal{O}}\left(\dfrac{\sqrt{n}}{N^{\nicefrac{{1}}{{4}}}}\right)$
Akhavan, Pontil, Tsybakov (2020), special case $\beta=2$ akhavan2020exploiting  | $\tilde{\mathcal{O}}\left(\dfrac{n}{\sqrt{\gamma N}}\right)$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\mathcal{O}}\left(\dfrac{\sqrt{n}}{N^{\nicefrac{{1}}{{4}}}}\right)}$
Zhang and al. (2020) zhang2020boosting  | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{O}\left(\dfrac{n}{\sqrt{\gamma N}}\right)}$ | $\mathcal{O}\left(\dfrac{\sqrt{n}}{N^{\nicefrac{{1}}{{4}}}}\right)$
Table 2: The dependence of $N$ (number of iterations) on $\varepsilon$, $n$ (dimension), $\gamma$, $\beta$ |  strongly convex | convex
---|---|---
lower bound akhavan2020exploiting  | $\mathcal{O}\left(\min\left(\dfrac{n^{1+\frac{1}{\beta-1}}}{(\gamma\varepsilon)^{\frac{\beta}{\beta-1}}},\dfrac{n^{2}}{\varepsilon^{2}}\right)\right)$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{O}\left(\min\left(\dfrac{n^{1+\frac{1}{\beta-1}}}{\varepsilon^{2+\frac{2}{\beta-1}}},\dfrac{n^{2}}{\varepsilon^{2}}\right)\right)}$
this work (2020) | $\tilde{\mathcal{O}}\left(\dfrac{n^{2+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{\beta-1}}}}{(\gamma\varepsilon)^{\frac{\beta}{\beta-1}}}\right)$ | $\tilde{\mathcal{O}}\left(\dfrac{n^{2+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{\beta-1}}}}{\varepsilon^{2+\frac{2}{\beta-1}}}\right)$
Akhavan, Pontil, Tsybakov (2020) akhavan2020exploiting  | $\tilde{\mathcal{O}}\left(\dfrac{n^{2+\frac{2}{\beta-1}}}{(\gamma\varepsilon)^{\frac{\beta}{\beta-1}}}\right)$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\mathcal{O}}\left(\dfrac{n^{2+\frac{2}{\beta-1}}}{\varepsilon^{2+\frac{2}{\beta-1}}}\right)}$
Bach, Perchet (2016) bach2016highly  | $\mathcal{O}\left(\dfrac{n^{2+\frac{2}{\beta-1}}}{\gamma\varepsilon^{\frac{\beta+1}{\beta-1}}}\right)$ | $\mathcal{O}\left(\dfrac{n^{2+\frac{2}{\beta-1}}}{\varepsilon^{2+\frac{2}{\beta-1}}}\right)$
Gasnikov and al. (2015), $\beta=2$ gasnikov2017stochastic  | $\tilde{\mathcal{O}}\left(\dfrac{n^{2}}{\gamma\varepsilon^{2}}\right)$ | $\tilde{\mathcal{O}}\left(\dfrac{n^{2}}{\varepsilon^{3}}\right)$
Akhavan, Pontil, Tsybakov (2020), special case $\beta=2$ akhavan2020exploiting  | $\tilde{\mathcal{O}}\left(\dfrac{n^{2}}{\gamma\varepsilon^{2}}\right)$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{\mathcal{O}}\left(\dfrac{n^{2}}{\varepsilon^{3}}\right)}$
Zhang and al. (2020) zhang2020boosting  | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{O}\left(\dfrac{n^{2}}{\gamma\varepsilon^{2}}\right)}$ | $\mathcal{O}\left(\dfrac{n^{2}}{\varepsilon^{3}}\right)$

Comments on Table 1 and Table 2.

  1. 1.

Note that in Table 1 and Table 2 the right column equals to the central one by
$\gamma\sim\varepsilon$.

  2. 2.

Note that the results of this work have better dependency $\varepsilon(N)$ or
$N(\varepsilon)$ than Gasnikov’s one-point method only if $\beta>2$ else
another technique in Theorem 3.1 is better (see gasnikov2017stochastic  or
Theorem 5.1 in akhavan2020exploiting ). The result in this work is achieved
using both kernel smoothing technique and measure concentration inequalities.

  3. 3.

The lower bound for strongly convex case is got under conditions $\gamma\geq
N^{-\nicefrac{{1}}{{2}}+\nicefrac{{1}}{{\beta}}}$ (otherwise it is better to
use convex methods) and (see akhavan2020exploiting )
$2\gamma\leq\max\limits_{x\in Q}\|\nabla f(x)\|$ .

  4. 4.

The bounds marked in blue are not given in this article and in references but
they can be got.

  5. 5.

Too optimistic bounds
$\mathcal{O}\left(\dfrac{n^{2-\frac{4}{\beta+1}}}{(\gamma
N)^{\frac{\beta-1}{\beta+1}}}\right)$ and
$\mathcal{O}\left(\dfrac{n^{2}}{\varepsilon^{2+\frac{2}{\beta-1}}}\right)$
were claimed in bach2016highly  instead of
$\mathcal{O}\left(\dfrac{n^{2-\frac{2}{\beta+1}}}{(\gamma
N)^{\frac{\beta-1}{\beta+1}}}\right)$ and
$\mathcal{O}\left(\dfrac{n^{2+\frac{2}{\beta-1}}}{\varepsilon^{2+\frac{2}{\beta-1}}}\right)$,
but Akhavan, Pontil and Tsybakov akhavan2020exploiting  found error in Lemma 2
in bach2016highly  where factor $d$ of dimension ($n$ in our notation) is
missing.

##  2 Preliminaries

In this section we give the necessary notation, definitions and assumptions.

###  2.1 Notation

Let $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ be the standard inner product
and Euclidean norm on $\mathbb{R}^{n}$ respectively. For every closed convex
set $Q\subset\mathbb{R}^{n}$ and for every $x\in\mathbb{R}^{n}$ let
$\Pi_{Q}(x)$ denote the Euclidean projection of $x$ on $Q$.

###  2.2 Problem

We address the conditional minimization problem

$f(x)\rightarrow\min_{x\in Q},$

where $f:U_{\varepsilon_{0}}(Q)\rightarrow\mathbb{R}$ – function (convex or
strongly convex), $Q\subset\mathbb{R}^{n}$ – convex compact set (Euclidean
metrics).

The optimization problem can be formulated as follows: find the sequence
$\\{x_{k}\\}_{k=1}^{N}\subset Q$ minimizing the average regret:

$\dfrac{1}{N}\sum\limits_{k=1}^{N}\mathbb{E}\left[f(x_{k})-f(x^{*})\right].$

If the average regret is less than or equal to $\varepsilon$ then the
optimization error of averaged estimator
$\overline{x}_{N}=\frac{1}{N}\sum\limits_{k=1}^{N}x_{k}$ is also less than or
equal to $\varepsilon$:

$\mathbb{E}\left[f(\overline{x}_{N})-f(x^{*})\right]\leq\dfrac{1}{N}\sum\limits_{k=1}^{N}\mathbb{E}\left[f(x_{k})-f(x^{*})\right]\leq\varepsilon.$

###  2.3 Noise

The function values $f(x_{k}+\tau_{k}r_{k}e_{k})$ and
$f(x_{k}-\tau_{k}r_{k}e_{k})$ are given with additive noise $\xi_{k}$ and
$\xi^{\prime}_{k}$ respectively (see Algorithm 1). Recall that the Algorithm 1
is randomized: the scalars $r_{1},\dots,r_{N}$ are distributed uniformly on
$[-1,1]$ and the vectors $e_{1},\dots,e_{N}$ are distributed uniformly on the
Euclidean unit sphere $S_{n}=\\{e\in\mathbb{R}^{n}:\,\|e\|=1\\}$.

###### Assumption 1

For all $k=1,2,\dots,N$ it holds that

  1. 1.

$\mathbb{E}[\xi_{k}^{2}]\leq\sigma^{2}$ and
$\mathbb{E}[{\xi^{\prime}}_{k}^{2}]\leq\sigma^{2}$ where $\sigma\geq 0$;

  2. 2.

the random variables $\xi_{k}$ and $\xi^{\prime}_{k}$ are independent from
$e_{k}$ and $r_{k}$, the random variables $e_{k}$ and $r_{k}$ are independent.

We do not assume here neither zero-mean of $\xi_{k}$ and $\xi^{\prime}_{k}$
nor i.i.d of $\\{\xi_{k}\\}_{k=1}^{N}$ and $\\{\xi^{\prime}_{k}\\}_{k=1}^{N}$
as condtition 2 from assumption 1 allows to avoid that.

###  2.4 Higher order smoothness

Let $l$ denote maximal integer number strictly less than $\beta$. Let ${\cal
F}_{\beta}(L)$ denote the set of all functions
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ which are differentiable $l$ times and
for all $x,z\in U_{\varepsilon_{0}}(Q)$ satisfy Hölder condition:

$\Biggl{|}f(z)-\sum_{0\leq|m|\leq
l}\dfrac{1}{m!}D^{m}f(x)(z-x)^{m}\Biggr{|}\leq L\|z-x\|^{\beta},$ (1)

where $L>0$, the sum is over multi-index
$m=(m_{1},\dots,m_{n})\in\mathbb{N}^{n}$, we use the notation
$m!=m_{1}!\cdot\dots\cdot m_{n}!$, $|m|=m_{1}+\dots+m_{n}$ and we defined

$D^{m}f(x)z^{m}=\dfrac{\partial^{|m|}f(x)}{\partial^{m_{1}}x_{1}\dots\partial^{m^{n}}x_{n}}z_{1}^{m_{1}}\cdot\dots\cdot
z_{n}^{m_{n}},\;\forall z=(z_{1},\dots,z_{n})\in\mathbb{R}^{n}.$

Let ${\cal F}_{\gamma,\beta}(L)$ denote the set of $\gamma$-strongly convex
functions $f\in{\cal F}_{\beta}(L)$. Recall that $f$ is called
$\gamma$-strongly convex for some $\gamma>0$ if for all $x,z\in\mathbb{R}^{n}$
it holds that $f(z)\geq f(x)+\langle\nabla
f(x),z-x\rangle+\frac{\gamma}{2}\|x-z\|^{2}$.

###  2.5 Kernel

For gradient estimator $\widetilde{g_{k}}$ we use the kernel

$K:[-1,1]\rightarrow\mathbb{R},$

satisfying

$\mathbb{E}[K(r)]=0,\,\mathbb{E}[rK(r)]=1,\,\mathbb{E}[r^{j}K(r)]=0,\,j=2,\dots,l,\,\mathbb{E}\left[|r|^{\beta}|K(r)|\right]\leq\infty,$
(2)

where $r$ is a uniformly distributed on $[-1,1]$ random variable. This helps
us to get better bounds on the gradient bias $\|\widetilde{g_{k}}-\nabla
f(x_{k})\|$ (see Theorem 3.1 for details).

A weighted sum of Legendre polynoms is an example of such kernels:

$K_{\beta}(r):=\sum\limits_{m=0}^{l(\beta)}p_{m}^{\prime}(0)p_{m}(r),$ (3)

where $l(\beta)$ is maximal integer number strictly less than $\beta$ and
$p_{m}(r)=\sqrt{2m+1}L_{m}(r)$, $L_{m}(u)$ is Legendre polynom. We have

$\mathbb{E}\left[p_{m}p_{m^{\prime}}\right]=\delta(m-m^{\prime}).$

As $\\{p_{m}(r)\\}_{m=0}^{j}$ is a basis for polynoms of degree less than or
equal to $j$ we can represent $u^{j}:=\sum\limits_{m=0}^{j}b_{m}p_{m}(r)$ for
some integers $\\{b_{m}\\}_{m=0}^{j}$ (they depend on $j$).

Let’s calculate the expectation

$\mathbb{E}\left[r^{j}K_{\beta}(r)\right]=\sum\limits_{m=0}^{j}b_{m}p_{m}^{\prime}(0)=(r^{j})^{\prime}|_{r=0}=\delta(j-1),$

here $\delta(0)=1$ and $\delta(x)=1$ if $x\neq 0$. We proved that the
presented $K_{\beta}(r)$ satisfies (2). We have the following kernels for
different betas (see Figure 1):

$\displaystyle K_{\beta}(r)=3r,\quad$ $\displaystyle\beta\in[2,3],$
$\displaystyle K_{\beta}(r)=\dfrac{15r}{4}(5-7r^{2}),\quad$
$\displaystyle\beta\in(3,5],$ $\displaystyle
K_{\beta}(r)=\dfrac{105r}{64}(99r^{4}-126r^{2}+35),\quad$
$\displaystyle\beta\in(5,7].$

Figure 1: Examples of kernels from (3)

For Theorem 3.1 and Theorem 3.2 we need to introduce the constants

$\kappa_{\beta}=\int|u|^{\beta}|K(u)|\,du$ (4)

and

$\kappa=\int K^{2}(u)\,du.$ (5)

It is proved in bach2016highly  that $\kappa_{\beta}$ and $\kappa$ do not
depend on $n$, they depend only on $\beta$:

$\kappa_{\beta}\leq 2\sqrt{2}(\beta-1),$ (6)
$\kappa\leq\sqrt{3}\beta^{\nicefrac{{3}}{{2}}}.$ (7)

##  3 Theorems

In this section we prove upper bounds on the optimization error of Algorithm 1
for strongly convex function (Theorem 3.1) and for convex function (Theorem
3.2).

###### Theorem 3.1

Let $f\in{\cal F}_{\gamma,\beta}(L)$ with $\gamma$, $L>0$ and $\beta>2$. Let
Assumption 1 hold and let $Q$ be a convex compact subset of $\mathbb{R}^{n}$.
Let $f$ be $G$-Lipschitz on the Euclidean $\tau_{1}$-neighborhood of $Q$.

Then the optimization error of averaged estimator
$\overline{x}_{N}=\frac{1}{N}\sum\limits_{k=1}^{N}x_{k}$ where the points
$x_{k}$ are given by Algorithm 1 with parameters

$\tau_{k}=\left(\dfrac{3\kappa\sigma^{2}n}{2(\beta-1)(\kappa_{\beta}L)^{2}}\right)^{\frac{1}{2\beta}}k^{-\frac{1}{2\beta}},\quad\alpha_{k}=\dfrac{2}{\gamma
k},\quad k=1,\dots,N$

satisfies

$\mathbb{E}\left[f(\overline{x}_{N})-f(x^{*})\right]\leq\dfrac{1}{\gamma}\left(n^{2-\frac{1}{\beta}}\dfrac{A_{1}}{N^{\frac{\beta-1}{\beta}}}+A_{2}\dfrac{n(1+\ln
N)}{N}\right),$

where
$A_{1}=3\beta(\kappa\sigma^{2})^{\frac{\beta-1}{\beta}}(\kappa_{\beta}L)^{\frac{2}{\beta}}$,
$A_{2}=c^{*}\kappa G^{2}$, $\kappa_{\beta}$ and $\kappa$ are constants
depending only on $\beta$, see (4) and (5).

###### Proof

Step 1. Fix an arbitrary $x\in Q$. As $x_{k+1}$ is the Euclidean projection we
have $\|x_{k+1}-x\|^{2}\leq\|x_{k}-\alpha_{k}\widetilde{g_{k}}-x\|^{2}$ which
is equivalent to

$\langle\widetilde{g_{k}},x_{k}-x\rangle\leq\dfrac{\|x_{k}-x\|^{2}-\|x_{k+1}-x\|^{2}}{2\alpha_{k}}+\dfrac{\alpha_{k}}{2}\|\widetilde{g_{k}}\|^{2}.$
(8)

By the strong convexity assumption we have

$f(x_{k})-f(x)\leq\langle\nabla
f(x_{k}),x_{k}-x\rangle-\dfrac{\gamma}{2}\|x_{k}-x\|^{2}.$ (9)

Combining the last two inequations we obtain

$\begin{split}f(x_{k})-f(x)\leq&\langle\nabla
f(x_{k})-\widetilde{g_{k}},x_{k}-x\rangle+\dfrac{\|x_{k}-x\|^{2}-\|x_{k+1}-x\|^{2}}{2\alpha_{k}}\\\
&+\dfrac{\alpha_{k}}{2}\|\widetilde{g_{k}}\|^{2}-\dfrac{\gamma}{2}\|x_{k}-x\|^{2}.\\\
\end{split}$ (10)

Taking conditional expectation given $x_{k}$ with respect to $r_{k}$,
$\xi_{k}$ and $\xi^{\prime}_{k}$ we obtain

$\begin{split}f(x_{k})-f(x)\leq&\langle\nabla
f(x_{k})-\mathbb{E}\left[\widetilde{g_{k}}|x_{k}\right],x_{k}-x\rangle+\dfrac{\alpha_{k}}{2}\mathbb{E}\left[\|\widetilde{g_{k}}\|^{2}|x_{k}\right]\\\
&+\dfrac{\|x_{k}-x\|^{2}-\mathbb{E}\left[\|x_{k+1}-x\|^{2}|x_{k}\right]}{2\alpha_{k}}-\dfrac{\gamma}{2}\|x_{k}-x\|^{2}.\\\
\end{split}$ (11)

Step 2 (Bounding bias term). Our aim is to bound the first term in (11),
namely $\langle\nabla
f(x_{k})-\mathbb{E}\left[\widetilde{g_{k}}|x_{k}\right],x_{k}-x\rangle$. Using
the Taylor expansion we have

$\begin{split}f\left(x_{k}+\tau_{k}r_{k}e_{k}\right)=&f(x_{k})+\langle\nabla
f(x_{k}),\tau_{k}r_{k}e_{k}\rangle\\\ +&\sum_{2\leq|m|\leq
l}\dfrac{(\tau_{k}r_{k})^{|m|}}{m!}D^{(m)}f(x_{k})e_{k}^{m}+R(\tau_{k}r_{k}e_{k}),\end{split}$
(12)

where by assumption $|R(\tau_{k}r_{k}e_{k})|\leq
L\|\tau_{k}r_{k}e_{k}\|^{\beta}=L(\tau_{k}\cdot|r_{k}|)^{\beta}$. Thus,

$\begin{split}\widetilde{g_{k}}=&\Bigl{(}\langle\nabla
f(x_{k}),\tau_{k}r_{k}e_{k}\rangle+\sum_{2\leq|m|\leq l,|m|\text{
odd}}\dfrac{(\tau_{k}r_{k})^{|m|}}{m!}D^{(m)}f(x_{k})e_{k}^{m}\\\
&+\frac{1}{2}R(\tau_{k}r_{k}e_{k})-\frac{1}{2}R(-\tau_{k}r_{k}e_{k})+\xi_{k}-\xi^{\prime}_{k}\Bigl{)}\dfrac{n}{\tau_{k}}K(r_{k})e_{k}.\end{split}$
(13)

Using the properties of the smoothing kernel $K$, independence of $e_{k}$ and
$r_{k}$ (Assumption 1) and the fact that
$\mathbb{E}\left[e_{k}e_{k}^{T}\right]=\frac{1}{n}\mathbb{I}_{n\times n}$ we
obtain

$\mathbb{E}_{e_{k},r_{k}}\left[\left\langle\nabla
f(x_{k}),\tau_{k}r_{k}e_{k}\right\rangle\dfrac{n}{\tau_{k}}K(r_{k})e_{k}\big{|}x_{k}\right]=\nabla
f(x_{k}).$ (14)

Using the fact that $\mathbb{E}\left[r_{k}^{|m|}K(r_{k})\right]=0$ if
$2\leq|m|\leq l$ or $|m|=0$ and Assumption 1 we have

$\Bigl{(}\sum_{2\leq|m|\leq l,|m|\text{
odd}}\dfrac{(\tau_{k}r_{k})^{|m|}}{m!}D^{(m)}f(x_{k})e_{k}^{m}+\xi_{k}-\xi^{\prime}_{k}\Bigl{)}\dfrac{n}{\tau_{k}}K(r_{k})e_{k}=0.$
(15)

Combining (13), (14) and (15) and using the definition of $\kappa_{\beta}$ we
obtain

$\left|\langle\nabla
f(x_{k})-\mathbb{E}\left[\widetilde{g_{k}}|x_{k}\right],x_{k}-x\rangle\right|=\\\
=\left|\mathbb{E}\left[\left(\frac{1}{2}R(\tau_{k}r_{k}e_{k})-\frac{1}{2}R(-\tau_{k}r_{k}e_{k})\right)\dfrac{n}{\tau_{k}}K(r_{k})\langle
e_{k},x_{k}-x\rangle\middle|x_{k}\right]\right|\\\ \leq
L\tau_{k}^{\beta-1}\cdot\mathbb{E}_{r_{k}}\left[|r_{k}|^{\beta}K(r_{k})\right]\cdot
n\left|\mathbb{E}_{e_{k}}\left[\langle
e_{k},x_{k}-x\rangle\middle|x_{k}\right]\right|\\\
\leq\kappa_{\beta}L\sqrt{n}\tau_{k}^{\beta-1}\|x_{k}-x\|,\\\ $ (16)

where in the last inequality the fact that $\left|\mathbb{E}_{e}\left[\langle
e,s\rangle\right]\right|^{2}\leq\mathbb{E}_{e}\left[\langle
e,s\rangle^{2}\right]=\frac{\|s\|^{2}}{n}$ was used (the fact from
concentration measure theory). Applying the inequality
$ab\leq\nicefrac{{1}}{{2}}(a^{2}+b^{2})$ to the last expression in (16) we
finally get

$\left|\langle\nabla
f(x_{k})-\mathbb{E}\left[\widetilde{g_{k}}|x_{k}\right],x_{k}-x\rangle\right|\leq\dfrac{(\kappa_{\beta}L)^{2}}{\gamma}n\tau_{k}^{2(\beta-1)}+\dfrac{\gamma}{4}\|x_{k}-x\|^{2}.$
(17)

Step 3 (Bounding second moment of gradient estimator).  Our aim is to estimate
$\mathbb{E}\left[\|\widetilde{g_{k}}\|^{2}|x_{k}\right]$ which is the second
term in (11). The expectation here is with respect to $r_{k}$, $\xi_{k}$ and
$\xi^{\prime}_{k}$. To lighten the presentation and withous loss of generality
we drop the lower script $k$ in all quantities.

We have

$\begin{split}\|\widetilde{g}\|^{2}=&\dfrac{n^{2}}{4\tau^{2}}\|(f(x+\tau
re)-f(x-\tau re)+\xi-\xi^{\prime})K(r)e\|^{2}\\\
=&\dfrac{n^{2}}{4\tau^{2}}\left((f(x+\tau re)-f(x-\tau
re)+\xi-\xi^{\prime})\right)^{2}K^{2}(r).\\\ \end{split}$ (18)

Using the inequality $(a+b+c)^{2}\leq 3(a^{2}+b^{2}+c^{2})$ and Assumption 1
we get

$\mathbb{E}\left[\|\widetilde{g}\|^{2}|x\right]\leq\dfrac{3n^{2}}{4\tau^{2}}\left(\mathbb{E}\left[(f(x+\tau
re)-f(x-\tau re))^{2}K^{2}(r)\middle|x\right]+2\kappa\sigma^{2}\right).$ (19)

Lemma 9 in shamir2017optimal  states that for any function $f$ which is
$G-Lipschitz$ with respect to 2-norm, it holds that if $e$ is uniformly
distributed on the Euclidean unit sphere, then

$\sqrt{\mathbb{E}\left[(f(e)-\mathbb{E}[f(e)])^{4}\right]}\leq\dfrac{cG^{2}}{n},$
(20)

where $c<3$ is a positive numerical constant.

Using (20), symmetry of Euclidean unit sphere and the inequality $(a+b)^{2}\
\leq 2(a^{2}+b^{2})$ we obtain

$\mathbb{E}\left[\left(f(x+e)-f(x-e)\right)^{2}\middle|x\right]=\mathbb{E}_{e}\left[\left(f(x+e)-f(x-e)\right)^{2}\right]\\\
\leq\mathbb{E}_{e}\left[\left(\left(f(x+e)-\mathbb{E}_{e}[f(x+e)]\right)-\left(f(x-e)-\mathbb{E}_{e}[f(x-e)]\right)\right)^{2}\right]\\\
\leq
2\mathbb{E}_{e}\left[\left(f(x+e)-\mathbb{E}_{e}[f(x+e)]\right)^{2}\right]+2\mathbb{E}_{e}\left[\left(f(x-e)-\mathbb{E}_{e}[f(x-e)]\right)^{2}\right]\\\
\leq
2\sqrt{\mathbb{E}_{e}\left[\left(f(x+e)-\mathbb{E}_{e}[f(x+e)]\right)^{4}\right]}+2\sqrt{\mathbb{E}_{e}\left[\left(f(x-e)-\mathbb{E}_{e}[f(x-e)]\right)^{4}\right]}\\\
\leq\dfrac{4cG^{2}}{n},$ (21)

so we have

$\mathbb{E}\left[\left(f(x+\tau re)-f(x-\tau
re)\right)^{2}\middle|x\right]\leq\dfrac{4c(\tau
r)^{2}G^{2}}{n}\leq\dfrac{4c\tau^{2}G^{2}}{n}.$ (22)

By substituting (22) into (19), using independence of $e$ and $r$ and
returning the lower script $k$ we finally get

$\mathbb{E}\left[\|\widetilde{g_{k}}\|^{2}|x\right]\leq\kappa\left(c^{*}nG^{2}+\dfrac{3(n\sigma)^{2}}{2\tau_{k}^{2}}\right),$
(23)

where $c^{*}=3c$.

Step 4.  Let $\rho_{k}^{2}$ denote $\mathbb{E}[\|x_{k}-x\|^{2}]$. Substituting
(17) and (23) into (11), taking full expectation and summing over $k$ we
obtain

$\begin{split}\sum_{k=1}^{N}\mathbb{E}[f(x_{k})-f(x)]\leq&\sum_{k=1}^{N}\left(\dfrac{(\kappa_{\beta}L)^{2}}{\gamma}n\tau_{k}^{2(\beta-1)}+\dfrac{\alpha_{k}}{2}\kappa\left(c^{*}nG^{2}+\dfrac{3(n\sigma)^{2}}{2\tau_{k}^{2}}\right)\right)\\\
&+\sum_{k=1}^{N}\left(\dfrac{\rho_{k}^{2}-\rho_{k+1}^{2}}{2\alpha_{k}}-\left(\dfrac{\gamma}{2}-\dfrac{\gamma}{4}\right)\rho_{k}^{2}\right).\\\
\end{split}$ (24)

Let $\rho_{N+1}^{2}=0$. Then setting $\alpha_{k}=\dfrac{2}{\gamma k}$ yields

$\begin{split}\sum_{k=1}^{N}\left(\dfrac{\rho_{k}^{2}-\rho_{k+1}^{2}}{2\alpha_{k}}-\dfrac{\gamma}{4}\rho_{k}^{2}\right)&\leq\rho_{1}^{2}\left(\dfrac{1}{2\alpha_{1}}-\dfrac{\gamma}{4}\right)+\sum_{k=2}^{N+1}\rho_{k}^{2}\left(\dfrac{1}{2\alpha_{k}}-\dfrac{1}{2\alpha_{k-1}}-\dfrac{\gamma}{4}\right)\\\
&=\rho_{1}^{2}\left(\dfrac{\gamma}{4}-\dfrac{\gamma}{4}\right)+\sum_{k=2}^{N+1}\rho_{k}^{2}\left(\dfrac{\gamma}{4}-\dfrac{\gamma}{4}\right)=0.\end{split}$
(25)

Substituting (25) into (24) with $\alpha_{k}=\frac{2}{\gamma k}$ we obtain

$\begin{split}\sum_{k=1}^{N}\mathbb{E}[f(x_{k})-f(x)]&\leq\dfrac{1}{\gamma}\sum_{k=1}^{N}\left((\kappa_{\beta}L)^{2}n\tau_{k}^{2(\beta-1)}+\kappa\left(c^{*}nG^{2}+\dfrac{3(n\sigma)^{2}}{2\tau_{k}^{2}}\right)\dfrac{1}{k}\right)\\\
&=\dfrac{1}{\gamma}\sum_{k=1}^{N}\left(\left[n\cdot(\kappa_{\beta}L)^{2}\tau_{k}^{2(\beta-1)}+n^{2}\cdot\dfrac{3\kappa\sigma^{2}}{2k\tau_{k}^{2}}\right]+\dfrac{c^{*}\kappa
nG^{2}}{k}\right).\\\ \end{split}$ (26)

If $\sigma>0$ then
$\tau_{k}={\left(\dfrac{3\kappa\sigma^{2}n}{2(\beta-1)(\kappa_{\beta}L)^{2}}\right)}^{\frac{1}{2\beta}}k^{-\frac{1}{2\beta}}$
is the minimizer of square brackets. Plugging this $\tau_{k}$ in (26) and
using two inequalities: for the expression in square brackets
$\sum\limits_{k=1}^{N}k^{-1+\nicefrac{{1}}{{\beta}}}\leq\beta
N^{\nicefrac{{1}}{{\beta}}}$ (if $\beta>2$) and for the term after square
brackets $\sum\limits_{k=1}^{N}\frac{1}{k}\leq 1+\ln N$ we get

$\sum_{k=1}^{N}\mathbb{E}[f(x_{k})-f(x)]\leq\dfrac{1}{\gamma}\left(n^{2-\frac{1}{\beta}}A_{1}N^{\frac{1}{\beta}}+A_{2}n(1+\ln{N})\right)$
(27)

with $A_{1}$ and $A_{2}$ from the formulation of Theorem 3.1. Due to the
convexity of $f$ we finally prove the theorem

$\mathbb{E}\left[f(\overline{x}_{N})-f(x^{*})\right]\leq\dfrac{1}{\gamma}\left(n^{2-\frac{1}{\beta}}\dfrac{A_{1}}{N^{\frac{\beta-1}{\beta}}}+A_{2}\dfrac{n(1+\ln{N})}{N}\right).$
(28)

$\square$

We emphasize that the usage of kernel smoothing technique, measure
concentration inequalities and the assumption that $\xi_{k}$ is independent
from $e_{k}$ or $r_{k}$ (Assumption 1) lead to the results better than the
state-of-the-art ones for $\beta>2$ (see Table 1 and Table 2). The last
assumption also allows us not to assume neither zero-mean of $\xi_{k}$ and
$\xi^{\prime}_{k}$ nor i.i.d of $\\{\xi_{k}\\}_{k=1}^{N}$ and
$\\{\xi^{\prime}_{k}\\}_{k=1}^{N}$.

###### Theorem 3.2

Let $f\in{\cal F}_{\beta}(L)$ with $\gamma$, $L>0$ and $\beta>2$. Let
Assumption 1 hold and let $Q$ be a convex compact subset of $\mathbb{R}^{n}$.
Let $f$ be $G$-Lipschitz on the Euclidean $\tau_{1}$-neighborhood of $Q$. Let
$\overline{x}_{N}$ denote $\frac{1}{N}\sum\limits_{k=1}^{N}x_{k}$.

Then we achieve the optimization error
$\mathbb{E}\left[f(\overline{x}_{N})-f(x^{*})\right]\leq\varepsilon$ after
$N(\varepsilon)$ steps of Algorithm 1 with settings from Theorem 3.1 for the
regularized function: $f_{\gamma}(x):=f(x)+\frac{\gamma}{2}\|x-x_{0}\|^{2}$,
where $\gamma\leq\frac{\varepsilon}{R^{2}}$, $R=\|x_{0}-x^{*}\|$, $x_{0}\in Q$
– arbitrary point.

$N(\varepsilon)=\max\left\\{\left(R\sqrt{2A_{1}}\right)^{\frac{2\beta}{\beta-1}}\dfrac{n^{2+\frac{1}{\beta-1}}}{\varepsilon^{2+\frac{2}{\beta-1}}},\left(R\sqrt{2c^{\prime}A_{2}}\right)^{2(1+\rho)}\dfrac{n^{1+\rho}}{\varepsilon^{2(1+\rho)}}\right\\},$

where
$A_{1}=3\beta(\kappa\sigma^{2})^{\frac{\beta-1}{\beta}}(\kappa_{\beta}L)^{\frac{2}{\beta}}$,
$A_{2}=c^{*}\kappa G^{2}$ – constants from Theorem 3.1, $\rho>0$ – arbitrarily
small positive number.

###### Proof

Step 1. Let $x^{*}$ and $x_{\gamma}^{*}$ denote $\arg\min\limits_{x\in Q}f(x)$
and $\arg\min\limits_{x\in Q}f_{\gamma}(x)$ respectively. Setting
$\gamma=\frac{\varepsilon}{R^{2}}$ and using the inequality
$f_{\gamma}(x_{\gamma}^{*})\leq f_{\gamma}(x^{*})$ we obtain

$\begin{split}f(\overline{x}_{N})-f(x^{*})&=f_{\gamma}(\overline{x}_{N})-f_{\gamma}(x^{*})-\dfrac{\gamma}{2}\|\overline{x}_{N}-x_{0}\|^{2}+\dfrac{\gamma}{2}\|x^{*}-x_{0}\|^{2}\\\
&\leq
f_{\gamma}(\overline{x}_{N})-f_{\gamma}(x^{*})+\dfrac{\gamma}{2}\|x^{*}-x_{0}\|^{2}\\\
&\leq
f_{\gamma}(\overline{x}_{N})-f_{\gamma}(x_{\gamma}^{*})+\dfrac{\varepsilon}{2}.\\\
\end{split}$ (29)

Step 2. Now we apply Theorem 3.1 for $f_{\gamma}(x)$ and bound RHS by
$\frac{\varepsilon}{2}$:

$\mathbb{E}\left[f_{\gamma}(\overline{x}_{N})-f_{\gamma}(x^{*})\right]\leq\dfrac{1}{\gamma}\left(n^{2-\frac{1}{\beta}}\dfrac{A_{1}}{N^{\frac{\beta-1}{\beta}}}+A_{2}\dfrac{n(1+\ln{N})}{N}\right)\leq\dfrac{\varepsilon}{2}.$
(30)

The inequality (30) is done if ($\gamma=\frac{\varepsilon}{R^{2}}$)

$\max\left\\{n^{2-\frac{1}{\beta}}\dfrac{A_{1}}{N^{\frac{\beta-1}{\beta}}},A_{2}\dfrac{n(1+\ln{N})}{N}\right\\}\leq\dfrac{\gamma\varepsilon}{2}=\dfrac{\varepsilon^{2}}{2R^{2}}.$
(31)

It is true that $1+\ln N\leq c^{\prime}N^{\frac{\rho}{\rho+1}}$ for some
$c^{\prime}>0$. So the inequality (31) holds if

$N\geq\max\left\\{\left(R\sqrt{2A_{1}}\right)^{\frac{2\beta}{\beta-1}}\dfrac{n^{2+\frac{1}{\beta-1}}}{\varepsilon^{2+\frac{2}{\beta-1}}},\left(R\sqrt{2c^{\prime}A_{2}}\right)^{2(1+\rho)}\dfrac{n^{1+\rho}}{\varepsilon^{2(1+\rho)}}\right\\}.$
(32)

The inequalities (29) and (30) yield
$\mathbb{E}\left[f(\overline{x}_{N})-f(x^{*})\right]\leq\varepsilon$.

$\square$

##  4 Numerical Experiments

In our experiment novitskii  we compare the Algorithm 1 (with $\beta=3$ and
$\beta=5$) proposed in this paper with Gasnikov’s one-point method and with
Akhavan’s method for the special case $\beta=2$.

We consider the problem of the minimization of the quadratic function

$f(x)=\dfrac{1}{4}x_{1}^{2}+x_{2}^{2}+4x_{3}^{2}$

on the Euclidean ball $Q=\\{x\in\mathbb{R}^{3}:\,\|x\|\leq 1\\}$.

The starting point is $x_{0}$ with $\|x_{0}\|=\nicefrac{{1}}{{2}}$. The
dependency of $f(\overline{x}_{N})-f(x^{*})$ (optimization error) on $N$
(iteration number) is presented on the Figure 2. The optimization error has
its mean and 0.95-confidence interval. As the Lipshitz constants for the
quadratic oracle are equal to zero, for the Algorithm 1 we choose $L=0.01$.

Figure 2: Dependency of optimization error of Algorithm 1 on iteration

We see on the Figure 2 that the usage of higher-order smoothness by Algorithm
1 helps to overcome the methods which do not use this.

##  5 Discussion and related work

The results of this paper can be generalized for the saddle-point problems.
Recently GANs and Reinforcement Learning caused a big interest for saddle-
point problems, see sadiev2020zeroth .

Another possible genelization of this paper is obtaining the large probability
bounds for optimization error. We cannot obtain upper bounds in terms of of
large deviation probability (not in terms of expectation) under the Assumption
1. The exploiting of higher order smoothness with the help of kernels under
rather general noise assumptions (non-zero mean) causes big variation
$\|\widetilde{g_{k}}-\nabla f(x_{k})\|$ and this can causes the problems with
large deviation probability rates.

It remains an open question whether large deviation probability can be
obtained under non-zero mean noise. And also it remains an open question
whether better dependence of optimization error on the dimenstion $n$ and
strong convexity parameter $\gamma$ can be obtained.

Acknowledgements. We would like to thank Alexandre B. Tsybakov for helpful
remarks about Tables 1 and 2.

## References

  * (1) Akhavan, A., Pontil, M., Tsybakov, A.B.: Exploiting higher order smoothness in derivative-free optimization and continuous bandits.  arXiv preprint arXiv:2006.07862 (2020)
  * (2) Bach, F., Perchet, V.: Highly-smooth zero-th order online optimization.  In: Conference on Learning Theory, pp. 257–283 (2016)
  * (3) Bubeck, S., Lee, Y.T., Eldan, R.: Kernel-based methods for bandit convex optimization.  In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 72–85 (2017)
  * (4) Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization.  Society for Industrial and Applied Mathematics (2009).  DOI 10.1137/1.9780898718768
  * (5) Duchi, J.C., Jordan, M.I., Wainwright, M.J., Wibisono, A.: Optimal rates for zero-order convex optimization: The power of two function evaluations.  IEEE Transactions on Information Theory 61(5), 2788–2806 (2015)
  * (6) Gasnikov, A., Dvurechensky, P., Kamzolov, D.: Gradient and gradient-free methods for stochastic convex optimization with inexact oracle.  arXiv preprint arXiv:1502.06259 (2015)
  * (7) Gasnikov, A., Dvurechensky, P., Nesterov, Y.: Stochastic gradient methods with inexact oracle.  arXiv preprint arXiv:1411.4218 (2014)
  * (8) Gasnikov, A.V., Krymova, E.A., Lagunovskaya, A.A., Usmanova, I.N., Fedorenko, F.A.: Stochastic online optimization. single-point and multi-point non-linear multi-armed bandits. convex and strongly-convex case.  Automation and remote control 78(2), 224–234 (2017)
  * (9) Larson, J., Menickelly, M., Wild, S.M.: Derivative-free optimization methods.  Acta Numerica 28, 287–404 (2019).  DOI 10.1017/S0962492919000060
  * (10) Novitskii, V.: Zeroth-order algorithms for smooth saddle-point problems (2020).  URL https://cutt.ly/bjxQHRY
  * (11) Polyak, B.T., Tsybakov, A.B.: Optimal order of accuracy of search algorithms in stochastic optimization.  Problemy Peredachi Informatsii 26(2), 45–53 (1990)
  * (12) Sadiev, A., Beznosikov, A., Dvurechensky, P., Gasnikov, A.: Zeroth-order algorithms for smooth saddle-point problems.  arXiv preprint arXiv:2009.09908 (2020)
  * (13) Shamir, O.: An optimal algorithm for bandit and zero-order convex optimization with two-point feedback.  The Journal of Machine Learning Research 18(1), 1703–1713 (2017)
  * (14) Spall, J.C.: Introduction to Stochastic Search and Optimization, 1 edn.  John Wiley & Sons, Inc., New York, NY, USA (2003)
  * (15) Zhang, Y., Zhou, Y., Ji, K., Zavlanos, M.M.: Boosting one-point derivative-free online optimization via residual feedback.  arXiv preprint arXiv:2010.07378 (2020)


# Thin-Wall Monopoles in a False Vacuum

M. B. Paranjape1<EMAIL_ADDRESS>Yash Saxena2<EMAIL_ADDRESS>1Groupe de physique des particules, Département de physique, Centre de
recherche mathématiques and Institut Courtois, Université de Montréal, C.P.
6128, Succ. Centre-ville, Montréal, Québec, CANADA, H3C 3J7  2 Department of
Physics and Astrophysics, University of Delhi, Delhi, India 110007

###### Abstract

We study numerically the existence in a false vacuum, of magnetic monopoles
which are “thin-walled”, i.e., which correspond to a spherical region of
radius $R$ that is essentially trivial surrounded by a wall of thickness
$\Delta\ll R$, hence the name thin wall, and finally an exterior region that
essentially corresponds to a pure Abelian magnetic monopole. Such monopoles
were dubbed false monopoles and can occur in non-abelian gauge theories where
the symmetry-broken vacuum is actually the false vacuum. This idea was first
proposed in Kumar et al. (2010), however, their proof of the existence of
false monopoles was incorrect. Here we fill this lacuna and demonstrate
numerically the existence of thin-wall false monopoles. The decay via quantum
tunnelling of the false monopoles could be of importance to cosmological
scenarios which entertain epochs in which the universe is trapped in a
symmetry broken false vacuum.

###### pacs:

12.60.Jv,11.27.+d

††preprint: UdeM-GPP-TH-23-300

##  I Introduction

Magnetic monopoles correspond to soliton solutions of non-abelian gauge field
theories with scalar matter fields, where a simply connected simple gauge
group symmetry is spontaneously broken to a smaller gauge symmetry where the
gauge group has a $U(1)$ factor. The topological possibility of the existence
of monopoles requires that the homotopy group $\Pi_{2}(G/H)$ be non-trivial
Goddard and Olive (1978). It is well understood that,
$\Pi_{2}(G/H)=\Pi_{1}(H)=\mathbb{Z}$, and the integer characterizing the
configuration is the monopole charge. However, it can be that the symmetry
breaking is temporary, that the symmetry breaking vacuum is a false vacuum
that will eventually decay to the true vacuum. Such a scenario adds new
dynamical aspects to the monopole solution. The universe can be trapped in
this false vacuum phase for an interminably long period. Indeed, cosmological
solutions coming from string theory Kachru et al. (2003) in fact propose that
the present universe is in a false vacuum, that in principle has a very, very
long lifetime (hence the justification that it has survived for 13 billion
years, the age of the universe). However, topological solitons occurring in
the false vacuum will promote the decay of the false vacuum, as we have shown
in a series of papers Dupuis et al. (2019, 2017); Haberichter et al. (2016);
Dupuis et al. (2015); Lee et al. (2013a, b); Kumar et al. (2010).

In this paper, we concentrate on the case of magnetic monopoles in a false
vacuum. In Kumar et al. (2010) it was analyzed how thin wall monopoles could
arise in the gauge field/Higgs field dynamics. However, the analysis presented
there relied on the proposition that the gauge field dynamics would
essentially follow the Higgs field dynamics, which was not rigorously
established, and seems not to be the case. In this paper, we analyze
numerically the magnetic monopole solutions that exist in a false vacuum. The
main difference is that the core of the monopole is in the true vacuum, and
hence provides an outward pressure causing the monopole to be thin-walled.
Classically the monopole is stable, however, it can tunnel quantum
mechanically Paranjape (2017) to a classically unstable state, essentially
where the radius of the core is large enough so that the core will expand
uncontrollably, converting false vacuum in the exterior to the true vacuum in
the interior. This tunnelling amplitude was computed in Kumar et al. (2010)
the computation which relied only on the requirement that the monopole be
thin-walled.

The analysis in Kumar et al. (2010) assumed that the dynamics of the gauge
field simply followed the dynamics of the Higgs field. This does not seem to
be correct, and hence the proof of the existence of thin-walled monopoles
given there is not accurate. In this paper, we perform the required numerical
analysis to confirm the existence of thin-walled monopoles for an appropriate
choice of the scalar field potential.

##  II False magnetic monopoles and their tunnelling decay

The monopole is constructed with a variation of the standard ’tHooft-Polyakov
monopole where the field content consists of a triplet scalar
$(\vec{\phi})_{i}$, $i=1,2,3$ and the corresponding gauge field
$\vec{A}_{\mu}$ of local $SO(3)$ gauge invariance. The potential for the
scalar field, as treated in Kumar et al. (2010), was a 6th order potential,
writing $h=|\vec{\phi}|$

$V(\phi)=\lambda h^{2}(h^{2}-a^{2})^{2}+\gamma^{2}h^{2}-\epsilon.$ (1)

The potential has the form given in Fig.(1), and comprises of a true vacuum at
$h=0$ and a false vacuum of the symmetry broken type at $h=\eta$ where

$\eta=\sqrt{\frac{2{a}^{2}}{3}+\frac{\sqrt{{a}^{4}{\lambda}^{2}-3{\gamma}^{2}{\lambda}}}{3{\lambda}}}.$
(2)

The parameters of the potential are chosen so that the energy density
difference is $\epsilon$ which normalizes the potential so that the false
vacuum has energy density zero, evidently $V(h=0)=-\epsilon$. Our idea was to
mimic the thin-wall instanton analysis done by Coleman Coleman (1977) in his
study of false vacuum decay via the tunnelling formation of thin-walled
bubbles of true vacuum, within the false vacuum.

The spherically symmetric ansatz is:

$\displaystyle\phi_{a}$ $\displaystyle=$ $\displaystyle\hat{r}_{a}\,h(r)$
$\displaystyle A_{\mu}^{a}$ $\displaystyle=$ $\displaystyle\epsilon_{\mu
ab}\,\hat{r}_{b}\,\frac{1-K(r)}{er}$ $\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle 0$ (3)

and the equations of motion giving rise to the monopole solution are given by:

$\displaystyle
h^{\prime\prime}+\frac{2}{r}h^{\prime}-\frac{2h}{r^{2}}K^{2}-\frac{\partial
V}{\partial h}$ $\displaystyle=$ $\displaystyle 0$ (4) $\displaystyle
K^{\prime\prime}-\frac{K}{r^{2}}(K^{2}-1)-e^{2}h^{2}K$ $\displaystyle=$
$\displaystyle 0.$ (5)

For the original ’tHooft-Polyakov monopole configuration ’t Hooft (1974);
Polyakov (1974), the potential has a maximum at $h=0$ and a symmetry-breaking
minimum at $h=a\neq 0$, which is the global minimum of the potential. For the
false monopoles that we are considering, the potential has a global minimum at
$h=0$ which does not break the symmetry and is the true vacuum, and has a
local minimum at $h=\eta\neq 0$ which corresponds to the false vacuum.

We imagine that the universe is trapped in the false vacuum, which is a
standard scenarioTurner and Wilczek (1982). However, as the vacuum is not
given by a single value of the scalar field but in fact a whole manifold of
values, the manner in which the universe is trapped could vary from spatial
point to point. This allows for the possibility of topologically non-trivial
configurations, within the false vacuum. We have dubbed such monopoles “false
monopoles” and they are seen, through our analysis Kumar et al. (2010), to be
unstable. The 6th order potential (1) with the appropriate choice of
parameters, does correspond to a potential with a true minimum at $h=0$ and a
false minimum at $h=\eta$ where $\epsilon$, the difference of energy density,
can be taken to be arbitrarily small.

The thin-walled false monopole solution that we are looking for would be
classically stable. We informally divide space into the interior and the
exterior of the monopole. The exterior is where the fields are essentially
abelian, while the interior of the monopole contains the full non-abelian
structure. The topological twisting of the scalar field in the exterior region
at spatial infinity requires that the scalar field must vanish in the interior
of the monopole for the configuration to be non-singular. Hence there is a
region in the interior where the configuration is essentially in the true
vacuum. This region would naturally be unstable to expand, as it has negative
energy density, $-\epsilon$, compared to the false vacuum in the exterior
which has been normalized to have vanishing energy density. For a solution of
the thin-wall type, this expansion is arrested because the energy in the wall
increases as the radius squared, while the negative energy increases as the
radius cubed. For small radius, the positive contribution to the energy
dominates, but eventually, for large radius, it is clear that the negative
contribution to the energy is dominant. The size of the interior region is
stable under collapse due to the Coulomb energy of the monopole. The energy in
the abelian magnetic field will diverge inversely with the radius of the
interior region consequently arresting the collapse. The eventual instability
of the false monopole is due to quantum tunnelling. The radius of the bubble
can tunnel out to a larger radius at which the negative energy density of the
true vacuum becomes dominant and the monopole size expands without bound,
converting false vacuum into true vacuum.

The equations of motion (4) and (5) can be interpreted as the equations
governing the position of two particles, $h$ and $K$, as a function of “time”
corresponding to $r$. The $h$ dynamics correspond to movement in the presence
of the inverted potential $-V$, as depicted in Fig.(1), but also in the
presence of a positive impulse due to the $K$ dependent term $2hK^{2}/r^{2}$
and a “time” dependent “drag” term $-2h^{\prime}/r$. The drag term is called
so because normally friction is included, in an otherwise conservative
equation of motion, through just such a first derivative term. Both of these
non-standard terms disappear as $r$ becomes large. The $K$ equation of motion
does not directly depend on the potential, however, it does implicitly do so
through its dependence on $h$. Intuitively, we expect that the $h$ field will
remain at or near $h=0$ if it starts very near $h=0$ or starts directly at
$h=0$ with a very small initial velocity. By changing the values of the
parameters, we can adjust how long the field remains at $h$ nearly zero. If
$r$ becomes large, both the impulse term (that depends on the $K$ field) and
the drag term which depends on the derivative $h^{\prime}$ become negligible,
as they are multiplied $1/r^{2}$ and $1/r$ respectively. The subsequent motion
is conservative and can be easily analyzed.

Figure 1: Inverted potential of Eqn.(1) for the scalar field at generic values
of the parameters.

In Kumar et al. (2010), it was invoked that the $K$ evolution would simply
follow the $h$ evolution, hence if the $h$ field stayed for a long time at
$h=0$, then the $K$ field would stay at $K=1$ until the $h$ field started its
non-trivial evolution. This was apparently not true. Numerical studies show
that it is largely the boundary conditions for large $r$ that govern the
behaviour of the $K$ field, and even though the $h$ field remains zero, the
$K$ field begins its trajectory towards a vanishing value independently of the
value of the $h$ field. Furthermore, it was found that even though we can make
the initial potential as flat as we like near $h=0$, it still does not give
rise to a thin wall solution. The $h$ field simply starts to move up to its
asymptotic value $h=\eta$ from $r\approx 0$. The reason for this behaviour has
to do with the impulse term $\frac{2h}{r^{2}}K^{2}$ which pushes the $h$ field
along immediately at $r=0$. Therefore, the idea of obtaining a thin wall
monopole with the potential Eqn.(1) was not correct. The content of this paper
is to show how to modify the potential in order to obtain a thin-wall, false
monopole solution.

##  III Modified potential

The potential can be modified in an infinite number of ways, as we are not
concerned with renormalizability. The potential that we use should simply be
thought of as an effective potential. The potential should have an absolute
minimum at $h=0$ and additionally a local minimum (false vacuum) at $h=a$, the
energy density difference between the minima should be an adjustable
parameter. The energy density of the false vacuum is normalized to zero. We
consider the following form for the potential,

$V(h)=\lambda\Big{(}(h^{2}-a^{2})^{2}(h^{2}-\frac{\epsilon^{\frac{1}{2n+1}}}{a^{4}})\Big{)}^{2n+1},$
(6)

as shown in Fig(2).

Figure 2: Potential for the scalar field with $n=1$, $a=1.4$ and $\epsilon=.4$

This potential has a global minimum at $h=0$ where its value is
$-\lambda\epsilon$. The potential is symmetric under $h\to-h$ however we are
only interested in the range $h\geq 0$. Here, the potential has two roots at
$h_{i}={\frac{\sqrt{\epsilon^{\frac{1}{2n+1}}}}{a^{2}}}$ and at $h=a$. It has
four critical points at $h=0$, at the root $h_{i}$, at the point
$\bar{h}=\sqrt{\frac{1}{3}\left(\frac{2\epsilon^{\frac{1}{2n+1}}}{a^{4}}+a^{2}\right)}$
and finally at the root $a$.

For $\epsilon^{\frac{1}{2n+1}}<a^{6}$ the potential rises up from
$-\lambda\epsilon$ at $h=0$ to an inflection point (and root) at $h_{i}$
followed by a maximum at $\bar{h}$ and then followed by a local minimum (and
root) at $h=a$. Afterwards, it rises up to $+\infty$. This is depicted in
Fig.(2). However, the three critical points can exchange their order.

For $\epsilon^{\frac{1}{2n+1}}>a^{6}$ we get the order interchanged,
$\\{h_{i},\bar{h},a\\}\to\\{a,\bar{h},h_{i}\\}$ the first critical point then
corresponding to a maximum (and root) at $h=a$ followed by a local minimum at
$h=\bar{h}$ and the inflection point (and root) moving to be the largest at
$h_{i}$. This is depicted in Fig.(3). We are not interested in a potential
with this behaviour, its structure is not one with a false vacuum with energy
density normalized to zero, therefore we will make sure to restrict ourselves
to the region $\epsilon^{\frac{1}{2n+1}}<a^{6}$.

Figure 3: Potential for the scalar field with $n=1$, $a=.8$ and $\epsilon=1.5$

In order to confirm the described behaviour of the potential, we first compute
its derivative. Writing $\bar{\epsilon}=\epsilon^{\frac{1}{2n+1}}$ we have

$\displaystyle V^{\prime}(h)$ $\displaystyle=$
$\displaystyle\lambda(2n+1)\Big{(}(h^{2}-a^{2})^{2}(h^{2}-\frac{\bar{\epsilon}}{a^{4}})\Big{)}^{2n}$
(7) $\displaystyle\times$
$\displaystyle\big{(}(2(h^{2}-a^{2})(h^{2}-\frac{\bar{\epsilon}}{a^{4}})+(h^{2}-a^{2})^{2})\big{)}2h$
$\displaystyle=$
$\displaystyle\lambda(2n+1)(h^{2}-a^{2})^{4n+1}(h^{2}-\frac{\bar{\epsilon}}{a^{4}})^{2n}$
$\displaystyle\times$
$\displaystyle\big{(}3h^{2}-(\frac{2\bar{\epsilon}}{a^{4}}+a^{2})\big{)}2h$

and one can read off the critical points $V^{\prime}(h)=0$ at
$0,\,\,h_{i}=\sqrt{\bar{\epsilon}}/a^{2},\,\,\bar{h}=\sqrt{(2\bar{\epsilon}/a^{4}+a^{2})/3}$
and $a$.

  1. 1.

The global minimum at $h=0$ is simple with clearly
$V^{\prime\prime}(0)=\lambda(2n+1)a^{2}\bar{\epsilon}^{2n}({2\bar{\epsilon}}/{a^{4}}+a^{2})>0$.

  2. 2.

The critical point at $h_{i}$ is an inflection point and a root of order
$2n+1$.

$\hskip
5.69054ptV^{2n+1}(h_{i})=\lambda(2n+1)!(h_{i}^{2}-a^{2})^{4n+2}(2h_{i})^{2n+1}$

which gives the leading contribution in the Taylor expansion of the potential
as $\sim(h-h_{i})^{2n+1}$. This is an odd power $h-h_{i}$ changing sign on
either side of $h_{i}$, implying an inflection point.

  3. 3.

The critical point at $\bar{h}$ is a simple local maximum. We find
$V^{\prime\prime}(\bar{h})$ by differentiating the term
$3h^{2}-(\frac{2\bar{\epsilon}}{a^{4}}+a^{2})$ in $V^{\prime}(h)$ and setting
$h=\bar{h}$ afterwards. The contributions from differentiating the other terms
of course vanish as the term vanishing at $h=\bar{h}$ is intact. Thus we find

$\hskip
22.76219ptV^{\prime\prime}(\bar{h})=-\lambda(2n+1)2^{4n+1}((a^{2}-\bar{\epsilon}/a^{4})/3)^{6n+1}12\bar{h}.$

We observe that the sign of the first factor changes from positive to negative
as $\bar{\epsilon}$ passes from below $a^{6}$ to above as the power $6n+1$ is
odd. The other factors do not change sign. Hence for $\bar{\epsilon}<a^{6}$,
the region that we are interested in, we have a maximum at $\bar{h}$.

  4. 4.

The critical point at $h=a$ is also a $4n+2$ order root, i.e.
$V(h)\sim(h-a)^{(4n+2)}$ with first non-vanishing derivative is of even order,
$V^{(4n+2)}(a)\neq 0$. Differentiating $V(h)$, $4n+2$ times and then
evaluating at $h=a$ only gives a non-vanishing contribution when the
derivative acts on the $h-a$ factor. We easily find
$V^{4n+2}(a)=\lambda(4n+2)!(2a)^{4n+2}(a^{2}-\bar{\epsilon}/a^{4})^{2n+1}$
which, if $\bar{\epsilon}<a^{6}$, is positive signifying a local minimum.

##  IV Numerical results

The equations of motion Eqns.(4) and (5) correspond to dynamics in minus the
potential, and as if $r$ is the time. We have experimented with various values
of $\epsilon$, $a$ and $n$. We are able to find thin-wall type (false)
monopole solutions.

The work is numerical, using a Matlab code to find the solution, which is
included in the supplementary materials. We find the monopole profiles have a
thin-wall behaviour as defined by the energy density of the configuration. The
field profiles for $h$ and $K$ are given in the Figs. (4).

Figure 4: The profiles for $h/a$ and $K$ for various values of $\epsilon$ and
for $n=4$.

These profiles correspond to thin-wall solutions when we look at the total
energy density shown in Fig.(5).

Figure 5: The total energy density for the profiles in Fig.(4)for various
values of $\epsilon$ and for $n=4$.

The total energy density starts at zero and descends as $\sim-r^{2}$, which we
confirm numerically. This is expected from the potential energy in the scalar
field is $-\lambda\epsilon$ when $h\approx 0$. Then it should behave simply as
$-\lambda\epsilon r^{2}$ from the contribution of the volume element. There is
a competing positive contribution from the magnetic field energy density as
$K$ starts to descend from $K=1$, however, in the interior, the total energy
density is dominated by the negative scalar potential energy. The descent of
the total energy density is arrested when the scalar field begins to move away
from $h=0$. This is where the wall starts, in Fig.(5) and implicitly from the
profiles in Fig.(4) we see that the increase in the energy density is rather
brusque. This allows for a well-defined value for the inner radius of the
wall. The outer radius of the wall starts once the total energy density begins
to get its dominant contribution from the Coulomb energy of the exterior
abelian magnetic field. This magnetic field is constructed from the non-
abelian magnetic field and the covariant derivatives of the scalar field. The
contribution to the total energy density from the scalar field potential
becomes negligible as the scalar field assumes its value at the false vacuum
(which has been normalized to zero energy density) in the exterior. The
magnetic field behaves as $\sim 1/r^{2}$, hence the total energy density
behaves as $\sim r^{2}(1/r^{2})^{2}=1/r^{2}$. To exhibit this numerically, we
consider the log of the total energy density squared, $\ln E^{2}$. We take the
square so that the log acts on a positive function. The slope of this function
should behave as $\approx 4$ in the interior region and as $\approx-4$ in the
exterior region, with a complicated interpolation between these values within
the wall region. In Fig.(6), we plot the derivative (with respect to $\ln r$)
of the log of the total energy density squared. We see clearly that the
interior value is indeed about +4 while the exterior value is -4 as expected.
Additionally, we see that the wall thickness $\Delta$, over which the
behaviour of the energy density interpolates from $\sim-r^{2}$ to $\sim
1/r^{2}$, is relatively small compared to the wall radius.

Figure 6: The derivative of the log of the total energy density squared for
$\epsilon=15$ and for $n=4$.

The non-abelian magnetic energy density behaves in a complicated way in the
interior and as a $1/r^{2}$ contribution in the exterior, as can be seen from
Fig.(7) and Fig.(8). The magnetic field energy density rises up to a peak at
the wall and then descends downwards to the expected $\sim 1/r^{2}$ behaviour
from its Coulomb tail. The derivative by $\ln r$ of the log of the magnetic
field energy density squared asymptotes to -4 confirming that the magnetic
field energy density behaves like $\sim 1/r^{2}$ in the exterior.

Figure 7: The magnetic energy density for $\epsilon=15$ and for $n=4$. Figure
8: The log of the magnetic energy density squared and its derivative (by $\ln
r$) for $\epsilon=15$ and for $n=4$.

In Fig.(9) we plot the energy density in the covariant derivative of the
scalar field and in Fig.(10) the derivative by $\ln r$ of the log of its
contribution to the energy density squared. We observe a vanishing
contribution in the interior, then a brusque rise up at the inner wall radius,
and then an interpolation to the exterior energy density contribution which
behaves exactly like $\sim 1/r^{2}$ as seen from the exterior behaviour in
Fig.(10) which assumes the corresponding asymptotic value of -4.

Figure 9: The covariant derivative energy density for $\epsilon=15$ and for
$n=4$. Figure 10: The derivative by $\ln r$ of the log of the covariant
derivative energy density squared for $\epsilon=15$ and for $n=4$.

We end this section with Fig.(11) of multiple profiles for the solutions for
$h$ and $K$ for various values of $\epsilon$.

Figure 11: Multiple solutions for $h$ and$K$ for various values of $\epsilon$
and for $n=4$.

##  V Conclusions

We will make several observations. First, the idea that $\epsilon$ should be
small is not the appropriate criterion. Indeed, we get thin-wall type
solutions as we take $\epsilon$ quite large, as can be seen in Fig.(11),
although we always remain in the region $\bar{\epsilon}<a^{6}$. Second, the
thin-wall nature of the solution is quite different than in the case of thin-
wall instantons that give rise to vacuum bubbles Coleman (1977). With the
vacuum bubbles, the interior of the bubble was a quiescent true vacuum. In our
case, the energy density of the true vacuum is dominant, however, the gauge
field does give a non-zero contribution. This is reminiscent of the false
vortices and cosmic strings that have been studied in a similar context Lee et
al. (2013b); Dupuis et al. (2017) where the interior contains non-trivial
magnetic fields. The profiles show an energy density that behaves as $\sim
r^{2}$ in the interior. Here the scalar field remains in the true vacuum,
$h\approx 0$ until it starts its trajectory up to the false vacuum, where the
wall nominally begins. The gauge field however immediately begins its descent
from $K=1$ giving a contribution to the energy density which behaves as $\leq
r^{2}$. However, the negative energy density of the true vacuum of the scalar
field gives a contribution to the energy density that behaves as
$\sim-\lambda\epsilon r^{2}$ and it dominates over the contribution to the
energy density from the gauge fields, for $\epsilon$ sufficiently large. The
wall region has a sharply defined inner radius, essentially where the scalar
field begins to move away from $h=0$, but the exact outer boundary is not as
sharp. However, eventually, in the exterior, the behaviour of the energy
density is that of the Coulomb energy of the abelian magnetic field.

With this understanding of the existence of the thin-wall false monopoles, we
can imagine that the further analysis done in Kumar et al. (2010) is perfectly
justified. The interior of the monopole will give a contribution to the total
energy that behaves as three terms, the interior negative energy proportional
to the volume, the positive wall energy proportional to the area and the
exterior Coulomb energy proportional to the inverse of the radius.

$E(R)=-\alpha R^{3}+4\pi\sigma R^{2}+\frac{C}{R}$ (8)

where $\alpha$ $\sigma$ and $C$ are calculable parmeters. The function $E(R)$
descends from $+\infty$ at $R=0$ to $-\infty$ at $R=+\infty$ but it will have
a classically stable minimum corresponding to a thin-wall false monopole as
long its derivative has two positive critical points. The first critical point
corresponds to a local minimum which gives the radius of the false thin-wall
monopole, while the second corresponds to the position at the height of the
barrier under which the wall must tunnel to render the false vacuum unstable.
$E^{\prime}(R)=0$ at the critical points which gives the equation $-3\alpha
R^{2}+8\pi\sigma R-\frac{C}{R^{2}}=0$. Multiplying by $R^{2}$ and dividing
through by $-C$ yields the quartic

$\frac{3\alpha}{C}R^{4}-\frac{8\pi\sigma}{C}R^{3}+1=0.$ (9)

For a small $C$, which is reasonable for small gauge coupling, the equation is
equivalent to

$AR^{4}-BR^{3}+1=0$ (10)

with $A,B\gg 1$. This equation is easily, analytically solved by symbolic
algebraic manipulation software (Mathematica) and yields 4 solutions. A clear
discriminant must be positive to have any real solution

$-256A^{3}+27B^{4}\geq 0$ (11)

i.e.

$C\leq\frac{(4\pi\sigma)^{3}}{16\alpha^{3}}.$ (12)

Furthermore, if Eqn.(12) is satisfied, it can be easily seen that in the limit
$\frac{8\pi\sigma}{C},\,\,\frac{3\alpha}{C}\gg 1$, there are always two real
positive solutions.

##  VI ACKNOWLEDGEMENTS

We thank NSERC, Canada for financial support and Nick Manton, Richard
MacKenzie and Urjit Yajnik for useful discussions and critical comments.

## References

  * Kumar et al. (2010) B. Kumar, M. B. Paranjape, and U. A. Yajnik, Phys. Rev. D82, 025022 (2010), eprint 1006.0693.
  * Goddard and Olive (1978) P. Goddard and D. I. Olive, Rept. Prog. Phys. 41, 1357 (1978).
  * Kachru et al. (2003) S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, Phys. Rev. D68, 046005 (2003), eprint hep-th/0301240.
  * Dupuis et al. (2019) E. Dupuis, M. Haberichter, R. MacKenzie, M. B. Paranjape, and U. A. Yajnik, Phys. Rev. D 99, 016016 (2019), eprint 1805.08038.
  * Dupuis et al. (2017) E. Dupuis, Y. Gobeil, B.-H. Lee, W. Lee, R. MacKenzie, M. B. Paranjape, U. A. Yajnik, and D.-h. Yeom, JHEP 11, 028 (2017), eprint 1709.03839.
  * Haberichter et al. (2016) M. Haberichter, R. MacKenzie, M. B. Paranjape, and Y. Ung, J. Math. Phys. 57, 042303 (2016), eprint 1506.05838.
  * Dupuis et al. (2015) E. Dupuis, Y. Gobeil, R. MacKenzie, L. Marleau, M. B. Paranjape, and Y. Ung, Phys. Rev. D 92, 025031 (2015), eprint 1506.05091.
  * Lee et al. (2013a) B.-H. Lee, W. Lee, R. MacKenzie, M. B. Paranjape, U. A. Yajnik, and D.-h. Yeom, Phys. Rev. D 88, 105008 (2013a), eprint 1310.3005.
  * Lee et al. (2013b) B.-H. Lee, W. Lee, R. MacKenzie, M. B. Paranjape, U. A. Yajnik, and D.-h. Yeom, Phys. Rev. D88, 085031 (2013b), eprint 1308.3501.
  * Paranjape (2017) M. Paranjape, _The Theory and Applications of Instanton Calculations_ , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2017).
  * Coleman (1977) S. R. Coleman, Phys. Rev. D15, 2929 (1977), [Erratum: Phys. Rev.D16,1248(1977)].
  * ’t Hooft (1974) G. ’t Hooft, Nucl. Phys. B79, 276 (1974).
  * Polyakov (1974) A. M. Polyakov, JETP Lett. 20, 194 (1974).
  * Turner and Wilczek (1982) M. S. Turner and F. Wilczek, Nature 298, 633 (1982), URL https://doi.org/10.1038/298633a0.