A Randomized Nonmonotone Block Proximal Gradient Method for a Class of Structured Nonlinear Programming

Introduction

Nowadays first-order (namely, gradient-type) methods are the prevalent tools for solving large-scale problems arising in science and engineering. As the size of problems becomes huge, it is, however, greatly challenging to these methods because gradient evaluation can be prohibitively expensive. Due to this reason, block coordinate descent (BCD) methods and their variants have been studied for solving various large-scale problems (see, for example, ). Recently, Nesterov proposed a randomized BCD (RBCD) method, which is promising for solving a class of huge-scale convex optimization problems, provided the involved partial gradients can be efficiently updated. The iteration complexity for finding an approximate optimal solution is analyzed in . More recently, Richtárik and Takáč extended Nesterov’s RBCD method to solve a more general class of convex optimization problems in the form of

where $f$ is convex differentiable in $\Re^{N}$ and $\Psi$ is a block separable convex function. More specifically,

where each $x_{i}$ denotes a subvector of $x$ with cardinality $N_{i}$, $\{x_{i}:i=1,\ldots,n\}$ form a partition of the components of $x$, and each $\Psi_{i}:\Re^{N_{i}}\to\Re\cup\{+\infty\}$ is a closed convex function.

Given a current iterate $x^{k}$, the RBCD method picks $i\in\{1,\ldots,n\}$ uniformly, solves a block-wise proximal subproblem in the form of

and sets $x^{k+1}_{i}=x^{k}_{i}+d_{i}(x^{k})$ and $x^{k+1}_{j}=x^{k}_{j}$ for all $j\neq i$, where $\nabla_{i}f\in\Re^{N_{i}}$ is the partial gradient of $f$ with respect to $x_{i}$ and $L_{i}>0$ is the Lipschitz constant of $\nabla_{i}f$ with respect to the norm $\|\cdot\|$ (see Assumption 1 for details). The iteration complexity of finding an approximate optimal solution with high probability is established in and has recently been improved by Lu and Xiao . Very recently, Patrascu and Necoara extended this method to solve problem (1) in which $F$ is nonconvex, and they studied convergence of the method under the assumption that the block is chosen uniformly at each iteration.

One can observe that for $n=1$, the RBCD method becomes a classical proximal (full) gradient method with a constant stepsize $1/L$. It is known that the latter method tends to be practically much slower than the same type of methods but with variable stepsizes, for example, spectral-type stepsize ) that utilizes partial local curvature information of the smooth component $f$. The variable stepsize strategy shall also be applicable to the RBCD method and improve its practical performance dramatically. In addition, the RBCD method is a monotone method, that is, the objective values generated by the method are monotonically decreasing. As mentioned in the literature (see, for example, ), nonmonotone methods often produce solutions of better quality than the monotone counterparts for nonconvex optimization problems. These motivate us to propose a randomized nonmonotone block proximal gradient method with variable stepsizes for solving a class of (possibly nonconvex) structured nonlinear programming problems in the form of (1) satisfying Assumption 1 below.

Throughout this paper we assume that the set of optimal solutions of problem (1), denoted by $X^{*}$, is nonempty and the optimal value of (1) is denoted by $F^{\star}$. For simplicity of presentation, we associate $\Re^{N}$ with the standard Euclidean norm, denoted by $\|\cdot\|$. We also make the following assumption.

$f$ is differentiable (but possibly nonconvex) in $\Re^{N}$. Each $\Psi_{i}$ is a (possibly nonconvex nonsmooth) function from $\Re^{N_{i}}$ to $\Re\cup\{+\infty\}$ for $i=1,\ldots,n$. The gradient of function $f$ is coordinate-wise Lipschitz continuous with constants $L_{i}>0$ in $\Re^{N}$, that is,

This paper is organized as follows. In Section 2 we propose a RNBPG method for solving structured nonlinear programming problem (1) and analyze its convergence. In Section 3 we analyze the convergence of RNBPG for solving structured convex problem. In Section 4 we conduct numerical experiments to compare RNBPG method with the RBCD method with fixed or variable stepsizes.

Before ending this section we introduce some notations that are used throughout this paper and also state some known facts. The domain of the function $F$ is denoted by ${\rm dom}(F)$. $t^{+}$ stands for $\max\{0,t\}$ for any real number $t$. Given a closed set $S$ and a point $x$, ${\rm dist}(x,S)$ denotes the distance between $x$ and $S$. For symmetric matrices $X$ and $Y$, $X\preceq Y$ means that $Y-X$ is positive semidefinite. Given a positive definite matrix $\Theta$ and a vector $x$, $\|x\|_{\Theta}=\sqrt{x^{T}\Theta x}$. In addition, $\|\cdot\|$ denotes the Euclidean norm. Finally, it immediately follows from Assumption 1 that

By Lemma 2 of Nesterov and Assumption 1, we also know that $\nabla f$ is Lipschitz continuous with constant $L_{f}:=\sum_{i}L_{i}$, that is,

Randomized nonmonotone block proximal gradient method

In this section we propose a RNBPG method for solving structured nonlinear programming problem (1) and analyze its convergence.

We start by presenting a RNBPG method as follows. At each iteration, this method randomly picks a block according to any prescribed (not necessarily uniform) probability distribution and solves typically several associated proximal subproblems in the form of (2) with $L_{i}$ replaced by some $\theta_{k}$ until a certain progress on objective value is achieved.

Randomized nonmonotone block proximal gradient (RNBPG) method Choose $x^{0}\in{\rm dom}(F)$, $\eta>1$, $\sigma>0$, $0<\underline{{\theta}}\leq\bar{\theta}$, integer $M\geq 0$, and $0<p_{i}<1$ for $i=1,\ldots,n$ such that $\sum^{n}_{i=1}p_{i}=1$. Set $k=0$.

Set $d^{k}=0$. Pick $i_{k}=i\in\{1,\ldots,n\}$ with probability $p_{i}$. Choose ${\theta}_{k}^{0}\in[\underline{{\theta}},\bar{\theta}]$.

Let ${\theta}_{k}={\theta}_{k}^{0}\eta^{j}$. Compute

Set $x^{k+1}=x^{k}+d^{k}$, $k\leftarrow k+1$ and go to step 1).

The above method becomes a monotone method if $M=0$.

Before studying convergence of RNBPG, we introduce some notations and state some facts that will be used subsequently.

Let ${\bar{d}}^{k,i}$ denote the vector $d^{k}$ obtained in Step (2) of RNBPG if $i_{k}$ is chosen to be $i$. Define

One can observe that $({\bar{d}}^{k,i})_{t}=0$ for $t\neq i$ and there exist $\theta^{0}_{k,i}\in[\underline{{\theta}},\bar{\theta}]$ and the smallest nonnegative integer $j$ such that $\theta_{k,i}=\theta^{0}_{k,i}\eta^{j}$ and

Let $\Theta_{k}$ denote the block diagonal matrix $(\theta_{k,1}I_{1},\ldots,\theta_{k,n}I_{n})$, where $I_{i}$ is the $N_{i}\times N_{i}$ identity matrix. By the definition of ${\bar{d}}^{k}$ and (9), we observe that

After $k$ iterations, RNBPG generates a random output $(x^{k},F(x^{k}))$, which depends on the observed realization of random vector

We define ${\bf E}_{\xi_{-1}}[F(x^{0})]=F(x^{0})$. Also, define

The following lemma establishes some relations between the expectations of $\|d^{k}\|$ and $\|{\bar{d}}^{k}\|$.

Let $d^{k}$ be generated by RNBPG and ${\bar{d}}^{k}$ defined in (7). There hold

Proof. By (13) and the definitions of $d^{k}$ and ${\bar{d}}^{k}$, we can observe that

The conclusion of this lemma follows by taking expectation with respect to $\xi_{k-1}$ on both sides of the above inequalities.

We next show that the inner loops of the above RNBPG method must terminate finitely. As a byproduct, we provide a uniform upper bound on $\Theta_{k}$.

Let $\{\theta_{k}\}$ be the sequence generated by RNBPG, $\Theta_{k}$ defined above, and $c$ defined in (14). There hold

$\underline{{\theta}}\leq\theta_{k}\leq c\quad\quad\forall k$.

$\underline{{\theta}}I\preceq\Theta_{k}\preceq cI\quad\quad\forall k$.

Proof. (i) It is clear that $\theta_{k}\geq\underline{{\theta}}$. We now show $\theta_{k}\leq c$ by dividing the proof into two cases.

Case (i) $\theta_{k}=\theta^{0}_{k}$. Since $\theta^{0}_{k}\leq\bar{\theta}$, it follows that $\theta_{k}\leq\bar{\theta}$ and the conclusion holds.

Case (ii) $\theta_{k}=\theta^{0}_{k}\eta^{j}$ for some integer $j>0$. Suppose for contradiction that $\theta_{k}>c$. By (13) and (14), we then have

Let $d\in\Re^{N}$ such that $d_{i}=0$ for $i\neq i_{k}$ and

On the other hand, using (3), (10), (17), (18) and the definition of $d$, we have

which is a contradiction to (19). Hence, $\theta_{k}\leq c$ and the conclusion holds.

(ii) Let $\theta^{k,i}$ be defined above. It follows from statement (i) that $\underline{{\theta}}\leq\theta_{k,i}\leq c$, which together with the definition of $\Theta_{k}$ implies that statement (ii) holds.

The next result provides some bound on the norm of a proximal gradient, which will be used in the subsequent analysis on convergence rate of RNBPG.

Let $\{x^{k}\}$ be generated by RNBPG, ${\bar{d}}^{k}$ and $c$ defined in (11) and (14), respectively, and

Assume that $\Psi$ is convex. There holds

We note that by the definition in (14), we have $c\geq\overline{{\theta}}>\underline{{\theta}}$, which implies

Therefore, the expression under the square root in (21) is always positive.

In this subsection we show that the sequence of expected objective values generated by the method converge to the expected limit of the objective values obtained by a random single run of the method.

The following lemma studies uniform continuity of the expectation of $F$ with respect to random sequences.

Suppose that $F$ is uniform continuous in some $S\subseteq{\rm dom}(F)$. Let $y^{k}$ and $z^{k}$ be two random vectors in $S$ generated from $\xi_{k-1}$. Assume that there exists $C>0$ such that $|F(y^{k})-F(z^{k})|\leq C$ for all $k$, and moreover,

Proof. Since $F$ is uniformly continuous in $S$, it follows that given any $\epsilon>0$, there exists $\delta_{\epsilon}>0$ such that $|F(x)-F(y)|<\epsilon/2$ for all $x,y\in S$ satisfying $\|x-y\|<\delta_{\epsilon}$. Using these relations, the Markov inequality, and the assumption that $|F(y^{k})-F(z^{k})|\leq C$ for all $k$ and $\lim_{k\to\infty}{\bf E}_{\xi_{k-1}}[\|\Delta^{k}\|]=0$, where $\Delta^{k}=y^{k}-z^{k}$, we obtain that for sufficiently large $k$,

Due to the arbitrarily of $\epsilon$, we see that the first statement of this lemma holds. The second statement immediately follows from the first statement and the well-known inequality

We are ready to establish the first main result, that is, the expected objective values generated by the RNBPG method converge to the expected limit of the objective values obtained by a random single run of the method.

Let $\{x^{k}\}$ and $\{d^{k}\}$ be the sequences generated by the RNBPG method. Assume that $F$ is uniform continuous in $\Omega(x^{0})$, where $\Omega(x^{0})$ is defined in (12). Then the following statements hold:

$\lim_{k\to\infty}[\|d^{k}\|]=0$ and $\lim_{k\to\infty}F(x^{k})=F^{*}_{\xi_{\infty}}$ for some $F^{*}_{\xi_{\infty}}\in\Re$, where $\xi_{\infty}=\{i_{1},i_{2},\cdots\}$.

$\lim_{k\to\infty}{\bf E}_{\xi_{k}}[\|d^{k}\|]=0$ and

We first show by induction that the following relations hold for all $j\geq 1$:

It follows from this relation and (28) that

One can also observe that $F(x^{k})\leq F(x^{0})$ and hence $\{x^{k}\}\subset\Omega(x^{0})$. Using this fact, (24), (30), Lemma 2.5, and uniform continuity of $F$ over $\Omega(x^{0})$, we obtain that

Using (24), (31), (32), the induction hypothesis, and a similar argument as above, we can obtain that

These relations, together with Lemma 2.5, uniform continuity of $F$ over $\Omega(x^{0})$ and the induction hypothesis, yield

Hence, (25) and (26) hold for $j+1$, and the proof of (25) and (26) is completed.

These, together with (25), (26), (34), Lemma 2.5 and uniform continuity of $F$ over $\Omega(x^{0})$, imply that

It follows from (35) that $\lim\limits_{k\to\infty}F(x^{k})=F^{*}_{\xi_{\infty}}$. Using this, (23) and (24), one can see that $\lim_{k\to\infty}\|d^{k}\|=0$. Hence, statement (i) holds. Notice that ${\bf E}_{\xi_{k-M-2}}[F(x^{k-M-1})]={\bf E}_{\xi_{k-1}}[F(x^{k-M-1})]$. Combining this relation with (36), we have

Using (37) and (38), we conclude that $\lim_{k\to\infty}{\bf E}_{\xi_{k}}[\|d^{k}\|]=0$.

Finally, we claim that $\lim_{k\to\infty}{\bf E}_{\xi_{k-1}}[F(x^{k})]={\bf E}_{\xi_{\infty}}[F^{*}_{\xi_{\infty}}]$. Indeed, we know that $\{x^{k}\}\subset\Omega(x^{0})$. Hence, $F^{*}\leq F(x^{k})\leq F(x^{0})$, where $F^{*}=\min_{x}F(x)$. It follows that

Using this relation and dominated convergence theorem (see, for example, [2, Theorem 5.4]), we have

which, together with $\lim_{k\to\infty}{\bf E}_{\xi_{k-1}}[F(x^{k})]=\lim_{k\to\infty}{\bf E}_{\xi_{\infty}}[F(x^{k})]$, implies that $\lim_{k\to\infty}{\bf E}_{\xi_{k-1}}[F(x^{k})]={\bf E}_{\xi_{\infty}}[F^{*}_{\xi_{\infty}}]$. Hence, statement (ii) holds.

2 Convergence to stationary points

In this subsection we show that when $k$ is sufficiently large, $x^{k}$ is an approximate stationary point of (1) with high probability.

Let $\{x^{k}\}$ be generated by RNBPG, and ${\bar{d}}^{k}$ and ${\bar{x}}^{k}$ defined in (7). Assume that $F$ is uniformly continuous and $\Psi$ is locally Lipschitz continuous in $\Omega(x^{0})$, where $\Omega(x^{0})$ is defined in (12). Then there hold

where $\partial\Psi$ denotes the Clarke subdifferential of $\Psi$.

Any accumulation point of $\{x^{k}\}$ is a stationary point of problem (1) almost surely.

Suppose further that $F$ is uniformly continuous in

Then $\lim_{k\to\infty}{\bf E}_{\xi_{k-1}}[|F(x^{k})-F({\bar{x}}^{k})|]=0$. Moreover, for any $\epsilon>0$ and $\rho\in(0,1)$, there exists $K$ such that for all $k\geq K$,

Proof. (i) We know from Theorem 2.6 (ii) that $\lim_{k\to\infty}{\bf E}_{\xi_{k}}[\|d^{k}\|]=0$, which together with (16) implies $\lim_{k\to\infty}{\bf E}_{\xi_{k-1}}[\|{\bar{d}}^{k}\|]=0$. Notice that ${\bar{d}}^{k}$ is an optimal solution of problem (11). By the first-order optimality condition (see, for example, Proposition 2.3.2 of ) of (11) and ${\bar{x}}^{k}=x^{k}+{\bar{d}}^{k}$, one can have

Using this relation along with Lemma 2.3 (ii) and (41), we obtain that

which together with the first relation of (39) implies that the second relation of (39) also holds.

(ii) Let $x^{*}$ be an accumulation point of $\{x^{k}\}$. There exists a subsequence ${\cal K}$ such that $\lim_{k\in{\cal K}\to\infty}x^{k}=x^{*}$. Since ${\bf E}_{\xi_{k-1}}[\|{\bar{d}}^{k}\|]\to 0$, it follows that $\{{\bar{d}}^{k}\}_{k\in{\cal K}}\to 0$ almost surely. This together with the second relation of (39) and outer semi-continuity of $\partial\Psi$ yields

almost surely. Hence, $x^{*}$ is a stationary point of problem (1) almost surely.

(iii) Recall that ${\bar{x}}^{k}=x^{k}+{\bar{d}}^{k}$. It follows from (4) that

Using this relation and Lemma 2.3 (ii), we have

Using this relation and the fact that $F({\bar{x}}^{k})\geq F^{*}$ and $F(x^{k})\leq F(x^{0})$, one can obtain that

In addition, since $F^{l(k)}\leq F(x^{0})$ and $F({\bar{x}}^{k})\geq F^{*}$, it follows from (8) that $\|{\bar{d}}^{k,i}\|^{2}\leq 2(F(x^{0})-F^{*})/\sigma$. Hence, one has

This inequality together with (43) yields

and hence $\{|F({\bar{x}}^{k})-F(x^{k})|\}$ is bounded. Also, this inequality together with $F(x^{k})\leq F(x^{0})$ and the definition of ${\cal S}$ implies that ${\bar{x}}^{k}$, $x^{k}\in{\cal S}$ for all $k$. In addition, by statement (i), we know ${\bf E}_{\xi_{k-1}}[\|x^{k}-{\bar{x}}^{k}\|]\to 0$. In view of these facts and invoking Lemma 2.5, one has

Using these inequalities, (44) and statement (i), we see that

The rest of statement (iii) follows from this relation and the Markov inequality.

3 Convergence rate analysis

In this subsection we establish a sublinear rate of convergence of RNBPG in terms of the minimal expected squared norm of certain proximal gradients over the iterations.

Let ${\bar{g}}^{k}=-\Theta_{k}{\bar{d}}^{k}$, $p_{\min}$, ${\hat{g}}^{k}$ and $c$ be defined in (13), (20) and (14), respectively, and $F^{*}$ the optimal value of (1). The following statements hold

Assume further that $\Psi$ is convex. Then

Proof. (i) Using ${\bar{g}}^{k}=-\Theta_{k}{\bar{d}}^{k}$, Lemma 2.3 (ii), and (15), one can observe that

Let $j(t)=l((M+1)t)-1$ and ${\bar{j}}(t)=(M+1)t-1$ for all $t\geq 0$. One can see from (29) that

Summing up the above inequality over $t=1,\ldots,s$, we have

which together with ${\bf E}_{\xi_{{\bar{j}}(s)}}[F(x^{j(s)+1})]\geq F^{*}$ implies that

Given any $k\geq M$, let $s_{k}=\lfloor(k+1)/(M+1)\rfloor$. Observe that

which together with (45) implies that statement (i) holds.

Using this relation and a similar argument as above, one has

Statement (ii) immediately follows from this inequality and (21).

Convergence analysis for structured convex problems

In this section we study convergence of RNBPG for solving structured convex problem (1). To this end, we assume throughout this section that $f$ and ${\Psi}$ are both convex functions.

The following result shows that $F(x^{k})$ can be arbitrarily close to the optimal value $F^{*}$ of (1) with high probability for sufficiently large $k$.

Let $\{x^{k}\}$ be generated by the RNBPG method, and let $F^{*}$ and $X^{*}$ the optimal value and the set of optimal solutions of (1), respectively. Suppose that $f$ and ${\Psi}$ are convex functions and $F$ is uniformly continuous in ${\cal S}$, where ${\cal S}$ is defined in (40). Assume that there exists a subsequence ${\cal K}$ such that $\{{\bf E}_{\xi_{k-1}}[{\rm dist}(x^{k},X^{*})]\}_{{\cal K}}$ is bounded. Then there hold:

For any $\epsilon>0$ and $\rho\in(0,1)$, there exists $K$ such that for all $k\geq K$,

Proof. (i) Let ${\bar{d}}^{k}$ be defined in(7). Using the assumption that $F$ is uniformly continuous in ${\cal S}$ and Theorem 2.7, one has

which, together with (47) and the assumption that $\{{\bf E}_{\xi_{k-1}}[{\rm dist}(x^{k},X^{*})]\}_{{\cal K}}$ is bounded, implies that

Using this relation, (48) and (49), we obtain that

(ii) Statement (ii) immediately follows from statement (i), the Markov inequality, and the fact $F(x^{k})\ \geq\ F^{*}$.

In the rest of this section we study the rate of convergence of a monotone version of RNBPG, i.e., $M=0$, or equivalently, (6) is replaced by

The following lemma will be subsequently used to establish a sublinear rate of convergence of RNBPG with $M=0$.

Suppose that a nonnegative sequence $\{\Delta_{k}\}$ satisfies

Proof. We divide the proof into two cases.

Case (i): Suppose $\Delta_{k}>0$ for all $k\geq 0$. Let ${\bar{\Delta}}_{k}=1/\Delta_{k}$. It follows from (51) that

which together with ${\bar{\Delta}}_{k}>0$ implies that

where $\beta=\min\left\{\alpha/2,{\bar{\Delta}}_{0}\right\}$. By the definition of $\beta$, one can see that (53) holds for $k=0$. Suppose it holds for some $k\geq 0$. We now need to show (53) also holds for $k+1$. Indeed, since $\beta\leq\alpha/2$, we have

Using this inequality, (52) and the induction hypothesis ${\bar{\Delta}}_{k}\geq\beta(k+1)$, we obtain that

namely, (53) holds for $k+1$. Hence, the induction is completed and (53) holds for all $k\geq 0$. The conclusion of this lemma follows from (53) and the definitions of ${\bar{\Delta}}_{k}$ and $\beta$.

We next establish a sublinear rate of convergence on the expected objective values for the RNBPG method with $M=0$ when applied to problem (1), where $f$ and $\psi$ are assumed to be convex. Before proceeding, we define the following quantities

where $X^{*}$ denotes the set of optimal solutions of (1) and $\Omega(x^{0})$ is defined in (12).

Let $c,r,q$ be defined in (14), (54), (55), respectively. Assume that $r$ and $q$ are finite. Suppose that $\Psi$ is $L_{\Psi}$-Lipschitz continuous in ${\rm dom}(\Psi)$, namely,

for some $L_{\Psi}>0$. Let $\{x^{k}\}$ be generated by RNBPG with $M=0$. Then

Proof. Let ${\bar{x}}^{k}$ be defined in (7). For each $x^{k}$, let $x^{k}_{*}\in X^{*}$ such that $\|x^{k}-x^{k}_{*}\|={\rm dist}(x^{k},X^{*})$. Due to $x^{k}\in\Omega(x^{0})$ and (54), we know that $\|x^{k}-x^{k}_{*}\|\leq r$. By the definition of ${\bar{x}}^{k+1}$ and (11), one can observe that

Using this inequality, (55), and (56), we have

where the first inequality follows from convexity of $f$ and (56), the second inequality is due to (55), the third inequality follows from (58), and the last inequality is due to $\|x^{k}-x^{k}_{*}\|\leq r$. The preceding inequality, (16) and the fact $F(x^{k+1})\leq F(x^{k})$ yield

In addition, using $\left({\bf E}_{\xi_{k-1}}[\|d^{k}\|]\right)^{2}\leq{\bf E}_{\xi_{k-1}}[\|d^{k}\|^{2}]$ and (50), one has

Let $\Delta_{k}={\bf E}_{\xi_{k-1}}[F(x^{k})]-F^{*}$. Combining the preceding two inequalities, we obtain that

where $\alpha$ is defined in (57). Notice that $\Delta_{0}=F(x^{0})-F^{*}$. Using this relation, the definition of $\Delta_{k}$, and Lemma 3.2, one can see that the conclusion of this theorem holds.

The next result shows that under an error bound assumption the RNBPG method with $M=0$ is globally linearly convergent in terms of the expected objective values.

Let $\{x^{k}\}$ be generated by RNBPG. Suppose that there exists $\tau>0$ such that

where ${\hat{g}}^{k}$ is given in (20) and $X^{*}$ denotes the set of optimal solutions of (1). Then there holds

Proof. For each $x^{k}$, let $x^{k}_{*}\in X^{*}$ such that $\|x^{k}-x^{k}_{*}\|={\rm dist}(x^{k},X^{*})$. Let ${\bar{d}}^{k}$ be defined in (7), and

Using this inequality, (11) and Lemma 2.3 (ii), we have that

where $\gamma=c+L_{f}$. Using this relation and (59), one can obtain that

It follows from this inequality and (21) that

In addition, by (3) and the definition of ${\bar{d}}^{k,i}$, we have

Using these two inequalities, we can obtain that

where the first inequality follows from (61) and the second inequality is due to (62). Taking expectation with respect to $\xi_{k-1}$ on both sides of the above inequality gives

Using this inequality and (60), we obtain that

where $\varpi$ is defined above. In addition, it follows from (50) that

Combining these two inequalities, we obtain that

and the conclusion of this theorem immediately follows.

The error bound condition (59) holds for a class of problems, especially when $f$ is strongly convex. More discussion about this condition can be found, for example, in .

Numerical experiments

where $A\in\Re^{m\times N}$, $b\in\Re^{m}$, and $\lambda>0$ is a regularization parameter. Clearly, this problem is a special case of the general model (1) with $f(x)=\|Ax-b\|_{2}^{2}/2$ and $\Psi(x)=\lambda\|x\|_{1}$ and thus our proposed RNBPG method can be suitably applied to solve it.

We generated a random instance with $m=1000$ and $N=2000$ following the procedure described in [19, Section 6]. The advantage of this procedure is that an optimal solution $x^{*}$ is generated together with $A$ and $b$, and hence the optimal value $F^{*}$ is known. We generated an instance where the optimal solution $x^{*}$ has only $200$ nonzero entries, so this can be considered as a sparse recovery problem. We compare RNBPG with the following methods:

RBCD: The RBCD method with constant step sizes $1/L_{i}$ determined by the Lipschitz constants $L_{i}$. Here, $L_{i}=\|A_{:,i}\|_{2}^{2}$ where $A_{:,i}$ is the $i$th column block corresponding to the block partitions of $x_{i}$ and $\|\cdot\|_{2}$ is the matrix spectral norm.

RBCD-LS: A variant of RBCD method with variable stepsizes that are determined by a block-coordinate-wise backtracking line search scheme. This method can also be regarded as a variant of RNBPG with $M=0$, but which has the property of monotone descent.

As discussed in , the structure of the least-squares function $f(x)=\|Ax-b\|_{2}^{2}/2$ allows efficient computation of coordinate gradients, with cost of $O(mN_{i})$ operations for block $i$ as opposed to $O(mN)$ for computing the full gradient. We note that the same structure also allows efficient computation of the function value, which costs the same order of operations as computing coordinate gradients. Therefore the backtracking line search used in RBCD-LS as well as the nonmonotone line search used in RNBPG (both relies on computing function values), have the same order computational cost as evaluating coordinate gradients at each iteration. Therefore we can focus on comparing their required number of iterations to obtain the same accuracy in reducing the objective value.

We run each algorithm with four different block coordinate sizes $N_{i}=1,20,200,2000$ for all $i$. For each blocksize, we pick the block coordinates uniformly at random at each iteration. Note that $N_{i}=2000=N$ gives the full gradient versions of the methods considered, which are deterministic algorithms. We choose the same initial point $x^{0}=0$ for all three methods.

For the RNBPG method, we used the parameters $M=10$, $\eta=1.1$, $\underline{{\theta}}=10^{-8}$, $\bar{\theta}=10^{8}$ and $\sigma=10^{-4}$. In addition, we used the Barzilai-Borwein spectral method to compute the initial estimate $\theta_{k}^{0}$. That is, we choose

Figure 1 shows the behavior of different algorithms with the four different block coordinate sizes. For $N_{i}=1$ in Figure 1, RBCD has slightly better convergence speed than RBCD-LS and RNBPG. The reason is that in this case, along each block $f$ becomes an one-dimensional quadratic function, and the value $L_{i}=\|A_{:,i}\|_{2}^{2}$ gives the accurate second partial derivative of $f$ along each dimension. Therefore in this case the RBCD method essentially uses the best step size, which is generally better than the ones used in RBCD-LS and RNBPG.

When the blocksize $N_{i}$ is larger than one, the value $L_{i}=\|A_{:,i}\|_{2}^{2}$ is the magnitude of second derivative along the most curved direction. Line search based methods may take advantage of the possibly much smaller local curvature along the search direction by taking larger step sizes. Figure 1 , and show that RBCD-LS converges much faster that RBCD while RNBPG (with $M=10$) converges substantially faster than RBCD-LS.

Figure 2 shows more comprehensive study of the performance of the three methods: RBCD, RBCD-LS, and RNBPG. Figure 2 shows the number of iterations of different methods required to reach the precision $F(x^{k})-F(x^{*})\leq 10^{-6}$, when using 10 different block sizes ranging from 1 to 2000 with equal logarithmic spacing. Figure 2 shows the number of epochs required to reach the same precision, where each epoch corresponds to one equivalent pass over the dataset $A\in\Re^{m\times N}$, that is, equivalent to $N/N_{i}$ iterations. For each method and each block size, we record the results of 10 runs with different random sequences to pick the block coordinates, and plot the mean with the standard deviation as error bars. As we can see, the number of iterations in general decreases when we increase the block size, because each iteration involves more coordinates and more computation. On the other hand, the number of epochs required increases with the block size, meaning that larger block size updates are less efficient than small block size updates.

The above observations suggest that using larger block sizes is less efficient in terms of the overall computation work (e.g., measured in total flops). However, this does not mean longer computation time. In particular, using larger block sizes may better take advantage of modern multi-core computers for parallel computing, thus may take less computation time. Figure 2 shows the computation time required to reach the same precision on a 12 core Intel Xeon computer. We used the Intel Math Kernel Library (MKL) to carry out parallel dense matrix and vector operations. The results suggest that using appropriate large block size may take the least amount of computation time. We note that such timing results heavily depend on the specific architecture of the computer, in particular its cache size for fast access, the relative size of the data matrix $A$, and other implementation details. For example, Figure 2 shows the timing results on the same computer for a different problem instance with $m=2000$ and $N=4000$. Here the best block size is smaller than one shown in Figure 2, because the size of each column of $A$ is doubled and the operations involved in each coordinate (corresponding to a column of the matrix) has increased. However, for any fixed block size, the relative performance of the three algorithms are consistent; in particular, RNBPG substantially outperforms the other two methods in most cases.

We also conducted experiments on using randomized block coordinate methods to solve a dual SVM problem in machine learning (specifically, the dual of a smoothed SVM problem described in [27, Section 6.2]). We used two real datasets from the LIBSVM web site , whose characteristics are summarized in Table 1. In the dual SVM problem, the dimension of the dual variables are the same as the number of samples $N$, and we partition the dual variables into blocks to apply the three randomized block coordinate gradient methods. Figure 3 shows the reduction of the objective value with the three methods on the two datasets, each illustrated with two block sizes: $N_{i}=100$ and $N_{i}=1000$. We observe that the RNBPG method converges faster than the other two methods, especially with relatively larger block sizes.

To conclude, our experiments on both synthetic and real datasets clearly demonstrate the advantage of the nonmonotone line search strategy (with spectral initialization) for randomized block coordinate gradient methods.

References