text:
Aria Property Group gets Green Light for AUD$350m Mooloolaba Project
Main Photo: The new Aria project – imagined
Location: Mooloolaba, Queensland's Sunshine Coast, Australia
Name: TBA – five-star
No. of Keys: 139 plus 66 apartments
Developer: Residential developer Aria Property Group has received approval for a landmark 5-star hotel development on Queensland's Sunshine Coast.
The developer, led by Tim Forrester, has identified the region for its growth, and lack of five-star hotel accommodation limiting the range of tourists visiting the Sunshine Coast.
The region is also set for massive change through the construction of a new AUD$225 million runway at the local airport which will facilitate direct flights to and from Asia and bolster development of the new CBD in neighbouring Maroochydore.
Aria's latest approval, located at 1 The Esplanade in Mooloolaba —opposite The Surf Club, draws on its previous projects on Brisbane's Edward Street and the emerging Fish Lane precinct.
In a tentative lending market, the five-star development will be tailored to owner-occupiers with 66 residential apartments, as well as an international standard hotel comprising of 139 guest rooms.
The developer tapped architecture firm Elenberg Fraser to design the tower, which is located on the last available section of the Mooloolaba Esplanade.
The integrated resort will feature 1,764sq m of retail and restaurants, a day spa, conference facility and publicly accessible roof-top bar and restaurant.
▲ The landmark project, designed by nationally acclaimed architects Elenberg Fraser incorporates an international standard hotel and resort. Image: Elenberg Fraser
Aria development director Michael Hurley said the project's rooftop would reference the iconic buildings like the Marina Bay Sands in Singapore, W Hotel Barcelona and One Hotel South Beach Miami.
"The development approval is the first step — now detailed design can continue with the architectural, hotel and specialist consultant team."
"We hope to be in a position to release the residential units for sale in 2020 ahead of a 2021 construction start."
Speaking at The Urban Developer's annual residential property summit, Hurley said Aria's brand reputation and philosophy for place-making had allowed it to press forward with high-end developments in challenging market conditions.
Recently, the group established Aria Living, its management rights division. The scheme provides long-term owner-occupied buildings with concierge services as well as communal events.
It also offers complimentary personal training, yoga classes, resident bikes and hotel-type services.
Earlier this year, Aria lodged plans for a 13-storey, 127-apartment residential tower in Brisbane's Kangaroo Point, where Aria Living will be enacted.
The developer also recently completed Valencia in Kangaroo Point and The Drapery in Woolloongabba. The developer also has plans for a 145-apartment tower in West End, designed by Rothelowman.
Last week, confirmation of the east coast's second submarine cable, which will be laid or buried under the ocean floor, with the opening of a new $7.5 million landing station was opened adjacent to the Maroochydore City Centre.
The new $35 million data and telecommunications cable — which will allow Queensland businesses to connect with Asia and the United States without going via the Sydney hub — will provide Australia's fastest data and telecommunication speeds to Asia and the second fastest to the United States.
The Sunshine Coast Council is using the new international cable as a lure for businesses to move to the region's "digital trade hub", which will be up and running by next year.
THPT Comment: Ambitious sounding project for this successful tourist destination on Queensland's Sunshine Coast. Be interested to see who the hotel operator will be.
First Seen:
HR Group of Germany Buys 5 Hotels for €100m
Date: March 2018 Location: Frankfurt - Sulzbach, Bad Neuenahr, Hannover, Bottrop, Flensburg Name: Dorint (2),…
London Hotel Group Given Green Light for 221-Room Aparthotel
Date: April 2018 Location: Tooting High Street, South West London, UK Name: TBA planned to…
citizenM Unveils Plans for Latest London Property
Main Photo: The new citizenM Victoria - imagined Date: August 2019 Location: Vauxhall Bridge Road, Victoria,…
← WertInvest Acquires the Hotel Ananas, Vienna from Amisola Immobilien PwC UK Hotels Forecast 2019-2020 Launched Today →


text:
Landmark Books in All Fields
ItemID: #115414
Abraham Lincoln: The War Years
"THE MOST POPULAR LINCOLN BIOGRAPHY YET WRITTEN"
(LINCOLN, Abraham) SANDBURG, Carl. Abraham Lincoln: The War Years. New York: Harcourt, Brace, (1939). Four volumes. Octavo, modern full navy morocco, raised bands, elaborately gilt-decorated spines, marbled endpapers, all edges gilt. $2800.
First trade edition of Sandburg's profusely illustrated four-volume biography covering Lincoln's Civil War presidency, very handsomely bound.
Sandburg's monumental six-volume life of Lincoln, which concludes with The War Years, began with two-volume Prairie Years (issued separately in 1926). Sandburg grew up in the Knox County neighborhoods with which Lincoln was familiar. The poet once wrote: "For 30 years and more I have planned to make a certain portrait of Abraham Lincoln. It would sketch the country lawyer and prairie politician who was intimate with the settlers of the Knox County neighborhoods where I grew up as a boy, and where I heard the talk of men and women who had eaten with Lincoln, given him a bed overnight, heard his jokes and lingo, remembered his silences and his mobile face… Of all the sources from which men are to gather impressions of the personality of Lincoln, the foremost singly important one is the collection of his letters and papers, the speeches and writings of the man himself. This is the high document, always, to be lived with and brooded over, to be scrutinized and forgotten and gone back to and searched again with all gifts of imagination, intuition, experience, prayer, silence, sacrifice, and the laughter next door to tears" (The Prairie Years). "The most popular Lincoln biography yet written… Lincoln publications increased after the appearance of this work" (Monaghan 2877). Preceded by signed limited edition of 525 copies. Howes S82.
Fine condition.
Main Office & Gallery: 1608 Walnut Street, 19th Floor .::. Philadelphia, PA 19103 .::. 215-546-6466 .::. fax 215-546-9064
web: www.baumanrarebooks.com .::. email: [email protected]

redpajama_set_name: RedPajamaCommonCrawl


The new Netflix original series "Rilakkuma & Kaoru" is based on a character produced by the Japanese company San-X. Rilakkuma is a custard loving stuffed bear that lives with Kaoru, a hardworking girl just trying to make it through the year. Rilakkuma is featured on stationary, accessories, clothing and much more in Japan. He's kind of an icon! The series will be released on Netflix in English and there is also a Japanese version, its only right.
Lana expressed her excitement to voice Kaoru via Instagram.
This show truly looks so adorable! "Rilakkuma and Kaoru" the stop motion, animation premieres on Netflix, April 19th.


text:
\section{Introduction}\label{sec:intro}
In this work, we consider a standard composite convex minimization problem of the form
\begin{equation}\label{eq:opt}
    \min_{x\in\mathbb{R}^d} \left\{F(x)\equiv f(x)+g(x) \right\},
\end{equation}
where $f:\mathbb{R}^d\rightarrow \mathbb{R}$ is a $L$-smooth convex function (with $0<L<\infty$), and $g:\mathbb{R}^d\rightarrow \mathbb{R}\cup\{+\infty\}$ is a proper closed convex function. In addition, we allow either $f$ or $g$ to be possibly $\mu$-strongly convex. In this setting, we propose an inexact accelerated forward-backward method for solving~\eqref{eq:opt} relying on the access to the gradient of $f$, and to an iterative routine for approximating the proximal operator of $g$.

\paragraph{Relation to previous works}  The main algorithms presented in this note were originally presented in~\citet{barre2020principled}, along with their worst-case analyses. It was removed from~\citet{barre2020principled} for exposition and length purposes. The same methods were then re-analyzed and used by~\citet{alves2021variants} for accelerating higher-order tensor algorithms.

When the proximal operator of $g$ is readily available, the method presented below becomes a variant of standard accelerated (or fast) forward-backward (or proximal gradient) methods for convex minimization, see e.g.,~\citet{nesterov2013gradient,beck2009fast}, and the introductory survey by~\citet{d2021acceleration}.

Purely backward versions ($f=0$) emerged earlier from the works of~\citet{guler1992new} and~\citet{monteiro2013accelerated,salzo2012inexact}, whereas the first purely forward version ($g=0$) was developed by~\citet{Nesterov:1983wy}. The first inexact versions of accelerated forward-backward methods that we are aware of were presented in~\citet{schmidt2011convergence,villa2013accelerated,jiang2012inexact}, whereas versions with relative errors appeared more recently in \citet{millan2019inexact,Bello2020}. In contrast, our method allows handling different types of error (namely absolute and relative errors of different types), while allowing to exploit strong convexity of $f$ or $g$---see e.g.~\citet{Book:Nesterov,nesterov2013gradient,chambolle2016introduction}, for original analyses in the strongly convex case, when the proximal operator of $g$ is readily available. The same developments allow obtaining a strongly convex version of the accelerated hybrid proximal extragradient method (A-HPE), in the spirit of~\citet{monteiro2013accelerated}.

The notion of an ``approximate proximal point'' used in this note (see \Cref{s:prox}) was used in a few previous works, starting with the hybrid extragradient method \citep{solodov1999hybrid,solodov2000comparison}. It was also used for its accelerated version~\citep{monteiro2013accelerated} and in the context of another forward-backward splitting method~\citep{millan2019inexact}. In these works, the primal-dual requirement is presented under a different formulation involving the notion of $\varepsilon$-subdifferentials \citep[Section 3]{brondsted1965subdifferentiability} (or $\varepsilon$-enlargement in the context of monotone operators \citep{burachik1997enlargement,burachik1998varepsilon}). A survey on common notions of ``approximate proximal point'' used in the literature can be found in \citet[Section 2]{barre2020principled_new}.

\paragraph{Paper organization and contribution} This note is organized as follows. First, we give some base results and notations in \cref{s:backg}. We provide the inexact accelerated forward-backward in \cref{s:accel_FB}, along with a worst-case analysis, relying on a standard Lyapunov argument (for which we provide symbolic notebooks, helping the reader reproducing the algrebraic part of the proof without pain). Numerical experiments illustrating the practical behavior of the method are then provided in \cref{s:numerics}. After that, \cref{s:ahpe} shows how to slightly improve the proof for obtaining an accelerated hybrid proximal extragradient method~\citep{monteiro2013accelerated}, specifically dealing with the case $f=0$. We draw some conclusions in \cref{s:ccl}.

\paragraph{Notations} We refer to classical textbooks~\citep{Book:Rockafellar,hiriart2013convex} for standard elements of convex analysis. We use the notation $\Fccp(\mathbb{R}^d)$ to denote the set of closed convex proper function on $\mathbb{R}^d$. The corresponding subset of closed convex proper functions that are $\mu$-strongly convex and $L$-smooth (with $0\leq \mu < L\leq \infty$) is denoted $\FmuL(\mathbb{R}^d)$. That is, $h\in\FmuL(\mathbb{R}^d)$ if and only if
\begin{itemize}
    \item ($\mu$-strong convexity) $\forall x,y\in\mathbb{R}^d$, $s_h(x)\in\partial h(x)$, $s_h(y)\in\partial h(y)$, it holds $\|s_h(x)-s_h(y)\|\geq \mu \|x-y\|$,
    \item ($L$-smoothness) $\forall x,y\in\mathbb{R}^d$, $s_h(x)\in\partial h(x)$, $s_h(y)\in\partial h(y)$, it holds $\|s_h(x)-s_h(y)\|\leq L \|x-y\|$,
\end{itemize}
where $\partial h(x)$ denotes the subdifferential of $h$ at $x\in\mathbb{R}^d$. When $h\in\FmuL(\mathbb{R}^d)$ with $L<\infty$, we use $h'(x)$ to denote the unique element $h'(x)\in\partial h(x)$ (i.e. the gradient of $h$ at $x$).

\paragraph{Codes} For helping the reader reproducing the analytical results (via Mathematica notebooks) as well as numerical experiments, our code is available at
\begin{center}
    \url{https://github.com/mathbarre/StronglyConvexForwardBackward}.
\end{center}

\section{Background results}\label{s:backg}
\subsection{Smooth strongly convex functions}\label{s:ssc_functions}
We recall some standard inequalities satisfied by smooth convex and strongly convex functions, which we use in the sequel for exploiting strong convexity and smoothness, see e.g.~\citet{Book:Nesterov}.
\begin{proposition}[$\mu$-strong convexity]\label{prop:strcvx}
Let $g\in\Fmu(\mathbb{R}^d)$. For all $x,y\in\mathbb{R}^d$ and all $s_g(x)\in \partial g(x)$ it holds that
\[g(y)\geq g(x) + \langle s_g(x),y-x\rangle + \tfrac{\mu}{2}\normsq{x-y}. \]
\end{proposition}
\begin{proposition}[$L$-smoothness \& convexity]\label{prop:smooth}
Let $f\in\mathcal{F}_{0,L}(\mathbb{R}^d)$ with $L <+\infty$. For all $x,y\in\mathbb{R}^d$ it holds that
\[f(y)\geq f(x) + \langle f'(x), y-x\rangle + \tfrac{1}{2L}\normsq{f'(x)-f'(y)}. \]
\end{proposition}

In the sequel, the use of the inequalities provided by~\Cref{prop:strcvx} and~\Cref{prop:smooth} are motivated by their \emph{interpolation} (or extension) properties; that is, the analyses provided below were obtained following a principled approach to worst-case analyses of first-order methods, see e.g.,~\citet{taylor2017smooth} or~\citet{barre2020principled_new} specifically for the cases of methods relying on approximate proximal operations.

\subsection{Proximal operations}\label{s:prox}
The proximal operation is a base primitive that is widely used in modern optimization methods; it is a central building blocks in many optimization algorithms, see e.g.,~\citet{parikh2014proximal,ryu2016primer}. The proximal operator of a function $g\in \Fccp(\mathbb{R}^d)$ with step size $\lambda \geq 0$ is defined as
\begin{equation}\label{eq:prox}
    \mathrm{prox}_{\lambda g}(z) = \underset{x\in\mathbb{R}^d}{\argmin}\,\left\{ \lambda g(x) + \tfrac{1}{2}\normsq{x-z}\right\},
\end{equation}
with $z\in \mathbb{R}^d$. When $g\in \Fccp(\mathbb{R}^d)$, the proximal operation is well defined, and its solution is unique. A comprehensive list of cases where~\eqref{eq:prox} has an analytical solution is provided in~\citet{chierchia2020proximity}. In other cases, the proximal operator has to be approximated. For doing that, one can define the following primal and dual problems associated to the proximal operation
\begin{align}
     \underset{x\in\mathbb{R}^d}{\min}&\,\{\Phi_\mathrm{p}(z;\,x)\equiv   \lambda g(x) + \tfrac{1}{2}\normsq{x-z}\}, \label{eq:P}\tag{P}\\
     \underset{v\in\mathbb{R}^d}{\max}&\,\{\Phi_\mathrm{d}(z;\,v)\equiv  -\lambda g^*(v) - \tfrac{1}{2}\normsq{z - \lambda v} + \tfrac{1}{2}\normsq{z}\},\label{eq:D} \tag{D}
\end{align}
where $g^*\in\Fccp(\mathbb{R}^d)$ is the Fenchel conjugate of $g$. Let us further note that $\mathrm{prox}_{\lambda g}(z)$ is the unique solution to~\eqref{eq:P}, and that $\mathrm{prox}_{g^*/\lambda}(z/\lambda)$ is the unique solution of \eqref{eq:D}. In this context, the primal and dual solutions are linked by the well-known Moreau's identity $\mathrm{prox}_{\lambda g}(z) +\lambda\,\mathrm{prox}_{{g^*}/{\lambda}}\left(z/{\lambda}\right)= z$.

Under relatively weak conditions (such as $\ri(\mathrm{dom} g) \neq \emptyset$, see e.g.,~\citet[Section 3.5]{chambolle2016introduction}), strong duality holds between~\eqref{eq:P} and~\eqref{eq:D} and hence
\[\underset{x\in\mathbb{R}^d}{\min}\,\Phi_\mathrm{p}(z;\,x) = \underset{v\in\mathbb{R}^d}{\max}\,\Phi_\mathrm{d}(z;\,v). \]
Motivated by those elements, we use the quantity
\begin{equation}\label{eq:PD}\tag{PD}
    \PDg_{\lambda g}(x,v;\,z) = \Phi_\mathrm{p}(z;\,x) - \Phi_\mathrm{d}(z;\,v),
\end{equation}
for quantifying how well $(x,v)$ approximates the pair $(\mathrm{prox}_{\lambda g}(z),\mathrm{prox}_{g^*/\lambda}(z/\lambda))$, in the sequel.

\subsection{A notion of approximate proximal point
In this section, we define the notion of approximate proximal point of $g\in\Fccp$ used throughout the paper (see \Cref{sec:intro}~\S``{Relation to previous works}'' for historical references for the case $\mu=0$). This notion features two parameters: a tolerance and a lower bound on the strong convexity parameter of $g$ (possibly $0$). The estimate of the strong convexity is used for relating proximal points of $g(\cdot)$ in terms of that of $g_\mu(\cdot)=g(\cdot)-\tfrac{\mu}{2}\|\cdot\|^2\in\Fccp(\mathbb{R}^d)$, and the tolerance is used for quantifying the quality of an approximate solution to the proximal problem on $g_\mu(\cdot)$, which simplifies the analyses below. More precisely, for $g\in\Fmu(\mathbb{R}^d)$, it is relatively straightforward to verify that
\[\mathrm{prox}_{\lambda g}(z) = \mathrm{prox}_{\tfrac{\lambda}{1+\lambda\mu}g_\mu}\left( \tfrac{z}{1+\lambda\mu}\right),\]
with $g_\mu(x) = g(x) - \tfrac{\mu}{2}\normsq{x}$. This observation motivates the introduction of the following inexactness criterion.
\begin{definition}\label{def:criterion} Let $\mu> 0$, $g\in \Fmu(\mathbb{R}^d)$, and let $\lambda>0$ be a step size and $\varepsilon\geq0$ be a tolerance. For a triplet $(x,v,y)\in \mathbb{R}^d\times \mathbb{R}^d\times\mathbb{R}^d$ we use the notation
\[(x,v) \approx_{\varepsilon,\mu} \left(\mathrm{prox}_{\lambda g}(y),\mathrm{prox}_{\tfrac{g*}{\lambda}}(\tfrac{y}{\lambda}) \right), \]
for denoting that
\[\PDg_{\tfrac{\lambda}{1+\mu\lambda} g_\mu }\left(x,v-\mu x;\tfrac{y}{1+\mu\lambda}\right) \leq \varepsilon,\]
with $g_\mu(x) = g(x)-\tfrac{\mu}{2}\|x\|^2$ and $\PDg$ the primal-dual gap of the proximal problem defined in \eqref{eq:PD}.
\end{definition}
In the following technical lemma, we provide an explicit expression for quantifying the quality of a triplet $(x,v,y)\in \mathbb{R}^d\times \mathbb{R}^d\times\mathbb{R}^d$ in light of \Cref{def:criterion}.
\begin{lemma}
Let $\mu\geq 0$, $g\in \Fmu(\mathbb{R}^d)$, and let $\lambda>0$ be a step size and $(x,v,z)\in\mathbb{R}^d\times\mathbb{R}^d\times\mathbb{R}^d$. The following equality holds
\begin{equation}\label{eq:pdgap-2}
\begin{aligned}
    \PDg_{\tfrac{\lambda}{1+\mu\lambda} g_\mu}(x,v-\mu x;\tfrac{z}{1+\mu\lambda})
    =& \;\tfrac{1}{2(1+\lambda\mu)^2}\normsq{x - z +\lambda v}\\
    &+ \tfrac{\lambda}{1+\lambda\mu}\left(g(x)-g(w)+\tfrac{\mu}{2}\normsq{x-w} - \langle x-w,v \rangle \right),
\end{aligned}
\end{equation}
with $g_\mu(\cdot) = g(\cdot)-\tfrac{\mu}{2}\|\cdot\|^2$ and $w \in \mathbb{R}^d$ satisfying $v-\mu x + \mu w \in \partial g (w)$ (i.e., $w \in \partial g_\mu^*(v-\mu x)$).
\end{lemma}
\begin{proof}
    \begin{equation}\label{eq:pdgap-1}
\begin{aligned}
    \PDg_{\tfrac{\lambda}{1+\mu\lambda} g_\mu}(x,v-\mu x;\tfrac{z}{1+\mu\lambda}) =& \;\tfrac{1}{2}\normsq{x-\tfrac{z}{1+\lambda \mu}+\tfrac{\lambda}{1+\lambda\mu}(v-\mu x)}\\
    &+ \tfrac{\lambda}{1+\lambda\mu}\left(g_\mu(x) + g_\mu^*(v-\mu x) - \langle x,v-\mu x \rangle \right)\\
    =& \;\tfrac{1}{2(1+\lambda\mu)^2}\normsq{x - z +\lambda v}\\
    &+ \tfrac{\lambda}{1+\lambda\mu}\left(g(x)+\tfrac{\mu}{2}\normsq{x} + g_\mu^*(v-\mu x) - \langle x,v \rangle \right).
\end{aligned}
\end{equation}
In particular
\begin{equation*}
    g_\mu^*(v-\mu x) = \max_y\, \langle y,v-\mu x \rangle - g(y)+\tfrac{\mu}{2}\normsq{y},
\end{equation*}
and by choosing $w \in \mathbb{R}^d$ such that $v-\mu x+\mu w \in \partial g(w)$ we get
\begin{equation}
    g_\mu^*(v-\mu x) = \langle w,v-\mu x \rangle - g(w)+\tfrac{\mu}{2}\normsq{w}.
\end{equation}
Finally, using the expression of $g_\mu^*(v-\mu x)$ in \eqref{eq:pdgap-1} leads to the desired results.
\end{proof}

In the next section, we present an inexact accelerated forward-backward method where inexactness in proximal computations are measured using the primal-dual criterion from \cref{def:criterion}.

\section{An inexact accelerated forward-backward method}\label{s:accel_FB}
In this section, we provide the main contribution of this work, namely \Cref{eq:algo2}. This method aims at solving~\eqref{eq:opt} when the gradient of $f$ is readily available and the proximal operator of $g$ can be efficiently approximated within a target precision (e.g., by an iterative method). It further allows to exploit $g$ to be $\mu$-strongly convex. In the case where $f$ is strongly convex, one can shift this strong convexity to $g$ instead (by removing the corresponding quadratic of $f$ and adding it to $g$). Of course, any under-approximation of $\mu$ can be used within the method.

The worst-case analysis is based on a simple Lyapunov (or potential) argument, following the now standard template for accelerated schemes as in~\citet{Nesterov:1983wy}, for which surveys are provided in e.g.,~\citet{bansal2019potential,wilson2021lyapunov}, and~\citet[Chapter 4]{d2021acceleration}. As a byproduct of the analysis, the method does not require an accurate estimate of the smoothness constant $L$, whose estimation is improved on the fly using  standard backtracking tricks, similar in spirit with~\citet{Nesterov:1983wy,beck2009fast}.

The algorithm below builds on approximations of the forward-backward operator (with step sizes $\lambda_k$) of problem~\eqref{eq:opt}. More precisely, it relies on primal-dual pairs $(x_{k+1},\,g_{k+1})$ approximating the forward-backward operator evaluated at some iterates $y_k$, and satisfying
\[ (x_{k+1},\;g_{k+1})\approx_{\varepsilon_k,\mu}\left(\mathrm{prox}_{\lambda_{k} h}(y_{k}-\lambda_k f'(y_k)),\mathrm{prox}_{\tfrac{h^*}{\lambda_{k}}}(\tfrac{y_{k}-\lambda_k f'(y_k)}{\lambda_k})\right), \]
where $\varepsilon_k$ encodes some approximation level. In this work, this error term is parameterized by three sequences of nonnegative scalars $\{\sigma_k\}_k$, $\{\zeta_k\}_k$, $\{\xi_k\}_k$ that can be chosen by the user for possibly mixing both \emph{relative} (or multiplicative) and \emph{absolute} (or additive) error terms
\[ \varepsilon_k =\tfrac{\sigma_k^2}{2(1+\lambda_k \mu)^2}\|x_{k+1}-y_k\|^2+\tfrac{\zeta_k^2\lambda_k^2}{2(1+\lambda_k \mu)^2}\|g_{k+1}+f'(y_k)\|^2+\tfrac{\lambda_k\xi_k}{2(1+\lambda_k \mu)^2},\]
where $\{\xi_k\}_k$ parameterizes the absolute error term, and where $\{\sigma_k\}_k$ and $\{\zeta_k\}_k$ parametrize two types of relative errors. Of course, convergence properties of the algorithm depend on the choice of those sequences of parameters, as provided in \cref{cor:speed_strcvx} and \cref{cor:speed_cvx} below. Examples of simple rules for $\{\sigma_k\}_k$, $\{\zeta_k\}_k$, $\{\xi_k\}_k$ are provided in \cref{s:numerics} (typically, $\{\sigma_k\}_k$, $\{\zeta_k\}_k$ can be chosen constant, whereas $\{\xi_k\}_k$ should be either identically $0$ or decreasing fast enough).

Before going into the algorithm itself, let us mention that the backtracking line-search strategy \eqref{eq:backtracking} for estimating the smoothness constant builds on the condition
\begin{equation}\label{eq:smoothness}
    f(y_{k})  \geq f(x_{k+1}) + \langle f'(x_{k+1}),y_{k} - x_{k+1}\rangle + \tfrac{\lambda_k}{2(1-\sigma_k^2)}\| f'(y_{k}) -  f'(x_{k+1})\|^2, \tag{Smooth}
\end{equation}
where $x_k$'s and $y_k$'s are some iterates. In particular, picking $\lambda_k \in (0,\tfrac{1-\sigma_k^2}{L}]$ (hence depending on the true smoothness constant $L$) guarantees~\eqref{eq:smoothness} to be satisfied without backtracking, as, when $f \in \mathcal{F}_{0,L}(\mathbb{R}^d)$, \cref{prop:smooth} holds.

\begin{remark}[Related methods]
When the objective function is not strongly convex (i.e. $\mu=0$), the update rules of \Cref{eq:algo2} are very similar to those of the accelerated inexact forward-backward methods from \citet[Algorithm 3]{millan2019inexact} (when $\zeta_k=0$ and $\xi_k=0$) or \citet[Algorithm 2]{Bello2020} (when $\sigma_k=0$ and $\xi_k=0$). Compared to those works in this setup, \Cref{eq:algo2} allows using both relative and absolute errors while having a backtracking strategy. Note also the similarities with some inexact FISTA \citep{villa2013accelerated,schmidt2011convergence}, although these methods do not re-use explicitly the dual direction $g_{k+1}$ and focus only on absolute error terms (i.e., $\sigma_k=\zeta_k=0$). Finally, when the computation of the proximal operator is exact, we recover one of the many variants of an accelerated forward-backward method; see for example~\citet{nesterov2013gradient,tseng2008accelerated,beck2009fast}; we refer to~\citet[Chapter 4]{d2021acceleration} and the references therein for further discussions on this topic.\end{remark}

\newpage
\subsection{Algorithm}\label{subsec:FB}
\begin{oframed}
	\textbf{An inexact accelerated forward-backward method } (\Cref{eq:algo2}) \label[algorithm]{eq:algo2}
	\begin{itemize}
		\item[] {\bf{}Input:}
		\begin{itemize}
		    \item Objective function: $f\in\mathcal{F}_{0,L}(\mathbb{R}^d)$, $g\in\mathcal{F}_{\mu,\infty}(\mathbb{R}^d)$, and $\mu\geq 0$.
		    \item Initial point: $x_0\in\mathbb{R}^d$.
		    \item Initial step size: $\lambda_0 >0$.
		    \item Tolerance parameters: sequences $\{\sigma_k\}_k,\,\{\zeta_k\}_k$ with $\sigma_k,\,\zeta_k\in[0,1)$, and $\{\xi_k\}_k$ with $\xi_k\geq 0$.
		    \item Backtracking parameters $0<\alpha <1$ and $\beta \geq 1$.
		\end{itemize}
		\item[] {\bf{}Initialization:} $z_0=x_0$, $A_0 = 0$.
		\item[] {\bf{}Run:} For $k=0,1,\hdots$:
		\begin{align}
		    \eta_k &= (1-\zeta_k^2)\lambda_k\label{eq:init}\\
			A_{k+1} &= A_k + \tfrac{\eta_k + 2 A_{k}\mu\eta_k +\sqrt{ \eta_k^2 + 4\eta_k A_{k}(1+\eta_k\mu)(1+A_{k}\mu)}}{2} \nonumber\\
			y_{k} &= x_{k}+\tfrac{(A_{k+1}-A_{k})(A_{k}\mu+1)}{A_{k+1}+A_k(2A_{k+1}-A_k)\mu }(z_{k}-x_{k}) \nonumber\\
			\varepsilon_k &=\tfrac{\sigma_k^2}{2(1+\lambda_k \mu)^2}\|x_{k+1}-y_k\|^2+\tfrac{\zeta_k^2\lambda_k^2}{2(1+\lambda_k \mu)^2}\|g_{k+1}+f'(y_k)\|^2+\tfrac{\lambda_k\xi_k}{2(1+\lambda_k \mu)^2} \nonumber\\
			(x_{k+1},\;g_{k+1})&\approx_{\varepsilon_k,\mu}\left(\mathrm{prox}_{\lambda_{k} h}(y_{k}-\lambda_k f'(y_k)),\mathrm{prox}_{\tfrac{h^*}{\lambda_{k}}}(\tfrac{y_{k}-\lambda_k f'(y_k)}{\lambda_k})\right)\nonumber\\
		 \mathbf{\big[}\text{If } &\eqref{eq:smoothness} \text{ is not satisfied, set }
		 \lambda_k \leftarrow \alpha \lambda_k \tag{Btr}\label{eq:backtracking} \text{ and go back to step~\eqref{eq:init}\big]}\nonumber\\%\\

			z_{k+1} &= z_{k}+\tfrac{A_{k+1}-A_{k}}{1+\mu A_{k+1}}\left(\mu(x_{k+1}-z_{k}) - (g_{k+1} + f'(y_{k}))\right)\nonumber\\
			\lambda_{k+1} &= \beta \lambda_{k} \nonumber
			\end{align}
			\item[] {\bf{}Output:} $x_{k+1}$
	\end{itemize}
\end{oframed}

The following theorem contains the main (Lyapunov-based) ingredient of the worst-case analysis.
\begin{theorem}\label{thm:deacres-potential}
Let $f\in\mathcal{F}_{0,L}$, $g \in \Fmu$, $F\equiv f+g$, $k\geq 0$, parameters $\sigma_k,\zeta_k \in [0,1)$, $\xi_k \geq 0$ and some $\lambda_k>0$ such that \eqref{eq:smoothness} is satisfied. For any $x_k,z_k\in\mathbb{R}^d$, and $A_k\geq0$,
it holds that
\begin{equation}\label{eq:deacres-potential}
\begin{aligned}
A_{k+1}&(F(x_{k+1})-F(x_\star))+\tfrac{1+\mu A_{k+1}}{2}\normsq{z_{k+1}-x_\star}\\&\leq A_k (F(x_k)-F(x_\star))+\tfrac{1+\mu A_k}{2}\normsq{z_k-x_\star}+ \tfrac{A_{k+1}}{2}\xi_k,
\end{aligned}
\end{equation}
with $x_\star\in\argmin_x F(x)$, and where $z_{k+1}$ and $x_{k+1}$ are constructed by one iteration of \Cref{eq:algo2}.
\end{theorem}

The proof of this Theorem is deferred to \cref{subsec:proof-FB}. The following (classical) corollary establishes that the growth rate of the sequence $\{A_k\}_k$ drives the convergence rate of the worst-case guarantee.  Those factors $A_{k+1}$, controlling the convergence rate, were greedily chosen (as large as possible) while enforcing \eqref{eq:deacres-potential} to hold.

\begin{corollary}\label{cor:FB}
Let $f\in\mathcal{F}_{0,L}$, $g \in \Fmu$ and $
F\equiv f+g$. Let $x_0\in\mathbb{R}^d$, $\lambda_0$ be a positive initial step size, $\alpha\in(0,1)$ and $\beta\geq 1$ be some backtracking parameters, sequences (relative error parameters) $\{\sigma_k\}_k$, $\{\zeta_k\}_k$, satisfying $\sigma_k,\zeta_k\in[0,1)$ and a sequence (absolute error parameters) $\{\xi_k\}_k$ with $\xi_k\geq 0$. Let $x_N\in\mathbb{R}^d$ be the output after $N\in\mathbb{N}^*$ iterations of \Cref{eq:algo2} on $F$ initiated at $x_0\in\mathbb{R}^d$, it holds that
\[ F(x_N) - F(x_\star) \leq \tfrac{1}{2A_N}\normsq{x_0-x_\star} + \sum_{i=0}^{N-1}\tfrac{A_{i+1}}{2A_N}\xi_i,\]
where $x_\star \in \argmin_x F(x)$.
\end{corollary}
\begin{proof}
We denote by $\Phi_k$ the quantity (a.k.a., the Lyapunov/potential function)
\[\Phi_k = A_k(F(x_k)-F(x_\star)) + \tfrac{1+\mu A_k}{2}\normsq{z_k-x_\star}, \]
for $k \geq 0$. \cref{thm:deacres-potential} allows nesting the $\Phi_k$'s together as
\[\Phi_N \leq \Phi_{N-1} + \tfrac{A_{N}}{2}\xi_{N-1} \leq \hdots \leq \Phi_1 + \sum_{i=1}^{N-1}\tfrac{A_{i+1}}{2}\xi_i\leq \Phi_0 +\sum_{i=0}^{N-1}\tfrac{A_{i+1}}{2}\xi_i.\]
We reach the target conclusion using $A_N(F(x_N)-F(x_\star)) \leq \Phi_N$, together with $z_0=x_0$ and $A_0 =0$.
\end{proof}

Let us note that when $\mu=0$, we recover a composite version of the A-HPE method~\citep{monteiro2013accelerated}. In that case, we can bound $A_k\geq\tfrac14\left(\sum_{i=0}^{k-1}\sqrt{\eta_k}\right)^2 \geq \tfrac{\eta_\mathrm{min}}{4}k^2$, assuming the existence of some $\eta_\mathrm{min}\leq \eta_k$ for all $k\geq 0$. Such a lower bound on $\eta_k$ exists as soon as the parameters $\{\sigma_k\}_k$, $\{\zeta_k\}_k$ are well chosen (see for example \cref{cor:speed_strcvx} and \cref{cor:speed_cvx} below), and due to the $L$-smoothness of the function. Similarly, when $\mu>0$, $A_k$'s are growing exponentially as
\begin{align*}
    A_{k+1} &= A_{k} + \tfrac{\eta_k+2A_k \eta_k \mu + \sqrt{4 \eta_k A_k (A_k \mu +1) (\eta_k \mu +1)+\eta_k^2}}{2}\\
    & \geq A_{k}(1 + \eta_k\mu) +A_k \sqrt{\eta_k\mu(1+\eta_k\mu)}\\
    & = A_{k}/\left(1-\sqrt{\tfrac{\eta_k  \mu}{1+\eta_k \mu}}\right),
\end{align*}
with $A_1 > 0$, reaching $1/A_k \leq \eta_{\mathrm{min}}\left(1-\sqrt{\tfrac{\eta_{\mathrm{min}}\mu}{1+\eta_{\mathrm{min}}\mu}}\right)^{k-1}$ assuming again the existence of some $\eta_\mathrm{min}\leq \eta_k$ for all $k\geq 0$. The following corollaries provide more precise convergence bounds for \Cref{eq:algo2}, by quantifying the growth rate of the $A_k$'s, for some particular choices of parameters $\{\sigma_k\}_k$~(constant), $\{\zeta_k\}_k$~(constant), and $\{\xi_k\}_k$ (parameterized function of $k$), linking the behavior of the decrease rate of the absolute errors $\xi_k$ with the convergence bound.
\begin{corollary}\label{cor:speed_strcvx}
Let $f\in\mathcal{F}_{0,L}$, $g \in \Fmu$ and $F\equiv f+g$. Let $x_0\in\mathbb{R}^d$, $\lambda_0$ be an initial positive step size, $\alpha\in(0,1)$ and $\beta\geq 1$ be some backtracking parameters, sequences (relative error parameters) $\sigma_k = \sigma$, $\zeta_k =\zeta$ with $\sigma,\zeta\in[0,1)$ and a sequence (absolute error parameters) $\xi_k=C \rho^k$ with $C,\rho>0$. Let $x_N\in\mathbb{R}^d$ be the output after $N\in\mathbb{N}^*$ iterations of \Cref{eq:algo2} on $F$ initiated at $x_0\in\mathbb{R}^d$, it holds that
\[F(x_N) - F(x_\star)\leq \tfrac{1}{2\eta}\left(1-\sqrt{\tfrac{\eta\mu}{1+\eta\mu}}\right)^{N-1}\|x_0-x_\star\|^2+\left\{\begin{array}{ll}
   \medmath{\frac{C}{ 2(1-\sqrt{\tfrac{\eta\mu}{1+\eta\mu}}-\rho)}\left(1-\sqrt{\tfrac{\eta\mu}{1+\eta\mu}} \right)^{N}}\hspace{-.1cm}& \textrm{if } \medmath{\rho  < 1-\sqrt{\tfrac{\eta\mu}{1+\eta\mu}}}, \\
     \medmath{\frac12 C N \left(1-\sqrt{\tfrac{\eta\mu}{1+\eta\mu}} \right)^{N-1}} & \textrm{if } \medmath{\rho  = 1-\sqrt{\tfrac{\eta\mu}{1+\eta\mu}}}, \\
     \medmath{\frac{C}{2(\rho -  1+\sqrt{\tfrac{\eta\mu}{1+\eta\mu}})}\rho^N} &\textrm{if } \medmath{\rho  > 1-\sqrt{\tfrac{\eta\mu}{1+\eta\mu}}},\end{array} \right.\]
     for some $\eta = \underset{i=0,\hdots,N-1}{\min}\,\eta_i\geq \eta_{\mathrm{min}} = (1-\zeta^2)\min\left(\lambda_0, \tfrac{\alpha(1-\sigma^2)}{L}\right)$.
\end{corollary}
\begin{proof} Starting from the conclusion of \cref{cor:FB}, we obtain the desired result using classical properties of geometric sums along with $A_k \leq \left(1-\sqrt{\tfrac{\eta\mu}{1+\eta\mu}}\right)^{N-k}A_N$ where $\eta = \underset{i=0,\hdots,N-1}{\min}\,\eta_i\geq\eta_{\mathrm{min}}$.
\end{proof}
When $\mu = 0$, the proof is still valid, and $\tfrac{1}{A_N}=O(N^{-2})$. In particular, we recover the same rates as those of~\citet[Theorem 4.4]{villa2013accelerated} (who used the particular choice $g_{k+1}=\tfrac{y_k-\lambda_k f'(y_k)-x_{k+1}}{\lambda_k}$).
\begin{corollary}\label{cor:speed_cvx}
Let $f\in\mathcal{F}_{0,L}$, $g \in \Fmu$ and $F\equiv f+g$. Let $x_0\in\mathbb{R}^d$, $\lambda_0$ be an initial positive step size and sequences (relative error parameters) $\sigma_k = \sigma$, $\zeta_k =\zeta$ with $\sigma,\zeta\in[0,1)$. Let $x_N\in\mathbb{R}^d$ denote the output after $N\in\mathbb{N}^*$ iterations of \Cref{eq:algo2} on $F$ initiated at $x_0\in\mathbb{R}^d$.
\begin{itemize}
    \item We further let $\alpha\in(0,1)$ and $\beta = 1$ be the backtracking parameters, and a sequence (absolute error parameters) $\xi_k=C (k+1)^{-q}$ with $C,q\geq 0$. It holds that
    \[F(x_N) - F(x_\star)\leq \tfrac{2}{\eta_{\mathrm{min}} N^2}\|x_0-x_\star\|^2+ \left\{\begin{array}{ll}
     2C \tfrac{\eta_{\mathrm{max}}}{\eta_{\mathrm{min}}}\tfrac{\left(\sum_{k=0}^\infty (k+1)^{2-q}\right)}{N^2} & \textrm{if } q > 3,\\
     2C\tfrac{\eta_{\mathrm{max}}}{\eta_{\mathrm{min}}}\tfrac{(1+\ln(N))}{N^2} & \textrm{if } q = 3, \\
    2C\tfrac{\eta_{\mathrm{max}}}{\eta_{\mathrm{min}}}\left(\tfrac{1}{N^2} + \tfrac{1}{(3-q)N^{q-1}} \right) & \textrm{if } 1 < q < 3,\end{array} \right. \]
    with $\eta_{\mathrm{min}} = (1-\zeta^2)\min(\lambda_0,\tfrac{\alpha(1-\sigma^2)}{L})$ and $\eta_{\mathrm{max}} = (1-\zeta^2)\max(\lambda_0,\tfrac{(1-\sigma^2)}{L})$.
    \item We further let $\alpha\in(0,1)$ and $\beta \geq 1$ be the backtracking parameters, and a sequence $\xi_k=0$ (no absolute error). It holds that
    \[F(x_N) - F(x_\star)\leq \tfrac{2}{\eta_{\mathrm{min}} N^2}\|x_0-x_\star\|^2,\]
    where $\eta_{\mathrm{min}} = (1-\zeta^2)\min(\lambda_0,\tfrac{\alpha(1-\sigma^2)}{L})$.
\end{itemize}
\end{corollary}
\begin{proof}
Starting from the conclusion of \cref{cor:FB}, we obtain the desired result in the case $\beta =1$ using comparisons of sums with integrals along with the bounds $\tfrac{\eta_{\mathrm{min}}}{4} k^2 \leq A_k \leq \eta_{\mathrm{max}} k^2 $. In the second case, where $\beta\geq 1$ and $\xi_k = 0$, the target result follows from $\tfrac{\eta_{\mathrm{min}}}{4} k^2 \leq A_k$.
\end{proof}

\subsection{Proof of \cref{thm:deacres-potential}}\label{subsec:proof-FB}

The following proof is presented in a purely algebraic form consisting in a weighted sum of inequalities satisfied by the functions $f$ and $g$ as well as inexactness requirements. Indeed, it has been obtained from a dual certificate of a performance estimation problem (see \citet[Section 3]{barre2020principled_new} for more details on performance estimation in the context of inexact proximal operations). As mentioned in \Cref{sec:intro}, the algebraic equivalences stated below can be verified either by hand or with help of Mathematica notebooks (see \Cref{sec:intro}, \S ``Codes'').
\begin{proof}
    Let $w_{k+1} \in \mathbb{R}^d$ such that $g_{k+1}-\mu x_{k+1} + \mu w_{k+1} \in \partial g(w_{k+1})$. Using \eqref{eq:pdgap-2}, this leads to
    \begin{align*}
        \PDg_{\tfrac{\lambda_{k}}{1+\mu\lambda_{k}} \left(g(\cdot)-\tfrac{\mu}{2}\|\cdot\|^2 \right)}(x_{k+1},g_{k+1}-\mu x_{k+1};\tfrac{y_k-\lambda_kf'(y_k)}{1+\mu\lambda_{k}}) =& \;\tfrac{1}{2(1+\lambda_k\mu)^2}\normsq{x_{k+1} - y_k +\lambda_k(g_{k+1} +f'(y_k))}\\
    &+ \tfrac{\lambda_k}{1+\lambda_k\mu}\Big(g(x_{k+1})-g(w_{k+1}) \\
    &+\tfrac{\mu}{2}\normsq{x_{k+1}-w_{k+1}} - \langle x_{k+1}-w_{k+1},g_{k+1} \rangle \Big).
    \end{align*}
    The proof consists in performing a weighted sum of the following inequalities:
     \begin{itemize}
        \item strong convexity of $g$ between $w_{k+1}$ and $x_\star$ with weight $\nu_1 = A_{k+1}-A_k$
        \[g(x_\star)\geq g(w_{k+1}) + \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_\star-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_\star},\]
        \item strong convexity of $g$ between $w_{k+1}$ and $x_{k}$ with weight $\nu_2 = A_k$
        \[g(x_k)\geq g(w_{k+1}) + \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_k-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_{k}},\]
        \item strong convexity of $g$ between $w_{k+1}$ and $x_{k+1}$ with weight $\nu_3 = A_{k+1}\lambda_k\mu$
        \[g(x_{k+1})\geq g(w_{k+1}) + \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_{k+1}-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_{k+1}},\]
        \item convexity of $f$ between $y_k$ and $x_\star$ with weight $\nu_4 = A_{k+1}-A_k$
        \[f(x_\star)\geq f(y_{k}) + \langle f'(y_k), x_\star-y_{k}\rangle ,\]
        \item convexity of $f$ between $y_k$ and $x_k$ with weight $\nu_5 = A_k$
        \[f(x_k)\geq f(y_{k}) + \langle f'(y_k), x_k-y_{k}\rangle ,\]
        \item convexity and $\tfrac{1-\sigma_k^2}{\lambda_k}$-smoothness of $f$ between $x_{k+1}$ and $y_k$ required by \eqref{eq:smoothness} with weight $\nu_6=A_{k+1}$
        \[f(y_k)\geq f(x_{k+1}) + \langle f'(x_{k+1}), y_k-x_{k+1}\rangle  + \tfrac{\lambda_k}{2(1-\sigma_k^2)}\normsq{f'(y_k) - f'(x_{k+1})},\]
        \item approximation requirement on $x_{k+1}$ with weight $\nu_7 = \tfrac{A_{k+1}}{\lambda_k}$
        \[\begin{aligned}
        \tfrac{\sigma_k^2}{2}\normsq{x_{k+1}-y_k} +\tfrac{\zeta_k^2\lambda_k^2}{2}\|g_{k+1}
		 +f'(y_k)\|^2 +\tfrac{\lambda_k}{2}\xi_k\geq&\,\lambda_k(1+\lambda_k\mu)\Big(g(x_{k+1})-g(w_{k+1})\\
		 &+\tfrac{\mu}{2}\normsq{x_{k+1}-w_{k+1}}
         - \langle x_{k+1}-w_{k+1},g_{k+1} \rangle \Big)\\
         &+\tfrac{1}{2}\normsq{x_{k+1} - y_k +\lambda_k(g_{k+1}+f'(y_k))}.
        \end{aligned}\]
    \end{itemize}
    The weighted sum can be written as
    \begin{equation}\label{eq:wsum}
    \hspace{-0.1cm}\begin{aligned}
        0 \geq&\, \nu_1\left[  g(w_{k+1})-g(x_\star) + \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_\star-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_\star}\right]\\
        &+ \nu_2\left[g(w_{k+1}) -g(x_k)+ \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_k-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_{k}} \right]\\
        &+\nu_3\left[ g(w_{k+1})-g(x_{k+1}) + \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_{k+1}-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_{k+1}}\right]\\
        &+\nu_4 \left[ f(y_{k})-f(x_\star) + \langle f'(y_k), x_\star-y_{k}\rangle\right]+\nu_5\left[  f(y_{k})-f(x_k) + \langle f'(y_k), x_k-y_{k}\rangle \right]\\
        &+\nu_6\left[ f(x_{k+1}) -f(y_k)+ \langle f'(x_{k+1}), y_k-x_{k+1}\rangle  + \tfrac{\lambda_k}{2(1-\sigma_k^2)}\normsq{f'(y_k) - f'(x_{k+1})}\right]\\
        &+\nu_7\left[ \lambda_k(1+\lambda_k\mu)\Big(g(x_{k+1})-g(w_{k+1})
		 +\tfrac{\mu}{2}\normsq{x_{k+1}-w_{k+1}}
         - \langle x_{k+1}-w_{k+1},g_{k+1} \rangle \Big)\right.\\
         &\left.\quad\quad+\tfrac{1}{2}\normsq{x_{k+1} - y_k +\lambda_k(g_{k+1}+f'(y_k))} -  \tfrac{\sigma_k^2}{2}\normsq{x_{k+1}-y_k} -\tfrac{\zeta_k^2\lambda_k^2}{2}\|g_{k+1}
		 +f'(y_k)\|^2 -\tfrac{\lambda_k}{2}\xi_k\right].
    \end{aligned}
    \end{equation}
    Substituting $y_k$ and $z_{k+1}$ in the weighted sum, that is
    \begin{equation*}
        \begin{array}{ccc}
             y_k &=& x_k+\tfrac{(A_{k+1}-A_k)(1+\mu A_k)}{A_{k+1}+\mu A_k(2 A_{k+1}-A_k)}(z_k-x_{k})  \\
             z_{k+1} &=& z_{k}+\tfrac{A_{k+1}-A_k}{1+\mu A_{k+1}}\left(\mu(x_{k+1}-z_k) -g_{k+1}\right),
        \end{array}
    \end{equation*}
    \eqref{eq:wsum} is equivalently reformulated as
    \begin{align*}
        &A_{k+1}(F(x_{k+1}) - F(x_\star)) + \tfrac{1+\mu A_{k+1}}{2}\normsq{z_{k+1}-x_\star} \\
        &\leq A_{k}(F(x_{k}) - F(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star} +\tfrac{A_{k+1}}{2}\xi_k\\
        &\,\,\,\,\,- \tfrac{A_k(A_{k+1}-A_k)\mu (1 + A_k \mu)}{2(A_{k+1} + A_k(2A_{k+1}-A_k)\mu)}\normsq{x_k-z_k}\\
        &\,\,\,\,\,- \tfrac{A_{k+1}\lambda_k}{2(1-\sigma_k^2)}\normsq{f'(x_{k+1})-f'(y_k)+(1-\sigma_k^2)\tfrac{y_k-x_{k+1}}{\lambda_k}}\\
        &\,\,\,\,\,- \tfrac{\mu\left(A_{k+1}+A_k(2A_{k+1}-A_k)\mu \right)}{2(1+A_{k+1}\mu)}\normsq{x_{k+1}-y_k + \tfrac{(A_{k+1}-A_k)^2}{A_{k+1}+A_k(2A_{k+1}-A_k)\mu}(g_{k+1}+f'(y_k))}\\
        &\,\,\,\,\,+ A_{k+1}\tfrac{A_{k+1}(A_{k+1}-\eta_k)-A_k(1+\eta_k\mu)(2A_{k+1}-A_k)}{2(A_{k+1}+A_k(2A_{k+1}-A_k)\mu)}\normsq{g_{k+1}+f'(y_k)}\\
        &\leq A_{k}(F(x_{k}) - F(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star}+\tfrac{A_{k+1}}{2}\xi_k\\
        &\,\,\,\,\,+ A_{k+1}\tfrac{A_{k+1}(A_{k+1}-\eta_k)-A_k(1+\eta_k\mu)(2A_{k+1}-A_k)}{2(A_{k+1}+A_k(2A_{k+1}-A_k)\mu)}\normsq{g_{k+1}+f'(y_k)}\\
        &=A_{k}(F(x_{k}) - F(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star}+\tfrac{A_{k+1}}{2}\xi_k,
    \end{align*}
    where the inequality in the second to last line comes from the fact that factors in front of three squared Euclidean norms are nonpositive. In addition, the last equality follows from the particular choice of $A_{k+1}$ satisfies
    \[A_{k+1}(A_{k+1}-\eta_k) - A_k(1+\eta_k\mu)(2A_{k+1}-A_k) = 0,\]
    which implies that the factors in front of the last squared Euclidean norm vanishes.
\end{proof}

\section{Numerical examples}\label{s:numerics}
\input{numerical_examples}
\section{An accelerated hybrid proximal extragradient method}\label{s:ahpe}
In this section, we provide an improved analysis for the specific case $f=0$ (no smooth convex term in~\eqref{eq:opt}). This type of methods is often used as a \emph{globalization} strategy for higher-order methods, see~\citet{monteiro2013accelerated}. The version presented in this section allows exploiting the possible strong convexity of the objective, which was not incorporated in previous versions of the method, to the best of our knowledge.

\newpage
\subsection{Algorithm}
\begin{oframed}
	\textbf{Strongly convex accelerated hybrid proximal extragradient method} (\Cref{eq:ahpe}) \label[algorithm]{eq:ahpe}
	\begin{itemize}
		\item[] {\bf{}Input:} \begin{itemize}
		    \item Objective function: $g\in\Fmu(\mathbb{R}^d)$.
		    \item Initial point: $x_0\in\mathbb{R}^d$.
		    \item Step sizes: $\{\lambda_k\}_k$ with $\lambda_k >0$.
		    \item Tolerance parameters: sequence $\{\sigma_k\}_k$ with  $\sigma_k\in[0,1]$.
		\end{itemize}
		\item[] {\bf{}Initialization:} $z_0=x_0$, $A_0=0$.
		\item[] {\bf{}Run:} For $k=0,1,\hdots$:
		\begin{equation}\notag
			\begin{aligned}
			A_{k+1}&=A_k+(2(1-\sigma_k)+\lambda_{k}\mu)\lambda_{k}\tfrac{1 + 2 A_k\mu+ \sqrt{1 +4A_k(1+A_k\mu)\tfrac{(1+\lambda_k\mu)^2 - \sigma_k(\sigma_k+\lambda_k\mu)}{(2(1-\sigma_k)+\lambda_{k}\mu)\lambda_{k}}}}{{2(1-\sigma_k^2 + \lambda_{k}\mu\sigma_k)}}\\
            y_{k}&=x_k+\tfrac{(A_{k+1}-A_k)(1+\mu A_k)}{A_{k+1}+\mu A_k(2 A_{k+1}-A_k)}(z_k-x_{k})\\
            \varepsilon_k &= \tfrac{\sigma_k^2}{2(1+\lambda_k \mu)^2} \|x_{k+1}-y_k\|^2\\
			(x_{k+1},\;g_{k+1})&\approx_{\varepsilon_k,\mu}\left(\mathrm{prox}_{\lambda_{k} g}(y_{k}),\mathrm{prox}_{g^*/\lambda_{k}}(\tfrac{y_{k}}{\lambda_k})\right) \\
			{z}_{k+1}&=z_{k}+\tfrac{A_{k+1}-A_k}{1+\mu A_{k+1}}\left(\mu(x_{k+1}-z_k) -g_{k+1} \right)\\
			\end{aligned}
			\end{equation}
			\item[] {\bf{}Output:} $x_{k+1}$
		\end{itemize}
	\end{oframed}
When $\mu=0$ and $\sigma_k$ is fixed, this method actually reduces to the optimized relatively inexact proximal point algorithm from \citet[(ORIPPA)]{barre2020principled_new}. In this case, the growth rate of the sequence $\{A_k\}_k$ is essentially
\[ A_{k}\geq \tfrac{1}{4}\left(\sum_{i=0}^{k-1} \sqrt{\tfrac{2\lambda_i}{(1+\sigma_i)}}\right)^2 \geq O(k^2)\quad \text{for } k \geq 1,\]
when the parameters $\{\lambda_k\}_k$, $\{\sigma_k\}_k$ are well chosen (e.g., constant parameters). When $\mu>0$, the sequence $\{A_k\}_k$ grows as
\begin{align*}
A_{k+1} &\geq A_k\left(1 + \tfrac{2(1-\sigma_k)+\lambda_k\mu}{1-\sigma_k^2+\lambda_k\mu\sigma_k}\left(\lambda_k\mu + \sqrt{\lambda_k\mu\tfrac{(1+\lambda_k\mu)^2 - \sigma_k(\sigma_k+\lambda_k\mu)}{(2(1-\sigma_k)+\lambda_{k}\mu)}}\right) \right)\\
&= A_k\left( \tfrac{(1+\lambda_k\mu)^2 - \sigma_k(\sigma_k+\lambda_k\mu)+\sqrt{\lambda_k\mu(2(1-\sigma_k)+\lambda_{k}\mu)((1+\lambda_k\mu)^2 - \sigma_k(\sigma_k+\lambda_k\mu))}}{1-\sigma_k^2+\lambda_k\mu\sigma_k}\right)\\
&= A_k\left(\tfrac{(1+\lambda_k\mu)^2 - \sigma_k(\sigma_k+\lambda_k\mu) - \lambda_k\mu(2(1-\sigma_k)+\lambda_{k}\mu)}{1-\sigma_k^2+\lambda_k\mu\sigma_k} \right)\big/\left(1 - \sqrt{\tfrac{\lambda_k \mu (2(1-\sigma_k) + \lambda_k\mu)}{(1+\lambda_k \mu)^2 - \sigma_k(\sigma_k+\lambda_k \mu)}}\right)\\
    &= A_k\big/\left(1 - \sqrt{\tfrac{\lambda_k \mu (2(1-\sigma_k) + \lambda_k\mu)}{(1+\lambda_k \mu)^2 - \sigma_k(\sigma_k+\lambda_k \mu)}}\right),
\end{align*}
with $A_1 = \lambda_0\tfrac{2(1-\sigma_0)+\lambda_0\mu}{(1-\sigma_0^2+\lambda_0\mu\sigma_0)}\geq \lambda_0$.
In particular we recover the rate of the inexact accelerated forward-backward method when $\sigma_k = 1$. In addition, we notice that
\[ 1 - \sqrt{\tfrac{\lambda_k \mu (2 + \lambda_k\mu - 2\sigma_k)}{(1+\lambda_k \mu)^2 - \sigma_k(\sigma_k+\lambda_k \mu)}} \sim 1-\sqrt{\tfrac{2}{1+\sigma_k}\lambda_k \mu},\]
when $\lambda_k \mu \ll 0$.

\begin{theorem}\label{thm:ahpe}
    Let $g\in \Fmu(\mathbb{R}^d)$, $k\geq 0$, a parameter $\sigma_k\in[0,1]$ and some $\lambda_k > 0$. For any $x_k,z_k\in\mathbb{R}^d$ and $A_k\geq 0$, it holds that
    \[A_{k+1}(g(x_{k+1}) - g(x_\star))+\tfrac{1+\mu A_{k+1}}{2}\|z_{k+1}-x_\star\|^2 \leq  A_{k}(g(x_{k}) - g(x_\star))+\tfrac{1+\mu A_k}{2}\|z_{k}-x_\star\|^2\]
    with $x_\star\in \argmin_x g(x)$, and where $z_{k+1}$, $x_{k+1}$ are constructed by one iteration of \Cref{eq:ahpe}.
\end{theorem}
\begin{proof}
    The proof of this Theorem is deferred to \cref{subsec:proof-AHPE}
\end{proof}
Just as for the its forward-backward version, one can obtain a final worst-case guarantee driven by the growth rate of the sequence $\{A_k\}_k$.
\begin{corollary}
Let $g \in \Fmu$, $\{\lambda_k\}_k$ be a sequence of positive parameters, and a sequence (relative error parameters) $\{\sigma_k\}_k$ satisfying $\sigma_k\in[0,1]$. Let $x_N\in\mathbb{R}^d$ be the output after $N\in\mathbb{N}^*$ iterations of \Cref{eq:ahpe} on $g$ initiated at $x_0\in\mathbb{R}^d$, it holds that
\[ g(x_N) - g(x_\star) \leq \tfrac{1}{2A_N}\normsq{x_0-x_\star},\]
where $x_\star \in \argmin_x g(x)$.
\end{corollary}
\begin{proof}
    The proof follows from the same lines as that of~\cref{cor:FB}, using \cref{thm:ahpe} instead of \cref{thm:deacres-potential}.
\end{proof}
\subsection{Proof of \cref{thm:ahpe}}\label{subsec:proof-AHPE}
The proof follows the same structure as that of in \cref{subsec:proof-FB}, and simply consists in reformulating a weighted sum of inequalities.
\begin{proof}
    First we consider $\sigma_k \in (0,1]$ as the case $\sigma_k=0$ requires a particular treatment.
    Let $w_{k+1} \in \mathbb{R}^d$ such that $g_{k+1}-\mu x_{k+1} + \mu w_{k+1} \in \partial g(w_{k+1})$. Using \eqref{eq:pdgap-2}, this leads to
    \begin{align*}
        \PDg_{\tfrac{\lambda_{k}}{1+\mu\lambda_{k}} \left(g(\cdot)-\tfrac{\mu}{2}\|\cdot\|^2 \right)}(x_{k+1},g_{k+1}-\mu x_{k+1};\tfrac{y_k}{1+\mu\lambda_{k}}) =& \;\tfrac{1}{2(1+\lambda_k\mu)^2}\normsq{x_{k+1} - y_k +\lambda_k g_{k+1}}\\
    &+ \tfrac{\lambda_k}{1+\lambda_k\mu}\Big(g(x_{k+1})-g(w_{k+1})+\tfrac{\mu}{2}\normsq{x_{k+1}-w_{k+1}} \\
    & - \langle x_{k+1}-w_{k+1},g_{k+1} \rangle \Big).
    \end{align*}
    The proof consists in performing a weighted sum of the following inequalities:
    \begin{itemize}
        \item strong convexity between $w_{k+1}$ and $x_\star$ with weight $\nu_1 = A_{k+1} - A_k$
        \[g(x_\star)\geq g(w_{k+1}) + \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_\star-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_\star},\]
        \item strong convexity between $w_{k+1}$ and $x_{k}$ with weight $\nu_2 = A_k$
        \[g(x_k)\geq g(w_{k+1}) + \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_k-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_{k}},\]
        \item strong convexity between $w_{k+1}$ and $x_{k+1}$ with weight $\nu_3 = \tfrac{A_{k+1}(1-\sigma_k+\lambda_k\mu)}{\sigma_k}$
        \[g(x_{k+1})\geq g(w_{k+1}) + \langle g_{k+1}-\mu x_{k+1}+\mu w_{k+1}, x_{k+1}-w_{k+1}\rangle + \tfrac{\mu}{2}\normsq{w_{k+1}-x_{k+1}},\]
        \item approximation requirement on $x_{k+1}$ with weight $\nu_4 = \tfrac{A_{k+1}}{\lambda_k\sigma_k}$
        \[\begin{aligned}
        \tfrac{\sigma_k^2}{2}\normsq{x_{k+1}-y_k}\geq&\,\tfrac{1}{2}\normsq{x_{k+1} - y_k +\lambda_k g_{k+1}}+ \lambda_k(1+\lambda_k\mu)\Big(g(x_{k+1})-g(w_{k+1})\\
        &\,+\tfrac{\mu}{2}\normsq{x_{k+1}-w_{k+1}}
    - \langle x_{k+1}-w_{k+1},g_{k+1} \rangle \Big).
        \end{aligned}\]
    \end{itemize}
    Substituting $y_k$ and $z_{k+1}$ in the weighted sum, that is
    \begin{equation*}
        \begin{array}{ccc}
             y_k &=& x_k+\tfrac{(A_{k+1}-A_k)(1+\mu A_k)}{A_{k+1}+\mu A_k(2 A_{k+1}-A_k)}(z_k-x_{k})  \\
             z_{k+1} &=& z_{k}+\tfrac{A_{k+1}-A_k}{1+\mu A_{k+1}}\left(\mu(x_{k+1}-z_k) -g_{k+1}\right),
        \end{array}
    \end{equation*}
     the weighted sum is equivalently reformulated as
    \begin{align*}
        &A_{k+1}(g(x_{k+1}) - g(x_\star)) + \tfrac{1+\mu A_{k+1}}{2}\normsq{z_{k+1}-x_\star} \\
        &\leq A_{k}(g(x_{k}) - g(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star}\\
        &\,\,\,\,\,- \tfrac{A_k(A_{k+1}-A_k)\mu (1 + A_k \mu)}{2(A_{k+1} + A_k(2A_{k+1}-A_k)\mu)}\normsq{x_k-z_k}\\
        &\,\,\,\,\,-\tfrac{\lambda_k}{2}\left(\tfrac{\lambda_k\mu(A_{k+1}+A_k(2A_{k+1}-A_k)\mu)}{1+A_{k+1}\mu}  \right)\normsq{\tfrac{y_k-x_{k+1}}{\lambda_k} - \tfrac{\mu(A_{k+1}^2+A_k^2\sigma_k)+A_{k+1}(1-\sigma_k -2A_k\mu\sigma_k)}{A_{k+1}(1+A_{k+1\mu})(1-\sigma_k^2) +\lambda_k \mu\sigma_k(A_{k+1}+A_k(2A_{k+1}-A_k)\mu)}g_{k+1}}\\
        &\,\,\,\,\,- \tfrac{\lambda_k(1-\sigma_k^2)A_{k+1}}{2\sigma_k}\normsq{\tfrac{y_k-x_{k+1}}{\lambda_k} - \tfrac{\mu(A_{k+1}^2+A_k^2\sigma_k)+A_{k+1}(1-\sigma_k -2A_k\mu\sigma_k)}{A_{k+1}(1+A_{k+1\mu})(1-\sigma_k^2) +\lambda_k \mu\sigma_k(A_{k+1}+A_k(2A_{k+1}-A_k)\mu)}g_{k+1}}\\
        &\,\,\,\,\,+ \tfrac{A_{k+1}((2A_kA_{k+1}-A_{k}^2)(1-\sigma_k^2+\lambda_k^2\mu^2+\lambda_k\mu(2-\sigma_k)) - A_{k+1}^2(1-\sigma_k^2+\lambda_k\mu\sigma_k)+A_{k+1}\lambda_k(2(1-\sigma_k)+\lambda_k\mu))}{2(A_k^2\lambda_k\mu^2\sigma_k-A_{k+1}^2\mu(1-\sigma_k^2)-A_{k+1}(1-\sigma_k^2+\lambda_k\mu\sigma_k(1+2A_k\mu)))}\normsq{g_{k+1}}\\
        &\leq A_{k}(g(x_{k}) - g(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star}\\
        &\,\,\,\,\,+ \tfrac{A_{k+1}((2A_kA_{k+1}-A_{k}^2)(1-\sigma_k^2+\lambda_k^2\mu^2+\lambda_k\mu(2-\sigma_k)) - A_{k+1}^2(1-\sigma_k^2+\lambda_k\mu\sigma_k)+A_{k+1}\lambda_k(2(1-\sigma_k)+\lambda_k\mu))}{2(A_k^2\lambda_k\mu^2\sigma_k-A_{k+1}^2\mu(1-\sigma_k^2)-A_{k+1}(1-\sigma_k^2+\lambda_k\mu\sigma_k(1+2A_k\mu)))}\normsq{g_{k+1}}\\
        &=A_{k}(g(x_{k}) - g(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star},
    \end{align*}
    where the inequality in the second to last line comes from the fact that factors in front of squared Euclidean norms are nonpositive, and the last equality from the fact that $A_{k+1}$ is chosen such that it satisfies
    \[(2A_kA_{k+1}-A_{k}^2)(1-\sigma_k^2+\lambda_k^2\mu^2+\lambda_k\mu(2-\sigma_k)) - A_{k+1}^2(1-\sigma_k^2+\lambda_k\mu\sigma_k)+A_{k+1}\lambda_k(2(1-\sigma_k)+\lambda_k\mu) = 0.\]

        Note that the intermediary expressions largely simplifies when choosing this $A_{k+1}$, as the last term disappears, and the two other squared Euclidean norms become (up to nonpositive multiplicative factors) $\normsq{x_k-z_k}$ and $\normsq{\tfrac{y_k-x_{k+1}}{\lambda_k} - \tfrac{1-\sigma_k + \lambda_k\mu}{1-\sigma_k^2+\lambda_k\mu}g_{k+1}}$.

    For the case $\sigma_k = 0$ (i.e., exact proximal computations) we proceed as previously by performing the following weighted sum of inequalities:
    \begin{itemize}
        \item strong convexity between $x_{k+1}$ and $x_\star$ with weight $\nu_1 = A_{k+1}-A_k$
        \[g(x_\star)\geq g(x_{k+1}) + \langle g_{k+1}, x_\star-x_{k+1}\rangle + \tfrac{\mu}{2}\normsq{x_{k+1}-x_\star},\]
        \item strong convexity between $x_{k+1}$ and $x_{k}$ with weight $\nu_2 = A_k$
        \[g(x_k)\geq g(x_{k+1}) + \langle g_{k+1}, x_k-x_{k+1}\rangle + \tfrac{\mu}{2}\normsq{x_{k+1}-x_{k}}.\]
    \end{itemize}
    Substituting $y_k$, $x_{k+1}$ and $z_k$ in the weighted sum, that is
        \begin{equation*}
        \begin{array}{ccc}
             y_k &=& x_k+\tfrac{(A_{k+1}-A_k)(1+\mu A_k)}{A_{k+1}+\mu A_k(2 A_{k+1}-A_k)}(z_k-x_{k})  \\
             x_{k+1} &=& y_k - \lambda_kg_{k+1}\\
             z_{k+1} &=& z_{k}+\tfrac{A_{k+1}-A_k}{1+\mu A_{k+1}}\left(\mu(x_{k+1}-z_k) -g_{k+1}\right),
        \end{array}
    \end{equation*}
    the weighted sum is equivalently reformulated as
    \begin{align*}
        &A_{k+1}(g(x_{k+1}) - g(x_\star)) + \tfrac{1+\mu A_{k+1}}{2}\normsq{z_{k+1}-x_\star} \\
        &\leq A_{k}(g(x_{k}) - g(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star}\\
        &\,\,\,\,\,- \tfrac{A_k(A_{k+1}-A_k)\mu (1 + A_k \mu)}{2(A_{k+1} + A_k(2A_{k+1}-A_k)\mu)}\normsq{x_k-z_k}\\
        &\,\,\,\,\,+ \tfrac{A_{k+1}(A_{k+1}-\lambda_k(2+\lambda_k\mu)) - A_k(2A_{k+1}-A_k)(1+\lambda_k\mu)^2}{2(1+A_{k+1}\mu)}\normsq{g_{k+1}}\\
        &\leq A_{k}(g(x_{k}) - g(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star}\\
        &\,\,\,\,\,+ \tfrac{A_{k+1}(A_{k+1}-\lambda_k(2+\lambda_k\mu)) - A_k(2A_{k+1}-A_k)(1+\lambda_k\mu)^2}{2(1+A_{k+1}\mu)}\normsq{g_{k+1}}\\
        & = A_{k}(g(x_{k}) - g(x_\star)) + \tfrac{1+\mu A_{k}}{2}\normsq{z_{k}-x_\star},
    \end{align*}
     where the inequality in the second to last line comes from the fact that factor in front of squared Euclidean norm is nonpositive, and the last equality from the fact that $A_{k+1}$ is chosen such that it satisfies
    \[A_{k+1}(A_{k+1}-\lambda_k(2+\lambda_k\mu)) - A_k(2A_{k+1}-A_k)(1+\lambda_k\mu)^2 = 0,\]
    when $\sigma_k = 0$.
\end{proof}

\section{Conclusion}\label{s:ccl}
In this note, we proposed an inexact accelerated forward-backward method for solving composite convex minimization problems, along with some worst-case guarantees. The method supports inexact evaluations of the proximal subproblems, backtracking line-search on the smoothness parameter, and allows exploiting the possible strong convexity of one of the component in the objective function. The analysis relies on a now standard Lyapunov argument of the same type as that of~\citet{Nesterov:1983wy}, and the numerical behavior is illustrated on a factorization and a total variation problem.

We further provide a strongly convex version of the hybrid proximal extragradient method~\citep{monteiro2013accelerated} allowing to exploit strong convexity of the objective.

\bibliographystyle{plainnat}

\subsection{Factorization problem} Our first numerical experiment is a CUR-like factorization problem, introduced in~\citet{mairal2011convex}. It consists, given a matrix $W \in \mathrm{R}^{m\times p}$, in solving the minimization problem
\[ \min_X\,F(X) \equiv \underbrace{\tfrac{1}{2}\|W-WXW\|^2_{F}}_{f(X)} + \underbrace{\lambda_{\mathrm{row}}\sum_{i=1}^{n_r}\|X^i\|_2 + \lambda_{\mathrm{col}}\sum_{j=1}^{n_c}\|X_j\|_2 + \tfrac{\mu_\mathrm{reg}}{2}\|X\|^2_F}_{g(X)},  \]
where $\|\cdot\|_{F}$ is the Frobenius norm, and where $X^i$ and $X_j$ respectively denote the $i$th row and the $j$th column of the matrix $X$. This problem has already been used in~\citet{schmidt2011convergence} for illustrating convergence guarantees of an inexact accelerated proximal gradient method with absolute errors. As in~\citet{schmidt2011convergence}, we use an inexact version of the proximal operator of the regularization part, which we solve via a dual block coordinate ascent method~\citep{jenatton2010proximal} (i.e., we solve the dual of the proximal problem). Our implementation (see link in \S Codes from Section~\ref{sec:intro}) is based on that of~\citet{schmidt2011convergence}, and our experiments are done on the ``a1a'' dataset from the LIBSVM library \citep{CC01a}.
The corresponding matrix $W$ is normalized for having zero mean and unit norm. We also impose $\lambda_{\text{col}} = \sqrt{\tfrac{p}{m}}\lambda_{\text{row}}$ for having a similar scaling for the row and column regularization parameters. The choice of the error criteria and regularization parameters is detailed in \cref{fig:CUR} where we plot gaps between the objective function values at the iterates of \Cref{eq:algo2} and the optimal objective value versus the number of iteration of \Cref{eq:algo2} (left) and versus the total number of dual block coordinate ascent iterations (right).

\begin{figure}[!ht]
\centering
\begin{tabular}{ll}
\begin{tikzpicture}
			\begin{semilogyaxis}[legend pos=outer north east, legend style={draw=none},legend cell align={left},plotOptions, ymin=1e-13, ymax=2e-1,xmin=-1,xmax=700,width=.33\linewidth, ylabel={$F(x_{k})-F_\star$}, xlabel={iterations $k$}]

			\addplot [colorP1] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000_no_mu.txt};
			\addplot [colorP2] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000_no_mu_backtrack.txt};
			\addplot [colorP3] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP4] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000_backtrack.txt};
			\end{semilogyaxis}
\end{tikzpicture}
    & \begin{tikzpicture}
			\begin{semilogyaxis}[legend pos=outer north east, legend style={draw=none},legend cell align={left},plotOptions, ymin=1e-13, ymax=2e-1,xmin=-100,xmax=15000,width=.33\linewidth, xlabel={total iterations}]

			\addplot [colorP1] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000_no_mu.txt};
			\addplot [colorP2] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000_no_mu_backtrack.txt};
			\addplot [colorP3] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP4] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000_backtrack.txt};
			\addlegendentry{$\mu = 0$}
			\addlegendentry{$\mu = 0 \,+ $ (Btr)}
			\addlegendentry{$\mu = \mu_{\mathrm{reg}} $}
			\addlegendentry{$\mu = \mu_{\mathrm{reg}} \,+ $ (Btr)}
			\end{semilogyaxis}
\end{tikzpicture}
\\
\begin{tikzpicture}
			\begin{semilogyaxis}[legend pos=outer north east, legend style={draw=none},legend cell align={left},plotOptions, ymin=1e-13,ymax=2e-1,xmin=-1,xmax=400,width=.33\linewidth, ylabel={$F(x_{k})-F_\star$}, xlabel={iterations $k$}]

			\addplot [colorP1] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_0.000000e+00_zeta_0.000000e+00_xi_poly2_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP2] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_0.000000e+00_zeta_8.000000e-01_xi_zero_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP3] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_5.000000e-01_zeta_5.000000e-01_xi_zero_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP4] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP5] table [x=N,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_0.000000e+00_zeta_0.000000e+00_xi_lin_maxiter_20000_inneriter_20000.txt};
			\end{semilogyaxis}
\end{tikzpicture}
    & \begin{tikzpicture}
			\begin{semilogyaxis}[legend pos=outer north east, legend style={draw=none},legend cell align={left},plotOptions, ymin=1e-13, ymax=2e-1,xmin=-100,xmax=10000,width=.33\linewidth, xlabel={total iterations}]

			\addplot[color=colorP1] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_0.000000e+00_zeta_0.000000e+00_xi_poly2_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP2] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_0.000000e+00_zeta_8.000000e-01_xi_zero_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP3] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_5.000000e-01_zeta_5.000000e-01_xi_zero_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP4] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_20000_inneriter_20000.txt};
			\addplot [colorP5] table [x=cs,y=f,col sep=space] {data/CUR/CUR_a1a_L_1.331492e-01_mu_2.662985e-04_sigma_0.000000e+00_zeta_0.000000e+00_xi_lin_maxiter_20000_inneriter_20000.txt};
			\addlegendentry{$\xi_k = \tfrac{C}{(k+1)^2}$}
			\addlegendentry{$\zeta_k = 0.8$}
			\addlegendentry{$\sigma_k = 0.8$}
			\addlegendentry{$\sigma_k = 0.5$, $\zeta_k=0.5$}
			\addlegendentry{$\xi_k = C(1-\sqrt{\mu_{\mathrm{reg}}/L})^{k}$}
			\end{semilogyaxis}
\end{tikzpicture}

\end{tabular}
		\caption{\Cref{eq:algo2} on CUR factorization. The initial step size is set to $\lambda_0 = \tfrac{1-\sigma_0^2}{L}$, initial $L$ to $\norm{W}^4$, $\lambda_\mathrm{row} = \lambda_\mathrm{col}\sqrt{m/p} = 2.10^{-3}$ ($\sim 30\%$ nonzero coefficients in the solution) and $\mu_{\mathrm{reg}} = 2.10^{-3}L$. Top: $\sigma_k = 0.8$ , $\zeta_k = 0$ and $\xi_k = 0$. Bottom: accelerated inexact forward-backward with $\mu=\mu_{\mathrm{reg}}$ and no backtracking.  When backtracking is used, $\alpha$ is set to $\tfrac12$ and $\beta$ to $1.1$. ``Total iterations" refers to total the number of block coordinate ascent iterations used in the subroutine that computes the proximal steps approximately. $F_*$ is approximated by the smallest objective values encountered in $2.10^4$ total iterations of block coordinate ascent.} \label{fig:CUR}
		\vspace{-0.0cm}
\end{figure}

\subsection{Total variation regularization} In this section, we compare the behaviors of the accelerated inexact forward backward method (\Cref{eq:algo2}) with different tunings, on the classical problem of deblurring through total variation regularization~\citep{rudin1992nonlinear,rudin1994total,wang2008new}. Given a blurred image $Y \in \mathrm{R}^{n\times n}$ and a blurring operator $A$, the problem consists in solving
\[\min_X \,F(X) \equiv \underbrace{\tfrac{1}{2}\|AX-Y\|_F^2}_{f(X)} + \underbrace{\lambda_{\text{reg}}\sum_{i,j = 0}^{n}\|(\nabla X)_{i,j}\|_2  + \tfrac{\mu_{\mathrm{reg}}}{2}\|X\|^2_F}_{g(X)},\]
where $\nabla$ is the discrete gradient of an image, see e.g.,~\citet[Equation (2.4)]{chambolle2016introduction}. One way of dealing with this problem is to approximate the proximal operator of the discrete total variation plus the Tikhonov regularization. As in~\citet{villa2013accelerated,millan2019inexact}, we apply FISTA~\citep{beck2009fast} on the dual of the proximal subproblem (which is provided e.g., in~\citet[Example 3.1]{chambolle2016introduction}), which we use in the accelerated inexact forward-backward method.

In the experiments we select $Y$ to be the popular $256\times256$ greyscale fishing boat image (that can be found at \url{http://sipi.usc.edu/database/}). We blur $Y$ via a $5 \times 5$ box blur kernel $A$, and add a Gaussian noise of standard deviation $0.01$ times the mean of the blurred image and zero mean to the picture. Some results are detailed in \cref{fig:TV} where we plot gaps between the objective function values at the iterates of \Cref{eq:algo2} and the optimal objective value versus the number of iterations of \Cref{eq:algo2} (left) and versus the total number iterations of FISTA on the dual subproblem (right).

\begin{figure}[!ht]
\centering
\begin{tabular}{ll}
\begin{tikzpicture}
			\begin{semilogyaxis}[legend pos=outer north east, legend style={draw=none},legend cell align={left},plotOptions, ymin=0.0001, ymax=1e7,xmin=-1,xmax=300,width=.33\linewidth, ylabel={$F(x_{k})-F_\star$}, xlabel={iterations $k$}]

			\addplot [colorP1] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000_no_mu.txt};
			\addplot [colorP2] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000_no_mu_backtrack.txt};
			\addplot [colorP3] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000.txt};
			\addplot [colorP4] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000_backtrack.txt};
			\end{semilogyaxis}
\end{tikzpicture}
    & \begin{tikzpicture}
			\begin{semilogyaxis}[legend pos=outer north east, legend style={draw=none},legend cell align={left},plotOptions, ymin=0.0001, ymax=1e7,xmin=-500,xmax=70000,width=.33\linewidth, xlabel={total iterations} ]

			\addplot [colorP1] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000_no_mu.txt};
			\addplot [colorP2] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000_no_mu_backtrack.txt};
			\addplot [colorP3] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000.txt};
			\addplot [colorP4] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000_backtrack.txt};

			\addlegendentry{$\mu = 0$}
			\addlegendentry{$\mu = 0 \,+ $ (Btr)}
			\addlegendentry{$\mu = \mu_{\mathrm{reg}} $}
			\addlegendentry{$\mu = \mu_{\mathrm{reg}} \,+ $ (Btr)}
			\end{semilogyaxis}
\end{tikzpicture}
\\
\begin{tikzpicture}
			\begin{semilogyaxis}[legend pos=outer north east, legend style={draw=none},legend cell align={left},plotOptions, ymin=0.0001, ymax=1e7,xmin=-1,xmax=300,width=.33\linewidth, ylabel={$F(x_{k})-F_\star$}, xlabel={iterations $k$}]

			\addplot [colorP1] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_0.000000e+00_zeta_0.000000e+00_xi_poly2_maxiter_70000_inneriter_10000_backtrack.txt};
			\addplot [colorP2] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_0.000000e+00_zeta_8.000000e-01_xi_zero_maxiter_70000_inneriter_10000_backtrack.txt};
			\addplot [colorP3] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000_backtrack.txt};
			\addplot [colorP4] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_5.000000e-01_zeta_5.000000e-01_xi_zero_maxiter_70000_inneriter_10000_backtrack.txt};

			\addplot [colorP5] table [x=N,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_0.000000e+00_zeta_0.000000e+00_xi_lin_maxiter_70000_inneriter_10000_backtrack.txt};
			\end{semilogyaxis}
\end{tikzpicture}
    & \begin{tikzpicture}
			\begin{semilogyaxis}[legend pos=outer north east, legend style={draw=none},legend cell align={left},plotOptions, ymin=0.0001, ymax=1e7,xmin=-500,xmax=70000,width=.33\linewidth, xlabel={total iterations}]

			\addplot [colorP1] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_0.000000e+00_zeta_0.000000e+00_xi_poly2_maxiter_70000_inneriter_10000_backtrack.txt};

			\addplot [colorP2] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_0.000000e+00_zeta_8.000000e-01_xi_zero_maxiter_70000_inneriter_10000_backtrack.txt};
			\addplot [colorP3] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_8.000000e-01_zeta_0.000000e+00_xi_zero_maxiter_70000_inneriter_10000_backtrack.txt};
			\addplot [colorP4] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_5.000000e-01_zeta_5.000000e-01_xi_zero_maxiter_70000_inneriter_10000_backtrack.txt};

			\addplot [colorP5] table [x=cs,y=f,col sep=space] {data/TV_boat_L_1.000000e+00_mu_1.000000e-02_sigma_0.000000e+00_zeta_0.000000e+00_xi_lin_maxiter_70000_inneriter_10000_backtrack.txt};
			\addlegendentry{$\xi_k = \tfrac{C}{(k+1)^2}$}
			\addlegendentry{$\zeta_k = 0.8$}
			\addlegendentry{$\sigma_k = 0.8$}
			\addlegendentry{$\sigma_k = 0.5$, $\zeta_k=0.5$}
			\addlegendentry{$\xi_k = C(1-\sqrt{\mu_{\mathrm{reg}}/L})^{k}$}
			\end{semilogyaxis}
\end{tikzpicture}

\end{tabular}
		\caption{\Cref{eq:algo2} on TV regularization. The initial step size is set to $\lambda_0 = \tfrac{1-\sigma_0^2}{L}$, the initial $L$ and $\lambda_{\mathrm{reg}}$ to $1$ and $\mu_{reg}$ to $10^{-2} L$. Top: $\sigma_k = 0.8$ , $\zeta_k = 0$ and $\xi_k = 0$. Bottom: accelerated inexact forward-backward with $\mu=\mu_{\mathrm{reg}}$ and backtracking. When backtracking is used, $\alpha$ is set to $\tfrac12$ and $\beta$ to $1.1$. ``Total iterations" refers to total the number of FISTA iterations used in the subroutine that computes the proximal steps approximately.  $F_*$ is approximated by the smallest objective values encountered in $2.10^4$ total FISTA iterations.} \label{fig:TV}
		\vspace{-0.5cm}
\end{figure}

redpajama_set_name: RedPajamaArXiv


Barrage Battle – Load Your Dice!
Barrage Battle is a strategy and tactical game for 2 players. Your goal is to destroy your opponent's King, by playing cards and dice against him and by using Units, Structures, War Machines and Spell Cards that you have at your disposal. The player who plays in a more strategic and dexterous way wins the game.
A significant characteristic of Barrage Battle is the fact that it contains dice of different types and number of sides, since dice determine to a great extent the attacks that you make against your opponent.
Barrage Battle has been launched on Kickstarter where you can back it.
So, load your dice and get ready for the battle!
Witless Wizards on Kickstarter – In Depth Overview!


From La Repubblica/Palermo edition of 22 March 2015.
Antica Focacceria San Francesco serves the best traditional Sicilian food at moderate prices.
Definitely an act of courage but what a way to live afterwards. So? Did you get your interview with his brother?


text:
Top 50 hip-hop/rap albums of 2020: #40-31
Nick Sligh on the best hip-pop/rap albums released in 2020.
Photo: Curren$y and Harry Fraud
40. Curren$y and Harry Fraud: "The OutRunners"
"The OutRunners" is a great fit between Harry Fraud's soulful production and Curren$y's style as an artist. With a short run time, under 25 minutes total, this project is still able to pack multiple worthwhile tracks from the frequently collaborating duo. Curren$y has had a career filled with many consistently solid projects, with this effort being no different. While remaining true to his sound, Curren$y's "The OutRunners" provides an interesting addition to his discography and provides a worthwhile listen for fans of either member of this rapper-producer duo.
Favorite Songs: "Cutlass Cathedrals," "Gold and Chrome" and "Offloading"
Photo: Statik Selektah
39. Statik Selektah: "The Balancing Act"
Statik has consistently been one of the most solid producers of the past decade, so "The Balancing Act" was a welcome curation from a great contributor in modern hip-hop. Although not quite living up to the star-studded track list, there are still many tracks that provide refreshing sounds and interesting collections of rappers over the pleasant production. The solid production throughout and the consistently intriguing feature list allow "The Balancing Act" to be an interesting and enjoyable album.
Favorite Songs: "Keep It Moving," "No More," "Off My Mind" and "Balance Beam"
Photo: Quelle Chris and Chris Keys
38. Quelle Chris and Chris Keys: "Innocent Country 2"
Quelle Chris and Chris Keys meet with an impressive rapper-producer chemistry on 2020's "Innocent Country 2," the sequel to 2015's "Innocent Country." Although the predecessor was a solid effort, the more recent addition to the series is an album that feels more cohesive, thoughtful and entertaining from front to back. Keys' production paints a warm and vivid backdrop with jazzy pianos and smooth melodies that allow Quelle Chris's narrative and flows to effortlessly go over. An album that is heavily personal and introspective, "Innocent Country 2" provides an uplifting and encouraging listen, something that we all know 2020 needed desperately.
Favorite Songs: "Mirage," "Graphic Bleed Outs," "Black Twitter" and "Living Happy"
Photo: Conway the Machine
37. Conway the Machine and Big Ghost Ltd: "No One Mourns the Wicked"
Conway easily had one of the best years out of anybody in rap. The Buffalo native and Griselda member seems to only put out quality music, and 2020 was no different in this trend. The second of three major projects in 2020 for Conway, "No One Mourns the Wicked" pairs the Griselda rapper with Big Ghost Ltd, whose dark and grimy production effortlessly fits the style and themes of the project. Conway's authenticity is a significant part of what makes him such a special artist, and this project speaks to that at great length.
Favorite Songs: "Fake Love," "Bricks to Murals," "Icon" and "S.D.L.N"
Photo: redveil
36. redveil: "Niagara"
"Niagara" is definitely the best rap project of last year released by anybody 16 years old or younger. The raw feel and pure talent shown throughout redveil's 2020 album make him one of the most exciting young figures in rap. This project helps to establish redveil as more than just an Earl Sweatshirt-esque prodigy in the lo-fi and abstract lanes of hip-hop, and it allows the young rapper to showcase a versatile selection of unique sounds. "Niagara" makes redveil a must-watch figure in rap for years to come.
Favorite Songs: "Campbell," "Weight," "Badnews," "Grass" and "Clench"
Photo: J Hus
35. J Hus: "Big Conspiracy"
One of the better non-American rap albums of 2020, J Hus's "Big Conspiracy" highlights the significant growth and development in his artistry, culminating in a coherent album that is thoughtful throughout. There is not necessarily anything groundbreaking on this project artistically, but with solid production and lyricism, J Hus makes a quality addition to his own discography and to the greater scene of 2020 hip-hop.
Favorite Songs: "Triumph," "Big Conspiracy," "Reckless" and "Play Play"
Photo: Westside Gunn
34. Westside Gunn: "WHO MADE THE SUNSHINE"
The Griselda trio of Westside Gunn, Conway the Machine and Benny the Butcher are on a current run as a group that is hard to compare across the recent history of rap. Though not as refined as his earlier 2020 release "Pray For Paris," "WHO MADE THE SUNSHINE" still highlights many of the strengths of Westside Gunn's signature style and delivery. The Griselda catalog is benefited by this project with consistent production, great guest appearances and the Griselda feel that has come to be a welcome presence in hip-hop.
Favorite Songs: "The Butcher and The Blade," "98 Sabres," "Ishkabibble's" and "All Praises"
Photo: Jay Electronica
33. Jay Electronica: "Act II: The Patents of Nobility (The Turn)"
"Act II: The Patents of Nobility (The Turn)" certainly wins the award for the most strange album release. After roughly a decade of scattered leaks, delayed releases and rumors, the entire album leaked in full on Oct. 4, followed by a formal release the next day on Jay-Z's Tidal platform. In late 2020, the album then again disappeared from Tidal, and it is currently not on any major streaming services (in its formal version). For the music itself, the album certainly has its highs and lows. It is clear that parts of the album were being presented in very raw versions, with parts feeling unfinished. However, there are a collection of fantastic songs and some incredible themes that are represented and explored for stretches across the album. The strongest songs on this album really compete with the best that are present on any recent major releases, and the inconsistencies in the album are the only things holding it back from being one of the very best albums of the year. Some of Jay Electronica's best solo work comes on "Act II," and it is hard to believe that a full, formal and refined version of this album would not make an incredible listen.
Favorite Songs: "Run and Hide," "Memories & Merlot," "Road to Perdition" and "Better in Tune"
Photo: Kenny Mason
32. Kenny Mason: "Angelic Hoodrat"
"Angelic Hoodrat" was one of the best debut projects in hip-hop of last year, and Kenny Mason establishes himself as a unique presence in the genre to follow for years to come. Following the lead single "Hit," Kenny was able to come through with a highly personal and introspective album covering a wide array of sounds. Fun tracks like "Chevron" and "Hit" blend with dark and introspective tracks like "Angels Calling // My Dad" in a bold way that create a dynamic listen. Kenny's debut shows that he has no fear of trying innovative sounds, blending refreshing production, unique flows and deliveries and a constantly fluctuating and evolving style. Although "Angelic Hoodrat" might not be the most flawless album in terms of coherency or thematic content, it is an exciting listen that makes Kenny Mason a rapper that is undeniably intriguing.
Favorite Songs: "Hit," "Chevron," "Angels Calling // My Dad," "Angelic Hoodrat" and "Once Again"
Photo: Duckwrth
31. Duckwrth: "SuperGood"
Duckwrth was able to create one of the sonically diverse albums of the year with his second full-length studio album. Blending funk, r&b, hip-hop, pop and soul seamlessly, such a solid album deserves appreciation for the creativity and talent that it takes to pull this off. Naturally, the style of the project lends itself to feeling scattered in parts and lacking some coherency. However, there are also obviously significant positives as a result, primarily that it is never a boring listen and that there is always a feeling of not knowing what exactly to expect next from Duckwrth.
Favorite Songs: "Super Good," "Super Bounce," "World on Wheels" and "Too Bad"
Click here for a Spotify Playlist of some of my favorite songs across the year 2020.
Contact Nick Sligh at nick1019 'at' stanford.edu.


text:
INTERVIEW-Wealthy nations have a responsibility to help rebuild storm-hit Barbuda: PM
by Sebastien Malo | @SebastienMalo | Thomson Reuters Foundation
"We are the injured party, we're not the ones who are emitting these greenhouse gases"
By Sebastien Malo
UNITED NATIONS, Nov 20 (Thomson Reuters Foundation) - Rich nations responsible for climate change need to help rebuild the Caribbean island of Barbuda where living conditions remain "primitive" more than two months after Hurricane Irma, Antigua and Barbuda Prime Minister Gaston Browne said on Monday.
Ahead of a United Nations summit where countries will be invited to pledge donations to Caribbean nations ravaged by storms, Browne said he hoped at least half of the estimated $220 million needed to rebuild Barbuda could be raised.
"You have people who are living there notwithstanding the fact that you don't have basic services. They're living there under very primitive conditions," Browne said in an interview.
"We are the injured party, we're not the ones who are emitting these greenhouse gases into the earth's atmosphere," he said at a press conference ahead of the interview.
Irma, packing 185 mph (295 kph) winds, ravaged a series of small islands in the northeast Caribbean in early September, with climate scientists saying warmer air and water resulting from climate change may have added to the storm's severity.
The dual-island nation of Antigua and Barbuda, with a population of 100,000, was especially hard hit.
The northernmost island, Barbuda, normally home to about 1,800 people, was "decimated" and temporarily uninhabitable, with 95 percent of all dwellings leveled, Browne said.
So far, promised donations to Antigua and Barbuda of some $25 million represent about 10 percent of funds needed to rebound, said Browne. Pledges have come from Britain, the United Arab Emirates, India, and China among others.
"There's no water, no electricity, and there's no functioning police station or ... fire station," said Browne.
"We are building a new Codrington," he added, referring to Barbuda's main town before the storm which will be rebuilt further inland to prevent flooding.
Browne said he was deeply saddened after surveying Irma's damage by helicopter but he has since been consumed by the daunting task of ensuring Barbuda recovers despite estimated costs equal to the country's entire governmental revenue.
CULPRITS AND VICTIMS
Browne said he hoped rich countries pledge half of what is needed at Tuesday's summit, which is held by CARICOM, the Caribbean's largest cooperation group, and the United Nations.
Countries that have said they would attend Tuesday's donors summit include the United States, Canada, China, Brazil and several European Union nations, said Browne.
The gathering will also seek to raise money for victims of Hurricane Maria, with winds near 165 mph (270 kph) that destroyed broad swathes of homes and infrastructure on the nearby island of Dominica only weeks after Irma.
Small island nations, including Antigua and Barbuda, have increased pressure for top fossil-fuel consumers whose greenhouse gas emissions drive climate change to help them cope with the devastation wrought by Irma and other recent storms.
Browne said the entire Caribbean produces less than 0.1 percent of global greenhouse gas emissions.
"It means, therefore, that the developed countries ... have an obligation to assist because we are suffering disproportionately from the effects of climate change," said Browne who has been prime minister since 2014.
He said he hoped plans to turn Barbuda into a laboratory of resilience to extreme weather events - with raised homes capable of sustaining winds of 220 mph (355 kph) and renewable energy farms - would represent a selling point to donors.
"We can show that you can sustain a totally green island," he said.
"But the exploitation of these opportunities will be based on our capacity to raise the necessary resources."
(Reporting by Sebastien Malo @sebastienmalo, Editing by Belinda Goldsmith; Please credit the Thomson Reuters Foundation, the charitable arm of Thomson Reuters, that covers humanitarian news, women's rights, trafficking, property rights, climate change and resilience. Visit http://news.trust.org)


text:
require 'spec_helper'

describe 'lvm::default' do
  let(:chef_run) do
    ChefSpec::ChefRunner.new(:platform => 'ubuntu', :version => '12.04').converge('lvm::default')
  end

  it 'installs lvm2' do
    expect(chef_run).to install_package('lvm2')
  end

  it 'installs `di-ruby-lvm` as a Ruby gem' do
    expect(chef_run).to install_chef_gem('di-ruby-lvm')
  end
end

redpajama_set_name: RedPajamaGithub


text:
Together, Saving Lives and Building Hope · UOSSM USA
Coronavirus in Syria Updates
Latest Updates From Syria
THE SYRIAN CRISIS
The Crisis in Yemen
Become A Support Member
Become a Fundraiser for UOSSM USA!
Kafr Zeta Specialized Surgical and Maternity Hospital Leveled to the Ground; Attacks Continue Throughout Syria
posted by Najah Allouch | 755.40sc
The specialized surgical and maternity hospital in Kafr Zeta, Hama,(picture above) was leveled to the ground this weekend after being attacked three times in 24 hours. The first attack happened on Friday, April 28 at 4 p.m Damascus time, causing damage to the facility and the outpatient clinics. The second attack happened on Saturday, April 29 at 8:45 a.m. Damascus time. This attack caused major damage to the emergency department and injured one staff member. The third attack happened at 2 p.m. Damascus time on April 29. This attack completely destroyed the facility. No fatalities or injuries were reported as the hospital was evacuated due to earlier attacks. The hospital provided 4,050 consultations, 100 natural birth deliveries, 40 C-sections, 280 major surgeries, and 66 war related trauma cases a month.
Today, at 11 a.m. Damascus time, the Arbeen Surgical Hospital in eastern Ghouta (picture below) was attacked and put out of service after the four-story building next to the hospital collapsed on the facility. Three civilians were killed (two children and a woman) and tens of civilians were wounded. The hospital provided an average of 1,400 consultations and 125 major surgeries a month.
Please click here to read the full press release.
Donate Our Programs Where We Work What We Do
Union of Medical Care and Relief Organizations - USA
UOSSM USA
18140 Zane St. NW, Suite 337
Elk River, MN 55330
connect@uossm.us
UOSSM USA is a federally tax-exempt non-profit 501(c)(3) charitable humanitarian organization. Tax ID 47-3403988
UOSSM participates in the Combined Federal Campaign for federal workers. Charity number: 83435
©UOSSM USA. All rights reserved.
Created with NationBuilder | Terms Of Use | Privacy Policy | Charitable Solicitation Disclosures


It is July. The heat is rising. The fireflies are out in droves to the accompaniment of a cricket orchestra every night. Corn on the cob. Grills. The summer season has really begun. Click on the Book Giveaway tab to grab yourself what I would consider to be the perfect summer read-- Lady of Devices by Shelley Adina. The tenth novel in the series is due this fall so get started now.
It is no secret that I am a massive Sanderson fan. A friend of mine recommended the "Mistborn" trilogy back in high school, and I have been reading ever since. When I first began the books, it was not apparent that all the different series and stand-alones he was producing would be connected. A decade later and readers have been informed by Sanderson that there is a bigger picture--Cosmere. It is all connected. The lands and magics. They all come from something larger than any individual series. That scope--that is amazing.
He has written middle grade fiction, young adult fiction, fantasy, epic fantasy, and he even completed the "Wheel of Time" series. Now, he is trying his hands at graphic novels. It seems impossible that any reader could not be a fan of Sanderson. He has a format for anyone. White Sand enters the graphic novel scene with a new set of characters plus some others previously introduced--but don't worry, you won't need to know any of it as all of his stories can stand alone despite their unity.
I'll be honest; it is not my favorite of his endeavors. Sanderson builds a mighty picture. He is a man of words. Graphic novels just do not have the space for the kind of set up I expect out of Sanderson. BUT; I was very aware of this when I requested to read the Advanced Copy.
The overall read was not disappointing. 5 stars for the layout, graphic style, color palette, and all that artsy stuff. The introduction of the characters and conspiracy were just a little crunched for me. For what is there, it is Sanderson level 100% and certainly worthy of 3 stars. Sanderson magic systems just hook you in with their depth and viability--sand mastery included. Total package is 3.5 stars and certainly on my recommended reading list. An absolute must for Brandon Sanderson fans.


text:
EU separates heavy metal from hard rock
THE EU's 'non-energy extractive industry' is usually divided into three sub-sectors, according to a European Commission communication on sustainable development:
Metallic minerals such as iron, copper, zinc;
construction minerals, such as natural stone, aggregates, sand and gravel, limestone, chalk, gypsum, and;
industrial minerals, divided into physical industrial minerals – for example kaolin, feldspar, talc, and chemical industrial minerals such as salt, potash and sulphur.
Coal, uranium, lignite, peat, brown coal and oil shale are addressed in separate EU legislation – hence the term 'non-energy' to set the rest of the industry apart.
The extraction of aggregates (crushed stone), sand and gravel is evenly spread out across member countries, the Commission states.
Finland, Greece, Ireland, Portugal, Spain and Sweden together account for some 75% of EU production in the metallic minerals sector while France, Greece, Italy, Portugal and Spain account for some 90% of EU production in the quarrying of natural stone. At global level, major producing countries include the United States, Canada, Australia, Russia and China.
Most EU mining companies are SMEs, but Europe also has several large multinationals active worldwide. The EU accounts for around 20% of world production of industrial and construction minerals, and is among the world's largest producers of natural stone, feldspar, kaolin and potash.
In metallic minerals, the EU delivers a mere 2-3% of world production. Many deposits in the EU are state-owned and thus subject to specific national laws on exploration and exploitation.
This is also the case in the countries due to join the Union next year, where big state-owned mines were the name of the game under communism. Poland, for instance, boasts the world's biggest copper mine.
Australia	Canada	China	Finland	France	Greece	Ireland	Italy	Poland	Portugal	Russia	Spain	Sweden	United States


MOSCOW, Aug 20 — RAPSI. Confiscated from corrupt officials billions are transferred to Pension Fund of Russia, said Monday the Secretary of the General Council "United Russia", Vice-speaker of the Federation Council Andrey Turchak.
"People ask at meetings, where billions Zakharchenko, where billions of Mr. Slabikova, and other corruption? Our party must insist that all funds seized from corrupt officials and bribe-takers, as well as from the sale of confiscated their property, sent directly to the Pension Fund," said Turchak at the joint meeting of the General Council and the Council of leaders of factions EP.
Next Post:Rutskoi and the company. What happened to the participants in the August coup?


text:
using System;
using System.Diagnostics;

namespace CSharpSixSamples
{
    public class NameOfExpressions
    {
        [Conditional("DEBUG")]
        public void ShowWhatCanNowBeDone(string someParam)
        {
            // No magic string literals or reflection needed for class names
            Console.WriteLine(nameof(NameOfExpressions));

            // No magic string literals or reflection needed for method names
            Console.WriteLine(nameof(ShowWhatCanNowBeDone));

            // No magic string literals or reflection needed for parameter names
            Console.WriteLine(nameof(someParam));

            // No magic string literals or reflection needed for Attribute names
            Console.WriteLine(nameof(ConditionalAttribute));

            // Can also work by addressing objects with dot syntax
            //Console.WriteLine(nameof(Customer.Address.Postcode));
            // Above would print "Postcode"
        }
    }
}


text:
Q: cron job creating folder + move files I've got an IP camera sending images to my FTP server when triggered a motion. The problem is that it only sending the the file to the folder and it's not categorized after date.
What I need to do is to create a folder with todays date (Y M D) and as soon as the camera has sent images to the root folder of /pics a cron job will move the files to the created date folder.
How can this be done?
Thank you!
Best regards,
Martin

redpajama_set_name: RedPajamaStackExchange


We are dedicated to providing investment management and strategic wealth planning to that is specific to you and your situation. Simply put, we strive to be our client's trusted advisor. As a financial advisory firm, our primary focus is to provide unbiased opinions that are designed to achieve long term investment results.
We are dedicated to professionally supporting, educating, and providing informed direction to each and every client and we do this with the highest levels of trust, integrity and respect while always collaborating using a team approach.


text:
Jidōsha Kōgyō Co. Ltd. (englisch Automobile Industries Co. Ltd.) war ein Hersteller von Kraftfahrzeugen aus Japan.

Unternehmensgeschichte
Das Unternehmen aus Tokio wurde 1933 als gemeinsames Unternehmen von Ishikawajima Jidōsha Seisakusho und Datsun gegründet, allerdings ohne die Pkw-Abteilung von Datsun. Eine Quelle nennt diesbezüglich das Datsun-Werk in Osaka. Eine Quelle gibt an, dass das Unternehmen für die Herstellung von Lastkraftwagen für den zivilen Markt gegründet wurde.

Es stellte Lastkraftwagen, Personenkraftwagen und Militärfahrzeuge her. Der Markenname lautete wie bereits zu Zeiten von Ishikawajima Sumida.

1937 kam es zur Fusion mit der Fahrzeugabteilung von Tōkyō Gas. Eine Quelle nennt als dritten Partner Kyodo Kokusan K.K. Das neue Unternehmen hieß entweder Tokyo Automobile Industry Co. Ltd. oder Tokyo Automobile Industries Co. Ltd. 1941 erfolgte die Umbenennung in Diesel Motor Industry Co. Ltd. und 1949 in Isuzu.  Eine andere Quelle gibt an, dass das neue Unternehmen Automotive Industries Co. Ltd. hieß, während das Werk in Hino als Diesel Motor Industry Company firmierte. Das Werk in Hino wurde 1942 abgespalten, woraus Hino Jidōsha entstand.

Fahrzeuge
Das Modell H war eine viertürige Limousine mit einer Ähnlichkeit zu den damaligen Modellen von LaSalle. Der K-93 war ein Tourenwagen mit sechs Rädern. Der JC 4x4 war ein Stabswagen.

Eine andere Quelle, die japanische Fahrzeuge beschreibt, bestätigt den Zeitraum von 1933 bis 1937 für einige Modelle. Der H hatte einen Motor mit 4398 cm³ Hubraum und 70 PS Leistung. Er war vorne montiert und trieb die Hinterachse an. Einzige Karosseriebauform war eine viertürige Limousine. Einführung war 1934 und Einstellung 1937. Der bereits 1933 erschienene J war ähnlich, hatte aber eine Karosserie als Phaeton mit vier Türen. Der K hatte sechs Räder, von denen vier angetrieben wurden. Da die zweite Hinterachse Platz beanspruchte, hatte der Phaeton nur zwei Türen.

Eine weitere Quelle meint, dass trotz der Fusion 1937 bis 1940 noch einige Pkw mit dem Markennamen Sumida entstanden. Genannt werden  Model JC, Model JD und Model K 10.

Die Breitspur-Triebwagen Typ 91 und Panzerwagen Typ 93 waren Militärfahrzeuge.

Literatur

 Jan P. Norbye: Autos made in Japan. Konzerne · Daten · Fakten. Alle Marken und Modelle 1912–1992. Bleicher Verlag, Gerlingen 1991, ISBN 3-88350-161-1, S. 31–32.

Weblinks

 d'Auto (niederländisch)
 Allcarindex (englisch)

Einzelnachweise

Ehemaliger Pkw-Hersteller (Japan)
Ehemaliger Lkw-Hersteller (Japan)
Unternehmen (Präfektur Tokio)
Gegründet 1933
Aufgelöst 1937
IHI
Wirtschaft (Japanisches Kaiserreich)

redpajama_set_name: RedPajamaWikipedia


text:
Gay "Marriage"
Made in 2008 or so, Gay "Marriage" exposes the dilemma of the maddening and unfair paperwork imposed on gay couples who want the same financial rights as heterosexual married couples. It is a tragicomedy that blends interview and animation to bring to life the horrifying effects of paperwork unfairly imposed on gay couples: JTROS, taxable gifts, powers of attorney, HIPAA, and more.
A star-crossed lesbian couple get the romance knocked out of them as they wake up to legal reality. Music by Mayra Casales. Running time, 12:13 minutes.
In July of 2006, when the NY Court of Appeals upheld the law against gay marriage, I was crushed. It was the final straw on the camel who had been limping since our beloved Bill signed the so-called "Defense of Marriage"act. I was crushed by anger and shame about benefiting from an unjust law just because I am in a heterosexual partnership. What if I had fallen for a woman?
So many of my friends were hurt by this state of affairs that I offered to publicly divorce my male partner, make a film of it, and start a sort-of boycott by moral heterosexuals. My gay friends then set me straight on the matter--told me I would be a fool to give up the legal and financial rights. I really didn't know what they were talking about.
Gay "Marriage", which screened at TrailDance Film Festival and many Living Rooms, shows you what I discovered.
Followup: It is fantabulous that states are starting to question the consitutionality of forbidding the marriage of gay people. Until there is Federal recognition of gay marriage, however, economic injustice prevails.
in Queer People
« Menopause is an Athletic Event: Insomnia! | Main | What is Dis? »


Google Analytics is a free tool which can show you everything you need to know about your web users, informing you which pages and online campaigns are most effective. This workshop focuses on training you on how to use Google Analytics to drive actionable insights from your data. Starting with a little background on how web analytics works, we'll go over and show you how to install Google Analytics on your website. We'll then end with a walk-through of everything you'll need to know about the Google Analytics tool and how to make it work for you. You'll leave class with both knowledge and resources for further learning.