A New Exact Worst-Case Linear Convergence Rate of the Proximal Gradient Method

TL;DR

This paper establishes a new exact worst-case linear convergence rate for the proximal gradient method based on the proximal gradient norm, refining existing results and improving convergence rate estimates under the Polyak-Lojasiewicz inequality.

Contribution

It introduces a new exact worst-case linear convergence rate for the proximal gradient method, enhancing understanding of its convergence behavior.

Findings

New exact worst-case linear convergence rate established

Improved convergence rate of objective function under Polyak-Lojasiewicz inequality

Refined descent lemma for the proximal gradient method

Abstract

In this note, we establish a new exact worst-case linear convergence rate of the proximal gradient method in terms of the proximal gradient norm, which complements the recent results in [1] and implies a refined descent lemma.descent lemma. Based on the new lemma, we improve the linear convergence rate of the objective function accuracy under the Polyak-Lojasiewicz inequality.

Equations

$${\iota }_C:C\to [-\infty ,+\infty ]:x\to \{\begin{array}{rl}0,& x\in C;\\ +\infty ,& otherwise.\end{array}$$

$$\underset{x\in {\mathbb{R}}^n}{\min }\{\phi (x):=f(x)+g(x)\}$$

$$x^{+}={\mathbf{prox}}_{tg}(x-t\nabla f(x))=x-t\cdot {\mathcal{G}}_t(x),t>0$$

$$x^{+}=x-t(\nabla f(x)+s^{+}).$$

$$\mu \Vert x-y\Vert \le \Vert \nabla f(x)-\nabla f(y)\Vert \le L\Vert x-y\Vert ,$$

$$⟨\nabla f(x)-\nabla f(y),x-y⟩\ge \frac{\mu L}{\mu +L}\Vert x-y{\Vert }^2+\frac{1}{\mu +L}\Vert \nabla f(x)-\nabla f(y){\Vert }^2,$$

$$f(x)\ge f(y)+⟨\nabla f(y),x-y⟩+\frac{1}{2L}\Vert \nabla f(x)-\nabla f(y){\Vert }^2+\frac{\mu L}{2(L-\mu )}\Vert x-y-\frac{1}{L}(\nabla f(x)-\nabla f(y)){\Vert }^2.$$

$$\Vert {\mathcal{G}}_t(x)\Vert \le d(0,\partial \phi (x)).$$

$$\Vert {\mathcal{G}}_t(x^{+})\Vert \le d(0,\partial \phi (x^{+}))\le \rho (t)\Vert {\mathcal{G}}_t(x)\Vert \le \rho (t)d(0,\partial \phi (x)).$$

$$\Vert {\mathcal{G}}_t(x^{+})\Vert \le d(0,\partial \phi (x^{+}))\le \Vert {\mathcal{G}}_t(x)\Vert \le d(0,\partial \phi (x)).$$

$$\Vert \nabla f(x^{+})+s^{+}{\Vert }^2$$

$$=$$

$$\le$$

$$=$$

$$\le$$

$$=$$

$$\Vert \nabla f(x^{+})+s^{+}{\Vert }^2\le {\rho }^2(t)\Vert \nabla f(x)+s{\Vert }^2,\forall s\in \partial g(x).$$

$$\phi (x)\ge \phi (x^{+})+\frac{t}{2}\Vert {\mathcal{G}}_t(x){\Vert }^2+\frac{t}{2(1-\mu t)}\Vert {\mathcal{G}}_t(x^{+}){\Vert }^2,0<t\le \frac{1}{L}.$$

$$\phi (x)\ge \phi (x^{+})+\frac{t}{2}\Vert {\mathcal{G}}_t(x){\Vert }^2+\frac{t}{2}\Vert {\mathcal{G}}_t(x^{+}){\Vert }^2,0<t\le \frac{1}{L}.$$

$$f(x)\ge f(x^{+})+\frac{t}{2}\Vert \nabla f(x){\Vert }^2+\frac{t}{2}\Vert \nabla f(x^{+}){\Vert }^2,0<t\le \frac{1}{L}.$$

$$f(x)\ge f(x^{+})+⟨\nabla f(x^{+}),x-x^{+}⟩+\frac{t}{2}\Vert \nabla f(x)-\nabla f(x^{+}){\Vert }^2+\frac{\mu }{2(1-\mu t)}\Vert x-x^{+}-t(\nabla f(x)-\nabla f(x^{+})){\Vert }^2.$$

$$\phi (x)\ge$$

$$+\frac{\mu }{2(1-\mu t)}\Vert x-x^{+}-t(\nabla f(x)-\nabla f(x^{+})){\Vert }^2$$

$$=$$

$$+\frac{t}{2}\Vert \nabla f(x)-\nabla f(x^{+}){\Vert }^2+\frac{\mu }{2(1-\mu t)}\Vert x-x^{+}-t(\nabla f(x)-\nabla f(x^{+})){\Vert }^2$$

$$\phi (x)\ge$$

$$+\frac{t}{2}\Vert \nabla f(x)-\nabla f(x^{+}){\Vert }^2+\frac{\mu t^2}{2(1-\mu t)}\Vert s^{+}+\nabla f(x^{+}){\Vert }^2$$

$$=$$

$$+\frac{1}{2t}\Vert x-x^{+}{\Vert }^2+\frac{\mu t^2}{2(1-\mu t)}\Vert s^{+}+\nabla f(x^{+}){\Vert }^2$$

$$=$$

$$\Vert s^{+}+\nabla f(x^{+})\Vert \ge d(0,\partial \phi (x^{+}))\ge \Vert {\mathcal{G}}_t(x^{+})\Vert .$$

$$\phi (x)\ge \phi (x^{+})+\frac{t}{2}\Vert {\mathcal{G}}_t(x){\Vert }^2+\frac{t}{2(1-\mu t)}\Vert {\mathcal{G}}_t(x^{+}){\Vert }^2.$$

$$\phi (x)\ge \phi (x^{+})+\frac{t}{2}\Vert g_Q(x,t){\Vert }^2,0<t\le \frac{1}{L}.$$

$$\phi (x)\ge \phi (x^{+})+\frac{L}{2}\Vert x^{+}-x{\Vert }^2.$$

$$\forall x\in \text{dom}f,\frac{1}{2}\Vert \nabla f(x){\Vert }^2\ge \eta (f(x)-\min f).$$

Full text

††journal: JOTA

1 Introduction

A well-known algorithm for minimizing the sum of a smooth function with a non-smooth convex one is the proximal gradient (PG) method. Recently, the authors of taylor2018exact studied the exact worst-case linear convergence rates of the PG method for three different standard performance measures: objective function accuracy, distance to optimality and residual gradient norm. However, the first and third measures rely on the minimizers and the optimal value, which are in general unknown; while the second measure is usually difficult to compute. On the other hand, the proximal gradient (also called stepsize in drusvyatskiy2016nonsmooth ) norm is suggested as a more appealing stopping criteria in drusvyatskiy2016nonsmooth . This motivates us to consider the proximal gradient norm as an alternative to the existing three performance measures.

As a result, we derive an exact worst-case linear convergence rate for the PG method in terms of the proximal gradient norm. The proof idea shares the same spirit of Theorem 2 in nutini2018active but is quite different from that in taylor2018exact . Our result not only complements the recent results in taylor2018exact , but also helps us refine the classic descent lemma for the PG method and further yields an improved linear convergence rate of the objective function accuracy for non-strongly convex case.

2 Notations and preliminaries

2.1 Notations and definitions

Throughout the paper, $\mathbb{R}^{n}$ will denote an $n$-dimensional Euclidean space associated with inner-product $\langle\cdot,\cdot\rangle$ and induced norm $\|\cdot\|$. For any nonempty $S\subset\mathbb{R}^{n}$, we define the distance function by $d(x,S):=\inf_{y\in S}\|x-y\|$. Besides, we define the indicator function of a set $C\subset\mathbb{R}^{n}$ as

[TABLE]

Recall some basic notions, the domain of the function $f:\mathbb{R}^{d}\rightarrow(-\infty,+\infty]$ is defined by $\text{dom}~{}f=\{x\in\mathbb{R}^{d}:f(x)<+\infty\}$. We say that $f$ is proper if $\text{dom}~{}f\neq\emptyset$.

The $L$-smoothness and $\mu$-strongly convexity are defined as:

$\Box$

$L$-smoothness: $\forall x\in\mathbb{R}^{n},\|\nabla f(x)-\nabla f(y)\|\leq L\|x-y\|$ holds.

$\Box$

$\mu$-strong convexity: $f(x)-\frac{\mu}{2}\|x\|^{2}$ is convex on $\mathbb{R}^{n}$.

For simplicity, we make the following notations:

$\bullet$

${\mathcal{F}}_{L}^{1,1}(\mathbb{R}^{n})$: the class of $L$-smooth convex functions from $\mathbb{R}^{n}$ to $\mathbb{R}$;

$\bullet$

${\mathcal{S}}_{\mu,L}^{1,1}(\mathbb{R}^{n})$: the class of $L$-smooth and $\mu$-strongly convex functions from $\mathbb{R}^{n}$ to $\mathbb{R}$;

$\bullet$

$\Gamma_{0}(\mathbb{R}^{n})$: the class of proper closed and convex functions from $\mathbb{R}^{n}$ to $(-\infty,+\infty]$.

Obviously, we have ${\mathcal{S}}_{\mu,L}^{1,1}(\mathbb{R}^{n})\subseteq{\mathcal{F}}_{L}^{1,1}(\mathbb{R}^{n})$.

2.2 The proximal gradient algorithm

In this note, we consider the composite convex minimization:

[TABLE]

where $f\in{\mathcal{F}}_{L}^{1,1}(\mathbb{R}^{n})$ and $g\in\Gamma_{0}(\mathbb{R}^{n})$.

We focus on the PG method with constant step size $t$ to solve (1). For simplicity, we use the superscript ”+” to denote the subsequent iterate. The PG method can be simply expressed by

[TABLE]

where ${\mathbf{prox}}_{tg}(x):=\arg\min_{u\in\mathbb{R}^{n}}\left\{tg(u)+\frac{1}{2}\|u-x\|^{2}\right\}$ and $\mathcal{G}_{t}(x)=t^{-1}\left(x-{\mathbf{prox}}_{tg}(x-t\nabla f(x))\right)$ is defined as the proximal gradient. By the equality ${\mathbf{prox}}_{tg}=(I+t\partial g)^{-1}$, we have $x-t\nabla f(x)\in x^{+}+t\partial g(x^{+}),$ which implies that there exists $s^{+}\in\partial g(x^{+})$ such that

[TABLE]

2.3 Two important lemmas

Our analysis will rely on the following two lemmas.

Lemma 1 (Theorem 2.1.12, nesterov2013introductory ; Theorem 4, taylor2017smooth )

If $f\in{\mathcal{S}}^{1,1}_{\mu,L}(\mathbb{R}^{n})$, then for any $x,y\in\mathbb{R}^{n}$ we have

[TABLE]

and

[TABLE]

and the smooth strongly convex interpolation formula

[TABLE]

Lemma 2 (Theorem 3.5, drusvyatskiy2018error )

Let $\varphi=f+g$, where $f\in{\mathcal{F}}^{1,1}_{L}(\mathbb{R}^{n})$ and $g\in\Gamma_{0}(\mathbb{R}^{n})$. For any $x\in\mathbb{R}^{n}$, it holds that

[TABLE]

3 Main result and implications

In this section, we present two new results for the PG method: one is an exact worst-case linear convergence rate in terms of the proximal gradient norm, and the other is a refined sufficient decrease property of the objective function value.

3.1 Main result

Now, we are ready to present the main result of this note.

Theorem 3.1

Let $\varphi=f+g$, where $f\in{\mathcal{S}}^{1,1}_{\mu,L}(\mathbb{R}^{n})$ and $g\in\Gamma_{0}(\mathbb{R}^{n})$. Denote $\rho(t):=\max\{|1-Lt|,|1-\mu t|\}$. Then, the PG method for minimizing $\varphi$ achieves the exact worst-case linear convergence rate in terms of the proximal gradient norm:

[TABLE]

In particular, for $f\in{\mathcal{F}}^{1,1}_{L}(\mathbb{R}^{n})$, $g\in\Gamma_{0}(\mathbb{R}^{n})$, and $0<t\leq\frac{2}{L}$, it holds that

[TABLE]

Proof

Note that $s^{+}\in\partial g(x^{+})$ and hence $d(0,\partial\varphi(x^{+}))\leq\|\nabla f(x^{+})+s^{+}\|$. Therefore, to show (2), it suffices to show that $\|\nabla f(x^{+})+s^{+}\|^{2}\leq\rho^{2}(t)\|\mathcal{G}_{t}(x)\|^{2}$ in view of Lemma 2. Using Lemma 1, we derive that

[TABLE]

Here, the factor $\rho(t)$ can not be improved; otherwise, it will contradict the following exact worst-case convergence rate, which was recently established in taylor2018exact :

[TABLE]

3.2 Implicated result

The second result is a refined version of the classic descent lemma(see (nesterov2013introductory, , Corollary 2.2.1)(beck2009fast, , Lemma 2.3)).

Lemma 3

Let $\varphi=f+g$, where $f\in{\mathcal{S}}^{1,1}_{\mu,L}(\mathbb{R}^{n})$ and $g\in\Gamma_{0}(\mathbb{R}^{n})$. Then, the PG method for minimizing $\varphi$ has the refined sufficient decrease property

[TABLE]

In particular,

• •

for $f\in{\mathcal{F}}^{1,1}_{L}(\mathbb{R}^{n})$, $g\in\Gamma_{0}(\mathbb{R}^{n})$, it holds that

[TABLE]

• •

for $f\in{\mathcal{F}}^{1,1}_{L}(\mathbb{R}^{n})$, $g\equiv 0$, it holds that

[TABLE]

Proof

Note that $0<t\leq L^{-1}$ implies $t^{-1}\geq L$ and the fact that ${\mathcal{S}}^{1,1}_{\mu,L}(\mathbb{R}^{n})\subset{\mathcal{S}}^{1,1}_{\mu,t^{-1}}(\mathbb{R}^{n})$. We can use the smooth strongly convex interpolation formula with $L=t^{-1}$ and $y=x^{+}$ in Lemma 1 to get

[TABLE]

The convexity of $g$ gives $g(x)\geq g(x^{+})+\langle s^{+},x-x^{+}\rangle$ since $s^{+}\in\partial g(x^{+})$. Adding these two inequalities, we derive that

[TABLE]

Using the expression $x^{+}=x-t(\nabla f(x)+s^{+})$, we can further derive that

[TABLE]

Note that $x-x^{+}=t\mathcal{G}_{t}(x)$ and

[TABLE]

We finally obtain

[TABLE]

This completes the proof.

Remark 1

In (nesterov2013introductory, , Corollary 2.2.1), for $\varphi=f+g$ with $f\in{\mathcal{S}}^{1,1}_{\mu,L}(\mathbb{R}^{n})$ and $g$ being the indicator function of a set $Q$, the descent lemma of the projected gradient method can be stated as

[TABLE]

where $g_{Q}(x,t):=t^{-1}(x-x^{+})$ is the gradient mapping of $f$ on $Q$.

In (beck2009fast, , Lemma 2.3), for $\varphi=f+g$ with $f\in{\mathcal{F}}^{1,1}_{L}(\mathbb{R}^{n})$ and $g\in\Gamma_{0}(\mathbb{R}^{n})$, the corresponding descent lemma of the PG method is:

[TABLE]

Remarkably, our result improves these existing descent lemmas.

With the refined descent lemma, we can show a better linear convergence rate in terms of the objective function accuracy for the gradient descent method under the classic Polyak-Łojasiewicz inequality polyak1963gradient lojasiewicz1963topological .

Corollary 1

Let $f\in{\mathcal{F}}^{1,1}_{L}(\mathbb{R}^{n})$, $g\equiv 0$. Assume that $f$ satisfies the Polyak-Łojasiewicz inequality for some $\eta>0$:

[TABLE]

Let $x^{+}=x-t\nabla f(x)$, $0<t\leq\frac{1}{L}$, then it holds that

[TABLE]

Proof

Using the Polyak-Łojasiewicz inequality and (5) in Lemma 3, we have

[TABLE]

Rearranging and subtracting $\min f$ from both sides yield

[TABLE]

Remark 2

The result (8) with $t=\frac{1}{L}$ improves the existing linear convergence rate in (karimi2016linear, , Theorem 1) from $(1-\frac{\eta}{L})$ to $\frac{L-\eta}{L+\eta}$.

Finally, we extend the result above to the PG method.

Corollary 2

Let $f\in{\mathcal{F}}^{1,1}_{L}(\mathbb{R}^{n})$, $g\in\Gamma_{0}(\mathbb{R}^{n})$. Assume that $\varphi=f+g$ satisfies the generalized Polyak-Łojasiewicz inequality for some $\eta>0$:

[TABLE]

Let $x^{+}=x-t\mathcal{G}_{t}(x)$, $0<t\leq\frac{1}{L}$, then it holds that

[TABLE]

Proof

Using the generalized Polyak-Łojasiewicz inequality and (4) in Lemma 3, we have

[TABLE]

Rearranging and subtracting $\min\varphi$ from both sides give us

[TABLE]

Acknowledgements

This work is supported by the National Science Foundation of China (No.61571008).