Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

Abstract

In a Hilbert space setting ${{\mathcal {H}}}$, we study the fast convergence properties as $t \rightarrow + \infty$ of the trajectories of the second-order differential equation

$$\begin{aligned} \ddot{x}(t) + \frac{\alpha }{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t), \end{aligned}$$

where $\nabla \Phi$ is the gradient of a convex continuously differentiable function $\Phi : {{\mathcal {H}}} \rightarrow {{\mathbb {R}}}, \alpha$ is a positive parameter, and $g: [t_0, + \infty [ \rightarrow {{\mathcal {H}}}$ is a small perturbation term. In this inertial system, the viscous damping coefficient $\frac{\alpha }{t}$ vanishes asymptotically, but not too rapidly. For $\alpha \ge 3$, and $\int _{t_0}^{+\infty } t \Vert g(t)\Vert dt < + \infty$, just assuming that ${{\mathrm{argmin\,}}}\Phi \ne \emptyset$, we show that any trajectory of the above system satisfies the fast convergence property

$$\begin{aligned} \Phi (x(t))- \min _{{{\mathcal {H}}}}\Phi \le \frac{C}{t^2}. \end{aligned}$$

Moreover, for $\alpha > 3$, any trajectory converges weakly to a minimizer of $\Phi$. The strong convergence is established in various practical situations. These results complement the ${{\mathcal {O}}}(t^{-2})$ rate of convergence for the values obtained by Su, Boyd and Candès in the unperturbed case $g=0$. Time discretization of this system, and some of its variants, provides new fast converging algorithms, expanding the field of rapid methods for structured convex minimization introduced by Nesterov, and further developed by Beck and Teboulle with FISTA. This study also complements recent advances due to Chambolle and Dossal.

Appendix: Some auxiliary results

In this section, we present some auxiliary lemmas to be used later on. The following result can be found in [1]:

Lemma 5.6

Let $\delta >0, 1\le p<\infty$ and $1\le r\le \infty$. Suppose $F\in L^{p}([\delta ,\infty [)$ is a locally absolutely continuous nonnegative function, $G\in L^{r}([\delta ,\infty [)$ and

$$\begin{aligned} \frac{d}{dt}F(t)\le G(t) \end{aligned}$$

for almost every $t>\delta$. Then $\lim _{t\rightarrow \infty }F(t)=0$.

To establish the weak convergence of the solutions of (1), we will use Opial’s Lemma [30], that we recall in its continuous form. This argument was first used in [16] to establish the convergence of nonlinear contraction semigroups.

Lemma 5.7

Let S be a nonempty subset of ${{\mathcal {H}}}$ and let $x:[0,+\infty )\rightarrow {{\mathcal {H}}}$. Assume that

• (i) for every $z\in S, \lim _{t\rightarrow \infty }\Vert x(t)-z\Vert$ exists;

• (ii) every weak sequential limit point of x(t), as $t\rightarrow \infty$, belongs to S.

Then x(t) converges weakly as $t\rightarrow \infty$ to a point in S.

Its discrete version is

Lemma 5.8

Let S be a non empty subset of ${{\mathcal {H}}}$, and $(x_k)$ a sequence of elements of ${{\mathcal {H}}}$. Assume that

• (i) for every $z\in S, \lim _{k\rightarrow +\infty }\Vert x_k-z\Vert$ exists;

• (ii) every weak sequential limit point of $(x_k)$, as $k\rightarrow \infty$, belongs to S.

Then $x_k$ converges weakly as $k\rightarrow \infty$ to a point in S.

The following allows us to establish the existence of a limit for a real-valued function, as $t\rightarrow +\infty$:

Lemma 5.9

Let $\delta >0$, and let $w: [\delta , +\infty [ \rightarrow \mathbb R$ be a continuously differentiable function which is bounded from below. Assume

$$\begin{aligned} t\ddot{w}(t) + \alpha {{\dot{w}}}(t) \le g(t), \end{aligned}$$

(81)

for some $\alpha > 1$, almost every $t>\delta$, and some nonnegative function $g\in L^1 (\delta , +\infty )$. Then, the positive part $[{{\dot{w}}}]_+$ of ${{\dot{w}}}$ belongs to $L^1(t_0,+\infty )$ and $\lim _{t\rightarrow +\infty }w(t)$ exists.

Proof

Multiply (81) by $t^{\alpha -1}$ to obtain

$$\begin{aligned} \frac{d}{dt} \big (t^{\alpha } {{\dot{w}}}(t)\big ) \le t^{\alpha -1} g(t). \end{aligned}$$

By integration, we obtain

$$\begin{aligned} {{\dot{w}}}(t) \le \frac{{\delta }^{\alpha }|{{\dot{w}}}(\delta )|}{t^{\alpha } } + \frac{1}{t^{\alpha } } \int _{\delta }^t s^{\alpha -1} g(s)ds. \end{aligned}$$

Hence,

$$\begin{aligned}{}[{{\dot{w}}}]_{+}(t) \le \frac{{\delta }^{\alpha }|\dot{w}(\delta )|}{t^{\alpha } } + \frac{1}{t^{\alpha } } \int _{\delta }^t s^{\alpha -1} g(s)ds, \end{aligned}$$

and so,

$$\begin{aligned} \int _{\delta }^{\infty } [{{\dot{w}}}]_{+}(t) dt \le \frac{{\delta }^{\alpha }|{{\dot{w}}}(\delta )|}{(\alpha -1) \delta ^{\alpha -1}} + \int _{\delta }^{\infty }\frac{1}{t^{\alpha }} \left( \int _{\delta }^t s^{\alpha -1} g(s) ds\right) dt. \end{aligned}$$

Applying Fubini’s Theorem, we deduce that

$$\begin{aligned} \int _{\delta }^{\infty }\frac{1}{t^{\alpha }} \left( \int _{\delta }^t s^{\alpha -1} g(s) ds\right) dt= & {} \int _{\delta }^{\infty } \left( \int _{s}^{\infty } \frac{1}{t^{\alpha }} dt\right) s^{\alpha -1} g(s) ds\\= & {} \frac{1}{\alpha -1} \int _{\delta }^{\infty }g(s) ds. \end{aligned}$$

As a consequence,

$$\begin{aligned} \int _{\delta }^{\infty } [{{\dot{w}}}]_{+}(t) dt \le \frac{{\delta }^{\alpha }|{{\dot{w}}}(\delta ) |}{(\alpha -1) \delta ^{\alpha -1}} + \frac{1}{\alpha -1} \int _{\delta }^{\infty }g(s) ds < + \infty . \end{aligned}$$

Finally, the function $\theta :[\delta ,+\infty )\rightarrow {\mathbb {R}}$, defined by

$$\begin{aligned} \theta (t)=w(t)-\int _{\delta }^{t}[{{\dot{w}}}]_+(\tau )\,d\tau , \end{aligned}$$

is nonincreasing and bounded from below. It follows that

$$\begin{aligned} \lim _{t\rightarrow +\infty }w(t)=\lim _{t\rightarrow +\infty }\theta (t)+\int _{\delta }^{+\infty }[{{\dot{w}}}]_+(\tau )\,d\tau \end{aligned}$$

exists. $\square$

In the study of the corresponding algorithms, we use the following result, which is a discrete version of Lemma 5.9:

Lemma 5.10

Let $\alpha \ge 3$, and let $(a_k)$ and $(\omega _k)$ be two sequences of nonnegative numbers such that

$$\begin{aligned} a_{k+1} \le \frac{k -1}{k + \alpha -1}a_k + \omega _k \end{aligned}$$

for all $k\ge 1$. If $\sum _k k\omega _{k\in {{\mathbb {N}}}} <+\infty$, then $\sum _{k \in {{\mathbb {N}}}} a_k < +\infty$.

Proof

Since $\alpha \ge 3$ we have $\alpha -1\ge 2$, and hence

$$\begin{aligned} a_{k+1} \le \frac{k -1}{k + 2}a_k + \omega _k. \end{aligned}$$

Multiplying this expression by $(k+1)^2$, we obtain

$$\begin{aligned} (k+1)^2 a_{k+1} \le \frac{(k -1)(k+1)^2}{k + 2}a_k + (k+1)^2\omega _k \le k^2a_k + (k+1)^2\omega _k. \end{aligned}$$

Summing this inequality for $j=1,2,\ldots ,k$, we obtain

$$\begin{aligned} k^2 a_{k} \le a_1 + \sum _{j=1}^{k-1}(j+1)^2\omega _j. \end{aligned}$$

Dividing by $k^2$, and summing for $k=2,\ldots ,K$, we obtain

$$\begin{aligned} \sum _{k=2}^K a_{k} \le a_1 \sum _{k=2}^K \frac{1}{k^2} + \sum _{k=2}^K \frac{1}{k^2}\sum _{j=1}^{k-1}(j+1)^2\omega _j. \end{aligned}$$

Applying Fubini’s Theorem to this last sum, and observing that $\sum _{k=j+1}^{\infty }\frac{1}{k^2} \le \int _{j}^{\infty }\frac{1}{t^2}dt = \frac{1}{j}$, we obtain

$$\begin{aligned} \sum _{k=2}^K a_{k}\le & {} a_1 \sum _{k=2}^K \frac{1}{k^2} + \sum _{j=1}^{K-1} \left( \sum _{k=j+1}^{\infty }\frac{1}{k^2} \right) (j+1)^2\omega _j \\\le & {} a_1 \sum _{k=2}^K \frac{1}{k^2} + \sum _{j=1}^{K-1}\frac{(j+1)^2}{j} \omega _j\\\le & {} a_1 \sum _{k=2}^\infty \frac{1}{k^2} + 4\sum _{j=1}^{\infty }j \omega _j <+\infty , \end{aligned}$$

and the result follows. $\square$

The following is a continuous version of Kronecker’s Theorem for series (see, for example, [23, page 129]):

Lemma 5.11

Take $\delta >0$, and let $f \in L^1 (\delta , +\infty )$ be nonnegative and continuous. Consider a nondecreasing function $\psi :(\delta ,+\infty )\rightarrow (0,+\infty )$ such that $\lim \limits _{t\rightarrow +\infty }\psi (t)=+\infty$. Then,

$$\begin{aligned} \lim _{t \rightarrow + \infty } \frac{1}{\psi (t)} \int _{\delta }^t \psi (s)f(s)ds =0. \end{aligned}$$

Proof

Given $\epsilon >0$, fix $t_\epsilon$ sufficiently large so that

$$\begin{aligned} \int _{t_\epsilon }^{\infty } f(s) ds \le \epsilon . \end{aligned}$$

Then, for $t \ge t_\epsilon$, split the integral $\int _{\delta }^t \psi (s)f(s) ds$ into two parts to obtain

$$\begin{aligned} \frac{1}{\psi (t)} \int _{\delta }^t \psi (s)f(s)ds= & {} \frac{1}{\psi (t)}\int _{\delta }^{t_\epsilon } \psi (s) f(s) ds + \frac{1}{\psi (t)}\int _{t_\epsilon }^t \psi (s) f(s) ds \\\le & {} \frac{1}{\psi (t)}\int _{\delta }^{t_\epsilon } \psi (s)f(s) ds + \int _{t_\epsilon }^t f(s) ds. \end{aligned}$$

Now let $t\rightarrow +\infty$ to deduce that

$$\begin{aligned} 0\le \limsup _{t\rightarrow +\infty }\frac{1}{\psi (t)}\int _{\delta }^t \psi (s)f(s)ds \le \epsilon . \end{aligned}$$

Since this is true for any $\epsilon >0$, the result follows. $\square$

Using the previous result, we also obtain the following vector-valued version of Lemma 5.9:

Lemma 5.12

Take $\delta >0$, and let $F \in L^1 (\delta , +\infty ; {{\mathcal {H}}})$ be continuous. Let $x:[\delta , +\infty [ \rightarrow {{\mathcal {H}}}$ be a solution of

$$\begin{aligned} t\ddot{x}(t) + \alpha \dot{x}(t) = F(t) \end{aligned}$$

(82)

with $\alpha >1$. Then, x(t) converges strongly in ${{\mathcal {H}}}$ as $t \rightarrow + \infty$.

Proof

As in the proof of Lemma 5.9, multiply (82) by $t^{\alpha -1}$ and integrate to obtain

$$\begin{aligned} \dot{x}(t) = \frac{{\delta }^{\alpha }\dot{x}(\delta )}{t^{\alpha }} + \frac{1}{t^{\alpha }} \int _{\delta }^t s^{\alpha -1}F(s)ds.\end{aligned}$$

Integrate again to deduce that

$$\begin{aligned} x(t) = x(\delta ) + {\delta }^{\alpha }\dot{x}(\delta ) \int _{\delta }^t \frac{1}{s^{\alpha }} ds + \int _{\delta }^t \frac{1}{s^{\alpha }} \left( \int _{\delta }^s {\tau }^{\alpha -1}F(\tau )d\tau \right) ds.\end{aligned}$$

Fubini’s Theorem applied to the last integral gives

$$\begin{aligned} x(t)= & {} x(\delta ) + \frac{{\delta }^{\alpha }\dot{x}(\delta )}{\alpha -1}\left( \frac{1}{\delta ^{\alpha -1}}-\frac{1}{t^{\alpha -1}}\right) \nonumber \\&+\,\frac{1}{\alpha -1} \left( \int _{\delta }^t F(\tau )d\tau - \frac{1}{t^{\alpha -1}} \int _{\delta }^t {\tau }^{\alpha -1} F(\tau )d\tau \right) . \end{aligned}$$

(83)

Finally, apply Lemma 5.11 to the last integral with $\psi (s)=s^{\alpha -1}$ and $f(s)=\Vert F(s)\Vert$ to conclude that all the terms in the right-hand side of (83) have a limit as $t \rightarrow + \infty$. $\square$

Finally, in the analysis of the solutions for the perturbed system, we shall use the following Gronwall-Bellman Lemma (see [15, Lemme A.5]):

Lemma 5.13

Let $m:[\delta ,T]\rightarrow [0,+\infty [$ be integrable, and let $c\ge 0$. Suppose $w:[\delta ,T]\rightarrow {\mathbb {R}}$ is continuous and

$$\begin{aligned} \displaystyle {\frac{1}{2}w^2 (t) \le \frac{1}{2} c^2 + \int _{\delta }^t m(\tau ) w(\tau ) d\tau } \end{aligned}$$

for all $t\in [\delta ,T]$. Then, $\displaystyle {|w(t)| \le c + \int _{\delta }^t m(\tau ) d\tau }$ for all $t\in [\delta ,T]$.

We shall also make use of the following discrete version of the preceding result:

Lemma 5.14

Let $(a_k)$ be a sequence of nonnegative numbers such that

$$\begin{aligned} a_k^2 \le c^2 + \sum _{j=1}^k \beta _j a_j \end{aligned}$$

for all $k\in {{\mathbb {N}}}$, where $(\beta _j )$ is a summable sequence of nonnegative numbers, and $c\ge 0$. Then, $\displaystyle a_k \le c + \sum \nolimits _{j=1}^{\infty } \beta _j$ for all $k\in {{\mathbb {N}}}$.

Proof

For $k\in {{\mathbb {N}}}$, set $A_k := \max _{1\le m \le k} a_m$. Then, for $1\le m\le k$, we have

$$\begin{aligned} a_m^2 \le c^2 + \sum _{j=1}^m \beta _j a_j \le c^2 + A_k \sum _{j=1}^{\infty } \beta _j. \end{aligned}$$

Taking the maximum over $1\le m\le k$, we obtain

$$\begin{aligned} A_k^2 \le c^2 + A_k \sum _{j=1}^{\infty } \beta _j. \end{aligned}$$

Bounding by the roots of the corresponding quadratic equation, we obtain the result. $\square$