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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 17
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 104190)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 17
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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string | meta
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|---|---|
\section{Introduction}
\subsection{Forward-backward splitting algorithms}
The forward-backward splitting (FBS) algorithm has become a standard solver for finding a zero point of the sum of a maximally monotone operator $\mathcal{A}: \mathcal{H} \mapsto 2^\mathcal{H}$ and a $\beta^{-1}$-cocoercive operator $\mathcal{B}: \mathcal{H} \mapsto \mathcal{H}$\footnote{Note that \eqref{inclusion} esssentially encompasses $\mathbf{v} \in (\mathcal{A}'+\mathcal{B})\mathbf{b}^\star$ for any constant vector $\mathbf{v}$. Indeed, ${\bf 0} \in (\mathcal{A} +\mathcal{B}) \mathbf{b}^\star$ is equivalent to $\mathbf{v} \in (\mathcal{A}'+\mathcal{B}) \mathbf{b}^\star$, if we define $\mathcal{A}': \mathcal{H} \mapsto 2^\mathcal{H}: \mathbf{b} \mapsto \mathcal{A} \mathbf{b}+\mathbf{v}$. It is easy to see that $\mathcal{A}'$ is monotone, if $\mathcal{A}$ is monotone.}:
\begin{equation} \label{inclusion}
{\bf 0} \in (\mathcal{A} +\mathcal{B}) \mathbf{b}^\star
\end{equation}
It reads as \cite{ywt_2017,lions,passty}:
\begin{equation} \label{fbs}
\mathbf{b}^{k+1} := (\mathcal{I} +\tau \mathcal{A})^{-1} (\mathcal{I} -\tau \mathcal{B}) \mathbf{b}^k
:=\mathcal{J}_{\tau \mathcal{A}} \circ (\mathcal{I} - \tau \mathcal{B}) \mathbf{b}^k
\end{equation}
where $\tau >0$ is a step size, $\mathcal{J}_{\tau \mathcal{A}}$ denotes a resolvent of $\tau \mathcal{A}$. Here, $\mathcal{I} -\tau \mathcal{B}$ performs the {\it forward} (explicit) computation w.r.t. the operator $\mathcal{B}$, followed by a {\it backward} (implicit) step of $(\mathcal{I} +\tau \mathcal{A})^{-1}$ w.r.t. the operator $\mathcal{A}$ \cite{attouch_2018}. The scheme \eqref{fbs} can be seen as a fixed point iteration of the classical FBS operator $\mathcal{T} := \mathcal{J}_{\tau \mathcal{A}} \circ (\mathcal{I} - \tau \mathcal{B})$. In this paper, we extend the operator \eqref{fbs} from scalar $\tau$ to arbitrary metric $\mathcal{Q}$:
\begin{equation} \label{t}
\mathcal{T} := (\mathcal{Q} + \mathcal{A})^{-1} (\mathcal{Q} - \mathcal{B})
:=\mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} \circ (\mathcal{I} - \mathcal{Q}^{-1} \mathcal{B})
\end{equation}
and study the associated generalized FBS (G-FBS) algorithm \cite{plc_vu_2014}:
\begin{equation} \label{gfbs}
\mathbf{b}^{k+1} := (\mathcal{Q} + \mathcal{A})^{-1} (\mathcal{Q} - \mathcal{B}) \mathbf{b}^k
:=\mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} \circ (\mathcal{I} - \mathcal{Q}^{-1} \mathcal{B}) \mathbf{b}^k
\end{equation}
If $\mathcal{A} = \partial g$, $\mathcal{B} = \nabla f$ for some functions $f$ and $g$, the problem \eqref{inclusion} becomes finding a minimizer of $f+g$: $\mathbf{b}^\star \in \Argmin (f+g) $. The algorithm \eqref{fbs} becomes the well-known {\it proximal FBS} (P-FBS) algorithm \cite{plc,plc_chapter}:
\begin{equation} \label{pfbs}
\mathbf{b}^{k+1} := (\mathcal{I} +\tau \partial g)^{-1} (\mathcal{I} - \tau \nabla f) \mathbf{b}^k = \prox_{\tau g} \big( \mathbf{b}^k - \tau \nabla f(\mathbf{b}^k) \big)
\end{equation}
where $\prox_{\tau g}$ is defined as: $\prox_{\tau g}: \mathcal{H} \mapsto \mathcal{H}: \mathbf{x} \mapsto \arg \min_\mathbf{u} $ $ g(\mathbf{u})+ \frac{1}{2\tau} \|\mathbf{u}-\mathbf{x}\|^2$ \cite[Definition 12.23]{plc_book},\cite[Eq. (2.13)]{plc}. The generalized FBS operator \eqref{t} yields the so-called {\it variable metric P-FBS algorithm}:
\begin{equation} \label{gpfbs}
\mathbf{b}^{k+1} : = (\partial g + \mathcal{Q})^{-1} (\mathcal{Q} - \nabla f ) \mathbf{b}^k = \prox_g^\mathcal{Q} \big( \mathbf{b}^k - \mathcal{Q}^{-1} \nabla
f(\mathbf{b}^k) \big)
\end{equation}
where the generalized proximity operator $\prox_g^\mathcal{Q}$ is defined as: $\prox_g^\mathcal{Q}: \mathcal{H} \mapsto \mathcal{H}: \mathbf{x} \mapsto \arg \min_\mathbf{u} $ $ g(\mathbf{u})+ \frac{1}{2 } \|\mathbf{u}-\mathbf{x}\|^2_\mathcal{Q}$ \cite{prox_acha}. The basic P-FBS \eqref{pfbs} has been extensively studied in the literature, e.g. \cite{plc,plc_chapter} in a convex setting, and further discussed in \cite{bolte_2009,bolte_2013,bolte_2014} for the nonconvex case. The generalized scheme \eqref{gpfbs} was recently studied in \cite{pesquet_2016,pesquet_2014} in a nonconvex setting. Though the above FBS schemes have been well understood, the nonexpansive properties of the G-FBS operator \eqref{t} are rarely discussed, which is the first focus of the paper.
\vskip.1cm
Nowadays, there has been a revived interest in the design and analysis of the first-order operator splitting algorithms \cite{teboulle_2018}, typically including the aforementioned P-FBS, Douglas-Rachford splitting (DRS) \cite{drs}, alternating direction method of multipliers (ADMM) \cite{glowinski_1,glowinski_2}, primal-dual splitting (PDS) \cite{pdhg,cp_2011} and Bregman methods \cite{osher_2005,yin_2008,sb,zxq}. The convergence analysis of these algorithms is often performed case-by-case. Though some unified frameworks and tools have recently been proposed, e.g. \cite{ljw_mapr,plc_vu,teboulle_2018,latafat_2017,latafat_chapter, unified_ieee,beck_unified}, these works do not connect the operator splitting algorithms with a simple and unified nonexpansive mapping. The second purpose of this paper is to show that most classes of splitting algorithms can be simply expressed by the G-FBS operator \eqref{t} or its relaxed version.
\subsection{Contributions}
We first study the nonexpansiveness of the G-FBS operator \eqref{t}. Then, for the associated fixed-point Banach-Picard and Krasnosel'skii-Mann iterations, we establish the global pointwise/nonergodic and ergodic convergence rates in terms of the solution distance and aysmptotic regularity. The convergence rates in terms of objective function value are further presented, when the operators $\mathcal{A}$ and $\mathcal{B}$ are associated with the functions $f$ and $g$.
The main results can be applied to many existing operator splitting methods. In particular, we show that a great variety of popular algorithms can be uniformly represented by the G-FBS operator \eqref{t}, by specifying the operators $\mathcal{A}$ and $\mathcal{B}$, variable metric $\mathcal{Q}$ (and relaxation matrix $\mathcal{M}$, if necessary). This unification and simplification helps to understand these algorithms
with substantially simplified analysis, compared to the original proofs in the literature.
\subsection{Related work}
\vskip.1cm
\paragraph{Nonexpansive mappings} The nonexpansive properties in a context of arbitrary variable metric have recently been revisited in \cite{fxue_1}, which lays the foundation for analyzing the G-FBS operator \eqref{t}. This work can also be seen an extension of the previous note \cite{fxue_2}, which focused on the metric resolvent $\mathcal{J}_{\mathcal{Q}^{-1}\mathcal{A}}$---a special case of G-FBS operator with $\mathcal{B}: \mathbf{b} \mapsto {\bf 0}, \forall \mathbf{b} \in \mathcal{H}$. It should be stressed that the metric resolvent cannot interpret the P-FBS algorithm \eqref{pfbs} or \eqref{gpfbs}, which, on the contrary, perfectly fits into the G-FBS operator \eqref{t}.
\paragraph{Unified frameworks} Several frameworks have recently been proposed to unify the existing optimization algorithms. Fej\'{e}r sequence \cite{plc_vu} is rather abstract, and not directly connected to the specific algorithms at hand. In a most recent paper \cite{plc_fixed}, various popular operator splitting methods were revisited by a fixed-point construction, including P-FBS, DRS, ADMM and PDS algorithms. The fixed-point iteration of nonexpansive mapping has also been investigated in \cite{ljw_mapr}, with more emphasis on the applications to the operator splitting algorithms. Even more complicated algorithms, e.g. three-operator splitting algorithm \cite{ywt_2017}, GFBS algorithm \cite{pfbs_siam}, can be viewed as a nonexpansive mapping. However, in these works, the nonexpansive mapping depends on specific algorithm, for which the nonexpansive properties have to be analyzed case-by-case. Our work differs from them in that: we represent the existing algorithms using a unified and explicit form of the G-FBS operator \eqref{t}, which provides a unified treatment of many classes of algorithms.
The AFBA framework was proposed in \cite{latafat_2017,latafat_chapter}, to generalize the classical splitting schemes, e.g. DRS and FBS. However, it fails to cover the multiple block algorithms, e.g. extended ADMM algorithms \cite{bai_2018,hbs_yxm_2018,cch_2016}, and does not simplify the algorithms into a simple nonexpansive mapping. The works of \cite{teboulle_2018,plc_bregman} discussed the Bregman proximal mapping and the associated Bregman proximal gradient algorithm. It remains unclear that how to use the framework to analyze the splitting algorithms.
\paragraph{Operator splitting algorithms} We show that most splitting algorithms essentially take the (relaxed) G-FBS structure (often equipped with a designed product space), despite of their seemingly different splitting strategies and algorithm structures. Consequently, the convergence behaviours can be analyzed by a unified treatment.
\paragraph{Variable metric proximal point algorithms (PPA)} The fixed-point iteration of the G-FBS operator can be rewritten in an inclusion form:
\begin{equation} \label{pfbs_it}
{\bf 0} \in \mathcal{A} \mathbf{b}^{k+1} + \mathcal{B} \mathbf{b}^k +\mathcal{Q}
(\mathbf{b}^{k+1} - \mathbf{b}^k)
\end{equation}
which encompasses the variable metric PPA:
${\bf 0} \in \mathcal{A} \mathbf{b}^{k+1} +\mathcal{Q}
(\mathbf{b}^{k+1} - \mathbf{b}^k)$ \cite{lotito,qian,bonnans} as a special case of \eqref{pfbs_it} with the operator $\mathcal{B}: \mathbf{b} \mapsto \bf 0, \forall \mathbf{b} \in \mathcal{H}$. The variable metric PPA were also discussed in \cite{hbs_siam_2012,hbs_jmiv_2017,mafeng_2018} using the tool of variational inequality. With the success of reinterpretations of DRS, ADMM and PDHG by the variable metric PPA, these works, however, fail to cover the P-FBS and many PDS algorithms, which can be addressed by incorporating the cocoercive operator $\mathcal{B}$.
\subsection{Notations and definitions} \label{sec_notation}
We use standard notations and concepts from convex analysis and variational analysis, which, unless otherwise specified, can all be found in the classical and recent monographs \cite{rtr_book,rtr_book_2,plc_book,beck_book}.
\vskip.1cm
A few more words about our notations are in order. The classes of positive semi-definite (PSD) and positive definite (PD) matrices are denoted by $\mathcal{M}^+$ and $\mathcal{M}^{++}$, respectively. The classes of symmetric, symmetric and PSD, symmetric and PD matrices are denoted by $\mathcal{M}_\mathcal{S}$, $\mathcal{M}_\mathcal{S}^+$, and $\mathcal{M}_\mathcal{S}^{++}$, respectively. For our specific use, the $\mathcal{Q}$-based inner product (where $\mathcal{Q}$ is an arbitrary square matrix) is defined as: $\langle \mathbf{a} | \mathbf{b} \rangle_\mathcal{Q} := \langle \mathcal{Q} \mathbf{a} | \mathbf{b} \rangle = \langle \mathbf{a} | \mathcal{Q}^\top \mathbf{b} \rangle$, $\forall (\mathbf{a},\mathbf{b}) \in \mathcal{H} \times \mathcal{H}$; the $\mathcal{Q}$-norm is defined as: $\|\mathbf{a}\|_\mathcal{Q}^2 := \langle \mathcal{Q}\mathbf{a} | \mathbf{a} \rangle$, $\forall \mathbf{a} \in \mathcal{H}$. Note that unlike the conventional treatment in the literature, $\mathcal{Q}$ is {\it not} assumed to be symmetric and PSD here (e.g. see Section \ref{sec_extension}), and hence, $\|\cdot\|_\mathcal{Q}$ is not always well--defined.
\vskip.1cm
Note that our expositions in Sections \ref{sec_operator} and \ref{sec_iteration} are largely based on the nonexpansive properties in the context of arbitrary variable metric $\mathcal{Q}$, which have been thoroughly discussed in \cite{fxue_1}. For sake of completeness and convenience, a key notion of {\it $\mathcal{Q}$--based $\xi$--Lipschitz $\alpha$--averaged} is restated here.
\begin{definition} \label{def_lip}
{\rm \cite[Definition 2.2]{fxue_1}
An operator $\mathcal{T}: \mathcal{H} \mapsto \mathcal{H}$ is said to be {\it $\mathcal{Q}$--based $\xi$--Lipschitz $\alpha$--averaged} with $\xi \in \ ]0, +\infty[$ and $\alpha\in \ ]0, 1[$, denoted by $\mathcal{T} \in \mathcal{F}^\mathcal{Q}_{\xi,\alpha}$, if there exists a $\mathcal{Q}$--based $\xi$--Lipschitz continuous operator $\mathcal{K}: \mathcal{H} \mapsto \mathcal{H}$, such that $\mathcal{T} = (1-\alpha) \mathcal{I} + \alpha \mathcal{K}$. In particular, if $\xi \in \ ]1, +\infty[$, $\mathcal{T}$ is {\it $\mathcal{Q}$--weakly averaged}; if $\xi \in\ ]0,1]$, $\mathcal{T}$ is {\it $\mathcal{Q}$--strongly averaged}.
}
\end{definition}
\begin{lemma} \label{l_cocoercive}
Let an operator $\mathcal{B}: \mathcal{H} \mapsto \mathcal{H}$ be $\beta^{-1}$--cocoercive. If $\mathcal{Q}\in \mathcal{M}_\mathcal{S}^{++}$ and $\mathcal{Q} \succeq \nu \mathcal{I}$ with $\nu >0$, then, $\mathcal{Q}^{-1} \mathcal{B}$ is $\mathcal{Q}$--based $ \frac{\nu} {\beta}$--cocoercive.
\end{lemma}
\begin{proof}
We deduce that:
\begin{eqnarray}
\big\langle \mathbf{b}_1 - \mathbf{b}_2 \big|
\mathcal{Q}^{-1} \mathcal{B} \mathbf{b}_1 - \mathcal{Q}^{-1} \mathcal{B} \mathbf{b}_2 \big\rangle_\mathcal{Q}
& = & \big\langle \mathbf{b}_1 - \mathbf{b}_2 \big|
\mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big\rangle
\ge \frac{1}{\beta} \big\| \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big\|^2
\nonumber\\
& = & \frac{1}{\beta} \big\|
\mathcal{Q}^{-1} \mathcal{B} \mathbf{b}_1 - \mathcal{Q}^{-1} \mathcal{B} \mathbf{b}_2 \big\|_{\mathcal{Q}^2}^2
\nonumber\\
& \ge & \frac{\nu} {\beta} \big\|
\mathcal{Q}^{-1} \mathcal{B} \mathbf{b}_1 - \mathcal{Q}^{-1} \mathcal{B} \mathbf{b}_2 \big\|_{\mathcal{Q}}^2
\nonumber
\end{eqnarray}
\phantom{s}
\end{proof}
\section{The nonexpansiveness of G-FBS operator}
\label{sec_operator}
In this part, we will focus on the variable metric $\mathcal{Q}$ satisfying the following assumption:
\begin{assumption} \label{assume_1}
$ \mathcal{Q} \in \mathcal{M}_\mathcal{S}^{++}, \quad
\nu_1 \mathcal{I} \preceq
\mathcal{Q} \preceq \nu_2 \mathcal{I},\quad (\nu_1,\nu_2) \in\ ]0, +\infty[^2$
\end{assumption}
\subsection{Nonexpansive properties}
\begin{lemma} \label{l_T}
Given the operator $\mathcal{T}: \mathcal{H}\mapsto \mathcal{H}$ defined in \eqref{t}, then, under Assumption \ref{assume_1}, the following hold.
{\rm (i)} $\big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2 \le
\langle \mathbf{b}_1 - \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \rangle_\mathcal{Q}
- \langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1-\mathcal{T}\mathbf{b}_2 \rangle$
{\rm (ii)} $\langle \mathbf{b}_1 - \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \rangle_\mathcal{Q} \ge
\big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2
- \frac{\beta}{4} \big\| (\mathcal{I} - \mathcal{T}) \mathbf{b}_1 -
(\mathcal{I} - \mathcal{T}) \mathbf{b}_2 \big\|^2$
{\rm (iii)} $ \big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2
+ \Big( 1-\frac{\beta}{2 \nu_1 } \Big)
\big\|(\mathcal{I}- \mathcal{T}) \mathbf{b}_1 - (\mathcal{I} -\mathcal{T}) \mathbf{b}_2 \big\|_\mathcal{Q}^2
\le \big\| \mathbf{b}_1 - \mathbf{b}_2 \big\|_\mathcal{Q}^2 $
\end{lemma}
\begin{proof}
(i) We develop:
\begin{eqnarray}
&& \big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2
\nonumber \\ &=& \big \langle \mathcal{Q} \mathcal{T} \mathbf{b}_1 - \mathcal{Q} \mathcal{T} \mathbf{b}_2 | \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big \rangle
\nonumber \\
& \le &\big \langle \mathcal{Q} \mathcal{T} \mathbf{b}_1 - \mathcal{Q} \mathcal{T} \mathbf{b}_2 | \mathcal{T} \mathbf{b}_1- \mathcal{T} \mathbf{b}_2 \big \rangle +
\big \langle \mathcal{A} \mathcal{T} \mathbf{b}_1 - \mathcal{A} \mathcal{T} \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1 - \mathcal{T}\mathbf{b}_2 \big \rangle \quad \text{by monotonicity of $\mathcal{A}$}
\nonumber \\ &= &
\big \langle (\mathcal{Q} - \mathcal{B}) \mathbf{b}_1 - (\mathcal{Q} - \mathcal{B}) \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1 - \mathcal{T}\mathbf{b}_2\big \rangle_\mathcal{Q}
\quad \text{since $\mathcal{Q} - \mathcal{B} \in \mathcal{Q} \mathcal{T} +\mathcal{A} \mathcal{T} $ by \eqref{t}}
\nonumber \\ & = &
\big \langle \mathbf{b}_1 - \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1-\mathcal{T}\mathbf{b}_2 \big \rangle_\mathcal{Q}
- \big \langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1-\mathcal{T}\mathbf{b}_2 \big \rangle
\nonumber
\end{eqnarray}
\vskip.1cm
(ii) Denoting $\mathcal{R} :=\mathcal{I} - \mathcal{T}$, we deduce that:
\begin{eqnarray}
&& \big \langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big \rangle
\nonumber \\ &=& \big \langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 | \mathbf{b}_1 - \mathbf{b}_2 \big \rangle -
\big \langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 | \mathcal{R}\mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big \rangle
\quad \text{by $\mathcal{T} = \mathcal{I} - \mathcal{R}$}
\nonumber \\
& \ge & \frac{1}{\beta} \big\| \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big\|^2
- \big \langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 | \mathcal{R}\mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big \rangle \quad \text{by cocoerciveness of $\mathcal{B}$}
\nonumber \\
& \ge & \frac{1}{\beta} \big\| \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big\|^2
- \frac{1}{\beta} \big\| \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big\|^2
- \frac{\beta}{4} \big\| \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big\|^2
\quad \text{by Young's inequality}
\nonumber \\
& = & - \frac{\beta}{4} \big\| \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big\|^2
\nonumber
\end{eqnarray}
Substituting it into (i) obtains (ii).
\vskip.1cm
(iii) We develop:
\begin{eqnarray}
&& \big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2
+ \big\|(\mathcal{I}- \mathcal{T}) \mathbf{b}_1 - (\mathcal{I} -\mathcal{T}) \mathbf{b}_2 \big\|_\mathcal{Q}^2
\nonumber \\
& = & 2 \big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2
+ \big\| \mathbf{b}_1 - \mathbf{b}_2 \big\|_\mathcal{Q}^2
-2 \langle \mathbf{b}_1 - \mathbf{b}_2 | \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \rangle_\mathcal{Q}
\nonumber \\
& \le & \big\| \mathbf{b}_1 - \mathbf{b}_2 \big\|_\mathcal{Q}^2
+\frac{\beta}{2} \big\| (\mathcal{I} - \mathcal{T}) \mathbf{b}_1 -
(\mathcal{I} - \mathcal{T}) \mathbf{b}_2 \big\|^2
\quad \text{by Lemma \ref{l_T}--(ii)}
\nonumber \\
& \le & \big\| \mathbf{b}_1 - \mathbf{b}_2 \big\|_\mathcal{Q}^2
+ \frac{\beta}{2 \nu_1 }
\big\| (\mathcal{I} - \mathcal{T}) \mathbf{b}_1 - (\mathcal{I} - \mathcal{T}) \mathbf{b}_2 \big\|_\mathcal{Q}^2
\nonumber
\end{eqnarray}
which yields (iii).
\end{proof}
\begin{proposition} \label{p_T}
Let $\mathcal{T}$ be defined as \eqref{t}. Under Assumption \ref{assume_1}, if $\beta < 2 \nu_1$, then, the following hold.
{\rm (i)} $\mathcal{T} \in \mathcal{F}^\mathcal{Q}_{1, \frac{2\nu_1 }
{4 \nu_1 - \beta } }$.
{\rm (ii)} $\mathcal{T}$ is $\mathcal{Q}$--strongly averaged and $\mathcal{Q}$--nonexpansive, but not $\mathcal{Q}$--firmly nonexpansive.
{\rm (iii)} $\mathcal{I} - \gamma(\mathcal{I} - \mathcal{T}) \in \mathcal{F}^\mathcal{Q}_{1, \frac{2 \gamma \nu_1 }
{4 \nu_1 - \beta } }$, if $\gamma \in \ ]0, 2-\frac{\beta}{2\nu_1 }[$.
{\rm (iv)} $\mathcal{I} - \gamma(\mathcal{I} - \mathcal{T})$ is $\mathcal{Q}$--strongly averaged and $\mathcal{Q}$--nonexpansive. In particular, $\mathcal{I} - \gamma(\mathcal{I} - \mathcal{T})$ is $\mathcal{Q}$--firmly nonexpansive, if $\gamma \in\ ]0, 1-\frac{\beta}{4 \nu_1 } [$.
\end{proposition}
\begin{proof}
(i) Lemma \ref{l_T}--(iii) and Definition \ref{def_lip}.
\vskip.1cm
(ii) Observe in (i) that the averagedness of $\mathcal{T}$ is $\alpha = \frac{2\nu_1 }
{4\nu_1 - \beta } > \frac{1}{2}$.
\vskip.1cm
(iii)--(iv): \cite[Lemma 2.7--(iii) and (iv)]{fxue_1}.
\end{proof}
\begin{remark}
The condition $\beta < 2\nu_1 $ in Proposition \ref{p_T} is equivalent to $\mathcal{Q} \succ \frac{\beta}{2} \mathcal{I}$ or $\mathcal{Q} - \frac{\beta}{2} \mathcal{I} \succ \bf 0$.
\end{remark}
\subsection{Another view on the averagedness of the operator}
Lemma \ref{l_T} can also be verified by the composition of $\mathcal{T}$ given by \eqref{t}.
\begin{lemma} \label{l_com}
Given $\mathcal{T}$ defined as \eqref{t}, then, under Assumption \ref{assume_1}, the following hold.
{\rm (i)} $\mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} \in \mathcal{F}^\mathcal{Q}_{1, \frac{1}{2}}$
{\rm (ii)} $\mathcal{I} - \mathcal{Q}^{-1} \mathcal{B} \in \mathcal{F}^\mathcal{Q}_{1,
\frac {\beta} {2\nu_1 } }$, if $\beta < 2 \nu_1$. In particular, if $\beta \le \nu_1$, $\mathcal{I} - \mathcal{Q}^{-1} \mathcal{B}$ is $\mathcal{Q}$--firmly nonexpansive.
{\rm (iii)} $\mathcal{T} \in \mathcal{F}^\mathcal{Q}_{1, \frac{2\nu_1 }
{4 \nu_1 -\beta } }$, if $\beta < 2 \nu_1 $.
\end{lemma}
\begin{proof}
{\rm (i)} Noting that
$ \mathcal{Q} \in (\mathcal{A} + \mathcal{Q}) \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} $, we have:
\begin{eqnarray}
&& \big\| \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_1) -
\mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_2 ) \big\|_\mathcal{Q}^2
\nonumber \\ &=& \big \langle \mathcal{Q} \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}}( \mathbf{b}_1) - \mathcal{Q} \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_2) | \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}}( \mathbf{b}_1) - \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_2) \big \rangle
\nonumber \\
& \le & \big \langle \mathcal{Q} \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}}( \mathbf{b}_1) - \mathcal{Q} \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_2) | \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}}( \mathbf{b}_1) - \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_2) \big \rangle
\nonumber \\
&+ & \big \langle \mathcal{A} \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}}( \mathbf{b}_1) - \mathcal{A} \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_2) | \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}}( \mathbf{b}_1) - \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_2) \big \rangle
\quad \text{by monotonicity of $\mathcal{A}$}
\nonumber \\ & = &
\big \langle \mathbf{b}_1 - \mathbf{b}_2 | \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}}( \mathbf{b}_1) - \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} (\mathbf{b}_2) \big \rangle_\mathcal{Q}
\nonumber
\end{eqnarray}
which implies that $\mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}}$ is $\mathcal{Q}$--firmly nonexpansive, i.e. $\mathcal{Q}$--based $\frac{1}{2}$--averaged.
\vskip.1cm
(ii) First, $\mathcal{Q}^{-1} \mathcal{B}$ is $\mathcal{Q}$--based $ \frac{\nu_1} {\beta}$--cocoercive, by Lemma \ref{l_cocoercive}. Then, if $\frac{\beta}{\nu_1} \in \ ]0,2[$, $\mathcal{Q}^{-1} \mathcal{B} \in \mathcal{F}^\mathcal{Q}_{\frac{\beta}{2\nu_1 - \beta},
\frac{2\nu_1 - \beta} {2\nu_1}}$, and $\mathcal{I} - \mathcal{Q}^{-1} \mathcal{B} \in \mathcal{F}^\mathcal{Q}_{1, \frac {\beta} {2\nu_1 } }$, by \cite[Lemma 2.11]{fxue_1}.
\vskip.1cm
(iii) Since $\mathcal{T} = \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} \circ (\mathcal{I} - \mathcal{Q}^{-1} \mathcal{B}) $, with $\alpha_1 = \frac{1}{2}$ and $\alpha_2 = \frac {\beta} {2\nu_1 }$, the averagedness of $\mathcal{T}$, by \cite[Theorem 3]{yamada} or \cite[Proposition 2.4]{plc_yamada}, is given as: $\alpha = \frac{\alpha_1+\alpha_2 -2\alpha_1\alpha_2}
{1-\alpha_1\alpha_2} = \frac{2\nu_1} {4 \nu_1 -\beta }$.
\end{proof}
\begin{remark}
Lemma \ref{l_com}--(iii) is an extended version of \cite[Proposition 4.14]{pfbs_siam}, from a scalar parameter $\gamma$ to any variable metric $\mathcal{Q}$. As observed in \cite[Remark 1]{ljw_mapr}, Lemma \ref{l_com}--(iii) is sharper than \cite[Proposition 4.32]{plc_book}, which gives the averagedness of $\mathcal{T}$ as:
\[
\alpha = \frac{2} {1+ \frac{1}{\max \{\frac{1}{2},
\frac{2\nu_1} {\beta} \} } } =
\frac{2\beta} {\beta + 2\times \min\{\beta,
\nu_1 \} } = \max \bigg\{ \frac{2}{3},
\frac{2\beta} {\beta+ 2\nu_1 } \bigg\}
\]
\end{remark}
\subsection{The nonexpansive properties of $\mathcal{I} - \mathcal{T}$}
\begin{proposition} \label{p_R}
Given $\mathcal{T}$ defined as \eqref{t}, denote $\mathcal{R} := \mathcal{I} -\mathcal{T}$. Under Assumption \ref{assume_1}, if $\beta < 2 \nu_1 $, then, the following hold.
{\rm (i)} $\mathcal{R}$ is $\mathcal{Q}$--based $(1-\frac{\beta}{4\nu_1})$--cocoercive;
{\rm (ii)} $\mathcal{R} \in \mathcal{F}^\mathcal{Q}_{\frac{2\nu_1}{2\nu_1 - \beta}, \frac{2\nu_1 - \beta} {4\nu_1 - \beta} }$;
{\rm (iii)} $\mathcal{R}$ is $\mathcal{Q}$--weakly averaged, with the averagedness $\alpha \in \ ]0, \frac{1}{2} [$;
{\rm (iv)} $\mathcal{I} - \gamma \mathcal{R} \in \mathcal{F}^\mathcal{Q}_{1, \frac{2 \gamma \nu_1 } {4\nu_1 - \beta} }$, if $\gamma \in \ ]0, 2-\frac{\beta}{2\nu_1} [$.
\end{proposition}
\begin{proof}
(i) By Lemma \ref{l_T}--(i), and $\mathcal{R} = \mathcal{I}-\mathcal{T}$, we have:
\[
0 \le \langle \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \rangle_\mathcal{Q}
- \langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1-\mathcal{T}\mathbf{b}_2 \rangle
\]
Adding $\|\mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \|_\mathcal{Q}^2$ on both sides, we obtain:
\begin{eqnarray}
\big\|\mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big\|_\mathcal{Q}^2
& \le & \big \langle \mathbf{b}_1 - \mathbf{b}_2 \big| \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big \rangle_{\mathcal{Q}}
- \big\langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big| \mathcal{T}\mathbf{b}_1-\mathcal{T}\mathbf{b}_2
\big \rangle
\quad \text{by $\mathcal{R} + \mathcal{T} = \mathcal{I}$}
\nonumber \\
& = & \big \langle \mathbf{b}_1 - \mathbf{b}_2 \big| \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big \rangle_{\mathcal{Q}}
+ \big\langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big| \mathcal{R}\mathbf{b}_1-\mathcal{R}\mathbf{b}_2
\big \rangle
- \big\langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big| \mathbf{b}_1 - \mathbf{b}_2
\big \rangle
\nonumber \\
& \le & \big \langle \mathbf{b}_1 - \mathbf{b}_2 \big| \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big \rangle_{\mathcal{Q}}
+ \frac{1}{ \beta} \big\| \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big\|^2 +
\frac{\beta}{4} \big\| \mathcal{R}\mathbf{b}_1-\mathcal{R}\mathbf{b}_2 \big \|^2
- \frac{1}{ \beta} \big\| \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 \big\|^2
\nonumber \\
& = & \big \langle \mathbf{b}_1 - \mathbf{b}_2 \big| \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big \rangle_{\mathcal{Q}}
+ \frac{\beta} {4} \big\| \mathcal{R}\mathbf{b}_1-\mathcal{R}\mathbf{b}_2 \big \|^2
\nonumber
\end{eqnarray}
which yields (i).
\vskip.1cm
(ii): \cite[Lemma 2.11]{fxue_1}.
\vskip.1cm
(iii): note that $\xi = \frac{2\nu_1} {2\nu_1 - \beta} >1$, $\alpha = \frac{2\nu_1 - \beta} {4\nu_1 - \beta} <\frac{1}{2}$.
\vskip.1cm
(iv) Lemma \ref{p_T}--(iii) or combine Proposition \ref{p_R}--(i) with \cite[Lemma 2.7--(vi)]{fxue_1}.
\end{proof}
\subsection{The case of strongly monotone $\mathcal{A}$}
Under the condition of $\mu$--strongly monotone $\mathcal{A}$, Lemma \ref{l_T} and Proposition \ref{p_R} can be strengthened as follows.
\begin{lemma} \label{l_TR_mu}
Define the operator $\mathcal{T}$ as \eqref{t}, denote $\mathcal{R} = \mathcal{I} - \mathcal{T}$. Under Assumption \ref{assume_1},
if $\mathcal{A}$ in \eqref{t} is $\mu$-strongly monotone, the following hold:
{\rm (i)} $\big( 1+ \frac{\mu}{\nu_2} \big)
\big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2 \le
\langle \mathbf{b}_1 - \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \rangle_\mathcal{Q}
- \langle \mathcal{B} \mathbf{b}_1 - \mathcal{B} \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1-\mathcal{T}\mathbf{b}_2 \rangle$
{\rm (ii)} $\langle \mathbf{b}_1 - \mathbf{b}_2 | \mathcal{T}\mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \rangle_\mathcal{Q} \ge \big( 1+ \frac{\mu}{\nu_2} \big)
\big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2
- \frac{\beta}{4} \big\| (\mathcal{I} - \mathcal{T}) \mathbf{b}_1 -
(\mathcal{I} - \mathcal{T}) \mathbf{b}_2 \big\|^2$
{\rm (iii)} $ \big \langle \mathbf{b}_1 - \mathbf{b}_2 \big| \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big \rangle_{\mathcal{Q}}
\ge \Big( 1-\frac{\beta}{4 \nu_1} \Big) \big\|\mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big\|_{\mathcal{Q}}^2 + \frac{\mu}{\nu_2} \|\mathcal{T}\mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \|_\mathcal{Q}^2 $
{\rm (iv)} $ \big( 1+ \frac{2\mu}{\nu_2} \big)
\big\| \mathcal{T} \mathbf{b}_1 - \mathcal{T} \mathbf{b}_2 \big\|_\mathcal{Q}^2
+ \Big( 1-\frac{\beta}{2 \nu_1} \Big)
\big\|(\mathcal{I}- \mathcal{T}) \mathbf{b}_1 - (\mathcal{I} -\mathcal{T}) \mathbf{b}_2 \big\|_\mathcal{Q}^2
\le \big\| \mathbf{b}_1 - \mathbf{b}_2 \big\|_\mathcal{Q}^2 $
\end{lemma}
\begin{proof}
Combining Lemma \ref{l_T} and Proposition \ref{p_R} with the $\mu$--strongly monotone condition of $\mathcal{A}$:
\[
\big \langle \mathcal{A} \mathbf{b}_1 - \mathcal{A} \mathbf{b}_2 \big | \mathbf{b}_1 - \mathbf{b}_2 \big \rangle \ge \mu \big \|\mathbf{b}_1 - \mathbf{b}_2 \big \|^2
\ge \frac{\mu} {\nu_2}
\big \|\mathbf{b}_1 - \mathbf{b}_2 \big \|_\mathcal{Q}^2
\]
completes the proof.
\end{proof}
\begin{proposition} \label{p_TR_mu}
Under the condition of Lemma \ref{l_TR_mu}, if $\beta < 2 \nu_1 $, the following hold.
{\rm (i)} $\mathcal{T} \in \mathcal{F}^\mathcal{Q}_{\xi_1, \alpha_1 }$ with $\xi_1 = \frac{\sqrt{1+\frac{\mu \beta} {\nu_1 \nu_2 } } } {1+\frac{2\mu}{\nu_2} } $ and $\alpha_1 = \frac {1+\frac{2\mu}{\nu_2}} {2-\frac{\beta}{2\nu_1 } + \frac{2\mu}{\nu_2}} $;
$\mathcal{R} \in \mathcal{F}^\mathcal{Q}_{\xi_2, \alpha_2}$ with $\xi_2 = \frac{\sqrt{1+\frac{\mu \beta} {\nu_1 \nu_2 } } } {1-\frac{\beta}{2 \nu_1 } }$ and
$\alpha_2 = \frac {1-\frac{ \beta} {2 \nu_1 }} {2-\frac{\beta}{2 \nu_1 } + \frac{2\mu}{\nu_2}} $.
{\rm (ii)} $\mathcal{T}$ is $\mathcal{Q}$--strongly averaged with $ \alpha_1 \in \ ]\frac{1}{2}, 1[$; $\mathcal{R}$ is $\mathcal{Q}$--weakly averaged with
$\alpha_2 \in \ ]0, \frac{1}{2}[ $.
{\rm (iii)} Neither $\mathcal{T}$ nor $\mathcal{R}$ is $\mathcal{Q}$--firmly nonexpansive.
{\rm (iv)} $\mathcal{T}$ is not $\mathcal{Q}$--cocoercive; $\mathcal{R}$ is $\mathcal{Q}$--based $(1- \frac{\beta} {4\nu_1 })$--cocoercive.
\end{proposition}
\begin{proof}
(i) The averagedness of $\mathcal{T}$ follows by Lemma \ref{l_TR_mu}--(iv) and \cite[Theorem 2.8--(i)]{fxue_1}.
\vskip.1cm
(ii) It is easy to see that $\xi_1 \in\ ]0,1[$ and $\xi_2 \in \ ]1, +\infty[ $, by recognizing that
\begin{eqnarray}
1+ \frac{\mu \beta} {\nu_1 \nu_2 }
&= & \Big(1 + \frac{2\mu}{\nu_2 } \Big)^2 - \frac{2\mu}{\nu_2 } \Big( 2-\frac{\beta}{2\nu_1} + \frac{2\mu}{\nu_2 } \Big)
\nonumber \\
&= & \Big(1 - \frac{\beta}{2 \nu_1} \Big)^2 + \frac{ \beta}{2\nu_1 } \Big( 2-\frac{\beta}{2\nu_1} + \frac{2\mu}{\nu_2} \Big)
\nonumber
\end{eqnarray}
\vskip.1cm
(iii) follows from \cite[Theorem 2.8--(ii)]{fxue_1}, by noting that $\xi_1 > \frac{1-\alpha_1} {\alpha_1}$ and $\xi_2 >1$.
\vskip.1cm
(iv) follows from \cite[Theorem 2.8--(iii)]{fxue_1}, by noting that
$ \xi_1 \in \ ] \frac{1-\alpha_1} {\alpha_1}, 1[$ and
$ \xi_2\in \ ]1, \frac{1-\alpha_2} {\alpha_2} [$. The cocoerciveness of $\mathcal{R}$ coincides with Proposition \ref{p_R}--(i).
\end{proof}
\section{The fixed-point iterations}
\label{sec_iteration}
\subsection{The Banach-Picard iteration \eqref{gfbs}}
\subsubsection{Convergence of metric distance}
The convergence properties of \eqref{gfbs} are given below.
\begin{theorem}[Convergence in terms of metric distance] \label{t_gfbs}
Let $\mathbf{b}^0\in \mathcal{H}$, $\{\mathbf{b}^k\}_{k \in \mathbb{N}}$ be a sequence generated by \eqref{gfbs}. Under Assumption \ref{assume_1}, if $\nu_1 > \frac{\beta} {2} $, then, the following hold.
{\rm (i)} $\mathcal{T}$ is $\mathcal{Q}$--asymptotically regular.
{\rm (ii) [Basic convergence]} There exists $\mathbf{b}^\star \in \mathsf{zer} (\mathcal{A} +\mathcal{B})$, such that $\mathbf{b}^k \rightarrow \mathbf{b}^\star$, as $k\rightarrow \infty$.
{\rm (iii) [Sequential error]} $\|\mathbf{b}^{k+1 } -\mathbf{b}^{k} \|_\mathcal{Q}$ has the pointwise sublinear convergence rate of $\mathcal{O}(1/\sqrt{k})$:
\[
\big\|\mathbf{b}^{k +1} -\mathbf{b}^{k} \big\|_\mathcal{Q}
\le \frac{1}{\sqrt{k+1}} \sqrt{ \frac{2\nu_1}{2\nu_1 - \beta} }
\big\|\mathbf{b}^{0} -\mathbf{b}^\star \big\|_\mathcal{Q},
\forall k \in \mathbb{N}
\]
\end{theorem}
\begin{proof}
Note that $\mathcal{T} \in \mathcal{F}^\mathcal{Q}_{1, \frac{2\nu_1 } {4 \nu_1 -\beta } }$, by Proposition \ref{p_T}--(i). Then, (i) and (iii) follows from \cite[Theorem 3.3]{fxue_1}.
(ii) follows by noting that $\mathsf{Fix} \mathcal{T} = \mathsf{zer} (\mathcal{A} + \mathcal{B})$. Indeed,
$\mathbf{b}^\star \in \mathsf{Fix} \mathcal{T} \Longleftrightarrow
\mathbf{b}^\star = (\mathcal{A} + \mathcal{Q})^{-1} (\mathcal{Q} - \mathcal{B}) \mathbf{b}^\star
\Longleftrightarrow (\mathcal{Q} - \mathcal{B}) \mathbf{b}^\star \in
(\mathcal{A} + \mathcal{Q}) \mathbf{b}^\star \Longleftrightarrow {\bf 0} \in (\mathcal{A} +\mathcal{B}) \mathbf{b}^\star \Longleftrightarrow \mathbf{b}^\star \in \mathsf{zer} (\mathcal{A} +\mathcal{B})$.
\end{proof}
\vskip.2cm
If $\mathcal{A}$ in \eqref{gfbs} is $\mu$--strongly monotone, then we have the following results.
\begin{proposition}[Convergence in terms of metric distance] \label{p_banach}
Let $\mathbf{b}^0\in \mathcal{H}$, $\{\mathbf{b}^k\}_{k \in \mathbb{N}}$ be a sequence generated by \eqref{gfbs}. Under Assumption \ref{assume_1}, if $\mathcal{A}$ is $\mu$--strongly monotone, $\nu_1 > \frac{\beta} {2} $, then, the following hold.
{\rm (i) [$q$--linear convergence]} Both $\|\mathbf{b}^{k} -\mathbf{b}^\star \|_\mathcal{Q}$ and $\|\mathbf{b}^{k} -\mathbf{b}^{k+1} \|_\mathcal{Q}$ are $q$--linearly convergent with the rate of $\frac{1}{ \sqrt{1+ \frac{2\mu}{\nu_2 } } } $.
{\rm (ii) [$r$--linear convergence]} If $\mu \ge \frac{\nu_2 }
{\sqrt{5}+1}$,
$\big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|_\mathcal{Q}$ is globally $r$--linearly convergent w.r.t. $\big\| \mathbf{b}^0 - \mathbf{b}^\star \big\|_\mathcal{Q}$:
\[
\big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|_\mathcal{Q}
\le \sqrt{\frac{2\mu}{\nu_2}} \sqrt{ \frac{1+ \frac{2\mu}{\nu_2 } } {1-\frac{\beta}{2\nu_1 } } }
\cdot \bigg(1+ \frac{2\mu}{\nu_2} \bigg)^{-\frac{k+1}{4} }
\big\| \mathbf{b}^0 - \mathbf{b}^\star \big\|_\mathcal{Q}
\]
The above inequality is also locally satisfied for $k \ge \frac{\ln((1+\sqrt{5})/2)} {\ln \sqrt{1+\frac{2\mu}{\nu_2}} } - 1$,
if $\mu < \frac{\nu_2} {\sqrt{5}+1}$.
\end{proposition}
\begin{proof}
Combine Proposition \ref{p_TR_mu}--(i) with \cite[Theorem 3.3--(iv) and (v)]{fxue_1}.
\end{proof}
\begin{remark}
If $\mathcal{Q} = \frac{1}{\tau} \mathcal{I}$, the result of \cite[Example 27.12]{plc_book} is recovered from Proposition \ref{p_banach}.
\end{remark}
\subsubsection{Convergence of objective value}
If $\mathcal{A} = \partial g$, $\mathcal{B} =\nabla f$, \eqref{gfbs} becomes the scheme \eqref{gpfbs} for minimizing the objective function $f+g$. We make the following assumptions on the functions $f$ and $g$.
\begin{assumption} \label{assume_2}
(i) $f: \mathcal{H} \mapsto \mathbb{R}$ and $g: \mathcal{H} \mapsto \mathbb{R} \cup \{ +\infty\}$ are proper, lower semi-continuous;
(ii) $f$ is differentiable with $\beta$-Lipschitz continuous gradient;
(iii) $g$ is convex.
\end{assumption}
The following lemma is important for proving the convergence, which extends the {\it sufficient decrease property} \cite[Lemma 2]{bolte_2014} to the case of arbitrary variable metric $\mathcal{Q}$.
\begin{lemma} [Sufficient decrease property] \label{l_decrease}
Let $\mathbf{b}^0\in \mathcal{H}$, $\{\mathbf{b}^k\}_{k \in \mathbb{N}}$ be a sequence generated by \eqref{gpfbs}. Then, the following hold.
(i) Under Assumptions \ref{assume_1} and \ref{assume_2}-(i) and (ii), we have:
\[
(f+g) (\mathbf{b}^k) - (f+g) (\mathbf{b}^{k+1}) \ge
\frac{1}{2} \Big(1- \frac{\beta} { \nu_1} \Big)
\big\| \mathbf{b}^{k+1} - \mathbf{b}^k\big\|_\mathcal{Q}^2
\]
(ii) Under Assumptions \ref{assume_1} and \ref{assume_2}, we have:
\[
(f+g) (\mathbf{b}^k) - (f+g) (\mathbf{b}^{k+1}) \ge
\Big(1- \frac{\beta}{2\nu_1} \Big)
\big\| \mathbf{b}^{k+1} - \mathbf{b}^k\big\|_\mathcal{Q}^2
\]
\end{lemma}
\begin{proof}
Denote $h := f + g $. Rewrite \eqref{gpfbs} in an inclusion form:
\begin{equation} \label{gpfbs_eq}
{\bf 0} \in \nabla f(\mathbf{b}^k) +\partial g(\mathbf{b}^{k+1})
+\mathcal{Q} (\mathbf{b}^{k+1} - \mathbf{b}^k)
\end{equation}
By $\beta$--Lipschitz continuity of $\nabla f$ and {\it Descent Lemma} \cite[Theorem 18.15]{plc_book}, we have:
\begin{eqnarray} \label{x1}
h(\mathbf{b}^{k+1}) & = & f(\mathbf{b}^{k+1}) + g(\mathbf{b}^{k+1} )
\nonumber \\
& \le & f(\mathbf{b}^k) + \big \langle \nabla f(\mathbf{b}^k) | \mathbf{b}^{k+1} - \mathbf{b}^k \big \rangle
+\frac{\beta}{2} \big \|\mathbf{b}^{k+1} - \mathbf{b}^k\big \|^2
+ g(\mathbf{b}^{k+1})
\end{eqnarray}
(i) By the definition of generalized proximity operator, we have:
\[
\mathbf{b}^{k+1} =\arg\min_\mathbf{b} g(\mathbf{b}) + \frac{1}{2} \big\|\mathbf{b} - \mathbf{b}^k \big\|_\mathcal{Q}^2 + \big \langle \mathbf{b} - \mathbf{b}^k | \nabla f(\mathbf{b}^k)
\big \rangle
\]
Taking $\mathbf{b} = \mathbf{b}^k$, we obtain:
\begin{equation} \label{x3}
g(\mathbf{b}^{k+1}) + \frac{1}{2} \big\|\mathbf{b}^{k+1} - \mathbf{b}^k \big\|_\mathcal{Q}^2 + \big \langle \mathbf{b}^{k+1} - \mathbf{b}^k | \nabla f(\mathbf{b}^k)
\big \rangle \le g(\mathbf{b}^k)
\end{equation}
Combining \eqref{x1} with \eqref{x3} yields:
\[
h( \mathbf{b}^{k+1}) \le h(\mathbf{b}^{k }) - \frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|^2_{\mathcal{Q} - \beta \mathcal{I}} \le
h(\mathbf{b}^{k }) - \frac{1}{2} \Big(1- \frac{\beta}{\nu_1} \Big) \big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|^2_{\mathcal{Q}}
\]
\vskip.2cm
(ii) By convexity $g$, we have:
\begin{equation} \label{x2}
h(\mathbf{b}^k) = f(\mathbf{b}^k) + g(\mathbf{b}^{k} ) \ge f(\mathbf{b}^k)
+ g(\mathbf{b}^{k+1}) + \big \langle \partial g (\mathbf{b}^{k+1}) | \mathbf{b}^k - \mathbf{b}^{k+1} \big \rangle
\end{equation}
Combining \eqref{x1} with \eqref{x2} yields:
\begin{eqnarray}
h( \mathbf{b}^k) - h(\mathbf{b}^{k+1}) & \ge & - \big \langle
\nabla f(\mathbf{b}^{k}) + \partial g (\mathbf{b}^{k+1}) | \mathbf{b}^{k+1} - \mathbf{b}^k \big \rangle - \frac{\beta}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|^2
\nonumber \\
& = & \big \langle \mathbf{b}^{k+1} - \mathbf{b}^k | \mathbf{b}^{k+1} - \mathbf{b}^k \big \rangle_\mathcal{Q} - \frac{\beta}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|^2 \quad \text{by \eqref{gpfbs_eq}}
\nonumber \\
& = & \big\| \mathbf{b}^{k+1} - \mathbf{b}^k\big\|_{\mathcal{Q}-\frac{\beta}{2}\mathcal{I}}^2 \ge \Big(1- \frac{\beta}{2\nu_1} \Big) \big\| \mathbf{b}^{k+1} - \mathbf{b}^k\big\|_\mathcal{Q}^2
\nonumber
\end{eqnarray}
\end{proof}
\begin{remark}
Without convexity of $g$ (i.e. Lemma \ref{l_decrease}--(i)), the sufficient decreasing requires $\nu_1 \ge \beta $ (i.e. $\mathcal{Q} \succeq \beta \mathcal{I}$). If $g$ is convex (i.e. Lemma \ref{l_decrease}--(ii)), this condition is relaxed to $\nu_1 \ge \frac{\beta}{2}$ (i.e. $\mathcal{Q} \succeq \frac{\beta}{2} \mathcal{I}$). This coincides with the observation in \cite[Remark 4--(iii)]{bolte_2014}. In addition, combining \eqref{gpfbs_eq} with \eqref{x2}, we obtain:
\[
g(\mathbf{b}^{k+1}) + \big\langle \mathbf{b}^{k+1} - \mathbf{b}^k | \nabla f(\mathbf{b}^k) \big \rangle + \big\|\mathbf{b}^{k+1} - \mathbf{b}^k \big\|_\mathcal{Q}^2 \le
g(\mathbf{b}^k)
\]
which is in agreement with the {\it sufficient decrease condition} \cite[Eq.(7a)]{pesquet_2014}.
\end{remark}
\begin{theorem}[Convergence in terms of objective value] \label{t_gpfbs}
Let $\mathbf{b}^0\in \mathcal{H}$, $\{\mathbf{b}^k\}_{k \in \mathbb{N}}$ be a sequence generated by \eqref{gpfbs}. Under Assumptions \ref{assume_1} and \ref{assume_2}, if $\nu_1 \ge \beta $, the following hold.
{\rm (i) [Basic convergence]} The sequence $\{(f+g) (\mathbf{b}^k)\}_{k \in \mathbb{N}}$ is non-increasing, and converges to its minimum, which is attained at some point $\mathbf{b}^\star \in \Arg \min (f+g)$.
{\rm (ii) [Non-ergodic rate]} The objective value $h(\mathbf{b}^k)$ converges to $h(\mathbf{b}^\star)$ with the {\it non-ergodic} rate of $\mathcal{O}(1/k)$, i.e.
\[
h(\mathbf{b}^{k}) - h(\mathbf{b}^\star) \le \frac{1}{2k}
\big\| \mathbf{b}^0 - \mathbf{b}^\star \big\|_\mathcal{Q}^2
\]
\end{theorem}
\begin{proof}
(i) Theorem \ref{t_gfbs}, Lemma \ref{l_decrease}-(ii).
\vskip.1cm
(ii) Denote $h := f + g $. By convexity of $f$ and $g$, we have:
\begin{equation} \label{x4}
h( \mathbf{b}^\star) = f( \mathbf{b}^\star) + g( \mathbf{b}^\star) \ge f(\mathbf{b}^{k}) + \big \langle
\nabla f (\mathbf{b}^{k}) | \mathbf{b}^\star - \mathbf{b}^{k} \big \rangle
+ g(\mathbf{b}^{k+1}) + \big \langle \partial g (\mathbf{b}^{k+1}) | \mathbf{b}^\star - \mathbf{b}^{k+1}\big \rangle
\end{equation}
Combining \eqref{x4} with \eqref{x1} yields:
\begin{eqnarray} \label{x15}
h( \mathbf{b}^\star) - h(\mathbf{b}^{k+1}) & \ge & - \big \langle
\nabla f(\mathbf{b}^{k}) + \partial g (\mathbf{b}^{k+1}) | \mathbf{b}^{k+1} - \mathbf{b}^\star \big \rangle - \frac{\beta}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|^2
\nonumber \\
& = & \big \langle \mathbf{b}^{k+1} - \mathbf{b}^k | \mathbf{b}^{k+1} - \mathbf{b}^\star \big \rangle_\mathcal{Q} - \frac{\beta}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|^2 \quad \text{by \eqref{gpfbs_eq}}
\nonumber \\
& = & \frac{1}{2} \big\| \mathbf{b}^{k+1} - \mathbf{b}^k\big\|_{\mathcal{Q}-\beta\mathcal{I}}^2 +
\frac{1}{2} \big\| \mathbf{b}^{k+1} - \mathbf{b}^\star \big\|_\mathcal{Q}^2
-\frac{1}{2} \big\|\mathbf{b}^k -\mathbf{b}^\star \big\|_\mathcal{Q}^2
\nonumber \\
& \ge & \frac{1}{2} \big\| \mathbf{b}^{k+1} - \mathbf{b}^\star \big\|_\mathcal{Q}^2
-\frac{1}{2} \big\|\mathbf{b}^k -\mathbf{b}^\star \big\|_\mathcal{Q}^2
\quad \text{by $\nu_1 \ge \beta$}
\end{eqnarray}
The rest of the proof adopts similar techniques with \cite[Theorem 3.1]{fista}. Multiplying Lemma \ref{l_decrease}--(ii) by $k$, adding \eqref{x15} from $k=0$ to $k=K-1$, and combining them together, we obtain:
\begin{eqnarray}
K (h(\mathbf{b}^K) - h(\mathbf{b}^\star) ) & \le &
\frac{1}{2} \big\| \mathbf{b}^{0} - \mathbf{b}^\star\big\|^2_\mathcal{Q}
- \frac{1}{2} \big\| \mathbf{b}^K - \mathbf{b}^\star\big\|^2_\mathcal{Q}
- \sum_{k=0}^{K-1} k \big\| \mathbf{b}^{k+1} - \mathbf{b}^{k } \big\|^2
_{\mathcal{Q} - \frac{\beta}{2} \mathcal{I}}
\nonumber \\
& \le & \frac{1}{2}
\big\| \mathbf{b}^{0} - \mathbf{b}^\star \big\|^2_\mathcal{Q}
\nonumber
\end{eqnarray}
which completes the proof.
\end{proof}
\vskip.2cm
We further develop the linear convergence in terms of objective value.
\begin{proposition}[Convergence in terms of objective value] \label{p_gpfbs_mu}
Under the conditions of Theorem \ref{t_gpfbs}, Assumptions \ref{assume_1} and \ref{assume_2}, if $g$ is $\mu$--strongly convex, then, the following hold.
{\rm (i) [Basic convergence]} $ h(\mathbf{b}^{k }) - h(\mathbf{b}^\star) \le \frac{\mu} { 2 \nu_2} \cdot
\frac{1}{ ( 1+ \frac{\mu} {\nu_2 } )^k - 1 } \cdot
\big\|\mathbf{b}^0-\mathbf{b}^\star \big\|_\mathcal{Q}^2 $.
{\rm (ii) [$r$-linear convergence]} If $\mu \ge
\frac{\sqrt{5} +1} {2} \nu_2 $, $ h(\mathbf{b}^{k })$ is globally $r$-linearly convergent to
$ h(\mathbf{b}^\star)$:
\[
h(\mathbf{b}^{k }) - h(\mathbf{b}^\star) \le \frac{\mu} { 2 \nu_2 } \cdot \Big( 1+\frac{\mu} {\nu_2} \Big)
^{-\frac{k}{2} }
\big\|\mathbf{b}^0-\mathbf{b}^\star \big\|_\mathcal{Q}^2
\]
The above $r$--linear convergence is also locally satisfied, for $k \ge \frac {\ln ((1+\sqrt{5}) /2) }
{ \ln \sqrt{ 1+\frac{\mu}{\nu_2}} } $, if $\mu \in \ \big] 0,
\frac{\sqrt{5} +1} {2} \nu_2 \big[$.
\end{proposition}
\begin{proof}
Considering the case of $\mu$-strongly monotone $\mathcal{A}$,
the above results are modified as:
\begin{equation} \label{xxa}
\left\{ \begin{array}{lll}
h(\mathbf{b}^\star) - h( \mathbf{b}^{k+1}) & \ge & \frac{1}{2} \| \mathbf{b}^{k+1} - \mathbf{b}^{k } \|^2_{\mathcal{Q} - \beta \mathcal{I}}
+ \frac{1}{2} (1+\frac{\mu} {\nu_2}) \| \mathbf{b}^{k+1} - \mathbf{b}^\star\|^2_{\mathcal{Q} } - \frac{1}{2} \| \mathbf{b}^{k} - \mathbf{b}^\star\|^2_\mathcal{Q} \\
h(\mathbf{b}^k ) - h( \mathbf{b}^{k+1}) & \ge & \| \mathbf{b}^{k+1} - \mathbf{b}^{k } \|^2_{\mathcal{Q} + \frac{\mu-\beta}{2} \mathcal{I} }
\end{array} \right.
\end{equation}
where the first inequality of \eqref{xxa} uses $ \|\cdot \|^2 \ge \frac{1} {\nu_2 } \|\cdot \|_\mathcal{Q}^2$.
Denoting $\Delta_k = h( \mathbf{b}^{k})- h(\mathbf{b}^\star) $, multiplying the second of \eqref{xxa} by $\theta_k \ge 0$, and adding the first of \eqref{xxa}, one obtains:
\begin{eqnarray}
\theta_k \Delta_k - (\theta_k +1) \Delta_{k+1} & \ge &
\| \mathbf{b}^{k+1} - \mathbf{b}^{k } \|^2_{ (\theta_k +\frac{1}{2})\mathcal{Q}+\frac{\theta_k (\mu-\beta) - \beta} {2} \mathcal{I}}
+ \frac{1}{2} \big( 1+ \frac{\mu}{\nu_2 } \big)
\| \mathbf{b}^{k+1} - \mathbf{b}^\star\|^2_\mathcal{Q}
- \frac{1}{2} \| \mathbf{b}^{k} - \mathbf{b}^\star\|^2_\mathcal{Q}
\nonumber \\
& \ge &\frac{1}{2} \big( 1+ \frac{\mu}{\nu_2 } \big)
\| \mathbf{b}^{k+1} - \mathbf{b}^\star\|^2_\mathcal{Q}
- \frac{1}{2} \| \mathbf{b}^{k} - \mathbf{b}^\star\|^2_\mathcal{Q}
\nonumber
\end{eqnarray}
Multiplying by $\zeta_k \ge 0$ obtains:
\begin{equation} \label{xa1}
2 \underbrace{ \zeta_k \theta_k}_{t_k} \Delta_k -
2 \underbrace{ \zeta_k (\theta_k +1)}_{t_{k+1}} \Delta_{k+1}
\ge \underbrace{ \zeta_k \big( 1+ \frac{\mu}{\nu_2 } \big) }
_{ \zeta_{k+1}} \| \mathbf{b}^{k+1} - \mathbf{b}^\star\|^2_\mathcal{Q}
- \zeta_k \| \mathbf{b}^{k} - \mathbf{b}^\star\|^2_\mathcal{Q}
\end{equation}
which yields:
\begin{equation} \label{xa2}
2 t_{k+1} \Delta_{k+1} +
\zeta_{k+1} \| \mathbf{b}^{k+1} - \mathbf{b}^\star\|^2_\mathcal{Q}
\le 2 t_k \Delta_k + \zeta_k \| \mathbf{b}^{k} - \mathbf{b}^\star\|^2_\mathcal{Q}
\le \cdots \le 2 t_1 \Delta_1 + \zeta_1 \| \mathbf{b}^{1} - \mathbf{b}^\star\|^2_\mathcal{Q}
\end{equation}
By the first inequality of \eqref{xxa}, we have:
\begin{equation} \label{xa3}
2 t_1 \Delta_1 = 2 t_1 (h(\mathbf{b}^1) - h(\mathbf{b}^\star))
\le t_1 \|\mathbf{b}^0-\mathbf{b}^\star\|_\mathcal{Q}^2 - \big(1+ \frac{\mu}{\nu_2}\big) t_1 \|\mathbf{b}^1 - \mathbf{b}^\star\|_\mathcal{Q}^2 - t_1
\|\mathbf{b}^1-\mathbf{b}^0\|_\mathcal{Q}^2
\end{equation}
Substituting \eqref{xa3} into \eqref{xa2} yields:
\begin{eqnarray} \label{xa5}
2t_k \Delta_k & \le & 2t_1 \Delta_1 +\zeta_1 \big\|\mathbf{b}^1 - \mathbf{b}^\star \big\|_\mathcal{Q}^2
\nonumber \\
& \le & t_1 \|\mathbf{b}^0-\mathbf{b}^\star\|_\mathcal{Q}^2 - \big(1+ \frac{\mu}{\nu_2 }\big) t_1 \|\mathbf{b}^1 - \mathbf{b}^\star\|_\mathcal{Q}^2 + \zeta_1
\|\mathbf{b}^1-\mathbf{b}^\star\|_\mathcal{Q}^2
\nonumber \\
& \le & t_1 \|\mathbf{b}^0-\mathbf{b}^\star\|_\mathcal{Q}^2 - \big(t_1+ \frac{\mu}{\nu_2 } t_1 - \zeta_1 \big) \|\mathbf{b}^1 - \mathbf{b}^\star\|_\mathcal{Q}^2
\end{eqnarray}
Now, we evaluate $t_{k}$. From \eqref{xa1}, we have:
\[
\zeta_{k+1} = \big( 1+ \frac{\mu}{\nu_2 } \big) \zeta_k; \quad
\theta_{k+1} = \frac{1}{ 1+ \frac{\mu}{\nu_2 } } (\theta_k+1)
\]
which leads to:
\[
\theta_k = \Big( \theta_0 - \frac{\eta}{1-\eta} \Big) \eta^k +
\frac{\eta}{1-\eta};\quad
\zeta_k = \zeta_0 \eta^{-k}
\]
where $\eta := (1+ \frac{\mu}{\nu_2 })^{-1}$.
Back to \eqref{xa5}. It is easy to check that $ t_1+ \frac{\mu}{\nu_2 } t_1 \ge \xi_1$, as long as $\theta_0 \ge 0$. Thus, \eqref{xa5} becomes:
\[
\Delta_k \le \frac{ t_1}{2t_k} \big\|\mathbf{b}^0-\mathbf{b}^\star \big\|_\mathcal{Q}^2
= \frac{ \theta_1 \zeta_1} {2 \theta_k \zeta_k} \big\|\mathbf{b}^0-\mathbf{b}^\star \big\|_\mathcal{Q}^2 = \underbrace{
\frac{ \theta_0+1 } {2 \big( \theta_0 + \frac{\nu_2 }{\mu} (\eta^{-k}-1) \big) } }_\kappa \big\|\mathbf{b}^0-\mathbf{b}^\star \big\|_\mathcal{Q}^2
\]
Since $\kappa$ is increasing with $\theta_0$, the best estimate of $\Delta_k$ follows by letting $\theta_0 = 0$.
\vskip.3cm
(ii) It is easy to check that $ (1+\frac{\mu}{\nu_2 })^k-1 \ge
(1+\frac{\mu}{\nu_2 })^{k/2}$, if $k \ge \frac {\ln ((1+\sqrt{5})/2 ) } { \ln \sqrt{ 1+\frac{\mu}{\nu_2 } } } $. Then, the $r$-linear convergence immediately follows from (i).
\end{proof}
\subsection{The Krasnosel'skii-Mann iteration}
The Krasnosel'skii-Mann iteration of $\mathcal{T}$ in \eqref{t} is given as:
\begin{equation} \label{rgfbs}
\mathbf{b}^{k+1} := \mathbf{b}^k + \gamma \Big( \mathcal{J}_{\mathcal{Q}^{-1} \mathcal{A}} \big( \mathbf{b}^k - \mathcal{Q}^{-1} \mathcal{B} \mathbf{b}^k \big) - \mathbf{b}^k \Big)
\end{equation}
where $\gamma$ is a relaxation parameter. The scheme \eqref{rgfbs} is also a Banach-Picard iteration of $\mathcal{T}_\gamma := \mathcal{I} +\gamma (\mathcal{T} - \mathcal{I})$. The convergence properties of \eqref{rgfbs} are given below.
\begin{theorem}[Convergence in terms of metric distance] \label{t_rgfbs}
Let $\mathbf{b}^0\in \mathcal{H}$, $\{\mathbf{b}^k\}_{k \in \mathbb{N}}$ be a sequence generated by \eqref{rgfbs}. Under Assumptions \ref{assume_1} and \ref{assume_2}, if $\nu_1 > \frac{ \beta}{ 2 } $, $\gamma \in\ ] 0, 2-\frac{\beta}{2\nu_1 } [$, then, the following hold.
{\rm (i) [Basic convergence]} There exists $\mathbf{b}^\star \in \mathsf{zer} (\mathcal{A} +\mathcal{B})$, such that $\mathbf{b}^k \rightarrow \mathbf{b}^\star$, as $k\rightarrow \infty$.
{\rm (ii) [Sequential error]} $\|\mathbf{b}^{k+1 } -\mathbf{b}^{k} \|_\mathcal{Q}$ has the pointwise sublinear convergence rate of $\mathcal{O}(1/\sqrt{k})$:
\[
\big\|\mathbf{b}^{k +1} -\mathbf{b}^{k} \big\|_\mathcal{Q}
\le \frac{1}{\sqrt{k+1}} \sqrt{ \frac{2\gamma \nu_1 } {(4-2\gamma) \nu_1 - \beta} }
\big\|\mathbf{b}^{0} -\mathbf{b}^\star \big\|_\mathcal{Q},
\forall k \in \mathbb{N}
\]
\end{theorem}
\begin{proof}
First, $\mathcal{T}_\gamma := \mathcal{I} - \gamma(\mathcal{I} - \mathcal{T}) \in \mathcal{F}^\mathcal{Q}_{1, \frac{2 \gamma \nu_1 }
{4 \nu_1 - \beta } }$, if $\gamma \in\ ] 0, 2-\frac{\beta}{2\nu_1 } [$, by Proposition \ref{p_T}--(iii). Then, the proof is completed by \cite[Theorem 3.3]{fxue_1}. Note that (ii) follows by noting that $\mathsf{Fix} \mathcal{T}_\gamma = \mathsf{Fix} \mathcal{T} = \mathsf{zer} (\mathcal{A} + \mathcal{B})$.
\end{proof}
\begin{remark}
While extending the existing result of the relaxed PFBS \cite[Theorem 25.8]{plc_book} to arbitrary variable metric $\mathcal{Q}$, the convergence condition is less restrictive than the above one. In \cite[Theorem 25.8]{plc_book}, the condition is $\gamma < \min\{1, \frac{\nu_1}{\beta} \} +\frac{1}{2}$, which is obtained by the rough estimate of averagedness of $\mathcal{T}_\gamma$: $\alpha = \gamma \cdot \max\{ \frac{2}{3}, \frac{2\beta}{\beta +2\nu_1} \}$ (see Remark 2). By contrast, our result corresponds to the sharper estimate of $\alpha = \frac{2\gamma \nu_1}
{4\nu_1-\beta}$ (i.e. Proposition \ref{p_T}--(iii)).
\end{remark}
\vskip.2cm
If $\mathcal{A}$ in \eqref{rgfbs} is $\mu$--strongly monotone, the linear rate is achieved.
\begin{proposition}[Convergence in terms of $\mathcal{Q}$--based distance] \label{p_rgfbs}
Let $\mathbf{b}^0\in \mathcal{H}$, $\{\mathbf{b}^k\}_{k \in \mathbb{N}}$ be a sequence generated by \eqref{rgfbs}. Under Assumptions \ref{assume_1} and \ref{assume_2}, if $\mathcal{A}$ is $\mu$--strongly monotone, $\beta < 2 \nu_1 $, $\gamma < 1+ \frac{1-\frac{\beta}{2\nu_1}}
{1+\frac{2\mu}{\nu_2 } } $, then, the following hold.
{\rm (i) [$q$--linear convergence]} Both $\|\mathbf{b}^{k} -\mathbf{b}^\star \|_\mathcal{Q}$ and $\|\mathbf{b}^{k} -\mathbf{b}^{k+1} \|_\mathcal{Q}$ are $q$--linearly convergent with the rate of $ \sqrt{1 - \frac{2\mu \gamma}
{ 2\mu + \nu_2 } } $.
{\rm (ii) [$r$--linear convergence]} If $\mu \in [\frac{\nu_2 }
{\sqrt{5}+1}, +\infty[$, $ \gamma \in \ ]\frac{3-\sqrt{5}}{2} ( 1+ \frac{\nu_2 } { 2\mu} ) , 1] $,
$\big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|_\mathcal{Q}$ is globally $r$--linearly convergent w.r.t. $\big\| \mathbf{b}^0 - \mathbf{b}^\star \big\|_\mathcal{Q}$:
\[
\big\| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|_\mathcal{Q}
\le \gamma \sqrt{\frac{2\mu}{\nu_2}} \sqrt{ \frac{1+ \frac{2\mu}{\nu_2 } } {1-\frac{\beta}{2\nu_1} } }
\cdot \bigg(1 - \frac{2\mu \gamma}
{ 2\mu + \nu_2 } \bigg)^{\frac{k+1}{4} }
\big\| \mathbf{b}^0 - \mathbf{b}^\star \big\|_\mathcal{Q}
\]
\end{proposition}
\begin{proof}
First, $\mathcal{T}_\gamma \in \mathcal{F}^\mathcal{Q}_{\xi, \alpha }$ with $\xi = \frac{\sqrt{1+\frac{\mu \beta} {\nu_1 \nu_2 } } } {1+\frac{2\mu}{\nu_2 } } $ and $\alpha = \frac { \gamma (1+\frac{2\mu}{\nu_2}) }
{2-\frac{\beta}{2\nu_1 } + \frac{2\mu}{\nu_2 }} $ by Proposition \ref{p_TR_mu}--(i) and \cite[Theorem 2.8--(iv)]{fxue_1}. Then, the results follow from \cite[Theorem 3.3]{fxue_1}.
\end{proof}
\section{Further extension: A relaxed version of G-FBS operator}
\label{sec_extension}
In this sequel, we further consider a relaxed version of G-FBS operator:
\begin{equation} \label{t_relaxed}
\tilde{\mathcal{T}} := \mathcal{I} + \mathcal{M} \Big( (\mathcal{A} +\mathcal{Q})^{-1} (\mathcal{Q} - \mathcal{B}) - \mathcal{I} \Big) = \mathcal{I} + \mathcal{M} (\mathcal{T} - \mathcal{I})
\end{equation}
where $\mathcal{M}$ is a relaxation matrix. Then, the fixed-point iteration $\mathbf{b}^{k+1} : = \tilde{\mathcal{T}} \mathbf{b}^k$ is equivalent to the following {\it generalized variable metric FBS} algorithm:
\begin{equation} \label{gppa}
\left\lfloor \begin{array}{llll}
\bf 0 & \in & \mathcal{A} \tilde{\mathbf{b}}^k + \mathcal{B} \mathbf{b}^k +
\mathcal{Q} (\tilde{\mathbf{b}}^k - \mathbf{b}^k) & \text{variable metric FBS step} \\
\mathbf{b}^{k+1} & := & \mathbf{b}^k +\mathcal{M} (\tilde{\mathbf{b}}^k - \mathbf{b}^k)
& \text{relaxation step}
\end{array} \right.
\end{equation}
To the best of our knowledge, \eqref{gppa} has never been discussed before in the literature. The applications of \eqref{gppa} will be illustrated in Section \ref{sec_eg}.
Lemma \ref{l_gppa} presents several key ingredients, which are the `recipe' for proving the convergence of \eqref{gppa}.
\begin{lemma} \label{l_gppa}
Let $\mathcal{T}$ defined as \eqref{t}. Let $\mathbf{b}^\star \in \mathsf{zer} \mathcal{A}$ and $\{\mathbf{b}^k\}_{k\in\mathbb{N}}$ be a sequence generated by \eqref{gppa}. Denote $\mathcal{S}: = \mathcal{Q} \mathcal{M}^{-1}$, $\mathcal{G} := \mathcal{Q} + \mathcal{Q}^{\top} - \mathcal{M}^\top \mathcal{Q}$, and the operator $\mathcal{R} := \mathcal{I} - \mathcal{T}$. If $\mathcal{A}$ is maximally monotone and $\mathcal{S} \in \mathcal{M}_\mathcal{S}$. Then, the following hold.
{\rm (i)} $ \big\| \mathbf{b}^{k+1} - \mathbf{b}^\star \big\|_\mathcal{S}^2
\le \big\| \mathbf{b}^k - \mathbf{b}^\star \big\|_\mathcal{S}^2
- \big\| \mathbf{b}^k - \mathbf{b}^{k+1}\big\|_{\mathcal{M}^{-\top}
(\mathcal{G} - \frac{\beta}{2 }\mathcal{I} ) \mathcal{M}^{-1} }^2 $
{\rm (ii)} $ \big \langle \mathcal{R} \mathbf{b}^k \big|\mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1}
\big \rangle_{ \mathcal{M}^\top \mathcal{S} \mathcal{M}} \ge \frac{1}{2}
\big( 1 - \frac{\beta}{4\nu_1} \big)
\big\| \mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1} \big\|
_{\mathcal{Q}+\mathcal{Q}^\top}^2$
{\rm (iii)} $ \big \| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|_\mathcal{S}^2 -
\big\| \mathbf{b}^{k+1} - \mathbf{b}^{k+2} \big\|_\mathcal{S}^2
\ge \big\| \mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1} \big\|_{
( 1 - \frac{\beta}{4\nu_1} )
(\mathcal{Q} +\mathcal{Q}^\top) - \mathcal{M}^\top \mathcal{S}\mathcal{M} }^2 $
\end{lemma}
\begin{proof}
(i) First, we have:
\begin{eqnarray} \label{x5}
0 & \le & \langle \mathcal{A} \tilde{\mathbf{b}}^k - \mathcal{A} \mathbf{b}^\star \big|
\tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\nonumber \\
& =& \langle -\mathcal{B} \mathbf{b}^k +\mathcal{Q}(\mathbf{b}^k - \tilde{\mathbf{b}}^k) +\mathcal{B} \mathbf{b}^\star
\big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\quad \text{by \eqref{gppa}}
\nonumber \\
& =& \langle \mathcal{Q}(\mathbf{b}^k - \tilde{\mathbf{b}}^k) \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle - \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\nonumber \\
& =& \langle \mathcal{Q} \mathcal{M}^{-1} (\mathbf{b}^k - \mathbf{b}^{k+1}) \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle - \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\quad \text{by \eqref{gppa}}
\nonumber \\
& =& \langle \mathcal{S} (\mathbf{b}^k - \mathbf{b}^{k+1}) \big| \mathbf{b}^k - \mathbf{b}^\star +\mathcal{M}^{-1} (\mathbf{b}^{k+1} - \mathbf{b}^k) \rangle - \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\quad \text{by \eqref{gppa}}
\nonumber \\
& =& \langle \mathcal{S} (\mathbf{b}^k - \mathbf{b}^{k+1}) \big| \mathbf{b}^k - \mathbf{b}^\star \rangle
+ \langle \mathcal{S} (\mathbf{b}^k - \mathbf{b}^{k+1}), \mathcal{M}^{-1} (\mathbf{b}^{k+1} - \mathbf{b}^k) \rangle - \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\nonumber \\
& =& \frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1}\big\|^2_\mathcal{S}
+\frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^\star \big\|_\mathcal{S}^2
-\frac{1}{2} \big\| \mathbf{b}^{k+1} - \mathbf{b}^\star \big\|_\mathcal{S}^2
- \frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1}\big\|_{\mathcal{M}^{-\top} \mathcal{S}
+\mathcal{S} \mathcal{M}^{-1}}^2
\nonumber \\
& - & \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\nonumber \\
& =& \frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^\star \big\|_\mathcal{S}^2
-\frac{1}{2} \big\| \mathbf{b}^{k+1} - \mathbf{b}^\star \big\|_\mathcal{S}^2
- \frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1}\big\|_{\mathcal{M}^{-\top} \mathcal{S}
+\mathcal{S} \mathcal{M}^{-1} -\mathcal{S} }^2
\nonumber \\
& - & \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\nonumber \\
& =& \frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^\star \big\|_\mathcal{S}^2
-\frac{1}{2} \big\| \mathbf{b}^{k+1} - \mathbf{b}^\star \big\|_\mathcal{S}^2
- \frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1}\big\|_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2
- \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star, \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
\end{eqnarray}
where $\mathcal{G} = \mathcal{S}\mathcal{M} + \mathcal{M}^\top \mathcal{S} -\mathcal{M}^\top \mathcal{S} \mathcal{M} = \mathcal{Q} + \mathcal{Q}^\top - \mathcal{M}^\top \mathcal{Q}$. By adopting some techniques in \cite[Theorem 1]{lorenz}, the last term becomes:
\begin{eqnarray}
- \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k - \mathbf{b}^\star \rangle
& = & - \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k -
\mathbf{b}^k +\mathbf{b}^k - \mathbf{b}^\star \rangle
\nonumber \\
& = & - \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k -
\mathbf{b}^k \rangle
- \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \mathbf{b}^k - \mathbf{b}^\star \rangle
\nonumber \\
& \le & - \langle \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big| \tilde{\mathbf{b}}^k -
\mathbf{b}^k \rangle
- \frac{1}{ \beta} \big\| \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big\|^2
\nonumber \\
& \le & \frac{1}{ \beta} \big\| \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big\|^2 + \frac{\beta}{4} \big\| \tilde{\mathbf{b}}^k -
\mathbf{b}^k \big\|^2
- \frac{1}{ \beta} \big\| \mathcal{B} \mathbf{b}^k - \mathcal{B} \mathbf{b}^\star \big\|^2
\nonumber \\
& = & \frac{\beta}{4} \big\| \tilde{\mathbf{b}}^k -
\mathbf{b}^k \big\|^2 = \frac{\beta}{4 } \big\| \mathbf{b}^k -
\mathbf{b}^{k+1} \big\|_{\mathcal{M}^{-\top} \mathcal{M}^{-1}}^2
\quad \text{by \eqref{gppa}}
\nonumber
\end{eqnarray}
which is substituted into \eqref{x5} yields:
\[
\frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^\star \big\|_\mathcal{S}^2
-\frac{1}{2} \big\| \mathbf{b}^{k+1} - \mathbf{b}^\star \big\|_\mathcal{S}^2
- \frac{1}{2} \big\| \mathbf{b}^k - \mathbf{b}^{k+1}\big\|_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2
+ \frac{\beta}{4 } \big\| \mathbf{b}^k -
\mathbf{b}^{k+1} \big\|_{\mathcal{M}^{-\top} \mathcal{M}^{-1}}^2 \ge 0
\]
which is equivalent to (i).
\vskip.2cm
(ii) Note that Lemma \ref{l_T}--(i) is valid for any $\mathcal{Q}$, not limited to $\mathcal{Q} \in \mathcal{M}_\mathcal{S}^+$. Proposition \ref{p_R}--(i) can be modified to:
\[
\big\|\mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big\|_{\mathcal{Q}^\top}^2
\le \big \langle \mathbf{b}_1 - \mathbf{b}_2 \big| \mathcal{R} \mathbf{b}_1 - \mathcal{R} \mathbf{b}_2 \big \rangle_{\mathcal{Q}^\top}
+ \frac{\beta} {4} \big\| \mathcal{R}\mathbf{b}_1-\mathcal{R}\mathbf{b}_2 \big \|^2
\]
which leads to:
\[
\big \langle \mathbf{b}^k - \mathbf{b}^{k+1} \big|
\mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1} \big \rangle_{\mathcal{Q}^\top} \ge
\big( 1 - \frac{\beta}{4\nu_1} \big)
\big\| \mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1} \big\|_{\mathcal{Q}^\top}^2
= \frac{1}{2} \big( 1 - \frac{\beta}{4\nu_1} \big)
\big\| \mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1} \big\|_{\mathcal{Q} + \mathcal{Q}^\top}^2
\]
Then, (ii) follows from $\mathbf{b}^k - \mathbf{b}^{k+1} = \mathcal{M}\mathcal{R} \mathbf{b}^k$ by \eqref{gppa} and $\mathcal{Q}^\top = \mathcal{M}^\top \mathcal{S}$.
\vskip.2cm
(iii) From Lemma \ref{l_gppa}-(ii), we have:
\begin{eqnarray} \label{dd}
& & \big \| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|_\mathcal{S}^2 -
\big\| \mathbf{b}^{k+1} - \mathbf{b}^{k+2} \big\|_\mathcal{S}^2
\nonumber \\
& = & \big \|\mathcal{M} \mathcal{R} \mathbf{b}^k \big\|_\mathcal{S}^2 -
\big\| \mathcal{M} \mathcal{R}\mathbf{b}^{k+1} \big\|_\mathcal{S}^2
\quad \text{by \eqref{gppa} }
\nonumber \\
&=& 2 \big \langle \mathcal{R} \mathbf{b}^k \big|
\mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1} \big \rangle_{ \mathcal{M}^\top \mathcal{S} \mathcal{M}}
- \big\| \mathcal{R}\mathbf{b}^k - \mathcal{R}\mathbf{b}^{k+1} \big\|^2
_{\mathcal{M}^\top \mathcal{S} \mathcal{M} }
\nonumber \\
& \ge & \big\| \mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1} \big\|_{
( 1 - \frac{\beta}{4\nu_1} )
(\mathcal{Q} +\mathcal{Q}^\top) - \mathcal{M}^\top \mathcal{S}\mathcal{M} }^2 \qquad
\text{by Lemma \ref{l_gppa}-(ii) }
\nonumber
\end{eqnarray}
This completes the proof.
\end{proof}
The following theorem gives the convergence result.
\begin{theorem}[Convergence in terms of metric distance] \label{t_gppa}
Let $\mathbf{b}^0 \in \mathcal{H}$, $\{\mathbf{b}^k\}_{k\in\mathbb{N}}$ be a sequence generated by \eqref{gppa}, where $\mathcal{A}$ is maximally monotone, $\mathcal{B}$ is $\beta^{-1}$-cocoercive. If $\mathcal{Q}\mathcal{M}^{-1} \in \mathcal{M}_\mathcal{S}^{++}$ and $\mathcal{Q} + \mathcal{Q}^\top - \mathcal{M}^\top \mathcal{Q} - \frac{\beta}{2 }\mathcal{I} \in \mathcal{M}_\mathcal{S}^{++}$, then the following hold.
{\rm (i) [Basic convergence]} There exists $\mathbf{b}^\star\in \mathsf{zer} \mathcal{A}$, such that $\mathbf{b}^k \rightarrow \mathbf{b}^\star$, as $k \rightarrow \infty$.
{\rm (ii) [Sequential convergence]} If $ ( 1 - \frac{\beta}{4\nu_1} )
(\mathcal{Q} +\mathcal{Q}^\top) \succeq \mathcal{M}^\top \mathcal{Q}$, then, $\| \mathbf{b}^{k } - \mathbf{b}^{k+1 } \|_\mathcal{S}$ has the non-ergodic convergence rate of $\mathcal{O}(1/\sqrt{k})$, i.e.
\[
\big\| \mathbf{b}^{k+1 } - \mathbf{b}^{k } \big\|_\mathcal{S}
\le \frac{1}{ \sqrt{k+1 } } \sqrt{\frac
{\lambda_{\max}(\mathcal{S} )}
{\lambda_{\min}(\mathcal{M}^{-\top} (\mathcal{G} -\frac{\beta}{2} \mathcal{I}) \mathcal{M}^{-1})} }
\big\|\mathbf{b}^{0} -\mathbf{b}^\star \big\|_\mathcal{S},
\quad k \in \mathbb{N}
\]
\end{theorem}
\begin{proof}
In view of Lemma \ref{l_gppa}--(i), by the similar arguments of \cite[Theorem 5.3]{fxue_2}, invoking Opial's lemma \cite{opial}, the convergence of \eqref{gppa} is guaranteed, if $\mathcal{S}, \mathcal{G} - \frac{\beta}{2} \mathcal{I} \in \mathcal{M}_\mathcal{S}^{++}$.
\vskip.2cm
(ii) In view of Lemma \ref{l_gppa}--(i), we have:
\[
\big\| \mathbf{b}^{k+1} -\mathbf{b}^\star \big\|_\mathcal{S}^2 \le
\big\| \mathbf{b}^{k} - \mathbf{b}^\star \big\|_\mathcal{S}^2 - \frac{\lambda_{\min}( \mathcal{M}^{-\top} ( \mathcal{G} -\frac{\beta}{2} \mathcal{I}) \mathcal{M}^{-1})}
{\lambda_{\max}(\mathcal{S} )} \big\| \mathbf{b}^{k } - \mathbf{b}^{k+1} \big\|_\mathcal{S}^2
\]
where $\lambda_{\max}$ and $\lambda_{\max}$ denote the largest and smallest eigenvalues of a matrix.
On the other hand, the sequence $\{\| \mathbf{b}^{k} - \mathbf{b}^{k+1} \|_\mathcal{S}\}_{k\in\mathbb{N}} $ is non-increasing, if $ ( 1 - \frac{\beta}{4\nu_1} ) (\mathcal{Q} +\mathcal{Q}^\top) \succeq \mathcal{M}^\top \mathcal{Q}$, by Lemma \ref{l_gppa}-(iii). Finally, (ii) is obtained, following the similar proof of \cite[Theorem 3.3--(iii)]{fxue_1}.
\end{proof}
\begin{remark}
In particular, if $\mathcal{M}=\gamma \mathcal{I}$, Lemma \ref{l_gppa} is simplified as:
\[
\big\| \mathbf{b}^{k+1} - \mathbf{b}^\star \big\|_\mathcal{Q}^2
\le \big\| \mathbf{b}^k - \mathbf{b}^\star \big\|_\mathcal{Q}^2
- \frac{1}{\gamma} \big( 2 - \gamma - \frac{\beta}{2\nu_1} \big) \big\| \mathbf{b}^k - \mathbf{b}^{k+1}\big\|_\mathcal{Q}^2
\]
and
\[
\big \| \mathbf{b}^k - \mathbf{b}^{k+1} \big\|_\mathcal{Q}^2 -
\big\| \mathbf{b}^{k+1} - \mathbf{b}^{k+2} \big\|_\mathcal{Q}^2
\ge \gamma \big( 2-\gamma - \frac{\beta}{2\nu_1} \big)
\big\| \mathcal{R} \mathbf{b}^k - \mathcal{R} \mathbf{b}^{k+1} \big\|_\mathcal{Q}^2
\]
Then, Theorem \ref{t_rgfbs} is exactly recovered.
\end{remark}
\section{Applications to the first-order operator splitting algorithms}
\label{sec_eg}
The focus of this part is to show that many popular operator splitting algorithms fall into the G-FBS category. In each example, we only show $\mathcal{A}$, $\mathcal{B}$, $\mathcal{Q}$ and $\mathcal{M}$ associated with the specific algorithm, without further presenting the convergence propoerties, which can be readily obtained by the results in Sections \ref{sec_iteration} and \ref{sec_extension}.
\subsection{The ADMM algorithms}
ADMM is one of the most commonly used algorithms for solving the structured constrained optimization \cite{boyd_admm}:
\begin{equation} \label{problem1}
\min_{\mathbf{x},\mathbf{u}} f(\mathbf{x}) +g(\mathbf{u}),\quad
\text{s.t.}\ \ \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u} = \mathbf{c}
\end{equation}
where $\mathbf{x} \in \mathbb{R}^N$, $\mathbf{u} \in \mathbb{R}^L$, $\mathbf{A}: \mathbb{R}^N \mapsto \mathbb{R}^M$, $\mathbf{B}: \mathbb{R}^L \mapsto \mathbb{R}^M$, $f: \mathbb{R}^N \mapsto \mathbb{R}\cup\{+\infty\}$ and $g: \mathbb{R}^L \mapsto \mathbb{R}\cup\{+\infty\}$ are proper, lower semi-continuous and convex.
Two typical ADMM algorithms are listed here to show the corresponding PPA interpretations.
\begin{example} [Relaxed-ADMM] \label{e_admm_1}
The relaxed ADMM, or equivalent relaxed DRS applied to the dual problem \cite{boyd_control}, is given as \cite[Eq.(3)]{fang_2015}\footnote{The standard ADMM can be recovered by letting $\gamma=1$ \cite{boyd_admm}.}:
\[
\left\lfloor \begin{array}{lll}
\mathbf{x}^{k+1} & := & \arg \min_\mathbf{x} f(\mathbf{x}) +
\frac{\tau}{2} \big\| \mathbf{A} \mathbf{x} +\mathbf{B} \mathbf{u}^k - \mathbf{c} - \frac{1}{\tau} \mathbf{s}^k \big\|^2 \\
\mathbf{u}^{k+1} & := & \arg \min_\mathbf{u} g(\mathbf{u}) + \frac{\tau}{2} \big\|\mathbf{B} (\mathbf{u} - \mathbf{u}^k) + \gamma (\mathbf{A} \mathbf{x}^{k+1}
+ \mathbf{B} \mathbf{u}^{k} - \mathbf{c} ) - \frac{1}{\tau} \mathbf{s}^k \big\|^2 \\
\mathbf{s}^{k+1} & := & \mathbf{s}^k -\tau \mathbf{B} (\mathbf{u}^{k+1} - \mathbf{u}^k)
- \tau\gamma (\mathbf{A} \mathbf{x}^{k+1} + \mathbf{B} \mathbf{u}^{k} - \mathbf{c})
\end{array} \right.
\]
which fits into the relaxed G-FBS operator \eqref{t_relaxed} as:
\[
\mathbf{b}^k = \begin{bmatrix}
\mathbf{x}^k \\ \mathbf{u}^k \\ \mathbf{s}^k \end{bmatrix},
\quad \mathcal{A}: \mathbf{b} \mapsto \begin{bmatrix}
\partial f & \bf 0 & -\mathbf{A}^\top \\
\bf 0 & \partial g & -\mathbf{B}^\top \\
\mathbf{A} & \mathbf{B} & \bf 0 \end{bmatrix} \mathbf{b}
- \begin{bmatrix}
\bf 0 \\ \bf 0 \\ \mathbf{c}
\end{bmatrix}, \quad
\mathcal{B}: \mathbf{b} \mapsto {\bf 0}
\]
\[
\mathcal{Q} = \begin{bmatrix}
\bf 0 & \bf 0 & \bf 0 \\
\bf 0 & \tau \mathbf{B}^\top \mathbf{B} & (1-\gamma)\mathbf{B}^\top \\
\bf 0 & - \mathbf{B} & \frac{1}{\tau} \mathbf{I}_M
\end{bmatrix};\quad
\mathcal{M} = \begin{bmatrix}
\mathbf{I}_N & \bf 0 & \bf 0 \\
\bf 0 & \mathbf{I}_L & \bf 0 \\
\bf 0 & -\tau\mathbf{B} & \gamma \mathbf{I}_M \\
\end{bmatrix}
\]
\end{example}
\begin{example} [Proximal-ADMM] \label{e_admm_2}
The proximal-ADMM is given as \cite[modified SPADMM]{lxd_2016}, \cite[Eq.(10)]{admm_tmi}:
\[
\left\lfloor \begin{array}{lll}
\mathbf{x}^{k+1} & := & \arg \min_\mathbf{x} f(\mathbf{x}) +
\frac{\tau}{2} \big\| \mathbf{A} \mathbf{x} +\mathbf{B} \mathbf{u}^k - \mathbf{c} - \frac{1}{\tau} \mathbf{s}^k \big\|^2 + \frac{1}{2} \big\| \mathbf{x} - \mathbf{x}^k \big\|_{\mathbf{P}_1}^2 \\
\mathbf{u}^{k+1} & := & \arg \min_\mathbf{u} g(\mathbf{u}) + \frac{\tau}{2} \big\|\mathbf{A} \mathbf{x}^{k+1} + \mathbf{B} \mathbf{u} - \mathbf{c} - \frac{1}{\tau} \mathbf{s}^k \big\|^2
+ \frac{1}{2} \big\| \mathbf{u} - \mathbf{u}^k \big\|_{\mathbf{P}_2}^2 \\
\mathbf{s}^{k+1} & := & \mathbf{s}^k -\tau (\mathbf{A} \mathbf{x}^{k+1} +
\mathbf{B} \mathbf{u}^{k+1} - \mathbf{c})
\end{array} \right.
\]
which fits into the relaxed G-FBS operator \eqref{t_relaxed} as:
\[
\mathcal{Q} = \begin{bmatrix}
\mathbf{P}_1 & \bf 0 & \bf 0 \\
\bf 0 & \mathbf{P}_2 + \tau \mathbf{B}^\top \mathbf{B} & \bf 0 \\
\bf 0 & - \mathbf{B} & \frac{1}{\tau} \mathbf{I}_M
\end{bmatrix};\quad
\mathcal{M} = \begin{bmatrix}
\mathbf{I}_N & \bf 0 & \bf 0 \\
\bf 0 & \mathbf{I}_L & \bf 0 \\
\bf 0 & -\tau\mathbf{B} & \mathbf{I}_M \\
\end{bmatrix}
\]
with the same $\mathbf{b}^k$, $\mathcal{A}$ and $\mathcal{B}$ as Example \ref{e_admm_1}.
\end{example}
\begin{remark}
Other ADMM algorithms proposed in \cite{hbs_2014,hbs_yxm_2015,hbs_yxm_2018,
mafeng_2018,fang_2015,cch_2016,handeren_2013,
bai_2018} can also be interpreted by the (relaxed) G-FBS operator, with the same $\mathbf{b}$, $\mathcal{A}$ and $\mathcal{B}$ as Example \ref{e_admm_1}, but with different $\mathcal{Q}$ and $\mathcal{M}$. Interested readers can verify it case-by-case, which is omitted here.
Also notice that the monotone operator $\mathcal{A}$ bears the typical (diagonal) monotone + (off-diagonal) skew structure:
\[
\mathcal{A} = \begin{bmatrix}
\partial f & \bf 0 & - \mathbf{A}^\top \\
\bf 0 & \partial g & -\mathbf{B}^\top \\
\mathbf{A} & \mathbf{B} & \bf 0
\end{bmatrix} = \underbrace{
\begin{bmatrix}
\partial f & \bf 0 & \bf 0 \\
\bf 0 & \partial g & \bf 0 \\
\bf 0 & \bf 0 & \bf 0
\end{bmatrix} }_\text{monotone} +
\underbrace{ \begin{bmatrix}
\bf 0 & \bf 0 & - \mathbf{A}^\top \\
\bf 0 & \bf 0 & -\mathbf{B}^\top \\
\mathbf{A} & \mathbf{B} & \bf 0
\end{bmatrix} }_\text{skew-symmetric}
\]
which coincides with the observations in \cite{arias_2011,bredies_2017,plc_fixed}.
\end{remark}
\subsection{Steepest descent, P-FBS and PDS algorithms}
Consider the primal problem \cite[Problem 4.1]{vu_2013}:
\begin{equation} \label{p}
\min_\mathbf{x} f(\mathbf{x}) + \sum_{i=1}^m
(g_i \square l_i ) (\mathbf{A}_i \mathbf{x} - \mathbf{r}_i)
+ h(\mathbf{x}) + \langle \mathbf{x} | \mathbf{z} \rangle
\end{equation}
where $\mathbf{x} \in \mathbb{R}^N$, $\mathbf{A}_i: \mathbb{R}^N \mapsto \mathbb{R}^{M_i}$, $\mathbf{r}_i \in \mathbb{R}^{M_i}$, $\mathbf{z} \in \mathbb{R}^N$. The functions $f: \mathbb{R}^N \mapsto \mathbb{R}$, $g_i: \mathbb{R}^{M_i} \mapsto \mathbb{R}\cup\{+\infty\}$, $l_i: \mathbb{R}^{M_i} \mapsto \mathbb{R}\cup\{+\infty\}$, $h: \mathbb{R}^N \mapsto \mathbb{R}\cup\{+\infty\}$ are proper, lower semi-continuous and convex, $f$ is differentiable with $\beta$-Lipschitz continuous gradient, $l_i$ is $\mu_i$-strongly convex, and thus, by \cite[Theorem 18.15]{plc_book}, $l_i^*$ is differentiable with $\mu_i^{-1}$-Lipschitz continuous gradient.
There are various classes of algorithms for solving \eqref{p} or the special cases, listed below.
\begin{example} [Steepest descent]
For solving $\min_\mathbf{x} f(\mathbf{x})$, which is a special case of \eqref{p} with $m=1$, $l=\iota_{ \{\bf 0\} }:
\mathbf{a} \mapsto \left\{ \begin{array}{ll}
0, & \text{if \ } \mathbf{a} =\bf 0 \\
+\infty, & \text{otherwise}
\end{array} \right.$ (i.e. indicator function of the set $C=\{\bf 0\}$), $g: \mathbf{a} \mapsto 0$, $h: \mathbf{a} \mapsto 0$, $\mathbf{z} = \bf 0$, the steepest descent is given by \cite[Proposition 63]{plc_fixed}:
\[
\mathbf{x}^{k+1} := \mathbf{x}^k - \tau \nabla f(\mathbf{x}^k)
\]
which fits into the G-FBS operator \eqref{t} with:
\[
\mathbf{b} = \mathbf{x}, \quad \mathcal{A}: \mathbf{b} \mapsto {\bf 0}, \quad
\mathcal{B} = \nabla f, \quad \mathcal{Q} = \frac{1}{\tau} \mathbf{I}_N
\]
\end{example}
\begin{example} [Classical PPA]
For solving $\min_\mathbf{x} h(\mathbf{x})$, which is a special case of \eqref{p} with $m=1$, $l=\iota_{ \{\bf 0\} }$, $f: \mathbf{a} \mapsto 0$, $g: \mathbf{a} \mapsto 0$, $\mathbf{z} = \bf 0$, the classical PPA is given by \cite[Proposition 64]{plc_fixed}:
\[
\mathbf{x}^{k+1} := \mathbf{x}^k +\gamma \big( \prox_{\tau h}(\mathbf{x}^k) - \mathbf{x}^k \big)
\]
which fits the relaxed G-FBS operator \eqref{t_relaxed} with:
\[
\mathbf{b} = \mathbf{x}, \quad \mathcal{A}=\partial h, \quad
\mathcal{B}: \mathbf{b} \mapsto {\bf 0}, \quad \mathcal{Q} = \frac{1}{\tau} \mathbf{I}_N,
\quad \mathcal{M} = \gamma \mathbf{I}_N
\]
\end{example}
\begin{example} [Classical P-FBS \cite{plc,plc_chapter}]
For solving $\min_\mathbf{x} f(\mathbf{x}) + h(\mathbf{x})$, which is a special case of \eqref{p} with $m=1$, $l=\iota_{ \{\bf 0\} }$, $g: \mathbf{a} \mapsto 0$, $\mathbf{z} = \bf 0$, the error-free version of the classical P-FBS is given by \cite[Eq.(3.6)]{plc}:
\[
\mathbf{x}^{k+1} := \mathbf{x}^k + \gamma \big( \prox_{\tau h}(\mathbf{x}^k - \tau \nabla f(\mathbf{x}^k)) - \mathbf{x}^k \big)
\]
which fits the relaxed G-FBS operator \eqref{t_relaxed} with:
\[
\mathbf{b} = \mathbf{x}, \quad \mathcal{A}=\partial h, \quad
\mathcal{B} = \nabla f, \quad \mathcal{Q} = \frac{1}{\tau} \mathbf{I}_N,
\quad \mathcal{M} = \gamma \mathbf{I}_N
\]
\end{example}
\begin{example} [PDHG \cite{esser,cp_2011}] \label{pdhg}
For solving $\min_\mathbf{x} h(\mathbf{x}) + g(\mathbf{A} \mathbf{x})$, which is a special case of \eqref{p} with $m=1$, $l=\iota_{ \{\bf 0\} } $, $f: \mathbf{a} \mapsto 0$, $\mathbf{r} = \bf 0$, $\mathbf{z} = \bf 0$, the PDHG is given by \cite[PDHGMp]{esser}:
\[
\left\lfloor \begin{array}{llll}
\mathbf{s}^{k+1} & := & \prox_{\sigma g^*} \big( \mathbf{s}^k +\sigma
\mathbf{A} \mathbf{x}^k \big) & \text{\rm dual step} \\
\mathbf{x}^{k+1} & := & \prox_{\tau h} \big( \mathbf{x}^k - \tau
\mathbf{A}^\top (2\mathbf{s}^{k+1} - \mathbf{s}^k ) \big) &
\text{\rm primal step}
\end{array} \right.
\]
the associated G-FBS operator \eqref{t} is:
\[
\mathbf{b}^k = \begin{bmatrix} \mathbf{s}^{k} \\ \mathbf{x}^{k}
\end{bmatrix}, \quad
\mathcal{A}= \begin{bmatrix}
\partial g^* & -\mathbf{A} \\
\mathbf{A}^\top & \partial h \end{bmatrix},\quad
\mathcal{B}: \mathbf{b} \mapsto {\bf 0}, \quad
\mathcal{Q} = \begin{bmatrix}
\frac{1}{ \sigma} \mathbf{I}_M & \mathbf{A} \\
\mathbf{A}^\top & \frac{1}{ \tau} \mathbf{I}_N
\end{bmatrix}
\]
Another form of PDHG is \cite[PDHGMu]{esser}:
\[
\left\lfloor \begin{array}{llll}
\mathbf{s}^{k+1} & := & \prox_{\sigma g^*} \big( \mathbf{s}^k +\sigma
\mathbf{A} (2 \mathbf{x}^k - \mathbf{x}^{k-1} ) \big) & \text{\rm dual step} \\
\mathbf{x}^{k+1} & := & \prox_{\tau h} \big( \mathbf{x}^k - \tau
\mathbf{A}^\top \mathbf{s}^{k+1} \big) &
\text{\rm primal step}
\end{array} \right.
\]
The corresponding G-FBS operator is\footnote{Note that we use a mismatch of iteration indices between $\mathbf{x}$ and $\mathbf{s}$: $\mathbf{b}^k := (\mathbf{s}^k, \mathbf{x}^{k-1})$. This technique can also be found in \cite{bot_2015}.}:
\[
\mathbf{b}^k = \begin{bmatrix} \mathbf{s}^{k} \\ \mathbf{x}^{k-1}
\end{bmatrix}, \quad
\mathcal{A}= \begin{bmatrix}
\partial g^* & -\mathbf{A} \\
\mathbf{A}^\top & \partial h \end{bmatrix},\quad
\mathcal{B}: \mathbf{b} \mapsto {\bf 0}, \quad
\mathcal{Q} = \begin{bmatrix}
\frac{1}{ \sigma} \mathbf{I}_M & - \mathbf{A} \\
- \mathbf{A}^\top & \frac{1}{ \tau} \mathbf{I}_N
\end{bmatrix}
\]
\end{example}
\begin{example} [PDS algorithm \cite{arias_2011}]
For solving the primal problem
$\min_\mathbf{x} h(\mathbf{x}) + g(\mathbf{A} \mathbf{x} - \mathbf{r}) + \langle \mathbf{x}| \mathbf{z}\rangle$, which is a special case of \eqref{p} with $m=1$, $l=\iota_{ \{\bf 0\} } $, $f: \mathbf{a} \mapsto 0$, the error-free version of \cite[Proposition 4.2]{arias_2011} is given as:
\[
\left\lfloor \begin{array}{lll}
\tilde{\mathbf{x}}^{k} & := & \prox_{\tau h} \big( \mathbf{x}^k - \tau
\mathbf{A}^\top \mathbf{s}^k - \tau \mathbf{z} \big) \\
\tilde{\mathbf{s}}^{k} & := & \prox_{\tau g^*} \big( \mathbf{s}^k +\tau \mathbf{A} \mathbf{x}^k -\tau \mathbf{r} \big) \\
\mathbf{x}^{k+1} & := & \tilde{\mathbf{x}}^k - \tau \mathbf{A}^\top (\tilde{\mathbf{s}}^k - \mathbf{s}^k) \\
\mathbf{s}^{k+1} & := & \tilde{\mathbf{s}}^k +\tau \mathbf{A} (\tilde{\mathbf{x}}^k - \mathbf{x}^k)
\end{array} \right.
\]
which corresponds to the following relaxed G-FBS operator:
\[
\mathbf{b}^k =
\begin{bmatrix} \mathbf{x}^{k } \\ \mathbf{s}^{k }\end{bmatrix},
\quad \mathcal{A}: \mathbf{b} \mapsto \begin{bmatrix}
\partial f & \mathbf{A}^\top \\
-\mathbf{A} & \partial g^* \end{bmatrix} \mathbf{b}
+ \begin{bmatrix}
\mathbf{z} \\ \mathbf{r} \end{bmatrix},\quad
\mathcal{B}: \mathbf{b} \mapsto {\bf 0}, \quad
\mathcal{Q} = \begin{bmatrix}
\frac{1}{\tau}\mathbf{I}_N & -\mathbf{A}^\top \\
\mathbf{A} & \frac{1}{\tau} \mathbf{I}_M
\end{bmatrix} \quad
\mathcal{M} = \begin{bmatrix}
\mathbf{I}_N & -\tau \mathbf{A}^\top \\
\tau \mathbf{A} & \mathbf{I}_M
\end{bmatrix}
\]
\end{example}
\begin{example} [Generalized Dykstra-like algorithm \cite{plc_dual_2010}]
For solving \cite[Problem 1.2]{plc_dual_2010}:
\[
\min_\mathbf{x} h(\mathbf{x}) + g(\mathbf{A} \mathbf{x} -\mathbf{r})+
\frac{1}{2} \|\mathbf{x} - \mathbf{u}\|^2
\]
which is a special case of \eqref{p} with $m=1$, $l=\iota_{ \{\bf 0\} } $, $f = \frac{1}{2} \| \cdot - \mathbf{u}\|^2$, $\mathbf{z} = \bf 0$, \cite[Algorithm 3.5]{plc_dual_2010} is given as:
\begin{equation} \label{x45}
\left\lfloor \begin{array}{lll}
\mathbf{x}^{k} &: = & \prox_{f} \big( \mathbf{z} - \mathbf{L}^\top \mathbf{s}^k \big) \\
\mathbf{s}^{k+1}_i & := & \mathbf{s}^k +\gamma \big(
\prox_{\tau g^*} (\mathbf{s}^k + \tau (\mathbf{L} \mathbf{x}^k - \mathbf{r}) ) -\mathbf{s}^k \big)
\end{array} \right.
\end{equation}
Using Fenchel duality and Moreau's decomposition identity, \eqref{x45} is equivalent to a simple P-FBS for solving the dual problem:
\[
\mathbf{s}^{k+1} := \mathbf{s}^k+ \gamma \Big( \prox_{\tau g^*}
\big( \mathbf{s}^k - \tau \mathbf{r} - \tau \nabla q(\mathbf{s}^k) \big) - \mathbf{s}^k \Big)
\]
where the function $q$ is defined as: $q: \mathbf{s} \mapsto \frac{1}{2} \big\| \mathbf{u} - \mathbf{A}^\top \mathbf{s} \big\|^2 - \big( \min_\mathbf{x} h(\mathbf{x}) +\frac{1}{2} \big\| \mathbf{x} - \mathbf{u} + \mathbf{A}^\top \mathbf{s} \big\|^2 \big)$. This algorithm fits the relaxed G-FBS operator with:
\[
\mathbf{b} = \mathbf{s}, \quad \mathcal{A}: \mathbf{s} \mapsto \partial g^* (\mathbf{s}) + \mathbf{r}, \quad \mathcal{B}= \nabla q , \quad \mathcal{Q} = \frac{1}{\tau} \mathbf{I}_M,
\quad \mathcal{M} = \gamma \mathbf{I}_M
\]
\end{example}
\begin{example} [PAPC \cite{zxq_ip,teboulle_2015,gist}]
For solving $\min_\mathbf{x} f(\mathbf{x}) + g(\mathbf{A} \mathbf{x})$, which is
a special case of \eqref{p} with $m=1$, $h: \mathbf{b} \mapsto 0$, $l=\iota_{ \{\bf 0 \} }$, $\mathbf{r} = {\bf 0}$, $\mathbf{z} = \bf 0$, the PAPC scheme is given as:
\[
\left\lfloor \begin{array}{lll}
\mathbf{s}^{k} & := & \prox_{\sigma g^*} \Big(
(\mathbf{I} - \sigma \tau \mathbf{A} \mathbf{A}^\top ) \mathbf{s}^k +\sigma \mathbf{A}
(\mathbf{x}^k - \tau \nabla f(\mathbf{x}^k)) \Big) \\
\mathbf{x}^{k+1} & := & \mathbf{x}^k - \tau \nabla f(\mathbf{x}^k) - \tau \mathbf{A}^\top
\mathbf{s}^{k+1}
\end{array} \right.
\]
which can be interpreted by the G-FBS operator \eqref{t}:
\[
\mathbf{b}^k = \begin{bmatrix}
\mathbf{s}^{k } \\ \mathbf{x}^k \end{bmatrix}, \quad
\mathcal{A} = \begin{bmatrix}
\partial g^* & -\mathbf{A} \\
\mathbf{A}^\top & \bf 0 \end{bmatrix},\quad
\mathcal{B} = \begin{bmatrix}
\bf 0 & \bf 0 \\ \bf 0 & \nabla f \end{bmatrix},\quad
\mathcal{Q} = \begin{bmatrix}
\frac{1}{\sigma}\mathbf{I}_M - \tau \mathbf{A} \mathbf{A}^\top & \bf 0 \\
\bf 0 & \frac{1}{\tau} \mathbf{I}_N
\end{bmatrix}
\]
\end{example}
\begin{example} [AFBA \cite{latafat_2017,latafat_chapter}] \label{afba}
For solving $\min_\mathbf{x} f(\mathbf{x}) + h(\mathbf{x}) + g(\mathbf{A} \mathbf{x})$, which is
a special case of \eqref{p} with $m=1$, $l=\iota_{ \{\bf 0 \} }$, $\mathbf{r} = {\bf 0}$, $\mathbf{z} = \bf 0$, the AFBA scheme is given by:
\[
\left\lfloor \begin{array}{lll}
\mathbf{s}^{k+1} & := & \prox_{\sigma g^*} \big( \mathbf{s}^k
+ \sigma \mathbf{A} \mathbf{w}^k \big) \\
\mathbf{x}^{k+1} & := & \mathbf{w}^k - \tau \mathbf{A}^\top ( \mathbf{s}^{k+1} - \mathbf{s}^k) \\
\mathbf{w}^{k+1} & := & \prox_{\tau h} \Big( \mathbf{x}^{k+1} - \tau \nabla f (\mathbf{x}^{k+1}) -\tau \mathbf{A}^\top \mathbf{s}^{k+1} \Big)
\end{array} \right.
\]
To interpret it by the G-FBS operator, we remove $\mathbf{w}$ and obtain the equivalent form:
\[
\left\lfloor \begin{array}{lll}
\mathbf{s}^{k+1} & := & \prox_{\sigma h^*} \Big( \mathbf{s}^k
+ \sigma \mathbf{A} \big( \mathbf{x}^{k+1} +\tau \mathbf{A}^\top (\mathbf{s}^{k+1} - \mathbf{s}^k) \big) \Big) \\
\mathbf{x}^{k+1} & := & \prox_{\tau g } \Big( \mathbf{x}^{k } - \tau \nabla f (\mathbf{x}^{k }) -\tau \mathbf{A}^\top \mathbf{s}^{k } \Big) -\tau \mathbf{A}^\top (\mathbf{s}^{k+1} - \mathbf{s}^k)
\end{array} \right.
\]
which is exactly the relaxed G-FBS operator \eqref{t_relaxed}:
\[
\mathbf{b}^k = \begin{bmatrix}
\mathbf{s} ^{k } \\ \mathbf{x}^{k } \end{bmatrix},\quad
\mathcal{A} = \begin{bmatrix}
\partial g^* & -\mathbf{A} \\
\mathbf{A}^\top & \partial h \end{bmatrix}, \quad
\mathcal{B} = \begin{bmatrix}
\bf 0 & \bf 0 \\ \bf 0 & \nabla f \end{bmatrix},
\quad \mathcal{Q} = \begin{bmatrix}
\frac{1}{\sigma}\mathbf{I}_M & \bf 0 \\
-\mathbf{A}^\top & \frac{1}{ \tau} \mathbf{I}_N
\end{bmatrix},\quad
\mathcal{M} = \begin{bmatrix}
\mathbf{I}_M & \bf 0 \\ -\tau \mathbf{A}^\top & \mathbf{I}_N
\end{bmatrix}
\]
\end{example}
\begin{example} [PDS algorithm \cite{condat_2013}]
For solving the same problem as in Example \ref{afba}, \cite[Algorithm 3.1]{condat_2013} is given as:
\[
\left\lfloor \begin{array}{lll}
\tilde{\mathbf{x}}^{k} &: = & \prox_{\tau h} \big( \mathbf{x}^k - \tau
\nabla f(\mathbf{x}^k) - \tau \mathbf{A}^\top \mathbf{s}^k \big) \\
\tilde{\mathbf{s}}^{k} & := & \prox_{\sigma g^*} \big( \mathbf{s}^k +\sigma \mathbf{A} ( 2\tilde{\mathbf{x}}^k - \mathbf{x}^k ) \big) \\
\mathbf{x}^{k+1} & := & \mathbf{x}^k +\gamma_k (\tilde{\mathbf{x}}^k - \mathbf{x}^k) \\
\mathbf{s}^{k+1} & := & \mathbf{s}^k +\gamma_k (\tilde{\mathbf{s}}^k - \mathbf{s}^k)
\end{array} \right.
\]
which falls into the relaxed G-FBS operator \eqref{t_relaxed}:
\[
\mathbf{b}^k = \begin{bmatrix}
\mathbf{x}^k \\ \mathbf{s}^k \end{bmatrix},\quad
\mathcal{A} = \begin{bmatrix}
\partial h & \mathbf{A}^\top \\
-\mathbf{A} & \partial g^* \end{bmatrix},\quad
\mathcal{B} = \begin{bmatrix}
\nabla f & \bf 0 \\ \bf 0 & \bf 0 \end{bmatrix},\quad
\mathcal{Q} = \begin{bmatrix}
\frac{1}{\tau}\mathbf{I}_N & -\mathbf{A}^\top \\
-\mathbf{A} & \frac{1}{\sigma} \mathbf{I}_M
\end{bmatrix},\quad
\mathcal{M} = \begin{bmatrix}
\gamma \mathbf{I}_N & \bf 0 \\
\bf 0 & \gamma \mathbf{I}_M
\end{bmatrix}
\]
Another algorithm is \cite[Algorithm 3.2]{condat_2013}:
\[
\left\lfloor \begin{array}{lll}
\tilde{\mathbf{s}}^{k} & := & \prox_{\sigma g^*} \big( \mathbf{s}^k +
\sigma \mathbf{A} \mathbf{x}^k \big) \\
\tilde{\mathbf{x}}^{k} &: = & \prox_{\tau h} \big( \mathbf{x}^k -\tau
\nabla f(\mathbf{x}^k) - \tau \mathbf{A}^\top( 2\tilde{\mathbf{s}}^k - \mathbf{s}^k ) \big) \\
\mathbf{x}^{k+1} & := & \mathbf{x}^k +\gamma (\tilde{\mathbf{x}}^k - \mathbf{x}^k) \\
\mathbf{s}^{k+1} & := & \mathbf{s}^k +\gamma (\tilde{\mathbf{s}}^k - \mathbf{s}^k)
\end{array} \right.
\]
which falls into the relaxed G-FBS operator \eqref{t_relaxed}:
\[
\mathbf{b}^k = \begin{bmatrix}
\mathbf{s}^k \\ \mathbf{x}^k \end{bmatrix},\quad
\mathcal{A} = \begin{bmatrix}
\partial g^* & - \mathbf{A} \\
\mathbf{A}^\top & \partial h \end{bmatrix},\quad
\mathcal{B} = \begin{bmatrix}
\bf 0 & \bf 0 \\ \bf 0 & \nabla f \end{bmatrix},\quad
\mathcal{Q} = \begin{bmatrix}
\frac{1}{\sigma}\mathbf{I}_M & \mathbf{A} \\
\mathbf{A}^\top & \frac{1}{\tau} \mathbf{I}_N
\end{bmatrix},\quad
\mathcal{M} = \begin{bmatrix}
\gamma \mathbf{I}_M & \bf 0 \\
\bf 0 & \gamma \mathbf{I}_N
\end{bmatrix}
\]
\end{example}
\begin{example} [PDS algorithm \cite{condat_2013}]
For solving the problem $\min_\mathbf{x} f(\mathbf{x}) + h(\mathbf{x}) +\sum_{i=1}^m g_i(\mathbf{A}_i \mathbf{x})$, which is a special case of \eqref{p} with $l_i = \iota_{\{ \bf 0 \} }$, $\mathbf{r} = \bf 0$, $\mathbf{z} = \bf 0$,
\cite[Algorithm 5.1]{condat_2013} is given by:
\[
\left\lfloor \begin{array}{lll}
\tilde{\mathbf{x}}^{k} & = & \prox_{\tau h} \big( \mathbf{x}^k - \tau
\nabla f(\mathbf{x}^k) - \tau \sum_{i=1}^m
\mathbf{A}_i^\top \mathbf{s}_i^k \big) \\
\tilde{\mathbf{s}}_i^{k} & = & \prox_{\sigma g_i^*} \big( \mathbf{s}_i^k +\sigma \mathbf{A}_i ( 2\tilde{\mathbf{x}}^k - \mathbf{x}^k ) \big),
\quad i=1,2,...,m \\
\mathbf{x}^{k+1} & = & \mathbf{x}^k +\gamma (\tilde{\mathbf{x}}^k - \mathbf{x}^k) \\
\mathbf{s}_i^{k+1} & = & \mathbf{s}_i^k +\gamma (\tilde{\mathbf{s}}_i^k - \mathbf{s}_i^k), \quad i=1,2,...,m
\end{array} \right.
\]
which can be compactly expressed in a relaxed G-FBS form \eqref{t_relaxed}:
\[
\mathbf{b}^{k} = \begin{bmatrix}
\mathbf{x}^k \\ \mathbf{s}_1^k \\ \vdots \\ \mathbf{s}_m^k
\end{bmatrix},\
\mathcal{A} = \begin{bmatrix}
\partial h & \mathbf{A}_1^\top & \cdots & \mathbf{A}_m^\top \\
- \mathbf{A}_1 & \partial g_1^* & \cdots & \bf 0 \\
\vdots & \vdots & \ddots & \vdots \\
- \mathbf{A}_m & \bf 0 & \cdots & \partial g_m^* \\
\end{bmatrix},\
\mathcal{B} = \begin{bmatrix}
\nabla f & \bf 0 & \cdots & \bf 0 \\
\bf 0 & \bf 0 & \cdots & \bf 0 \\
\vdots & \vdots & \ddots & \vdots \\
\bf 0 & \bf 0 & \cdots & \bf 0
\end{bmatrix},
\]
\[
\mathcal{Q} = \begin{bmatrix}
\tau^{-1} \mathbf{I}_N & -\mathbf{A}_1^\top & \cdots & - \mathbf{A}_m^\top \\
- \mathbf{A}_1 & \sigma^{-1} \mathbf{I}_{M_1} & \cdots & \bf 0 \\
\vdots & \vdots & \ddots & \vdots \\
- \mathbf{A}_m & \bf 0 & \cdots & \sigma^{-1}\mathbf{I}_{M_m} \\
\end{bmatrix},\
\mathcal{M} = \begin{bmatrix}
\gamma \mathbf{I}_N & \bf 0 & \cdots & \bf 0 \\
\bf 0 & \gamma \mathbf{I}_{M_1} & \cdots & \bf 0 \\
\vdots & \vdots & \ddots & \vdots \\
\bf 0 & \bf 0 & \cdots & \gamma \mathbf{I}_{M_m}
\end{bmatrix}
\]
Another algorithm is given by \cite[Algorithm 5.2]{condat_2013}:
\[
\left\lfloor \begin{array}{lll}
\tilde{\mathbf{s}}_i^{k} & = & \prox_{\sigma g_i^*} \big( \mathbf{s}_i^k + \sigma \mathbf{A}_i \mathbf{x}^k \big),
\quad i=1,2,...,m \\
\tilde{\mathbf{x}}^{k} & = & \prox_{\tau h} \big( \mathbf{x}^k - \tau
\nabla f(\mathbf{x}^k) - \tau \sum_{i=1}^m
\mathbf{A}_i^\top (2\tilde{\mathbf{s}}_i^k - \mathbf{s}_i^k) \big) \\
\mathbf{x}^{k+1} & = & \mathbf{x}^k +\gamma (\tilde{\mathbf{x}}^k - \mathbf{x}^k) \\
\mathbf{s}_i^{k+1} & = & \mathbf{s}_i^k +\gamma (\tilde{\mathbf{s}}_i^k - \mathbf{s}_i^k), \quad i=1,2,...,m
\end{array} \right.
\]
whose corresponding relaxed G-FBS form is with the same $\mathbf{b}$, $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{M}$ as the above example, and $\mathcal{Q}$ is given as:
\[
\mathcal{Q} = \begin{bmatrix}
\tau^{-1} \mathbf{I}_N & \mathbf{A}_1^\top & \cdots & \mathbf{A}_m^\top \\
\mathbf{A}_1 & \sigma^{-1} \mathbf{I}_{M_1} & \cdots & \bf 0 \\
\vdots & \vdots & \ddots & \vdots \\
\mathbf{A}_m & \bf 0 & \cdots & \sigma^{-1}\mathbf{I}_{M_m} \\
\end{bmatrix}
\]
\end{example}
\subsection{Other examples}
Other classes of algorithms can also be expressed by the G-FBS operator. Let us now consider a typical optimization problem with a linear equality constraint:
\[
\min_\mathbf{x} h(\mathbf{x}),\qquad \text{s.t.\ } \mathbf{A}\mathbf{x} = \mathbf{c}
\]
where $\mathbf{A}: \mathbb{R}^N \mapsto \mathbb{R}^M$, $h: \mathbb{R}^N \mapsto \mathbb{R}\cup \{+\infty\}$ is proper, lower semi-continuous and convex.
\begin{example} [Basic ALM]
The augmented Lagrangian method (ALM) is (see \cite[Eq.(1.2)]{mafeng_2018} and \cite[Eq.(7.2)]{taomin_2018} for example):
\[
\left\lfloor \begin{array}{lll}
\mathbf{x}^{k+1} & := & \arg \min_\mathbf{x} h(\mathbf{x}) + \frac{\tau}{2}
\big\| \mathbf{A}\mathbf{x} - \mathbf{c} - \frac{1}{\tau} \mathbf{s}^k \big\|^2 \\
\mathbf{s}^{k+1} & := & \mathbf{s}^k - \tau ( \mathbf{A} \mathbf{x}^{k+1} - \mathbf{c} )
\end{array} \right.
\]
Its G-FBS interpretation is given as:
\[
\mathbf{b}^k = \begin{bmatrix}
\mathbf{x}^{k } \\ \mathbf{s}^{k } \end{bmatrix},\
\mathcal{A}: \mathbf{b} \mapsto \begin{bmatrix}
\partial h & -\mathbf{A}^\top \\ \mathbf{A} & \bf 0
\end{bmatrix} \mathbf{b} -
\begin{bmatrix} \bf 0 \\ \mathbf{c} \end{bmatrix},\
\mathcal{B}: \mathbf{b} \mapsto {\bf 0}, \
\mathcal{Q} = \begin{bmatrix}
\bf 0 & \bf 0 \\
\bf 0 & \frac{1}{\tau} \mathbf{I}_M \end{bmatrix}
\]
\end{example}
\begin{example} [Linearized ALM]
The linearlized ALM is given as \cite{yang_yuan_2013}:
\[
\left\lfloor \begin{array}{lll}
\mathbf{x}^{k+1} & := & \arg \min_\mathbf{x} h(\mathbf{x}) + \frac{\rho } {2}
\big\| \mathbf{x} - \mathbf{x}^k + \frac{1}{ \rho} \mathbf{A}^\top \big( \tau
(\mathbf{A} \mathbf{x}^k -\mathbf{c}) - \mathbf{s}^k \big) \big\|^2 \\
\mathbf{s}^{k+1} & := & \mathbf{s}^k - \tau
( \mathbf{A} \mathbf{x}^{k+1} - \mathbf{c})
\end{array} \right.
\]
Its G-FBS interpretation is given as:
\[
\mathbf{b}^k = \begin{bmatrix}
\mathbf{x}^{k } \\ \mathbf{s}^{k } \end{bmatrix},\
\mathcal{A}: \mathbf{b} \mapsto \begin{bmatrix}
\partial h & -\mathbf{A}^\top \\ \mathbf{A} & \bf 0
\end{bmatrix} \mathbf{b} -
\begin{bmatrix} \bf 0 \\ \mathbf{c} \end{bmatrix},\
\mathcal{B}: \mathbf{b} \mapsto {\bf 0}, \
\mathcal{Q} = \begin{bmatrix}
\rho \mathbf{I}_N - \tau \mathbf{A}^\top \mathbf{A} & \bf 0 \\
\bf 0 & \frac{1}{\tau} \mathbf{I}_M \end{bmatrix}
\]
\end{example}
\begin{example} [Linearized Bregman algorithm \cite{cjf_2}]
The scheme reads as (see also \cite[Eq.(1.11)]{zxq}):
\[
\left\lfloor \begin{array}{lll}
\mathbf{x}^{k+1} & := & \arg \min_\mathbf{x} \rho \tau h(\mathbf{x}) +
\frac{1}{2} \big\| \mathbf{x} - \rho \mathbf{A}^\top \mathbf{s}^k \big\|^2 \\
\mathbf{s}^{k+1} & := & \mathbf{s}^k - ( \mathbf{A}\mathbf{x}^{k+1} - \mathbf{c})
\end{array} \right.
\]
Its G-FBS interpretation is given as:
\[
\mathbf{b}^k = \begin{bmatrix}
\mathbf{x}^{k} \\ \mathbf{s}^{k} \end{bmatrix},\
\mathcal{A}: \mathbf{b} \mapsto \begin{bmatrix}
\tau \partial h + \frac{1}{\rho} \mathbf{I}_N - \mathbf{A}^\top \mathbf{A}
& -\mathbf{A}^\top \\ \mathbf{A} & \bf 0
\end{bmatrix} \mathbf{b} + \begin{bmatrix}
\mathbf{A}^\top \mathbf{c} \\ - \mathbf{c} \end{bmatrix},\
\mathcal{B}: \mathbf{b} \mapsto {\bf 0},\
\mathcal{Q} = \begin{bmatrix}
\bf 0 & \bf 0 \\ \bf 0 & \mathbf{I}_M \end{bmatrix}
\]
\end{example}
\begin{remark}
(1) It is the first time to show that most first-order operator splitting algorithms boil down to the simple G-FBS operator.
(2) The convergence conditions of these algorithms can be easily obtained by reformulating them as the G-FBS operator. It is much simpler than their original analysis in the literature.
(3) Many results given in Sections \ref{sec_iteration} and \ref{sec_extension} can be applied to the listed algorithms, and thus, more properties can be explored than the previous works. Interested readers can make further investigations along this direction.
(4) We did not enumerate all the examples here. One can verify more existing algorithms.
\end{remark}
\section{Conclusions}
In this paper, we considered the G-FBS operator and analyzed the nonexpansive properties. The fixed-point iterations were studied, and their convergence rates in terms of metric distance and objective value were established, which extended the existing results in some aspects. A great variety of operator splitting algorithms were illustrated as the concrete examples of the G-FBS operator.
Last, it seems interesting to further extend the proposed framework to the case when the variable metric $\mathcal{Q}$ and relaxation matrix $\mathcal{M}$ are allowed to vary over the iterations. Another limitation of this G-FBS operator is that it fails to cover Bregman proximal algorithms \cite{teboulle_2018,plc_bregman} and a few PDS algorithms \cite{plc_2012,bot_jmiv_2014}, since $\mathcal{Q}$ and $\mathcal{M}$ presented here denote the linear transforms only. It is worthwhile to extend the G-FBS to nonlinear operators $\mathcal{Q}$ and $\mathcal{M}$.
\bibliographystyle{siamplain}
|
{
"timestamp": "2021-09-07T02:18:14",
"yymm": "2109",
"arxiv_id": "2109.02064",
"language": "en",
"url": "https://arxiv.org/abs/2109.02064"
}
|
\section{Introduction}
In 1976, Schoenfeld \cite[Corollary 1]{schoenfeld1976sharper} proved that under assumption of the Riemann hypothesis,
\begin{equation}\label{schoeq}
|\pi(x)-\li(x)|<\frac{\sqrt{x}}{8\pi}\log x,
\end{equation}
for $x\geq 2657$. Although a complete proof of the Riemann hypothesis remains out of reach, partial results can be used to prove \eqref{schoeq} for a finite range. In this direction, we prove that \eqref{schoeq} holds provided $x\geq 2657$ and
\begin{equation}\label{djbound}
\frac{9.06}{\log\log x}\sqrt{\frac{x}{\log x}}\leq T.
\end{equation}
Here, $T$ is the largest known value such that the Riemann hypothesis is true for all zeros $\rho$ with $\Im(\rho)\in(0,T]$. A recent computation by Platt and Trudgian \cite{platt2021riemann} allows us to take $T=3\cdot 10^{12}$. Substituting this $T$ into \eqref{djbound} tells us that $\eqref{schoeq}$ holds for all $2657\leq x\leq 1.101\cdot 10^{26}$.
These results improve on earlier work by B{\"u}the \cite[Theorem 2]{buthe2016estimating}, who proved that \eqref{schoeq} holds provided $x\geq 2657$ and
\begin{equation}\label{buthebound}
4.92\sqrt{\frac{x}{\log x}}\leq T.
\end{equation}
In particular, provided $T\geq 46$, \eqref{djbound} holds for a wider range of $x$ than $\eqref{buthebound}$. So by comparison, one only obtains $x\leq 2.169\cdot 10^{25}$ using \eqref{buthebound} with $T=3\cdot 10^{12}$.
To prove \eqref{djbound} we use B{\"u}the's original method with an additional iterative argument and several other optimisations. Similar to B{\"u}the, we prove corresponding bounds for the prime counting functions $\theta(x)$, $\psi(x)$ and $\Pi(x)$.
In Section \ref{sectnot} we list all the definitions and lemmas that we will use from \cite{buthe2016estimating}. In Section \ref{sectmain} we prove the main result. Then, in Section \ref{sectimprov} we discuss possible improvements and variations. For instance, we show (Theorem \ref{weakthm}) that the weaker bound $|\pi(x)-\li(x)|<\sqrt{x}\log x$ holds for $2\leq x\leq 2.165\cdot 10^{30}$. Finally, in Section \ref{appsec} we discuss an application to an inequality of Ramanujan.
\section{Notation and setup}\label{sectnot}
Throughout this paper, we work with the normalised prime counting functions
\begin{align}\label{chebfuncs}
&\pi(x)=\sideset{}{^*}\sum_{p\leq x} 1, \qquad\qquad\ \ \Pi(x)=\sideset{}{^*}\sum_{p^m\leq x}\frac{1}{m},\notag\\
&\theta(x)=\sideset{}{^*}\sum_{p\leq x}\log p, \qquad\psi(x)=\sideset{}{^*}\sum_{p^m\leq x}\log p,
\end{align}
where $\sum^*$ indicates that the last term in the sum is multiplied by $1/2$ when $x$ is an integer. However, we note that our main results (Theorem \ref{mainthm} and Theorem \ref{weakthm}), will also hold for the standard (unnormalised) prime counting functions. Our main focus will be on the function $\psi(x)$ since the other functions in \eqref{chebfuncs} can be related to $\psi(x)$ by simple bounding and partial summation arguments.
Following \cite[Section 2]{buthe2016estimating}, we define
\begin{align*}
\ell_{c,\varepsilon}(\xi)=\frac{c}{\sinh(c)}\frac{\sin(\sqrt{(\xi\varepsilon)^2-c^2})}{\sqrt{(\xi\varepsilon)^2-c^2}},\quad a_{c,\varepsilon}(\rho)=\frac{1}{\ell_{c,\varepsilon}(i/2)}\ell_{c,\varepsilon}\left(\frac{\rho}{i}-\frac{1}{2i}\right)
\end{align*}
for $c,\varepsilon>0$. We will also make use of the auxiliary function $\psi_{c,\varepsilon}(x)$ defined on page 2484 of \cite{buthe2016estimating}. Notably, $\psi_{c,\varepsilon}(x)$ is a continuous approximation to $\psi(x)$. Moreover, for $x\geq 10$ and $0<\varepsilon\leq 10^{-4}$, we have \cite[Proposition 2]{buthe2018analytic}
\begin{equation}\label{vonmaneq}
x-\psi_{c,\varepsilon}(x)=\sum_{\rho}\frac{a_{c,\varepsilon}(\rho)}{\rho}x^\rho+\Theta(2).
\end{equation}
Here, $\Theta(2)$ indicates a constant with absolute value less than 2, and the sum is taken over all the non-trivial zeros of the Riemann zeta-function and computed as
\begin{equation*}
\lim_{T\to\infty}\sum_{|\Im(\rho)|<T}\frac{a_{c,\varepsilon}(\rho)}{\rho}x^\rho.
\end{equation*}
To obtain an expression for $|\psi(x)-x|$ we thus need bounds on $\sum_{\rho}\frac{a_{c,\varepsilon}(\rho)}{\rho}x^\rho$ and $|\psi(x)-\psi_{c,\varepsilon}(x)|$. We will make use of the following collection of lemmas taken from \cite{buthe2016estimating} with slight modifications.
\begin{lemma}[{\cite[Proposition 3]{buthe2016estimating}}]\label{butprop3}
Let $x>1$, $\varepsilon\leq 10^{-3}$ and $c\geq 3$. Then
\begin{equation*}
\sum_{|\Im(\rho)|>\frac{c}{\varepsilon}}\left|a_{c,\varepsilon}(\rho)\frac{x^\rho}{\rho}\right|\leq 0.16\frac{x+1}{\sinh(c)}e^{0.71\sqrt{c\varepsilon}}\log(3c)\log\left(\frac{c}{\varepsilon}\right).
\end{equation*}
Moreover, if $a\in (0,1)$ with $a\frac{c}{e}\geq 10^3$, and the Riemann hypothesis holds for all zeros $\rho$ with $\Im(\rho)\in(0,\frac{c}{\varepsilon}]$, then
\begin{equation*}
\sum_{a\frac{c}{\varepsilon}<|\Im(\rho)|\leq\frac{c}{\varepsilon}}\left|a_{c,\varepsilon}(\rho)\frac{x^\rho}{\rho}\right|\leq\frac{1+11c\varepsilon}{\pi ca^2}\log\left(\frac{c}{\varepsilon}\right)\frac{\cosh(c\sqrt{1-a^2})}{\sinh(c)}\sqrt{x}.
\end{equation*}
\end{lemma}
\begin{lemma}[{\cite[Lemma 3]{buthe2016estimating}}]\label{butlemma3}
If $t_2\geq 5000$ then
\begin{equation*}
\sum_{0<|\Im(\rho)|\leq t_2}\frac{1}{|\Im(\rho)|}\leq\frac{1}{2\pi}\log^2\left(\frac{t_2}{2\pi}\right).
\end{equation*}
\end{lemma}
\begin{lemma}[{\cite[Proposition 4]{buthe2016estimating}}]\label{butprop4}
Let $x>100$, $\varepsilon<10^{-2}$ and
\begin{equation*}
B_0=\frac{I_1(c)}{2\sinh(c)}\varepsilon xe^{-\varepsilon}>1,
\end{equation*}
where
\begin{equation*}
I_1(x)=\sum_{n=0}^\infty\frac{(x/2)^{2n+1}}{n!\Gamma(n+2)}
\end{equation*}
is a modified Bessel function of the first kind. We then have
\begin{equation*}
|\psi(x)-\psi_{c,\varepsilon}(x)|\leq e^{2\varepsilon}\log(e^{\varepsilon}x)\left[\frac{\varepsilon x}{\log B_0}\frac{I_1(c)}{\sinh(c)}+2.01\varepsilon\sqrt{x}+\frac{1}{2}\log\log(2x^2)\right].
\end{equation*}
\end{lemma}
\begin{lemma}[{\cite[Proposition 5]{buthe2016estimating}}]\label{butprop5}
For $c_0>0$ let
\begin{equation*}
D(c_0)=\sqrt{\frac{\pi c_0}{2}}\frac{I_1(c_0)}{\sinh(c_0)}.
\end{equation*}
Then
\begin{equation*}
\frac{D(c_0)}{\sqrt{2\pi c}}\leq \frac{I_1(c)}{2\sinh(c)}\leq\frac{1}{\sqrt{2\pi c}}
\end{equation*}
holds for all $c\geq c_0$.
\end{lemma}
In particular, note that for Lemma \ref{butprop4} we have taken the case $\alpha=0$ in \cite[Proposition 4]{buthe2016estimating}. Moreover, we remark that Brent, Platt and Trudgian \cite[Lemma 8]{brent2020mean} recently showed that Lemma \ref{butlemma3} holds more generally for $t_2\geq 4\pi e$.
\section{Proof of the main result}\label{sectmain}
We begin by stating the bounds obtained using B{\"u}the's result \cite[Theorem 2]{buthe2016estimating} and Platt and Trudgian's computation \cite{platt2021riemann}.
\begin{proposition}\label{buthethm}
The following estimates hold:
\begin{align}
|\psi(x)-x|&<\frac{\sqrt{x}}{8\pi}\log^2(x),&\text{for }59\leq x\leq 2.169\cdot10^{25},\notag\\
|\theta(x)-x|&<\frac{\sqrt{x}}{8\pi}\log^2(x),&\text{for }599\leq x\leq 2.169\cdot10^{25},\notag\\
|\psi(x)-x|&<\frac{\sqrt{x}}{8\pi}\log x(\log x-3),&\text{for }5000\leq x\leq 2.169\cdot10^{25},\label{psiminus3}\\
|\theta(x)-x|&<\frac{\sqrt{x}}{8\pi}\log x(\log x-2),&\text{for }5000\leq x\leq 2.169\cdot10^{25},\label{thetaminus2}\\
|\Pi(x)-\li(x)|&<\frac{\sqrt{x}}{8\pi}\log x,&\text{for }59\leq x\leq 2.169\cdot10^{25},\notag\\
|\pi(x)-\li(x)|&<\frac{\sqrt{x}}{8\pi}\log x,&\text{for }2657\leq x\leq 2.169\cdot10^{25}.\notag
\end{align}
\end{proposition}
\begin{proof}
The $2.169\cdot 10^{25}$ comes from substituting $T=3\cdot 10^{12}$ \cite{platt2021riemann} into \cite[Theorem 2]{buthe2016estimating}. Note that \eqref{psiminus3} and \eqref{thetaminus2} do not appear in the statement of B{\"u}the's theorem but are established as intermediary steps in the proof.
\end{proof}
We now prove the main result of this paper.
\begin{theorem}\label{mainthm}
Let $T>0$ be such that the Riemann hypothesis holds for zeros $\rho$ with $0<\Im(\rho)\leq T$. Then, under the condition $\frac{9.06}{\log\log x}\sqrt{\frac{x}{\log x}}\leq T$, the following estimates hold:
\begin{align}
|\psi(x)-x|&<\frac{\sqrt{x}}{8\pi}\log^2x,&\text{for }x\geq 59,\label{psiin}\\
|\theta(x)-x|&<\frac{\sqrt{x}}{8\pi}\log^2x,&\text{for }x\geq 599,\label{thetain}\\
|\Pi(x)-\li(x)|&<\frac{\sqrt{x}}{8\pi}\log x,&\text{for }x\geq 59\label{piin1},\\
|\pi(x)-\li(x)|&<\frac{\sqrt{x}}{8\pi}\log x,&\text{for }x\geq 2657\label{piin2}.
\end{align}
\end{theorem}
\begin{proof}
Throughout this proof we will label specific constants $A$, $B$, $C$, $D$ and $E$. This is done to make it clear where optimisations are being made and to allow us to perform an iterative argument.
Now, by Proposition \ref{buthethm} it suffices to consider $x>A$ where $A=2.169\cdot 10^{25}$. We also initially restrict ourselves to $x$ such that
\begin{equation}\label{beq}
\frac{B}{\log\log x}\sqrt{\frac{x}{\log x}}\leq T,
\end{equation}
where $B=9.65$ and later reduce the value of $B$. We first prove the bound
\begin{equation}\label{strongpsieq}
|\psi(x)-x|<\frac{\sqrt{x}}{8\pi}\log x(\log x-C),\quad\text{for }x>A,
\end{equation}
where $C=2.44$. Next, we define
\begin{equation}\label{ceeq}
c(x)=\frac{1}{2}\log x+D,\qquad \varepsilon(x)=\frac{\log^{3/2}x\log\log x}{E\sqrt{x}},
\end{equation}
where $D=6$ and $E=16$. To simplify notation we write $c=c(x)$ and $\varepsilon=\varepsilon(x)$. Note that for these choices of $D$ and $E$, we have $c>35$, $\varepsilon<4.9\cdot 10^{-11}$ and
\begin{equation*}
\frac{c}{\varepsilon}\leq\left(\frac{1}{2}E+\frac{DE}{\log A}\right)\frac{1}{\log\log x}\sqrt{\frac{x}{\log x}}\leq\frac{B}{\log\log x}\sqrt{\frac{x}{\log x}}\leq T.
\end{equation*}
Hence we may assume $\Re(\rho)=\frac{1}{2}$ for zeros $\rho$ with $|\Im(\rho)|\leq\frac{c}{\varepsilon}$.
Now, recall from \eqref{vonmaneq} that
\begin{equation*}
x-\psi_{c,\varepsilon}(x)=\sum_{\rho}\frac{a_{c,\varepsilon}(\rho)}{\rho}x^\rho+\Theta(2).
\end{equation*}
To prove \eqref{strongpsieq} we split $\sum_{\rho}\frac{a_{c,\varepsilon}(\rho)}{\rho}x^\rho$ into three parts and then bound $|\psi(x)-\psi_{c,\varepsilon}(x)|$. For $|\Im(\rho)|>c/\varepsilon$, Lemma \ref{butprop3} gives
\begin{align}
\sum_{|\Im(\rho)|>\frac{c}{\varepsilon}}\left|a_{c,\varepsilon}(\rho)\frac{x^{\rho}}{\rho}\right|&\leq 0.16\frac{x+1}{\sinh(c)}e^{0.71\sqrt{c\varepsilon}}\log(3c)\log\left(\frac{c}{\varepsilon}\right)\notag\\
&\leq\mathcal{E}_1(x),\label{eps1eq}
\end{align}
where
\begin{equation*}
\mathcal{E}_1(x)=0.000032\sqrt{x}\log x\log\log x.
\end{equation*}
The inequality in \eqref{eps1eq} follows by noticing that for $x>A$
\begin{equation*}
\left(\frac{x+1}{\sinh(c)}\right)/\sqrt{x},\ e^{0.71\sqrt{c\varepsilon}} \text{ and } \log(3c)/\log\log x
\end{equation*}
are all decreasing functions and $\log(\frac{c}{\varepsilon})/\log x\leq\frac{1}{2}$. Substituting $x=A$ into
\begin{equation*}
\frac{0.16\frac{x+1}{\sinh(c)}e^{0.71\sqrt{c\varepsilon}}\log(3c)}{\sqrt{x}\log x}\times\frac{1}{2}
\end{equation*}
then gives $0.0000316\ldots\leq 0.000032$.
When $\frac{\sqrt{2c}}{\varepsilon}<\Im(\rho)<\frac{c}{\varepsilon}$ we use the second part of Lemma \ref{butprop3} with $a=\sqrt{\frac{2}{c}}$ to obtain
\begin{align}
\sum_{\frac{\sqrt{2c}}{\varepsilon}<|\Im(\rho)|\leq\frac{c}{\varepsilon}}\left|a_{c,\varepsilon}(\rho)\frac{x^{\rho}}{\rho}\right|&\leq\frac{1+11c\varepsilon}{2\pi}\log\left(\frac{c}{\varepsilon}\right)\frac{\cosh(c\sqrt{1-a^2})}{\sinh(c)}\sqrt{x}\label{eps2eq1}\\
&\leq\mathcal{E}_2(x),\label{eps2eq2}
\end{align}
where
\begin{equation*}
\mathcal{E}_2(x)=0.0293\sqrt{x}\log x.
\end{equation*}
For the inequality in \eqref{eps2eq2} we note that similarly to before $\frac{1+11c\varepsilon}{2\pi}$ is decreasing and $\log\left(\frac{c}{\varepsilon}\right)/\log x\leq\frac{1}{2}$ for $x>A$. Then,
\begin{align*}
\frac{\cosh(c\sqrt{1-a^2})}{\sinh(c)}&=\frac{e^{\sqrt{\frac{1}{2}\log x+D}\sqrt{\frac{1}{2}\log x+D-2}}+e^{-\sqrt{\frac{1}{2}\log x+D}\sqrt{\frac{1}{2}\log x+D-2}}}{e^{\frac{1}{2}\log x+D}-e^{-\frac{1}{2}\log x-D}}\\
&\leq\frac{e^{\frac{1}{2}\log x+D-1}+e^{-\sqrt{\frac{1}{2}\log x+D}\sqrt{\frac{1}{2}\log x+D-2}}}{e^{\frac{1}{2}\log x+D}}\\
&=\frac{1}{e}+\frac{1}{e^{\sqrt{\frac{1}{2}\log x+D}\sqrt{\frac{1}{2}\log x+D-2}+\frac{1}{2}\log x+D}}
\end{align*}
which is also decreasing. Substituting $x=A$ into
\begin{equation*}
\frac{1}{2}\cdot\frac{1+11c\varepsilon}{2\pi}\cdot\left(\frac{1}{e}+\frac{1}{\exp\left(\sqrt{\frac{1}{2}\log x+D}\sqrt{\frac{1}{2}\log x+D-2}+\frac{1}{2}\log x+D\right)}\right)
\end{equation*}
then gives $0.0292\ldots\leq 0.0293$.
Next, we consider the range $0<|\Im(\rho)|\leq\frac{\sqrt{2c}}{\varepsilon}$. Note that
\begin{align}
a_{c,\varepsilon}(\rho)&=\frac{\sqrt{\varepsilon^2/4+c^2}}{\sinh(\sqrt{\varepsilon^2/4+c^2})}\times\frac{\sin(\sqrt{\Im(\rho)^2\varepsilon^2-c^2})}{\sqrt{\Im(\rho)^2\varepsilon^2-c^2}}\notag\\
&=\frac{\sqrt{\varepsilon^2/4+c^2}}{\sinh(\sqrt{\varepsilon^2/4+c^2})}\times\frac{\sinh(\sqrt{c^2-\Im(\rho)^2\varepsilon^2})}{\sqrt{c^2-\Im(\rho)^2\varepsilon^2}}\label{imeq}\\
&\leq\frac{c}{\sinh(c)}\times\frac{\sinh(c)}{c}=1\label{decreasingeq}.
\end{align}
In particular, \eqref{imeq} follows since $|\Im(\rho)|\leq\frac{c}{\varepsilon}\leq T$ and \eqref{decreasingeq} follows since $\frac{x}{\sinh(x)}$ is decreasing for $x>0$. Hence $|a_{c,\varepsilon}(\rho)/\rho|\leq 1/|\Im(\rho)|$ and so Lemma \ref{butlemma3} gives
\begin{align}
\sum_{0<|\Im(\rho)|\leq\frac{\sqrt{2c}}{\varepsilon}}\left|a_{c,\varepsilon}(\rho)\frac{x^{\rho}}{\rho}\right|&\leq\frac{\sqrt{x}}{2\pi}\log\left(\frac{\sqrt{2c}}{2\pi\varepsilon}\right)^2\notag\\
&=\frac{\sqrt{x}}{2\pi}\log\left(\frac{E\sqrt{x}\sqrt{\log x+2D}}{2\pi\log^{3/2} x\log\log x}\right)^2\notag\\
&\leq\frac{\sqrt{x}}{2\pi}\log\left(\frac{E\sqrt{x}\sqrt{\log x+2D\frac{\log x}{\log(A)}}}{2\pi\log^{3/2}x\log\log x}\right)^2\notag\\
&\leq\frac{\sqrt{x}}{2\pi}\left(\frac{1}{2}\log x+\log(2.8)-\log\log x-\log\log\log x\right)^2\notag\\
&\leq\frac{\sqrt{x}}{8\pi}\log^2x+\mathcal{E}_3(x),\label{e3eq}
\end{align}
where
\begin{equation*}
\mathcal{E}_3(x):=\frac{\sqrt{x}}{2\pi}\left(\frac{1}{2}\log x+\log(2.8)-\log\log x-\log\log\log x\right)^2-\frac{\sqrt{x}}{8\pi}\log^2x.
\end{equation*}
We now bound $|\psi(x)-\psi_{c,\varepsilon}(x)|$. By Lemma \ref{butprop5}
\begin{equation}\label{v0eq}
\frac{0.98}{\sqrt{2\pi c}}\leq \frac{I_1(c)}{2\sinh(c)}\leq\frac{1}{\sqrt{2\pi c}}.
\end{equation}
Combining \eqref{v0eq} and Lemma \ref{butprop4} with our definition \eqref{ceeq} of $\varepsilon$ then gives
\begin{align}\label{explicitpsibound}
|\psi(x)-\psi_{c,\varepsilon}(x)|\leq\frac{2.0001\sqrt{x}\log^{5/2}x\log\log x}{E\sqrt{\pi(\log x+2D)}}\log\left(\frac{0.97\sqrt{x}\log^{3/2}x\log\log x}{E\sqrt{\pi(\log x+2D)}}\right)^{-1}\notag\\
+\frac{2.02}{E}\log^{5/2}x\log\log x+0.51\log x\log\log(2x^2).
\end{align}
Since $x>A=2.169\cdot 10^{25}$, we have
\begin{align*}
\log\left(\frac{0.97\sqrt{x}\log^{3/2} x\log\log x}{E\sqrt{\pi(\log x+2D)}}\right)\geq\log(\sqrt{x})=\frac{1}{2}\log x.
\end{align*}
Hence, dividing the first summand in \eqref{explicitpsibound} by $\sqrt{x}\frac{\log^{3/2}x\log\log x}{\sqrt{\log x+2D}}$ gives
\begin{align*}
\frac{2.0001\log x}{E\sqrt{\pi}}\log\left(\frac{0.97\sqrt{x}\log^{3/2}x\log\log x}{E\sqrt{\pi(\log x+2D)}}\right)^{-1}&\leq\frac{2.0001\log x}{E\sqrt{\pi}}\times\frac{2}{\log x}\\
&=0.141\ldots\leq 0.142.
\end{align*}
So if we define
\begin{equation*}
\mathcal{E}_4(x)=0.142\sqrt{x}\frac{\log^{3/2}x\log\log x}{\sqrt{\log x+2D}}
\end{equation*}
and
\begin{equation*}
\mathcal{E}_5(x):=\frac{2.02}{E}\log^{5/2}x\log\log x+0.51\log x\log\log(2x^2)+2
\end{equation*}
then
\begin{equation}\label{e4e5eq}
|\psi(x)-\psi_{c,\varepsilon}(x)|\leq \mathcal{E}_4(x)+\mathcal{E}_5(x).
\end{equation}
Thus, by \eqref{vonmaneq}, \eqref{eps1eq}, \eqref{eps2eq2}, \eqref{e3eq} and \eqref{e4e5eq}
\begin{equation*}
|\psi(x)-x|\leq\frac{\sqrt{x}}{8\pi}\log^2x+\mathcal{E}_1(x)+\mathcal{E}_2(x)+\mathcal{E}_3(x)+\mathcal{E}_4(x)+\mathcal{E}_5(x).
\end{equation*}
Now consider the function
\begin{equation*}
\mathcal{E}(x)=\frac{1}{\sqrt{x}\log x}(\mathcal{E}_1(x)+\mathcal{E}_2(x)+\mathcal{E}_3(x)+\mathcal{E}_4(x)+\mathcal{E}_5(x)).
\end{equation*}
Differentiating $\mathcal{E}(x)$ with respect to $y=\log x$ we see that $\mathcal{E}(x)$ is decreasing for $x>A$. Moreover, $\mathcal{E}(A)=-0.0976\ldots<-\frac{C}{8\pi}=-0.0970\ldots$. This proves \eqref{strongpsieq}. Letting $T=3\cdot 10^{12}$ in \eqref{beq} and using Proposition \ref{buthethm} then gives
\begin{equation}\label{psifirstit}
|\psi(x)-x|<\frac{\sqrt{x}}{8\pi}\log x(\log x-C),\quad\text{for }5000\leq x\leq 9.68\cdot 10^{25}.
\end{equation}
From \eqref{psifirstit}, we also obtain
\begin{equation}\label{thetafirstit}
|\theta(x)-x|<\frac{\sqrt{x}}{8\pi}\log x(\log x-2),\quad\text{for }5000\leq x\leq 9.68\cdot 10^{25}.
\end{equation}
To see this, we use recent estimates by Broadbent et al. \cite[Corollary 5.1]{broadbent2021sharper} for $\psi(x)-\theta(x)$. Namely\footnote{This estimate is stated in \cite{broadbent2021sharper} for the unnormalised $\psi$ and $\theta$ functions. However, it also holds for the normalised functions whereby the difference $\psi(x)-\theta(x)$ is at most that in the unnormalised setting.}
\begin{equation*}
\psi(x)-\theta(x)<a_1x^{1/2}+a_2x^{1/3},
\end{equation*}
where for $x\geq e^{50}\approx 5.18\cdot 10^{21}$, we can take $a_1=1+1.93378\cdot 10^{-8}$ and $a_2=1.01718$. In particular, for $x>A$ we have $\psi(x)-\theta(x)\leq(C-2)\frac{\sqrt{x}}{8\pi}\log x$. Hence \eqref{thetafirstit} holds for $A<x\leq 9.68\cdot 10^{25}$ since
\begin{equation*}
|\theta(x)-x|\leq\psi(x)-\theta(x)+|\psi(x)-x|.
\end{equation*}
For the remaining values of $x$, we use Proposition \ref{buthethm}.
We now repeat the entire proof with
\begin{equation*}
(A,B,C,D,E)=(9.68\cdot 10^{25},9.34,2.43,5,16).
\end{equation*}
The error terms then update to (with more precision added this time):
\begin{align*}
\mathcal{E}_1(x)&=0.0000839\sqrt{x}\log x\log\log x,\\
\mathcal{E}_2(x)&=0.02928\sqrt{x}\log x,\\
\mathcal{E}_3(x)&=\frac{\sqrt{x}}{2\pi}\left(\frac{1}{2}\log x+\log(2.751)-\log\log x-\log\log\log x\right)^2-\frac{\sqrt{x}}{8\pi}\log^2x,\\
\mathcal{E}_4(x)&=0.1411\sqrt{x}\frac{\log^{3/2}x\log\log x}{\sqrt{\log x+10}},\\
\mathcal{E}_5(x)&=0.12625\log^{5/2}x\log\log x+0.51\log x\log\log(2x^2)+2,\\
\mathcal{E}(A)&=-0.0967\ldots
\end{align*}
and we get
\begin{align*}
|\psi(x)-x|<\frac{\sqrt{x}}{8\pi}\log x(\log x-C),\quad\text{for }5000\leq x\leq 1.03\cdot 10^{26},\\
|\theta(x)-x|<\frac{\sqrt{x}}{8\pi}\log x(\log x-2),\quad\text{for }5000\leq x\leq 1.03\cdot 10^{26}.
\end{align*}
Iterating again with
\begin{equation*}
(A,B,C,D,E)=(1.03\cdot 10^{26},9.08,2.42,2.4,16.8)
\end{equation*}
followed by
\begin{equation*}
(A,B,C,D,E)=(1.096\cdot 10^{26},9.06,2.42,2.34,16.8)
\end{equation*}
we get
\begin{align}
|\psi(x)-x|<\frac{\sqrt{x}}{8\pi}\log x(\log x-C),\label{finalpsi}\\
|\theta(x)-x|<\frac{\sqrt{x}}{8\pi}\log x(\log x-2)\label{finaltheta},
\end{align}
for $x\geq 5000$ and $\frac{9.06}{\log\log x}\sqrt{\frac{x}{\log x}}\leq T$. Combining \eqref{finalpsi} and \eqref{finaltheta} with Proposition \ref{buthethm} proves \eqref{psiin} and \eqref{thetain}. Certainly, one could perform further iterations but this would produce a minimal improvement.
Now, using integration by parts
\begin{equation*}
\li(x)-\li(a)=\frac{x}{\log x}+\int_a^x\frac{\mathrm{d}t}{\log^2t}-\frac{a}{\log a}
\end{equation*}
so that by partial summation
\begin{equation*}
\pi(x)-\pi(a)=\li(x)-\li(a)-\frac{x-\theta(x)}{\log x}+\frac{a-\theta(a)}{\log a}-\int_a^x\frac{t-\theta(t)}{t\log^2t}\mathrm{d}t.
\end{equation*}
Hence, for $5000\leq x$ and $\frac{9.06}{\log\log x}\sqrt{\frac{x}{\log x}}\leq T$,
\begin{align*}
|\pi(x)-\li(x)|&\leq\frac{\sqrt{x}}{8\pi}(\log x-2)+\left|\pi(5000)-\li(5000)-\frac{\theta(5000)-5000}{\log(5000)}\right|+\frac{\sqrt{x}}{4\pi}-\frac{\sqrt{5000}}{4\pi}\\
&=\frac{\sqrt{x}}{8\pi}\log x+4.91...-\frac{\sqrt{5000}}{4\pi}\\
&=\frac{\sqrt{x}}{8\pi}\log x+4.91...-5.62...\\
&<\frac{\sqrt{x}}{8\pi}\log x.
\end{align*}
Making use of \eqref{finalpsi} as opposed to \eqref{finaltheta} then gives $|\Pi(x)-\li(x)|<\frac{\sqrt{x}}{8\pi}\log x$.
Combined with Proposition \ref{buthethm} we obtain \eqref{piin1} and \eqref{piin2} thereby completing the proof of the theorem.
\end{proof}
Setting $T=3\cdot 10^{12}$ we obtain the following result.
\begin{corollary}\label{maincor}
The bounds \eqref{psiin}-\eqref{piin2} hold for $x\leq 1.101\cdot 10^{26}$.
\end{corollary}
\section{Possible improvements and variations}\label{sectimprov}
\subsection{Improvements for larger $T$}
The constant 9.06 appearing in \eqref{djbound} can be lowered if the Riemann hypothesis were verified to a higher height. This is because a higher value of $T$ means that the bounds \eqref{psiin}--\eqref{piin2} hold for larger values of $x$ thereby giving sharper error terms in the proof of Theorem \ref{mainthm}. Table \ref{ttable} lists improvements that one would get by increasing $T$ to $10^{13}$, $10^{14}$ and $10^{15}$. The values in the table were computed using the same iterative method as in the proof of Theorem \ref{mainthm}.
\def1.5{1.5}
\begin{table}[h]
\centering
\caption{Value of $K$ such that \eqref{psiin}--\eqref{piin2} hold for $\frac{K}{\log\log x}\sqrt{\frac{x}{\log x}}\leq T$ when $T\geq T_0$. Here, $x_{\text{max}}$ is the largest value of $x$ for which this inequality holds when $T=T_0$.}
\begin{tabular}{|c|c|c|}
\hline
$T_0$ & $K$ & $x_{\text{max}}$ \\
\hline
$ 10^{13} $& $8.94$ & $1.335\cdot 10^{27}$\\
\hline
$ 10^{14} $& $8.76$ & $1.550\cdot 10^{29}$\\
\hline
$ 10^{15} $& $8.64$ & $1.762\cdot 10^{31}$\\
\hline
\end{tabular}
\label{ttable}
\end{table}
\subsection{Weakening the constant}
Using the methods in Section \ref{sectmain}, we can obtain weaker bounds that hold for larger ranges of $x$. Here, the main idea is to alter the definition of $\mathcal{E}_3(x)$ and thereby change the leading term in \eqref{e3eq}. Doing this with the constant changed from $1/8\pi$ to a selection of larger values, we obtained the following result.
\begin{theorem}\label{weakthm}
Let $T>0$ be such that the Riemann hypothesis holds for zeros $\rho$ with $0<\Im(\rho)\leq T$. Then, for corresponding values of $a$ and $K$ in Table \ref{constanttable}, the following estimates hold:
\begin{align}
|\psi(x)-x|&<a\sqrt{x}\log^2x,&\text{for }x\geq 3,\label{psiin2}\\
|\theta(x)-x|&<a\sqrt{x}\log^2x,&\text{for }x\geq 3,\label{thetain2}\\
|\Pi(x)-\li(x)|&<a\sqrt{x}\log x,&\text{for }x\geq 2\label{piin12},\\
|\pi(x)-\li(x)|&<a\sqrt{x}\log x,&\text{for }x\geq 2\label{piin22},
\end{align}
provided $K\sqrt{\frac{x}{\log^3x}}\leq T$.
\end{theorem}
\begin{proof}
Let $(a,K)=(1,1.19)$. For other values of $a$ and $K$ the method of proof is essentially identical. We use the same general reasoning as in the proof of Theorem \ref{mainthm}. Hence we only describe the small modifications required in this setting.
Firstly, the minimum values for $x$ appearing in \eqref{psiin2}--\eqref{piin22} were obtained by checking each expression manually up to the minimum values appearing in \eqref{psiin}--\eqref{piin2}. We then let
\begin{equation}\label{ceeq2}
c(x)=\frac{1}{2}\log x+D,\qquad \varepsilon(x)=\frac{\log^{5/2}x}{E\sqrt{x}},
\end{equation}
initially setting $D=0$ and $E=2.4$. Each of the error terms $\mathcal{E}_1(x),\ldots,\mathcal{E}_5(x)$ changed slightly due to the new choice of $\varepsilon(x)$ in \eqref{ceeq2}. The main difference occurred with $\mathcal{E}_3(x)$ and $\mathcal{E}_4(x)$, now given by
\begin{align*}
\mathcal{E}_3(x)&:=\frac{\sqrt{x}}{2\pi}\left(\frac{1}{2}\log x+\log(\alpha)-2\log\log x\right)^2-a\sqrt{x}\log^2x,\\
\mathcal{E}_4(x)&:=\beta\sqrt{x}\log^2x
\end{align*}
for some computable constants $\alpha$ and $\beta$. In particular, this definition of $\mathcal{E}_3(x)$ gives
\begin{equation*}
\sum_{0<|\Im(\rho)|\leq\frac{\sqrt{2c}}{\varepsilon}}\left|a_{c,\varepsilon}(\rho)\frac{x^{\rho}}{\rho}\right|\leq a\sqrt{x}\log^2x+\mathcal{E}_3(x).
\end{equation*}
For the iterative process we started with $A=1.101\cdot 10^{26}$ (as per Corollary \ref{maincor}), $B=1.2$, $C=2.017$, $D=0$ and $E=2.4$. Here, $B$ was such that the inequalities \eqref{psiin2}--\eqref{piin22} held for $B\sqrt{\frac{x}{\log(x)^3}}\leq T$ and $C$ was such that
\begin{align*}
|\psi(x)-x|<a\sqrt{x}\log x(\log x-C),
\end{align*}
held for each $x$ in this range. For the second iteration we used
\begin{equation*}
(A,B,C,D,E)=(2.128\cdot 10^{30},1.19,2.015,0,2.38)
\end{equation*}
which gave the desired result.
\end{proof}
\begin{remark}
In the above proof we fixed $D=0$. A small improvement is possible if we allowed $D$ to be negative. However, this requires reworking several inequalities from the proof of Theorem \ref{mainthm} so we decided not to do so here.
\end{remark}
\def1.5{1.5}
\begin{table}[h]
\centering
\caption{Corresponding values of $a$ and $K$ for Theorem \ref{weakthm}. The value $x_{\text{max}}$ is the largest $x$ for which the inequalities \eqref{psiin2}--\eqref{piin22} hold upon setting $T=3\cdot 10^{12}$.}
\begin{tabular}{|c|c|c||c|c|c|}
\hline
$a$ & $K$ & $x_{\text{max}}$ & $a$ & $K$ & $x_{max}$\\
\hline
$ 1 $& $1.19$ & $2.165\cdot 10^{30}$ & $ 10^{4} $& $1.16\cdot 10^{-4}$ & $4.723\cdot 10^{38}$\\
\hline
$ 10 $& $0.117$ & $2.738\cdot 10^{32}$ & $10^5$ & $1.16\cdot 10^{-5}$ & $5.522\cdot 10^{40}$\\
\hline
$ 100 $& $0.0116$ & $3.360\cdot 10^{34}$ & $10^6$ & $1.16\cdot 10^{-6}$ & $6.404\cdot 10^{42}$\\
\hline
$ 1000 $& $0.00116$ & $4.004\cdot 10^{36}$ & $10^7$ & $1.16\cdot 10^{-7}$ &$7.375\cdot 10^{44}$\\
\hline
\end{tabular}
\label{constanttable}
\end{table}
\section{An inequality of Ramanujan}\label{appsec}
In one of his notebooks, Ramanujan proved that the inequality
\begin{equation}\label{rameq}
\pi(x)^2<\frac{ex}{\log x}\pi\left(\frac{x}{e}\right)
\end{equation}
holds for sufficiently large $x$ (see \cite[pp 112--114]{berndt2012ramanujan}). Several authors (\cite{dudek2015solving}, \cite{axler2017estimates}, \cite{platt2021error}, \cite{hassani2021remarks}) have attempted to make \eqref{rameq} completely explicit. It is widely believed that the last integer counterexample occurs at $x=38,358,837,682$. In fact, this follows under assumption of the Riemann hypothesis \cite[Theorem 1.3]{dudek2015solving}.
The best unconditional result is due to Platt and Trudgian \cite[Theorem 2]{platt2021error}. In particular, they show that \eqref{rameq} holds for both $38,358,837,683\leq x\leq\exp(58)$ and $x\geq\exp(3915)$. Our bounds on $\pi(x)$ allow for a significant improvement on the first of these results. To demonstrate this, we use a simple (but computationally intensive) method to verify \eqref{rameq}, obtaining the following result.
\begin{theorem}\label{ramthm}
For $38,358,837,683\leq x\leq\exp(103)$, Ramanujan's inequality \eqref{rameq} holds unconditionally.
\end{theorem}
\begin{proof}
For $38,358,837,682<x\leq\exp(43)$, the theorem follows from \cite[Theorem 3]{axler2017estimates}. Platt and Trudgian also prove \eqref{rameq} for $\exp(43)<x\leq\exp(58)$ but the author thought it would be instructive to re-establish their result.
So, let $x>\exp(43)$ and write $z=\log x$. Then \eqref{rameq} is equivalent to
\begin{equation}\label{zeq}
\frac{e^{z+1}}{z}\pi(e^{z-1})-\pi(e^z)^2>0.
\end{equation}
Set $a=1/8\pi$. By Theorem \ref{mainthm} we have that $|\pi(x)-\li(x)|<a\sqrt{x}\log x$ for $\exp(43)<x\leq\exp(59)$. Thus, \eqref{zeq} is true in this range provided
\begin{equation}\label{lieq}
\frac{e^{z+1}}{z}\li(e^{z-1})-\frac{a(z-1)}{ z}e^{\frac{3z+1}{2}}-\left(\li(e^z)+aze^{z/2}\right)^2>0
\end{equation}
for $43<z\leq 59$. We write
\begin{equation*}
f(z)=\frac{e^{z+1}}{z}\li(e^{z-1}),\quad g(z)=\frac{a(z-1)}{ z}e^{\frac{3z+1}{2}}+\left(\li(e^z)+aze^{z/2}\right)^2
\end{equation*}
so that \eqref{lieq} is equivalent to $f(z)-g(z)>0$. Note that $f(z)$ and $g(z)$ are both increasing for $z>1$. Hence, if $f(z_0)>g(z_0+\delta)$ for some $z_0>1$ and $\delta>0$, then $f(z)>g(z)$ for every $z\in(z_0,z_0+\delta)$. We thus performed a ``brute force" verification by setting $\delta=5\cdot 10^{-8}$ and showing that
\begin{equation*}
f(43)-g(43+\delta)>0,\ f(43+\delta)-g(43+2\delta)>0,\ \ldots,\ f(59-\delta)-g(59)>0.
\end{equation*}
This was achieved using a short algorithm written in Python. The computations took just under a day on a 2.4GHz laptop.
We then repeated the above argument using Theorem 4.1 with $a=1$ and a smaller $\delta=2.5\cdot 10^{-8}$. This proved \eqref{rameq} for $\exp(59)<x\leq\exp(69)$. Continuing in this fashion for each value of $a$ in Table 2 we see that \eqref{rameq} holds in the range $\exp(43)<x\leq\exp(103)$ as desired.
\end{proof}
Certainly one could extend Table \ref{constanttable} and the computations in the above proof. However this would require a large amount of computation time. Thus, to improve on Theorem \ref{ramthm} the author suggests switching to a more sophisticated and less computational method. For instance, one could attempt to modify the arguments in \cite[Section 6]{axler2017estimates} or \cite[Section 5]{platt2021error}.
\section{Future work}
There are several ways in which one could expand on the work in this paper, for instance:
\begin{enumerate}[label=(\arabic*)]
\item One could produce a wider range of weakened bounds similar to those in Theorem \ref{weakthm}. For example, one could provide a more general expression for $K$ as a function of $a$.
\item One could produce analogous results for primes in arithmetic progressions. To do this, one would need to rework the results in this paper and \cite{buthe2016estimating} using computations of zeros of Dirichlet $L$-functions (e.g.\ \cite{platt2016numerical}) and the explicit formula for $\psi(x,\chi)$ \cite[Chapter 19]{davenport2013multiplicative}. Then, if desired, one could also consider other types of $L$-functions.
\item As discussed in Section \ref{appsec}, it is possible to improve Theorem \ref{ramthm} with some work. It would be interesting to optimise the results of this paper and those in \cite{platt2021error} to see how close one could get to making Ramanujan's inequality completely explicit.
\end{enumerate}
\section*{Acknowledgements}
Thanks to my supervisor Tim Trudgian for all of his wonderful suggestions and insights on this project.
\printbibliography
\end{document}
|
{
"timestamp": "2022-06-15T02:16:08",
"yymm": "2109",
"arxiv_id": "2109.02249",
"language": "en",
"url": "https://arxiv.org/abs/2109.02249"
}
|
\section{}
\section{Introduction}
An ongoing activity of considerable interest is to investigate the physics out of equilibrium. While on one hand is the question about the late time fate of the non-equilibrium quantum state, on the other hand we want to uncover features in the dynamical evolution of the state. Under suitable conditions (e.g. eigenstate thermalization) the system equilibriates to certain ensembles characterized by a temperature and/or chemical potentials conjugate to the conserved charges. When energy is the only conserved quantity one expects the ensemble to be a thermal one, characterized by a temperature; to which all observables finally equilibriate. It is believed that a thermalizing non-equilibrium state also \emph{scrambles}. Here the notion of scrambling is in the context of operator growth due to chaotic dynamics. This is suitably measured by the Lyapunov exponent ($\lambda$) associated with the out of time ordered correlator (OTOC). In particular, for a maximally chaotic system in a thermal state, $\lambda$ is proportional to the temperature. For an out of equilibrium state that is thermalizing, we therefore expect the dynamical $\lambda$ to be related to the effective temperature to which this state equilibriates to. This temperature of course will depend on the energy generated during the quench, which further is governed by the particulars of the out of equilibrium scenario. Here we find evidences in some simple cases that the $\lambda$ evaluated in the non-equilibrium state shows aspects commensurate with energy generation, which can indeed be associated with thermalization in certain cases.
The non-equilibrium scenario is generated by the set up of quantum quench; wherein some coupling in the Hamiltonian is changed as a function of time. Various kinds of universalities are known to be associated with investigations of quenches. These emerge most notably whenever adiabaticity gets broken. This is guaranteed if the change in coupling is either sudden or if it crosses a critical point. Typically in a theory with cut-off $\Lambda$, the initial state is characterized by a mass gap $m_0$. The change of coupling $g(t)$ can be characterized by its rate of change, $\Gamma $ and / or the change in the value of the coupling $\Delta g$.
For slow, smooth quenches in the regime, $\Gamma < m_0 < \Lambda$, the breakdown of adiabaticity is associated with the emergence of a {Kibble-Zurek} scale which imbues quenched correlation functions with universal scaling behaviours. There are also universal scalings expected in the fast regime, $\Lambda > \Gamma > m_0$ and in the sudden case $\Gamma \rightarrow 0$. Evidences for these scaling behaviours have been limited to holography, solvable lattice models, two dimensional conformal field theories (CFTs), free theories and certain large $N$ theories. In this work, while remaining within this tractable set of illuminating theories we compute the OTOC during a quantum quench. Next, we outline the remaining sections along with the specific results from them.
In our first example we study the case of two harmonic oscillators $x$ and $y$ coupled by $g x^2 y^2$ (see \S\ref{s:osc}). This is a classically chaotic system, with classical Lyapunov index given by $\lambda_{cl} \sim E^{\nu}$, where $E$ is the classical energy. Interestingly, the quantum analog also exhibits a positive Lyapunov exponent $\lambda \sim T^{\nu}$ when the OTOC is computed in a thermal state with temperature $T$. The agreement with the classical exponent is not surprizing as in the thermal quantum case, the typical energy is given by the temperature. We consider quenching at zero temperature the coupling $g$, suddenly and compute the generated energy. Through the energy we are able to ascribe an effective temperature in terms of $\Delta g, \, T \sim (\Delta g)^\alpha$. Next, by computing the OTOC, we extract $\lambda$ as a power of $\Delta g$ which is consistent with the assigned effective temperature. The computations are carried out numerically, and although bound to errors due to various truncations, clearly illustrate that the OTOC prognosticates an effective thermalization resulting from the quench. In this set-up this statement is, $\lambda \sim (\Delta g)^{\nu \alpha}$.
The second example in \S \ref{s:cft} re-affirms the above observation of relating $\lambda$ and $\Delta g$ in a clean analytic example of sudden quench in 2D CFT in the limit of large central charge. We follow the Cardy-Calabrese prescription of approximating the quenched initial state with a regularized boundary state of the CFT. The regularization involves a parameter $\tau_0$ which is directly related to the mass gap of the initial theory. We then implement the three point BCFT set-up, in the semi-classical regime, to extract $\lambda = \pi/(2 \tau_0)$. The parameter $\tau_0$ is consistently related to the initial mass gap as well as the effective temperature of the quenched state.
The final example in \S \ref{s:ising} deals with the Ising chain in presence of both a transverse as well as a longitudinal field. We quench the transverse field smoothly (with rate $\Gamma$) while operating in a non-integrable regime. Under both the fast as well as slow quench, the quenched energy is known to scale differently as a function of $\Gamma$. We employ tensor network techniques to extract $\lambda$ in this set-up and discover same scalings as for the energy generated during the quench.
All the three examples are indicative of the fact that in out of equilibrium scenarios where scrambling takes place, the Lyapunov exponent associated with suitable OTOCs can be identified with the energy generated. Furthermore, in scenarios where an effective post-quench temperature maybe estimated, the Lyapunov exponent is directly related with this temperature.
\section{Semiclassical chaos by differently ordered correlators}
Classical chaos is quantified by exponential sensitivity to initial conditions, which is given by the following Poisson bracket (P.B.) : $$\frac{\delta x(t) }{\delta x(0)} = e^{\lambda_{cl } t } = \{ x(t) , p(0) \}_{P.B}.$$ The generalization in quantum mechancis is via different time commutators \cite{Swingle:2018ekw}, $$C = \vev{ [ V(t) , W(0) ]^2 }, $$ where the square is taken in order to prevent phase cancellations. When the commutator is expanded we find out of time ordered correlators of the form, $F(t) = \vev{ V(t) W(0) V(t) W(0) }$, make up $C$. In particular for unitary hermitian operators $C = 2 ( 1 - {\rm Re\hskip0.1em} F(t ) )$. Therefore when $C$ grows exponentially $\sim e^{2\lambda t}$ the quantity $F(t) \sim \text{const.} + \text{const.}' e^{\lambda t} $. The exponent $\lambda$ has been shown to satisfy a universal bound set by the temperature when the expectation is evaluated in a thermal state. In our set-ups, instead of computing the OTOC in a thermal state we quench a zero temperature low lying quantum state of the initial Hamiltonian and extract the corresponding Lyapunov exponent.
\subsection{Sudden quench of the coupled oscillator}\label{s:osc}
For our first example we consider two coupled harmonic oscillators described by the Hamiltonian,
\begin{align}
H &= p_x^2 + p_y^2 + \frac{1}{4} ( x^2 + y^2 ) + g x^2 y^2 .
\end{align}
Note that we avoid the simpler linear coupling, $g\, x y$, as via a global unitary transformation ( in normal coordinates ) the linear coupling problem can be reduced to two decoupled oscillators. This makes the model integrable, and OTOC only shows oscillatory behaviour with time.
Classically this has a non-zero $\lambda_{cl}$ which scales with energy as : $E^{1/4}$. In \cite{Akutagawa:2020qbj} the OTOC was computed in thermal state (temperature $T$) and it was found that $\lambda \sim T^{c}$ with $c \sim 0.25 - 0.31$. This is expected from the classical scaling since temperature plays the role of Energy. We compute the OTOC at zero temperature but under a sudden change of the coupling from $g_0$ to $g$. The quantity of interest is
\begin{align}
C &= -\vev{n_0 | [x(t) , p(0)]^2 | n_0 },
\end{align}
where $\ket{n_0}$ is a low lying energy eigenstate of the initial Hamiltonian $H(g_0) = H_0$. The time evolved position operator $x(t)$ is calculated with the quenched Hamiltonian $H(g)$ : $x(t) = \exp ( i t H(g) ) x(0) \exp ( -i t H(g) ) $. We evaluate $C$ by inserting complete set of eigenstates of the new Hamiltonian $H(g)$ that we denote as $\ket{m}$.
\begin{align}
C&= -\sum_m \vev{n_0 | [x(t) , p(0)]\ket{m}\bra{m} [x(t) , p(0)] | n_0 } = \sum_m b_m(t) b_m^*(t),\label{Ct}
\end{align}
the quantity $b_m(t) =- i \vev{n_0 | [x(t) , p(0) ] |m }$ is Hermitian. Using another set of completeness insertions and the relation between matrix elements of $p$ and $x$ : $p_{km} = \frac{i}{2} E_{km} x_{km}$ we find :
\begin{align}
b_{m}(t) &= \frac{1}{2} \sum_{l,k} \vev{n_0|l } x_{lk} x_{km} \left( E_{km} e^{i E_{lk} t } - E_{lk} e^{i E_{km} t } \right). \label{bm}
\end{align}
Here, $x_{km} = \vev{k | x |m }$ and $E_{km} = E_k -E_m$. This is the main quantity that we compute numerically, which is plugged into eq\eqref{Ct} to get the OTOC. The wavefunctions $\vev{x,y | n } = \psi_n(x,y) $ are obtained by solving the Schr\"odinger's equation:
\begin{align}
- \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2 } \right) \psi_n(x,y) + \left( \frac{1}{4} ( x^2 + y^2 ) + g x^2 y^2 \right) \psi_n(x,y) &= E_n \psi_n(x,y),\label{Hpsi}
\end{align}
with Dirichlet (wavefunction vanishing) boundary conditions in a square box.
\subsubsection{Numerics}
The equation \eqref{Hpsi} is solved for various values of the coupling $g$ that is relevant to us. The \texttt{Mathematica}\textsuperscript{\textregistered} package \texttt{NDEigensystem} has been used to get the wavefunctions. The initial state $\ket{n_0}$ is taken to be a low lying eigenstate of $H_0$. The box size is kept at $20 \times 20$. The energy levels are suitably truncated in the completeness relations. Once the wavefunctions $\psi_{n_0}(x,y)$ and $\psi_m (x,y)$ corresponding to $H_0$ and $H(g)$ respectively are available, we numerically integrate using the \texttt{NIntegrate} command to find the overlaps and the matrix elements as needed in \eqref{bm}. We show the behaviour of $C$ as a function of time for different changes in amplitude of the coupling, $\Delta g = g -g_0$, in Fig. \ref{fig-c}. Also as can be seen in the figure, when the quench amplitude is very small then $C(t)$ does not have any exponential growth. All the plots have a regime where $C(t)$ grows exponentially with time, before beginning to oscillate in some complicated manner. Fortunately, the early time behaviour is sufficient to extract a Lyapunov exponent $\lambda$, since in any case due to truncation of high energy levels, late time numerics is unreliable. To extract the Lyapunov exponent we fit $C(t)$ for different values of $\Delta g$ with $b \exp ( 2\lambda t)$. Next we plot $\lambda$ as a function of $\Delta g$ and find that $\lambda \sim (\Delta g)^\theta$. In Fig. \ref{fig-lambda}, we present the case when the initial state is the $10^{th}$ eigenstate, for which the exponent turns out to be close to $0.25$. For other initial states as well, we find numerical fits for the exponent to be close to a quarter. In the thermal ensemble $\lambda \sim T^c$ with $c$ extracted from a similar numerical regime and fitted to similar values. Therefore, this is encouraging as it suggests, that $\Delta g$ seems to play the role of temperature.
\begin{figure}[t!]
\centering
\includegraphics[width=.7\linewidth]{ct-cho}
\caption{The time development of the commutator squared value shown for different quench amplitudes.}
\label{fig-c}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=.5\linewidth]{lg4}
\caption{We fit the extracted $\lambda$ values (black) to $a + b (\Delta g)^{\theta}$ (red curve). When we restrict $\theta \in [0.1,0.4]$ the best fit is obtained for $\theta =0.255$.}
\label{fig-lambda}
\end{figure}
It turns out that there is further evidence for this ``thermal" interpretation of $\Delta g$. We extracted an effective temperature for a given $\Delta g$ by equalizing energy computed in the thermal ensemble of $H_0$ and that in the quenched state, {\it i.e.} we solve for $T$ from the equation:
\begin{align}
\vev{H_0}_T &= \frac{\sum_{k_0} E_{k_0} e^{-E_{k_0}/ T }}{\sum_{j_0} e^{-E_{j_0}/ T } } = \sum_m E_m | \vev{n_0 | m } |^2.
\end{align}
The R.H.S contains the information about the quench and uses numerical results that we already used to compute $b_m(t)$. In Fig. \ref{fig-T} we plot the effective temperature as a function of $\Delta g$ and obtain a straight line, this therefore strengthens our thermal explanation for the scaling of the Lyapunov exponent.
\begin{figure}[t!]
\centering
\includegraphics[width=.5\linewidth]{t-vs-dg}
\caption{The effective temperature fits linearly with the quench amplitude.}
\label{fig-T}
\end{figure}
\subsection{Sudden quench in 2D CFT at large central charge}\label{s:cft}
In this section we study the OTOC for sudden critical quenches in one spatial dimension. The initial state $\ket{\psi_0}$ is the ground state of a gapped Hamiltonian, with gap $\sim 1/\tau_0$. At time zero, the gap is closed suddenly as the Hamiltonian changes to $H_{CFT}$. From the perspective of the conformal theory, there is a boundary at Euclidean zero time, which is naturally associated with a boundary state of the CFT. Since in the infrared this state should be a conformal boundary state : $\ket{B}$, a good approximation to $\ket{\psi_0}$ is provided by an irrelevant deformation to $\ket{B}$. The lowest universal irrelevant operator is a single power of the Hamiltonian itself. This is the proposed prescription of Cardy and Calabrese (CC) \cite{ Calabrese:2006rx, Calabrese:2016xau} :
\begin{align}\label{cc}
\ket{\psi_{0}} \propto e^{-\tau_{0}H}\ket{B}
\end{align}
Note, that the RG distance between $\ket{\psi_0}$ and $\ket{B}$, which is, $1/\text{gap}= \tau_0$ acts also as a regularizing parameter for the non-normalizable boundary state. Thereafter, post critical quench observables can be calculated as Euclidean strip (of width $2\tau_0$) correlators, which are suitable analytically continued to get the Lorentzian answers. The correlators of primary fields $\phi_{i}(x_{i},\tau_{i}) (i=1,2,\dots n)$ in the strip can be mapped to that on upper half plane(UHP) using the following conformal transformation
\begin{align}\label{map}
\omega \rightarrow z = ie^{\frac{\pi \omega}{2\tau_{0}}}, \; \omega=x+i\tau.
\end{align}
This mapping allows us to relate:
\begin{align}
\vev{\prod_{i=1}^{n}\phi_{i}(\omega_{i},\bar{\omega}_{i})}_{\text{strip}} = \prod_{i=1}^{n}\omega'_{i}(z_{i})^{-h_{i}}\bar{\omega}'_{i}(\bar{z_{i}})^{-\bar{h_{i}}}\vev{\prod_{i=1}^{n}\phi_{i}(z_{i},\bar{z}_{i})}_{\text{UHP}}
\end{align}
Here $(z,\bar{z})$ are UHP coordinates. Two or higher point functions are not fixed on the UHP, as using the conformal ward identities one can show that $n$-point BCFT correlators are equivalent to $2n$-point holomorphic CFT correlators on the entire plane. One point functions, however get fixed. An important one point function is the expectation value of the post-quench energy : $\vev{\psi_0 | H_{CFT} | \psi_0 }$. The contribution comes only from Schwarzian derivative associated with the map \eqref{map}.
\begin{align}
\langle B|e^{-\tau_{0}H_{CFT}}H_{CFT}e^{-\tau_{0}H_{CFT}}|B\rangle = \frac{\pi c}{96 \tau_{0}^{2}}
\end{align}
This result can also be obtained from the $T_{00}$ expectation value in a thermal ensemble with inverse temperature $\beta = 4\tau_0$, since $$\frac{{\rm tr}(H_{CFT} e^{-\beta H_{CFT}})}{{\rm tr}( e^{-\beta H_{CFT}})} = \frac{\pi c}{6\beta^{2}}.$$ Hence, the quench gap / energy gets natually associated to an effective temperature.
\subsubsection*{Extracting the Lyapunov exponent from BCFT}
In \cite{Das:2019tga}, the authors used a certain three point bulk-boundary thermal OTOC which shows maximal Lyapunov behaviour in large central charge limit of the BCFT. Here we consider three operators placed in the following way. $W$ of dimension $h_{w}$ is sitting at $(x,0)$ and two boundary operators $V$ of dimensions $h_{v}$ are sitting at $(0,t)$ in the Lorentzian $\omega$ coordinate. The OTOC can be obtained from the following object:
\begin{align}
C(t) = \frac{\langle V(t) W(0) V(t) \rangle}{\langle W \rangle \langle VV \rangle}
\end{align}
Operationally, to get the above out of time ordered object from the Euclidean correlator (with all operators $O_i$ at Euclidean time $\tau_i$ ), we analytically continue the $\tau_{i}$'s to their original Lorentzian values {\it i.e.} $\tau_{i} = t_{i}+i\epsilon_{i}$. Ultimately, we take all $\epsilon_{i} \rightarrow 0$ and the ordering in time comes from the ordering of the taking the limits of different $\epsilon_{i}$'s.
Using once again the map \eqref{map} the Euclidean correlator, $C_E$ is equivalent to a four point holomorphic function on the full plane, since only the bulk operator $W$ gets mirrored on the lower half plane. $C_E$ is just a function of the Euclidean cross-ratio $z$, and in the complex $z$ plane possesses branch-cuts originating from points when operators enter into each others' light cones. After analytic continuation, different time ordering dictates how or whether the branch cuts are being crossed. Here in our case, the four points on the plane, after the analytic continuation, are the following:
\begin{align}
z_{0}=ie^{b(x+i\epsilon_{0})}, \; \bar{z}_{0}=-ie^{b(x-i\epsilon_{0})}, \;
z_{1}=ie^{b(t+i\epsilon_{1})}, \; z_{2}=ie^{b(t+i\epsilon_{2})}, \; b= \frac{\pi}{2\tau_{0}}.
\end{align}
Here we take $\epsilon_{1} = \tau_{0}$ and $\epsilon_{2} = -\tau_{0}$ such that $z_{1}$ and $z_{2}$ are placed on the boundary of the UHP. Hence the cross ratio $z$ is given by $z = \frac{(z_{0}-\bar{z}_{0})(z_{1}-z_{2})}{(z_{0}-z_{1})(\bar{z}_{0}-z_{2})}$. For different time regimes, we get the following asymptotic behaviours for the cross-ratio:
\begin{align}\label{z limit}
z_{t\rightarrow 0} &= \frac{2i\epsilon^{*}_{12}}{e^{bx}-e^{-bx}}, \; z_{t\gg x} = -2i\epsilon_{12}^{*}e^{-b(t-x)}, \nonumber \\
z_{t=x} &= -\frac{2i\epsilon_{12}^{*}}{(1-e^{ib(\epsilon_{0}-\epsilon_{1})})(1+e^{-ib(\epsilon_{0}+\epsilon_{1})})} \approx \frac{\epsilon_{12}^{*}}{\epsilon_{10}^{*}}.
\end{align}
Here we have defined $\epsilon_{ij} \equiv i(e^{ib\epsilon_{i}}-e^{ib\epsilon_{j}})$ and $\epsilon_{ij}^{*}$ is the corresponding complex conjugate. From the above analysis, we see that the cross ratio goes to zero from opposite direction at $t\rightarrow 0$ and $t\rightarrow \infty$ limit and this is independent of the $\epsilon_{i}$ ordering. However at $t=x$, the cross ratio shows interesting behavior since it is a ratio of $\epsilon_{ij}$s. In particular for the OTOC, we need the following ordering: $\epsilon_{1}>\epsilon_{0}>\epsilon_{2}$. In this case, from (\ref{z limit}) we could clearly see that $z_{t=x}\approx 1+\frac{\epsilon_{02}^{*}}{\epsilon_{10}^{*}} >1$. Diagrammatically the situation is described in the figure (\ref{fig:con}).
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[->] (-3.5,0) --+(0:4);
\draw[->] (-2.5,-1.1) --+(90:4);
\draw (-1,-0.009)--+(90:0.06);
\node (1) at (-1,-0.2) {1};
\node (z) at (0.1,2.1) {z};
\draw (-0.08,1.95) --+(90:.3);
\draw (-0.08,1.95) --+(0:.3);
\draw[thick,darkblue,->] (-2.5,0) to[out=80,in=180,looseness=1] (-1.5,0.75) to[out=0,in=90,looseness=0.8] (-0.65,0) to[out=275,in=0,looseness=1] (-1.4,-0.75)
to[out=180,in=-60,looseness=1] (-2.45,-0.05);
\draw[thick,lightblue,branch cut] (-1,0) to (0:1.2);
\end{tikzpicture}
\end{center}
\caption{During the analytic continuation for OTOC the cut from 1 to infinity is crossed. }\label{fig:con}
\end{figure}
In the large central charge limit, we may assume that the dominant contribution to $C_E(z)$ comes from the Virasoro identitiy block, $\mathcal{F}(z)$ which in the monodromy regime of $h_i/c$ fixed with: $h_v \ll h_w$ has the universal form:
\begin{align}
\mathcal{F}(z) \approx \left( \frac{z}{1-(1-z)^{1-12\frac{h_{w}}{c}}}\right)^{2h_{v}}
\end{align}
To get to our desired OTOC, we need to wind around the branch cut at $z=1$ and then as Lorentzian time increases, end up at small $|z|$. Therefore we implement $(1-z) \rightarrow (1-z)e^{2\pi i }$ and then expand in small $z$, which leads to:
\begin{align}
\mathcal{F}(z) \approx \left( \frac{1}{1-\frac{24i\pi h_{w}}{cz}}\right)^{2h_{v}}
\end{align}
In terms of the Lorentzian time, $t$ :
\begin{align}
C(t) \approx \left(\frac{1}{1+\frac{12\pi h_{w}}{\epsilon_{12}^{*}}e^{b(t-t_{*}-x)}}\right)^{2h_{v}}.
\end{align}
In the above expression, the scrambling time $t_{*} = \frac{1}{b}\log(c)$ and the Lyapunov exponent associated with $C(t)$ is
\begin{align}\label{cft-lya}
\lambda = b = \frac{\pi}{2\tau_{0}}
\end{align}
Comparing with the maximal Lyapunov exponent for the large $c$ thermal OTOC ($2\pi T$), we once again get the effective inverse temperature to be $\beta = 4\tau_{0}$. Once again, we see that the effective temperature associated with the quantum quench determines the scrambling behaviour. This result has also been obtained recently in \cite{das2021critical}.
\subsection{Smooth quench in the quantum Ising model}\label{s:ising}
We now explore the identification of the Lyapunov index with the energy generated during quench in the canonical quantum Ising chain. Intense quenching investigations has been carried out in this many-body lattice model. In particular, results exist for behaviour of the energy generated for smooth quenches. There are two distinct scaling regimes for smooth quenches, \emph{viz.} the slow or the Kibble-Zurek (KZ) regime \cite{RevModPhys.83.863} and the fast regime \cite{Das:2016lla}. They are separated by the mass-gap scale of the initial Hamiltonian. While KZ can be understood using the diabatic approximation, the fast scaling can be derived by using conformal perturbation theory.
The quantum Ising model in one spatial dimensions is described by
\begin{align}
H &= - \sum_i J Z_i \otimes Z_{i+1} + g X_i + \epsilon Z_i.
\end{align}
We turn on both a transverse as well as an infinitesimal longitudinal field in order to be in the non-integrable regime. When the longitudinal field is absent, fermionization takes the model to a free massive ( $m \propto 1- |g/J|$) fermionic theory which is integrable. We quench linearly the coupling $g = g_0 + \Gamma t$ which plays the role of the transverse magnetic field. The Pauli $X$ is bilinear in terms of the Jordan-Wigner fermions, $\bar{\psi}\psi$ and has scaling dimension $\Delta = 1$, which is also the dimension of the energy operator. For relativistic theories in the KZ regime ( $\Gamma < m$ ), one-point functions, $\vev{O_\Delta}\sim t_{KZ}^{-\Delta}$. Here $t_{KZ}$ is determined from the adiabaticity breakdown condition. For the Ising model $t_{KZ} \sim \Gamma^{-1/2}$. Therefore the energy generated in the KZ regime is expected to show a $\sqrt{\Gamma}$ scaling.
The fast scaling regime is naturally defined by $\Lambda_{UV} > \Gamma > m$. Since the theory is insensitive to any scales other than the $UV$ cut-off, it is approximately conformal. Therefore $\vev{O_{\Delta}} \sim \Gamma^{2-2\Delta}$. Thus the energy generated shows logarithmic scaling with $\Gamma$. During smooth quenches of the transverse field Ising model, these scaling laws were showcased in the $\vev{\bar{\psi}\psi}$ operator expectation value \cite{Das:2017sgp}.
Here, we compute the OTOC in the ground state of the initial $H(g_0)$: \begin{align}
C(t) &= \vev{\psi_0 | Z_j(t) Z_j (0)Z_j(t) Z_j(0) |\psi_0 }.
\end{align}
Note, that we choose the $Z$ Pauli matrices since these are non-local in terms of the Jordan-Wigner fermions and have been shown (i) to exhibit ``fast'' thermalization and, even in the integrable regime, (ii) to mimic exponential Lyapunov behaviour in thermal state \cite{motrunich}. We find that the extracted Lyapunov exhibits the same smooth quench scalings as the energy generated, see Fig. \ref{fig:ising}. This once again strongly indicates that scrambling gets related to effective thermalization, and in fact that the energy generated via quench can be interpreted as the `heat' generated \cite{PhysRevB.81.012303, Chandran_2012}.
\begin{figure}[t!]
\centering
\includegraphics[width=.5\linewidth]{ising-otoc}
\caption{The Lyapunov exponent as a function of the quench rate $\Gamma$, shows $\sqrt{\Gamma}$ scaling (red is fit to $a_1 + a_2 \sqrt{\Gamma}$ ) in the KZ regime which smoothly goes over to $\log \Gamma$ scaling (blue is fit to $a_1 + a_2 \log \Gamma$) in the fast regime. The orange dashed line demarcates the two regimes. }
\label{fig:ising}
\end{figure}
\subsubsection{Numerical details}
The OTOC has been evaluated with the help of tensor network (TN) techniques, see \S Appendix \ref{app:tn} for further details. This allows us to go upto system size $L=50$. We have chosen periodic boundary conditions on the chain, and computed OTOC for Pauli $Z$ operators at the middle $=25^{th}$ site. The parameters used for the numerics are: $g_0 = 0.5, J=1, \epsilon= 0.0001$. Starting with the ground state in the ferromagnetic phase, we evolve the state with the time-dependent Hamiltonian, for different rates : $\dot g = \Gamma$. Numerically we implement this by discretizing the time-step in units of $\delta t = 0.01$. In the Schr\"{o}dinger picture the OTOC can be represented as
\begin{equation}
OTOC(t)=\langle\psi_{0}\vert Z_{25} \mathcal{U}(-t)Z_{25}\mathcal{U}(t)Z_{25}\mathcal{U}(-t)Z_{25}\mathcal{U}(t)\vert\psi_{0}\rangle
\end{equation}
where the evolution operator is of the form $\mathcal{U}(t)=e^{-i\mathcal{H}(g(t))\delta t}\cdots e^{-i\mathcal{H}(g_0)\delta t}$. As described in the appendix since we use time-dependent variational principle algorithm (TDVP), the tensor network also gets optimized during the time-evolution. See Fig. \ref{fig:tdvp} for a schematic of the contracted network used to extract the OTOC.
Since the OTOC is for Hermitian operators, we extract the squared commutator using: $C(t) = 2 \left( 1 - {\rm Re\hskip0.1em}( OTOC(t) ) \right)$. For the quenches we observe $C(t)$ to behave similar to that of the coupled oscillator quench Fig. \ref{fig-c}, there is a clear transient period where the commutator squared grows exponentially before starting to oscillate, see Fig. \ref{fig:comm}. Fortunately for us, the OTOC characterization is for very early times, and thus we do not have to deal with the growing entanglement at late times which acts as a bottleneck for the TN techniques owing to finiteness of the bond dimension. As shown in the figure, we extract $\lambda$ by fitting an exponential between $0.15 \leq t \leq 0.70$, for different rates. From this data we find the scaling shown in Fig. \ref{fig:ising}.
\begin{figure}[t!]\
\centering
\includegraphics[width=.7\linewidth]{Commutator}
\caption{The time development of the commutator squared value shown for different rates of quench.}
\label{fig:comm}
\end{figure}
In theories like the Ising model, operators have bounded norms. Therefore only transient chaotic behaviour is observable, since the bounds get saturated very soon. This motivates the consideration of density of OTOCs of extensive sums of local operators, which can show longer regimes of scrambling behaviour \cite{weakchaos}. We expect that the scaling behaviours observed in the Lyapunov exponent also to hold even in these modified OTOCs.
\section{Conclusions}
In this work we have studied scrambling during quantum quenches. All the chosen models start from a gapped low lying eigenstate of an initial Hamiltonian and then during the quench evolution either gets coupled to another system, or experiences sudden criticality, or evolves with a smoothly changing coupling. In all the situations, for suitably chosen operators, signs of scrambling in the form of exponential OTOCs are observed. The extracted Lyapunov exponents have the characteristics of an effective temperature since it can be identified with the energy generated during the quench. Furthermore in the case of smooth quenches the extracted Lyapunov exhibits both a Kibble-Zurek as well as a fast scaling as expected in the quench energy.
The coupled oscillator studied in this work can be shown to arise from the dimensional reduction of $SU(2)$ Yang-Mills Higgs theory in the unitary gauge \cite{Akutagawa:2020qbj}. In the thermal context the scaling of the Lyapunov index with the temperature is expected to hold for the full Yang-Mills as well as its susy generalizations. A very interesting generalization is the $SU(N)$ Yang-Mills in $9+1$ dimensions with large $N$. When dimensionally reduced to zero dimensions, it is described by D0-brane matrix quantum mechanics. At high temperatures the Lyapunov index of this matrix model also shows the scaling $\lambda \sim T^{1/4}$ \cite{Gur-Ari:2015rcq, Berkowitz:2016znt}. It is then expected that the OTOC during a quench in these generalizations will also exhibit scalings. Using the gauge/gravity correspondence this will translate to universal scalings in genuinely non-equilibrium quantum gravity processes, which may include black hole formations as well as black hole transitions. It is to be noted, that equilibriation after quenches have already been studied in various matrix models \cite{Mandal:2013id}.
As pointed out here as well as in \cite{Akutagawa:2020qbj}, the exponent of $1/4$ has its origin in the behaviour of the classical Lyapunov exponent. However the Lyapunov index extracted from the OTOC can be distinct from the classical Lyapunov exponent in certain physical systems as shown in \cite{lya-d-1, lya-d-2}. Hence, it will be quite interesting to carry out the quench and investigate the nature of effective thermalization as determined from scrambling in these models.
In the context of the gauge/gravity duality, universal scalings of quenched correlation functions in the field theory can be understood from bulk zero mode dynamics in the dual gravity theory \cite{Basu:2011ft}. On the other hand, OTOCs in holography can be computed from the bulk using a shockwave set-up \cite{Shenker:2013pqa}. It will thereby be interesting to imbue the end of world brane geometry (dual to the Cardy-Calabrese quenched state \cite{Hartman:2013qma}) with shockwaves and investigate the role of zero modes in the Lyapunov scalings.
A common underlying theme in our examples is the emergence of a scale through an effective ``thermalization". It is a natural question to ask what happens to the Lyapunovian behaviour when there are additional conserved charges? In this case one expects the final state to equilibrate to a generalized Gibbs ensemble, with generically non-zero chemical potentials turned on for different charges. For a CFT with an additional global $U(1)$ scrambling has been explored in the context of local quenches \cite{David:2017eno}. We expect that signs of such effective equilibration will get reflected in the scrambling characterizations associated with quenches.
When there are as many conserved charges as the degrees of freedom, a theory becomes integrable. Integrable theories are known not to show exponential OTOCs, however in presence of non-integrable perturbations can have a rich phase diagram of Lyapunovian dynamics, as was shown using the quantum tangent space formalism in \cite{Goldfriend:2019iwy}. Though our analysis of the Ising model falls in this class, it will be interesting to explore more generally within this framework the case of scrambling under quench.
In context of time-dependence and the quantum Ising model, OTOCs have also been explored in \cite{heyl} and very recently in \cite{souvik}. While the former establishes OTOCs as an order parameter during sudden quenches, \footnote{Assuming eigenstate thermalization this late time behaviour follows very naturally, see \cite{diptarka}.} the latter shows how the information about dynamic phase transitions are present in the OTOC corresponding to non-local operators. Both these studies are based on late-time properties of the OTOC, whereas our investigation is complementary since it focuses on the early time behaviour. It will also be interesting to explore scrambling during smooth quenches in models with richer critical structures and also in presence of \emph{disorder}, {\it e.g.} , \cite{Ben-Zion:2017tor}.
Recently TN + Prony method has been used in the non-integrable Ising model at finite temperature to compute unequal time commutators \cite{Banuls:2019qrq} with some success. It seems only natural to adapt these techniques to explore non-equilibrium scrambling in future.
It will also be interesting to explore OTOCs in quenches for analytically controllable interacting field theories. At finite temperatures both in the $O(N)$ non-linear sigma model\cite{Chowdhury:2017jzb} as well as the Gross-Neveu model \cite{Jian:2018ies} the Lyapunov exponent scales linearly with temperature at large $N$. Furthermore, the Schwinger-Keldysh formalism for studying quantum quenches have already been explored in these models \cite{Das:2012mt, Das:2020dfe}.
\subsubsection*{Note Added :} While this work was nearing completion, the recent preprint \cite{das2021critical} has explored OTOCs during inhomogenous quenches in the context of two dimensional CFTs, wherein they also independently obtained \eqref{cft-lya}.
\subsection*{Acknowledgements}
It is a pleasure to thank Titas Chanda, Amit Dutta, Bobby Ezhuthachan, Michal P. Heller, Arijit Kundu and Arnab Kundu for several useful discussions. DD, SD, \& BD would like to acknowledge the support provided by the Max Planck Partner Group grant MAXPLA/PHY/2018577. DD would also like to acknowledge the support provided by the MATRICS grant {{SERB/PHY/2020334}}. ASA would like to thank Titas Chanda for Tensor Network codes. ASA would like to acknowledge the support provided by National Science Centre (Poland) under project 2019/35/B/ST2/00034. BD would like to thank Supratim Das Bakshi for useful comments on Mathematica code. BD also acknowledges MHRD, India for Research Fellowship.
|
{
"timestamp": "2021-09-07T02:20:48",
"yymm": "2109",
"arxiv_id": "2109.02132",
"language": "en",
"url": "https://arxiv.org/abs/2109.02132"
}
|
"\\section{Introduction :}\\label{section-1}\nPerturbations in black hole space-times has been an in(...TRUNCATED)
| {"timestamp":"2021-12-14T02:38:03","yymm":"2109","arxiv_id":"2109.02185","language":"en","url":"http(...TRUNCATED)
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"\n\\section{Introduction}\nIt is difficult to reason about the correctness of quantum programs. Sou(...TRUNCATED)
| {"timestamp":"2021-09-07T02:23:32","yymm":"2109","arxiv_id":"2109.02198","language":"en","url":"http(...TRUNCATED)
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"\\section*{Key words}\nDegraded DNA, skeletal remains, DNA extraction, Chelex-100, inhibitors, opti(...TRUNCATED)
| {"timestamp":"2021-09-07T02:22:11","yymm":"2109","arxiv_id":"2109.02172","language":"en","url":"http(...TRUNCATED)
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"\\section*{Acknowledgments}\nThe research work disclosed in this publication is funded by the ENDEA(...TRUNCATED)
| {"timestamp":"2021-10-12T02:22:37","yymm":"2109","arxiv_id":"2109.02096","language":"en","url":"http(...TRUNCATED)
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"\\section{Experiments and Results}\n\\label{section:experiments}\n\n\\begin{table*}[t]\n\\centering(...TRUNCATED)
| {"timestamp":"2021-09-07T02:25:04","yymm":"2109","arxiv_id":"2109.02227","language":"en","url":"http(...TRUNCATED)
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"\\section{ Introduction}\n\\label{intro}\n\n\n\n\n\n\n{Pions, neutrinos and eventually the light a(...TRUNCATED)
| {"timestamp":"2022-01-06T02:15:18","yymm":"2109","arxiv_id":"2109.02203","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\nRecently, with the increasing adoption of AI in many fields, data privacy (...TRUNCATED)
| {"timestamp":"2021-09-07T02:18:05","yymm":"2109","arxiv_id":"2109.02053","language":"en","url":"http(...TRUNCATED)
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