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Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Column() changed from object to string in row 0
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 1025)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 797, in wrapped
for item in generator(*args, **kwargs):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Column() changed from object to string in row 0
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1524, in compute_config_parquet_and_info_response
parquet_operations, partial, estimated_dataset_info = stream_convert_to_parquet(
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1099, in stream_convert_to_parquet
builder._prepare_split(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text string |
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\section{Introduction}
Linear least squares problem is a ubiquitous problem arising frequently in data analysis and scientific computing. Specifically, given a data matrix $A\in R^{m\times n}$ and a data vector $b\in R^{m}$, a linear least squares problem can be written as follows
\begin{equation}
\label{ls}
\min \limi... |
In \cite{Leventhal2010}, Leventhal and Lewis proved that the randomized Gauss--Seidel (RGS) method, also known as the randomized coordinate descent method, converges to the solution at a linear rate in expectation. This method works on the columns of the matrix $A$ at random with probability proportional to their norms... |
Ma et al. \cite{Ma2015} presented a variant of the RGS mehtod, i.e., randomized extended Gauss--Seidel (REGS) method, and proved that the REGS method converges to $x_{\star}$ regardless of whether the matrix $A$ has full column rank. After that, many variants of the RGS (or REGS) method were developed and studied exten... |
To the best of our knowledge, when studying the convergence properties of the RGS and REGS methods, people mainly pay attention to their convergence rates and usually give corresponding upper bounds, and no
work focuses on what determines their convergence rates, what drives their convergence directions, and what their... |
In 2017, Jiao, Jin and Lu \cite{jiao2017preasymptotic} analyzed the preasymptotic convergence of the RK method. Recently, Steinerberger \cite{steinerberger2021randomized} made a more detailed analysis of the convergence property of the RK method for overdetermined full rank linear system. The author showed that the rig... |
In this paper, we analyze the convergence properties of the RGS and REGS methods for linear least squares problem and show that the decay rates of the sequences $\{Ax_{k}\}_{k=1}^{\infty}$ and $\{ x_{k}\}_{k=1}^{\infty}$ (resp., the sequences $\{Az_{k}\}_{k=1}^{\infty}$ and $\{ z_{k}\}_{k=1}^{\infty}$ ) generated by th... |
The rest of this paper is organized as follows. We first introduce some notations and preliminaries in Section \ref{sec2} and then present our main results about the RGS and REGS methods in Section \ref{sec3} and Section \ref{sec4}, respectively. Numerical experiments are given in Section \ref{sec5}. |
\section{Notations and preliminaries }\label{sec2}
Throughout the paper, for a matrix $A$, $A^T$, $A^{(i)}$, $A_{(j)}$,
$\sigma_i(A)$, $\sigma_r(A)$, $\|A\|_F$, and $\mathcal{R}(A)$ denote its transpose, $i$th row (or $i$th entry in the case of a vector), $j$th column, $i$th singular value, smallest nonzero singular va... |
In the following, we use $x_{\star}=A^{\dag}b$ to denote the minimum Euclidean norm least squares solution of (\ref{ls}), where $A^{\dag}$ denotes the Moore--Penrose pseudoinverse of the matrix $A$. Because the SVD is the basic tool for the convergence analysis in next two sections, we denote the SVD \cite{golub2013mat... |
\section{Convergence directions of the RGS method}\label{sec3}
We first list the RGS method \cite{Leventhal2010, Ma2015} in Algorithm \ref{alg1} and restate its convergence bound in Theorem \ref{theorem0}.
\begin{alg}
\label{alg1}
The RGS method
\begin{enumerate}[
\item \mbox{INPUT:} ~$A$, $b$, $\ell$, $x_{0}\in R^{n ... |
\begin{thm}
\label{theorem0}\cite{Leventhal2010, Ma2015}
Let $A\in R^{m\times n}$, $b\in R^{m}$, $x_{\star}=A^{\dag}b$ be the minimum Euclidean norm least squares solution, and $x_k$ be the $k$th approximation of the RGS method generated by Algorithm \ref{alg1} with initial guess $x_{0}\in R^{n }$. Then
\begin{align}
\... |
\begin{rmk}
\label{rmk1}
Theorem \ref{theorem0} shows that $Ax_{k}$ converges linearly in expectation to $Ax_{\star}$ regardless of whether the matrix $A$ has full rank. Since $\| Ax_{k}- Ax_{\star}\|_2^2=\| x_{k}- x_{\star}\|_{A^TA}^2$, it follows from (\ref{th0}) that
\begin{align}
\mathbb{E}[\| x_{k}- x_{\star}\|... |
Now, we give our three main results of the RGS method.
\begin{thm}
\label{theorem1}
Let $A\in R^{m\times n}$, $b\in R^{m}$, $x_{\star}=A^{\dag}b$ be the minimum Euclidean norm least squares solution, and $x_k$ be the $k$th approximation of the RGS method generated by Algorithm \ref{alg1} with initial guess $x_{0}\in R^... |
\begin{pf}
Let $\mathbb{E}_{k-1}[\cdot]$ be the conditional expectation conditioned on the first $k-1$ iterations of the RGS method. Then, from Algorithm \ref{alg1}, we have
\begin{align}
& \mathbb{E}_{k-1}[\langle Ax_{k}- Ax_{\star}, u_{\ell} \rangle]\notag
\\
&= \sum\limits_{j=1}^{n}\frac{ \|A_{ (j )} \|_{2}^{2}}{\... |
\end{align}
which together with the facts $A^T(b-Ax_{\star})=0$ and $A^Tu_{\ell}=\sigma_{\ell}(A) v_{\ell} $ yields
\begin{align}
& \mathbb{E}_{k-1}[\langle Ax_{k}- Ax_{\star}, u_{\ell} \rangle]\notag
\\
&= \langle Ax_{k-1}- Ax_{\star}, u_{\ell} \rangle - \frac{ 1}{\|A\|_{F}^{2}} \langle A^T(Ax_{k-1}-Ax_{\star}) ... |
\\
&= (1- \frac{\sigma_{\ell}^2(A)}{\|A\|_{F}^{2}} ) \langle Ax_{k-1}-Ax_{\star}, u_{\ell}\rangle. \notag
\end{align}
Thus, by taking the full expectation on both sides, we have
\begin{align}
\mathbb{E}[\langle Ax_{k}- Ax_{\star}, u_{\ell} \rangle] = (1- \frac{\sigma_{\ell}^2(A)}{\|A\|_{F}^{2}} ) \mathbb{... |
\end{pf}
\begin{rmk}
\label{rmk2}
Theorem \ref{theorem1} shows that the decay rates of $\|Ax_k-Ax_{\star}\|_2$ are different in different left singular vectors spaces. Specifically, the decay rates are dependent on the singular values: the larger the singular value of $A$ is, the faster the error decays in the corres... |
\begin{rmk}
\label{rmk3}
Let $r_k=b-Ax_k$ be the residual vector with respect to the $k$-th approximation $x_k$, and $r_{\star}=b-Ax_{\star}$ be the true residual vector with respect to the minimum Euclidean norm least squares solution $x_{\star}$. It follows from (\ref{th1}) and $Ax_k-Ax_{\star}=-( r_{k}- r_{\star})$ ... |
\begin{rmk}
\label{rmk4}
Using the facts $\langle Ax_{k}- Ax_{\star}, u_{\ell} \rangle=\langle x_{k}- x_{\star},A^Tu_{\ell} \rangle$ and $A^Tu_{\ell}=\sigma_{\ell}(A) v_{\ell} $, from (\ref{th1}), we have
\begin{align}
\mathbb{E}[\langle x_{k}- x_{\star}, v_{\ell} \rangle]= (1-\frac{\sigma_\ell^2(A)}{\|A\|^2_F})... |
\begin{thm}
\label{theorem2}
Let $A\in R^{m\times n}$, $b\in R^{m}$, $x_{\star}=A^{\dag}b$ be the minimum Euclidean norm least squares solution, and $x_k$ be the $k$th approximation of the RGS method generated by Algorithm \ref{alg1} with initial guess $x_{0}\in R^{n }$. Then
\begin{align}
\mathbb{E}[\| Ax_{k}- Ax_{\s... |
\begin{thm}
\label{theorem3}
Let $A\in R^{m\times n}$, $b\in R^{m}$, $x_{\star}=A^{\dag}b$ be the minimum Euclidean norm least squares solution, and $x_k$ be the $k$th approximation of the RGS method generated by Algorithm \ref{alg1} with initial guess $x_{0}\in R^{n }$. Then
\begin{align}
\mathbb{E}[\langle \frac{Ax_... |
\begin{pf}
From Algorithm \ref{alg1}, we have
\begin{align}
& \mathbb{E}_{k}[\langle \frac{Ax_{k}- Ax_{\star}}{\|Ax_{k}- Ax_{\star}\|_2}, \frac{Ax_{k+1}- Ax_{\star}}{\|Ax_{k+1}- Ax_{\star}\|_2} \rangle^2]\notag
\\
&= \mathbb{E}_{k}[\langle \frac{Ax_{k}- Ax_{\star}}{\|Ax_{k}- Ax_{\star}\|_2}, \frac{Ax_{k}-\frac{A_{... |
\end{align}
Since $\langle Ax_{k}- Ax_{\star} , Ax_{k}-\frac{A_{ (j )}^T(Ax_{k}-b)}{\|A_{ (j )} \|_{2}^{2}}A_{ (j )}- Ax_{\star} \rangle=\|Ax_{k}-\frac{A_{ (j )}^T(Ax_{k}-b)}{\|A_{ (j )} \|_{2}^{2}}A_{ (j )}- Ax_{\star}\|_2^2$, we have
\begin{align}
& \mathbb{E}_{k}[\langle \frac{Ax_{k}- Ax_{\star}}{\|Ax_{k}- Ax_{... |
\\
&= \mathbb{E}_{k}[1-\frac{(A_{ (j )}^T \frac{Ax_{k}-Ax_{\star}}{\|Ax_{k}- Ax_{\star}\|_2 } )^2}{\|A_{ (j )} \|_{2}^{2}} ] \notag
\\
&= \sum\limits_{j=1}^{n}\frac{ \|A_{ (j )} \|_{2}^{2}}{\|A\|_{F}^{2}} (1-\frac{(A_{ (j )}^T \frac{Ax_{k}-Ax_{\star}}{\|Ax_{k}- Ax_{\star}\|_2 } )^2}{\|A_{ (j )} \|_{2}^{2}}) \n... |
\begin{rmk}
\label{rmk6} |
\label{rmk6}
Let $u$ and $v$ are two unit vectors, i.e., $\|u\|_2=1$ and $\|v\|_2=1$. We use inner quantity $\langle u, v\rangle^2$ to represent the angle between $u$ and $v$, and the bigger the angle is, the bigger the fluctuation becomes from $u$ to $v$. Theorem \ref{theorem3} shows the fluctuation of two adjacent it... |
Since $\|A^T\frac{Ax_{k}- Ax_{\star}}{\|Ax_{k}- Ax_{\star}\|_2}\|_2^2\geq\sigma_r^2(A)$, Theorem \ref{theorem3} implies that if $Ax_{k}-Ax_{\star}$ is mainly composed of left singular vectors corresponding to the small singular values of $A$, its direction hardly changes, which means that the RGS method finally converg... |
Now, we list the REGS method presented in \cite{ Dukui2019} in Algorithm \ref{alg2}, which is a equivalent variant of the original REGS method \cite{Ma2015}, and restate its convergence bound presented in \cite{Dukui2019} in Theorem \ref{theorem5}. From the algorithm we find that, in each iteration, $x_k$ is the $k$th ... |
\begin{alg}
\label{alg2}
The REGS method
\begin{enumerate}[
\item \mbox{INPUT:} ~$A$, $b$, $\ell$, $x_{0}\in R^{n }$, $z_{0}\in \mathcal{R}(A^T)$
\item For $k=1, 2, \ldots, \ell-1$ do
\item ~~~~Select $j\in [n]$ with probability $\frac{\|A_{(j)}\|^2_2}{\|A\|^2_F}$
\item ~~~~Set $x_{k}=x_{k-1}-\frac{A_{(j)}^T ( Ax_{k-1... |
\begin{thm}
\label{theorem5}\cite{Dukui2019}
Let $A\in R^{m\times n}$, $b\in R^{m}$, $x_{\star}=A^{\dag}b$ be the minimum Euclidean norm least squares solution, and $z_k$ be the $k$th approximation of the REGS method generated by Algorithm \ref{alg2} with initial $x_{0}\in R^{n }$ and $z_o\in \mathcal{R}(A^T)$. Then
\... |
\begin{thm}
\label{theorem6}
Let $A\in R^{m\times n}$, $b\in R^{m}$, $x_{\star}=A^{\dag}b$ be the minimum Euclidean norm least squares solution, and $z_k$ be the $k$th approximation of the REGS method generated by Algorithm \ref{alg2} with initial $x_{0}\in R^{n }$ and $z_o\in \mathcal{R}(A^T)$. Then
\begin{align}
\ma... |
\\
&= \mathbb{E}[\langle (I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), v_{\ell} \rangle] +\mathbb{E}[\langle \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}}( x_{k}-x_{\star}), v_{\ell} \rangle], \label{th65}
\end{align}
so we next consider $\mathbb{E}[\langle (I-\frac{(A^{(i)})^TA^{(i)}... |
We first consider$\mathbb{E}[\langle (I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), v_{\ell} \rangle]$. Let $\mathbb{E}_{k-1}[\cdot]$ be the conditional expectation conditioned on the first $k-1$ iterations of the REGS method. That is,
\begin{align}
\mathbb{E}_{k-1}[\cdot]= \mathbb{E}[\cdot... |
\end{align}
Then, by the law of total expectation, we have
\begin{align}
\mathbb{E}_{k-1}[\cdot]= \mathbb{E}_{k-1}^{j}[ \mathbb{E}_{k-1}^{i}[\cdot] ]. \notag
\end{align}
Thus, we obtain
\begin{align}
&\mathbb{E}_{k-1}[\langle (I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), v_{\ell} \rangle]... |
\end{align}
Further, by making use of $z_{k-1}-x_{\star}=\sum\limits_{i=1}^{n}\langle z_{k-1}-x_{\star}, v_i\rangle v_i$ and $Av_i=\sigma_i(A) u_i $, we get
\begin{align}
&\mathbb{E}_{k-1}[\langle (I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), v_{\ell} \rangle] \notag
\\
&=\langle z_{k-1}-x_... |
\\
&=\langle z_{k-1}-x_{\star}, v_{\ell} \rangle -\frac{\sigma_{\ell}^2(A)}{\|A\|^2_F} \langle z_{k-1}-x_{\star}, v_{\ell}\rangle \notag
\\
&=(1 -\frac{\sigma_{\ell}^2(A)}{\|A\|^2_F} ) \langle z_{k-1}-x_{\star}, v_{\ell}\rangle . \notag
\end{align}
As a result, by taking the full expectation on both sides, we ... |
We now consider $\mathbb{E}[\langle \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}}( x_{k}-x_{\star}), v_{\ell} \rangle]$. It follows from
\begin{align}
&\mathbb{E}_{k-1}[\langle \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}}( x_{k}-x_{\star}), v_{\ell} \rangle] \notag
\\
&= \mathbb{E}_{k-1}^{j}[ \mathbb{E}_{k-1... |
\end{align}
that
\begin{align}
\mathbb{E} [\langle \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}}( x_{k}-x_{\star}), v_{\ell} \rangle]=\mathbb{E} [\frac{1}{\|A\|^2_F} \langle A ( x_{k}-x_{\star}), A v_{\ell} \rangle].\label{th63}
\end{align}
Since
\begin{align}
\mathbb{E} [\frac{1}{\|A\|^2_F} \langle ... |
\begin{align}
\mathbb{E} [\langle \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}}( x_{k}-x_{\star}), v_{\ell} \rangle]
&=\frac{1}{\|A\|^2_F} (1-\frac{\sigma_\ell^2(A)}{\|A\|^2_F})^k \langle Ax_{0}- Ax_{\star}, A v_{\ell} \rangle \notag
\\
&=\frac{1}{\|A\|^2_F} (1-\frac{\sigma_\ell^2(A)}{\|A\|^2_F})^k \langle... |
Combining (\ref{th65}), (\ref{th62}) and (\ref{th64}) yields
\begin{align}
&\mathbb{E}[\langle z_{k}- x_{\star}, v_{\ell} \rangle] \notag
\\
&= \mathbb{E}[\langle (I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), v_{\ell} \rangle] +\mathbb{E}[\langle \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}... |
\\
&=\ldots=(1 -\frac{\sigma_{\ell}^2(A)}{\|A\|^2_F} )^k \langle z_{0}-x_{\star}, v_{\ell}\rangle +\frac{k}{\|A\|^2_F} (1-\frac{\sigma_\ell^2(A)}{\|A\|^2_F})^k \langle A^T(Ax_{0}- Ax_{\star}), v_{\ell} \rangle, \notag
\end{align}
which is the desired result (\ref{th6}). |
\end{pf}
\begin{rmk}
\label{rmk61}
Theorem \ref{theorem6} shows that the decay rates of $\|z_k-x_{\star}\|_2$ are different in different right singular vectors spaces and the smallest singular value will lead to the slowest rate of convergence, which is the one in (\ref{th5}). So, the convergence bound presented by Du... |
\begin{pf}
Following an analogous argument to Theorem 4 of \cite{Dukui2019}, we get
\begin{align}
\mathbb{E} [\|z_{k}- x_{\star}\|_2^2 ]&= \mathbb{E}[ \|(I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star})\|_2^2]+\mathbb{E} [\| \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}}( x_{k}-x_{\star})\|_... |
\end{align}
Combining the above three equations, we have
\begin{align}
\mathbb{E} [ \|z_{k}- x_{\star}\|_2^2]
\leq\mathbb{E}[(1-\frac{1}{\|A\|^2_F}\|A\frac{z_{k-1}-x_{\star}}{\|z_{k-1}-x_{\star}\|_2}\|_2^2)\|z_{k-1}-x_{\star}\|_2^2]+ \frac{1}{\|A\|^2_F}(1-\frac{\sigma_r^2(A)}{\|A\|^2_F})^k \| Ax_{0}-Ax_{\star} \|^2_2,... |
\begin{rmk}
\label{rmk71}
Since $\|A\frac{z_{k-1}-x_{\star}}{\|z_{k-1}-x_{\star}\|_2}\|_2^2\geq\sigma_r^2(A)$, Theorem \ref{theorem7} implies that $z_{k}$ of the REGS method actually converges faster if $z_{k-1}-x_{\star}$ is not close to right singular vectors corresponding to the small singular values of $A$ .
\end{... |
\begin{pf}
Similar to the proof of (\ref{th65}) in Theorem \ref{theorem6}, we obtain
\begin{align}
\mathbb{E}[\langle Az_{k}- Ax_{\star}, u_{\ell} \rangle]
= \mathbb{E}[\langle A(I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), u_{\ell} \rangle] +\mathbb{E}[\langle A \frac{(A^{(i)})^TA^{(i)}}{... |
We first consider $\mathbb{E}[\langle A(I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), u_{\ell} \rangle]$. It follows from
$$ \langle A(I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), u_{\ell} \rangle = \langle (I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k... |
\\
&=(1 -\frac{\sigma_{\ell}^2(A)}{\|A\|^2_F} ) \mathbb{E}[ \langle z_{k-1}-x_{\star},\sigma_{\ell}(A) v_{\ell}\rangle ]\notag
\\
&=(1 -\frac{\sigma_{\ell}^2(A)}{\|A\|^2_F} ) \mathbb{E}[ \langle Az_{k-1}-Ax_{\star},u_{\ell}\rangle ]. \label{th81}
\end{align} |
We now consider $\mathbb{E}[\langle A \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}}( x_{k}-x_{\star}), u_{\ell} \rangle]$. Exploiting (\ref{th64}), we have
\begin{align}
\mathbb{E}[\langle A \frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}}( x_{k}-x_{\star}), u_{\ell} \rangle]
&=\mathbb{E}[\langle \frac{(... |
Thus, combining (\ref{th80}), (\ref{th81}) and (\ref{th82}) yields
\begin{align}
&\mathbb{E}[\langle Az_{k}- Ax_{\star}, u_{\ell} \rangle] \notag
\\
&= \mathbb{E}[\langle A(I-\frac{(A^{(i)})^TA^{(i)}}{\| A^{(i)} \|_{2}^{2}})( z_{k-1}-x_{\star}), u_{\ell} \rangle] +\mathbb{E}[\langle A \frac{(A^{(i)})^TA^{(i)}}{\| A^... |
\\
&=\ldots=(1 -\frac{\sigma_{\ell}^2(A)}{\|A\|^2_F} )^k \langle Az_{0}-Ax_{\star}, v_{\ell}\rangle +\frac{k}{\|A\|^2_F} (1-\frac{\sigma_\ell^2(A)}{\|A\|^2_F})^k \langle AA^T(Ax_{0}- Ax_{\star}), u_{\ell} \rangle, \notag
\end{align}
which is the desired result (\ref{th8}).
\end{pf} |
\begin{rmk}
\label{rmk81}
Theorem \ref{theorem8} shows the decay rates of $\|Az_{k}- Ax_{\star}\|$ of the REGS method and suggests that small singular values lead to poor convergence rates and vice versa. We note similar issues arise for the RK, REK, and RGS methods discussed in \cite{steinerberger2021randomized}, \ci... |
\section{Numerical experiments }\label{sec5}
Now we present two simple examples to illustrate that the convergence directions of the RGS and REGS methods. To this end, let $G_0\in R^{500\times 500}$ be a Gaussian matrix with i.i.d. $N(0, 1)$ entries and $D\in R^{500\times 500}$ be a diagonal matrix whose diagonal eleme... |
We first consider convergence directions of $Ax_k-Ax_{\star}$ and $x_k-x_{\star}$ of the RGS method. We generate a vector $x \in R^{500}$ using the MATLAB function \texttt{randn}, set the full column rank coefficient matrix $A=A_1$ and set the right-hand side $b=A x +z$, where $z$ is a nonzero vector belonging to the n... |
\begin{figure}[ht]
\begin{center}
\includegraphics [height=5.5cm,width=8.5cm ]{RGS-Ax-u-600-500.eps}
\end{center}
\caption{A sample evolution of $ |\langle (Ax_k-Ax_{\star})/\|Ax_k-Ax_{\star}\|_2, u_{500} \rangle|$ of the RGS method. }\label{fig1}
\end{figure}
From Figure \ref{fig1}, we find that $|\langle (Ax_k-Ax_... |
\begin{figure}[ht]
\begin{center}
\includegraphics [height=5.5cm,width=8.5cm ]{RGS-x-600-500.eps}
\end{center}
\caption{A sample evolution of $\frac{\|A(x_k-x_{\star})\|_2 }{\|x_k-x_{\star}\|_2}$ of the RGS method. }\label{fig2}
\end{figure} |
From Figure \ref{fig2}, we observe that the values of $\frac{\|A(x_k-x_{\star})\|_2 }{\|x_k-x_{\star}\|_2}$ decreases with $k$ and finally approaches the small singular value. This phenomenon implies that the forward direction of $x_k-x_{\star}$ is mainly determined by the right singular vectors corresponding to the la... |
We now consider convergence directions of $Az_k-Ax_{\star}$ and $z_k-x_{\star}$ of the REGS method. We generate a vector $x \in R^{600}$ using the MATLAB function \texttt{randn}, set the coefficient matrix $A=A_2$ which does not have full column rank, and set the right-hand side $b=Ax $. With $x_0=0$ and $z_0=0$, we p... |
\begin{figure}[ht]
\begin{center}
\includegraphics [height=5.5cm,width=8.5cm ]{REGS-z-500-600.eps}
\end{center}
\caption{A sample evolution of $\frac{\|A(z_k-x_{\star})\|_2 }{\|z_k-x_{\star}\|_2}$ of the REGS method. }\label{fig4}
\end{figure}
Figure \ref{fig3} and Figure \ref{fig4} show the similar results obtaine... |
\section{Introduction}
This note discusses methods for the global solution of semi-infinite programs (SIP).
Specifically, the method from \cite{mitsos11} is considered, and it is shown with a counterexample that the lower bounds do not always converge.
Throughout we use notation as close as possible to that used in \ci... |
Global solution of \eqref{eq:sip} often involves the construction of convergent upper and lower bounds.
The approach in \cite{mitsos11} to obtain a lower bound is a modification of the constraint-generation/discretization method of \cite{blankenshipEA76}.
The claim is that the lower-level program may be solved approxim... |
The setting of the method is the following.
The method is iterative and at iteration $k$, for a given finite subset $Y^{LBD,k} \subset Y$, a lower bound of $f^*$ is obtained from the finite program
\begin{alignat}{2}
\label{eq:sip_lower}
f^{LBD,k} =
\inf_{x}\; & f(x) \\
\st
\notag & x \in X, \\
\notag & g(x,y) \... |
If $\sup_y \set{g(\bar{x}^k,y) : y \in Y} \le 0$,
then $\bar{x}^k$ is feasible in \eqref{eq:sip} and thus optimal (since it also solves a relaxation).
Otherwise, set $Y^{LBD,k+1} = Y^{LBD,k} \cup \set{\bar{y}^k}$ and we iterate. |
The precise statement of the claim is repeated here (again, with only minor embellishments to the notation to help keep track of iterations).
\begin{lemma}[Lemma~2.2 in \cite{mitsos11}]
\label{lem:claim}
Take any $Y^{LBD,0} \subset Y$.
Assume that $X$ and $Y$ are compact and that $g$ is continuous on $X \times Y$.
Sup... |
\subsection{Counterexample}
We now present a counterexample to the claim in Lemma~\ref{lem:claim}.
Consider
\begin{alignat}{2}
\tag{CEx}
\label{eq:counter_ex}
\inf_{x}\; & -x \\
\st
\notag & x \in [-1,1], \\
\notag & 2x - y \le 0,\; \forall y \in [-1,1],
\end{alignat}
thus we define
$X = Y = [-1,1]$,
$f : x \... |
\begin{figure}
\begin{center}
\begin{tikzpicture}[xscale=1.5,yscale=1.5]
\draw[fill=gray] (-0.5,-1) -- (0.5,1) -- (1,1) -- (1,-1) -- cycle;
\draw[latex-latex] (0,-1.3) -- (0,1.3) node[above]{$y$};
\draw[latex-latex] (-1.3,0) -- (1.3,0) node[right]{$x$};
\draw (-1,-1) rectangle (1,1);
\draw[dashed,domain=0:1] plot (\x, ... |
The next iteration, with $Y^{LBD,2} = \set{1}$, adds the constraint
$2x - 1 \le 0$
to the lower bounding problem;
the feasible set is $[-1,\sfrac{1}{2}]$ so the minimizer is $\bar{x}^2 = \sfrac{1}{2}$.
Again, assume that solving the lower-level program approximately yields $\bar{y}^2 = \sfrac{1}{2}$;
again we get
\[
2... |
Proceeding in this way, we construct $\bar{x}^k$ and $\bar{y}^k$ so that
$g(\bar{x}^k,\bar{y}^k) > 0$ and
the lower bounds satisfy
$f^{LBD,k} = -\bar{x}^k = -\frac{1}{2^{k-1}}$, for all $k$.
Consequently, they converge to $0$, which we note is strictly less than the infimum of $\sfrac{1}{2}$.
\subsection{Modified ... |
\begin{lemma}
\label{lem:claim_mod}
Choose any finite $Y^{LBD,0} \subset Y$, and $\alpha \in (0,1)$.
Assume that $X$ and $Y$ are compact and that $f$ and $g$ are continuous.
Suppose that at each iteration $k$ of the lower bounding procedure \eqref{eq:llp} is solved approximately for the solution $\bar{x}^k$ of the lowe... |
Otherwise, since $X$ is compact, $Y^{LBD,k}$ is finite, and $f$ and $g$ are continuous, for every iteration the lower bounding problem has a solution by Weierstrass' (extreme value) theorem.
If at some iteration $k$ the lower bounding problem furnishes a point $\bar{x}^k$ for which $g^*(\bar{x}^k) \le 0$, then $\bar{x}... |
Otherwise, we have an infinite sequence of solutions to the lower bounding problems.
Since $X$ is compact we can move to a subsequence $\seq[k \in \mbb{N}]{\bar{x}^{k}} \subset X$ which converges to $x^* \in X$.
By construction of the lower bounding problem we have
\[
g(\bar{x}^{\ell},\bar{y}^k) \le 0, \quad \forall \e... |
\end{equation}
Using \eqref{eq:tails}, we can substitute $x = \bar{x}^k$ in \eqref{eq:gee} to get that for any $\epsilon > 0$, there exists $K$ such that
\[
g(\bar{x}^k,\bar{y}^k) < \epsilon, \quad \forall k \ge K.
\]
By assumption $g(\bar{x}^k,\bar{y}^k) > 0$ for all $k$, and so combined with the above we have that
$g... |
Combining $g(\bar{x}^k,\bar{y}^k) \to 0$ with
$
g(\bar{x}^k,\bar{y}^k) \ge \alpha g^*(\bar{x}^k) > 0,
$
for all $k$,
we see
$
g^*(\bar{x}^k) \to 0.
$
Meanwhile $g^* : X \to \mbb{R}$ is a continuous function, by classic parametric optimization results like \cite[Theorem~1.4.16]{aubin_frankowska} (using continuity of $g... |
The main contribution of \cite{mitsos11} is a novel \emph{upper} bounding procedure, which still stands, and combined with the modified lower bounding procedure from Lemma~\ref{lem:claim_mod} or the original procedure from \cite{blankenshipEA76}, the overall global solution method for \eqref{eq:sip} is still effective.... |
\section{Introduction}
In finite dimensions, one of the simplest methods for proving the
Brouwer fixed point theorem is via a combinatorial result known as
Sperner's lemma \cite{Sper28}, which is a statement about labelled
triangulations of a simplex in $\ensuremath{\mathbb{R}} ^n$. In this paper, we use
Sperner's le... |
Since $\ensuremath{\mathbb{R}} ^\infty$ is locally convex, our theorem
is a consequence of Tychonoff's fixed point theorem \cite{Smar74}.
However, some notable advantages of our approach are:
(1) the constructive nature of Sperner's lemma provides a method for
producing approximate fixed points for functions on the
i... |
\section{Working in $\ensuremath{\mathbb{R}} ^\infty$}
Let $\ensuremath{\mathbb{R}} ^\infty$ and $I^\infty = \prod [0,1]$ be the product of
countably many copies of $\ensuremath{\mathbb{R}} $, and $I=[0,1]$, respectively.
We equip $\ensuremath{\mathbb{R}} ^\infty$ with the standard product topology, which is
metriz... |
The natural extension of this definition to $\ensuremath{\mathbb{R}} ^\infty$ is
to consider $\Delta^\infty$, the convex hull of the standard basis
vectors $\{e_i\}$ in $\ensuremath{\mathbb{R}} ^\infty$, where $(e_i)_j= \delta_{ij}$, the
Kronecker delta function. As convex combinations are finite
sums, this convex hu... |
0\leq x_i\leq 1\}.
\]
It is easy to see that $\ensuremath{\Delta^\infty_0} $ is convex. It is also the closure
of the convex hull of
the standard basis vectors $\{e_i\}$ and $\mathbf{0}$.
It is also compact because it is a closed subset of $I^\infty$,
which is compact by Tychonoff's Theorem.
\footnote{If... |
It will be important for our purposes later to consider $F^n$,
the $n$-dimensional face of $\ensuremath{\Delta^\infty_0} $ given by $F^n = conv\{e_1,e_2,\dots,
e_{n+1}\}$. Notice that each $F^n$ is closed and thus compact.
\section{Some Preliminary Machinery}
\begin{comment
Before stating Sperner's Lemma, which... |
\begin{sperner} (van Mill p.103) Let $\sigma^k$ be a $k$-simplex in $\ensuremath{\mathbb{R}} ^n$ and let $A$ be the set of simplexes in a triangulation of $\sigma^k$. If $h$ is an A-Sperner map for $\sigma^k$ then the number of full simplexes in $A$ is odd and hence non-zero. \end{sperner}
\end{comment |
\end{comment
Let $\sigma^k = conv(x_0,...,x_k)$ be a $k$-simplex in $\ensuremath{\mathbb{R}} ^n$.
Let $T$ be a triangulation of $\sigma^k$ and $V$ be the set of
vertices of $T$ (i.e., the vertices of simplices in $T$).
A \emph{Sperner labelling} of the triangulation $T$
is a labelling function $\ell:V\rightarro... |
Though we will not prove this theorem here, an exposition of such
proofs can be found in
\cite{Su99}. In particular, there are constructive
``path-following'' proofs that locate the full simplex by tracing a
path of simplices through the triangulation. Such path-following
proofs have formed the basis of
algorithms f... |
Another crucial theorem for our purposes states that, under
appropriate hypotheses, the existence of approximate fixed points
implies the existence of fixed points. On the metric space $(X,d)$,
we can quantify the notion of an approximate fixed point by
defining an \textit{$\epsilon$-fixed point}, which for a given ... |
\begin{proof} Let $\{a_n\}$ be a sequence of $1/n$-fixed points.
That is, $d(a_n,f(a_n)) < 1/n$ for all $n$. Since $A$ is compact
it is sequentially compact and thus $\{a_n\}$ has a convergent
subsequence, which we denote $\{a'_n\}$ with $a_n'\rightarrow x \in
A$. Let $\epsilon >0$. Since $a_n'\rightarrow... |
Later it will be desirable to have an isometry between
$\Delta^{n-1}$, the standard $(n-1)$-simplex in $\ensuremath{\mathbb{R}} ^{n}$, and
$F^{n-1}$. The easiest way to do this is to consider
$\ensuremath{\mathbb{R}} ^n$ as a subspace of $\ensuremath{\mathbb{R}} ^\infty$ by projection onto the first $n$
factors, and r... |
\begin{lemma}
\label{le metricequivalence}
Let $A$ be a bounded subset of the normed space $(\ensuremath{\mathbb{R}} ^{n}
, \| \cdot \|_\infty)$. On $A$, the metric $\bar d_n$ is
equivalent to the metric induced by the norm $\| \cdot \|_\infty$.
\end{lemma}
\begin{proof} Suppose that $x,y \in \ensuremath{\mathb... |
Lemma \ref{le metricequivalence} tells us that bounded subsets of $\ensuremath{\mathbb{R}} ^n$ have
the same continuous functions regardless of whether they are
considered as subsets of a normed space or as subsets of $(\ensuremath{\mathbb{R}} ^n , \bar
d_n)$. Importantly, notice that $\Delta^{n-1}$ is bounded.
Furthe... |
\begin{comment}****************
\begin{proof} As proved in (include reference number) the standard
$(n-1)$-simplex in the normed space $\ensuremath{\mathbb{R}} ^n$ as an
arbitrarily small barycentric subdivision, and thus Lemma 2 tells us
that the standard $(n-1)$-simplex in the metric space
$(\ensuremath{\math... |
\begin{proof}
Since $\ensuremath{\Delta^\infty_0} $ is compact, by Lemma \ref{le epsilonfixed}, it is sufficient to show that $f$
has an $\epsilon$-fixed point for each $\epsilon >0$. Let $\epsilon
>0$ be given. Choose $N \geq \log_2(2/\epsilon)+1$. Notice that for
$x,y \in \ensuremath{\Delta^\infty_0} $, this i... |
see that $g:F^N \rightarrow F^N$. Consequently, $g$ is continuous. |
Let $\epsilon_0 = \frac{\epsilon}{8(N+1)}$ and $\epsilon_1 =
\frac{\epsilon}{2^{N+5}(N+1)}$. Since $g$ is continuous on a compact
set, it is uniformly continuous. Thus there exists $\delta_1 >0$
such that $\bar d(x,y) < \delta_1$ implies that $\bar d(g(x),g(y)) <
\epsilon_1$. Let $\delta = \min ( \delta_1 , \epsi... |
Observe that $\ell(x)$
produces a Sperner labeling on the vertices of $\mathscr T$. Thus by
Sperner's Lemma, there exists a fully-labeled simplex in $\mathscr T$.
This simplex can be found using the path-following method described in
\cite{Su99}. Let $\{x^1,x^2, \dots x^{N+1}\}$ be the
vertices of this simplex w... |
we find that for all $i, j$,
\begin{equation}
\label{eq:delta-bound}
|x^1_j - x^i_j| < 2^{N+2}\delta \leq 2^{N+2} \epsilon_1 \leq \epsilon_0.
\end{equation}
By the same logic, we have that for all $i, j$,
\begin{equation}
\label{eq:delta-eps-bound}
|g_j(x^1) - g_j(x^i)| < 2^{N+2}\epsilon_1 \leq \epsilon_0.
\end{equatio... |
g_{N+1}(x^{N+1})& \geq 0 .
\end{array}
\]
Summing down each column yields the following inequality.
\[2\epsilon_0(N+1) + \sum_{i=1}^{N+1} x^1_i - \sum_{i=1}^{N+1}
g_i(x^1) > \sum_{i=1}^{N+1}\left( x_i^i-g_i(x^i)\right) \geq 0.
\]
Now we recall that for all $i$, $x_i^i-g_i(x^i) \geq 0$ and
\[
\sum_{i=1}^{N+1} x^1_i -... |
< 2 \epsilon_0 + |x^i_i - g_i(x^i)|
$.
Hence,
\[ \begin{split} \bar d(x^1 , g(x^1)) = \sum_{i=1}^{N+1} \frac{|x^1_i - g_i(x^1)|}{2^i(1+|x^1_i - g_i(x^1)|)} & \leq \sum_{i=1}^{N+1} |x^1_i - g_i(x^1)| \\
& < \sum_{i=1}^{N+1} \left(2 \epsilon_0 + |x^i_i - g_i(x^i)|\right) \\
& < 4(N+1)\epsilon_0 \\
& = \frac{\epsi... |
& =\sum_{i=1}^{N} \frac{|y_i - f_i(y)|}{2^i(1+|y_i - f_i(y)|)}
+\sum_{i=N+1}^\infty \frac{|y_i - f_i(y)|}{2^i(1+|y_i - f_i(y)|)} \\
& < \frac{\epsilon}{2}+\frac{\epsilon}{2}\\
&=\epsilon .
\end{split}
\]
Therefore, $y$ is the desired $\epsilon$-fixed point.\end{proof}
Notice that the construction of the $\epsi... |
\section{Schauder's Theorem}
A well-known infinite-dimensional fixed point theorem that holds for
normed spaces is Schauder's theorem \cite{DuGr82, Smar74}:
\begin{sch} Suppose that $X$ is a compact convex subset of the normed
space $G$. If $f:X\rightarrow X$ is continuous, then $f$ has a
fixed point. \end{sch}... |
Define the vector space $H$ to be
\[H = \{x \in \ensuremath{\mathbb{R}} ^\infty | \sum_{i=1}^\infty \frac{|x_i|}{2^i} < \infty \}.\]
It is not difficult to see that $H$ is indeed a vector space.
Furthermore, we see that $\|x\| = \sum_{i=1}^\infty \frac{|x_i|}{2^i}$
defines a norm on this space and the closure of the st... |
\begin{prop} Every infinite-dimensional compact convex subset of a
normed space is homeomorphic to the Hilbert Cube.
\end{prop}
The significance of these propositions is that \emph{every}
infinite-dimensional compact convex subset of a normed space is
homeomorphic to $\ensuremath{\Delta^\infty_0} $.
Thus Theorem \r... |
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