Perturbed Fenchel duality and first-order methods

TL;DR

This paper introduces a unified framework based on perturbed Fenchel duality that explains the convergence rates of various first-order optimization algorithms, including gradient and proximal methods.

Contribution

It establishes a canonical perturbed Fenchel duality inequality that unifies the analysis of multiple first-order methods, providing new insights into their convergence behavior.

Findings

Unified derivation of convergence rates for popular first-order algorithms

Applicable to gradient, proximal, and universal gradient methods

Simplifies analysis through a canonical duality inequality

Abstract

We show that the iterates generated by a generic first-order meta-algorithm satisfy a canonical perturbed Fenchel duality inequality. The latter in turn readily yields a unified derivation of the best known convergence rates for various popular first-order algorithms including the conditional gradient method as well as the main kinds of Bregman proximal methods: subgradient, gradient, fast gradient, and universal gradient methods.

Equations

$$\underset{x\in \mathbb{E}}{\min }f(Ax)+\Psi (x)$$

$$\underset{u\in {\mathbb{F}}^{\ast }}{\max }-f^{\ast }(u)-{\Psi }^{\ast }(-A^{\ast }u),$$

$$f^{\ast }(u)=\underset{y\in \mathbb{F}}{\sup }\{⟨u,y⟩-f(y)\},{\Psi }^{\ast }(v)=\underset{x\in \mathbb{E}}{\sup }\{⟨v,x⟩-f(x)\}.$$

$$f(Ax)+\Psi (x)\ge -f^{\ast }(u)-{\Psi }^{\ast }(-A^{\ast }u).$$

$$f(A\bar{x})+\Psi (\bar{x})+f^{\ast }(\bar{u})+{\Psi }^{\ast }(-A^{\ast }\bar{u})=0.$$

$$f(A{x}_k)+\Psi (x_k)+f^{\ast }(u_k)+(\Psi +d_k{)}^{\ast }(-A^{\ast }{u}_k)\le {\delta }_k.$$

$$f^{\ast }(u_k)+(\Psi +d_k{)}^{\ast }(-A^{\ast }{u}_k)$$

$$=-f(Ax)-\Psi (x)-d_k(x).$$

$$f(A{x}_k)+\Psi (x_k)-(f(Ax)+\Psi (x))\le d_k(x)+{\delta }_k$$

$$f(A{x}_k)+(\Psi +d_k)(x_k)+f^{\ast }(u_k)+(\Psi +d_k{)}^{\ast }(-A^{\ast }{u}_k)\le d_k(x_k)+{\delta }_k.$$

$$\underset{x\in \mathbb{E}}{\min }f(Ax)+(\Psi +d_k)(x)$$

$$\underset{u\in {\mathbb{F}}^{\ast }}{\max }-f^{\ast }(u)-(\Psi +d_k{)}^{\ast }(-A^{\ast }u).$$

$$d_k(x)=\frac{D_h(x,x_0)}{{\sum }_{i=0}^{k-1}{t}_i}$$

$$(s_{-},g)\mapsto \underset{s}{\mathrm{argmin}}\{t(⟨A^{\ast }g,s⟩+\Psi (s))+D_h(s,s_{-})\}.$$

$$D_h(s,z)=h(s)-h(z)-⟨\nabla h(z),s-z⟩.$$

$$t_k(A^{\ast }{g}_k+g_k^{\Psi })+\nabla h(s_k)-\nabla h(s_{k-1})=0$$

$$x_k:=\frac{{\sum }_{i=0}^{k-1}{t}_i{s}_i}{{\sum }_{i=0}^{k-1}{t}_i},z_k:=\frac{{\sum }_{i=0}^{k-1}{t}_i{y}_i}{{\sum }_{i=0}^{k-1}{t}_i},u_k:=\frac{{\sum }_{i=0}^{k-1}{t}_i{g}_i}{{\sum }_{i=0}^{k-1}{t}_i}.$$

$$d_k(s):=\frac{D_h(s,s_{-1})}{{\sum }_{i=0}^{k-1}{t}_i},w_k:=\frac{{\sum }_{i=0}^{k-1}{t}_i(A^{\ast }{g}_i+g_i^{\Psi })}{{\sum }_{i=0}^{k-1}{t}_i}.$$

$$\begin{gathered} \frac{{\sum }_{i=0}^{k-1}{t}_i(f(A{y}_i)+\Psi (y_i)+f^{\ast }(g_i)+{\Psi }^{\ast }(g_i^{\Psi }))}{{\sum }_{i=0}^{k-1}{t}_i}+d_k^{\ast }(-w_k) \\ =\frac{{\sum }_{i=0}^{k-1}{t}_i(\Psi (y_i)-\Psi (s_i)-⟨A^{\ast }{g}_i,s_i-y_i⟩)-D_h(s_i,s_{i-1})}{{\sum }_{i=0}^{k-1}{t}_i}. \end{gathered}$$

$$f(A{z}_k)+\Psi (z_k)+f^{\ast }(u_k)+(\Psi +d_k{)}^{\ast }(-A^{\ast }{u}_k)\le {\delta }_k$$

$${\delta }_k=\frac{{\sum }_{i=0}^{k-1}{t}_i(\Psi (y_i)-\Psi (s_i)-⟨A^{\ast }{g}_i,s_i-y_i⟩)-D_h(s_i,s_{i-1})}{{\sum }_{i=0}^{k-1}{t}_i}.$$

$${\theta }_k:=\frac{t_k}{{\sum }_{i=0}^k{t}_i},k=0,1,\dots .$$

$$x_{k+1}=(1-{\theta }_k)x_k+{\theta }_k{s}_k,k=0,1,2,\dots$$

$$\begin{gathered} \mathcal{D}(x,y,s,\theta ):=(f\circ A+\Psi )(x+\theta (s-x))-(1-\theta )(f\circ A+\Psi )(x)-\theta (f\circ A+\Psi )(s) \\ +\theta D_{f\circ A}(s,y). \end{gathered}$$

$$\begin{gathered} f(A{x}_k)+\Psi (x_k)+\frac{{\sum }_{i=0}^{k-1}{t}_i(f^{\ast }(g_i)+{\Psi }^{\ast }(g_i^{\Psi }))}{{\sum }_{i=0}^{k-1}{t}_i}+d_k^{\ast }(-w_k) \\ =\frac{{\sum }_{i=0}^{k-1}\left(t_i\mathcal{D}(x_i,y_i,s_i,{\theta }_i)/{\theta }_i-D_h(s_i,s_{i-1})\right)}{{\sum }_{i=0}^{k-1}{t}_i}. \end{gathered}$$

$$f(A{x}_k)+\Psi (x_k)+f^{\ast }(u_k)+(\Psi +d_k{)}^{\ast }(-A^{\ast }{u}_k)\le {\delta }_k$$

$${\delta }_k=\frac{{\sum }_{i=0}^{k-1}\left(t_i\mathcal{D}(x_i,y_i,s_i,{\theta }_i)/{\theta }_i-D_h(s_i,s_{i-1})\right)}{{\sum }_{i=0}^{k-1}{t}_i}.$$

$$t_k\mathcal{D}(x_k,y_k,s_k,{\theta }_k)/{\theta }_k-D_h(s_k,s_{k-1})\le {ϵ}_k,k=0,1,2,\dots .$$

$$f(A{x}_k)+\Psi (x_k)+f^{\ast }(u_k)+(\Psi +d_k{)}^{\ast }(-A^{\ast }{u}_k)\le \frac{{\sum }_{i=0}^{k-1}{ϵ}_i}{{\sum }_{i=0}^{k-1}{t}_i},k=1,2,\dots .$$

$$f(A{x}_k)+\Psi (x_k)-(f(Ax)+\Psi (x))\le \frac{D_h(x,x_0)+{\sum }_{i=0}^{k-1}{ϵ}_i}{{\sum }_{i=0}^{k-1}{t}_i}$$

$$\underset{x\in \mathbb{E}}{\min }(f+\Psi )(x)\iff \underset{x\in \mathbb{E}}{\min }f(x)+\Psi (x).$$

Full text

1 Introduction

The central goal of this paper is to highlight an interesting connection between two major threads in convex optimization, namely first-order methods and Fenchel duality. More precisely, we show that a generic first-order meta-algorithm generates iterates that satisfy a canonical perturbed Fenchel duality property. The meta-algorithm relies on some flexibly selected sequence of points $\{y_{k}:k=0,1,\dots\}$. Specific choices of the sequence $\{y_{k}:k=0,1,\dots\}$ turn our first-order meta-algorithm into each of the following popular first-order methods: conditional gradient, Bregman proximal gradient, Bregman proximal subgradient, fast Bregman proximal gradient, and universal Bregman proximal gradient. The perturbed Fenchel duality property in turn yields a generic convergence result for the meta-algorithm and thereby a unified approach to derive the best known convergence rates for all of the above popular first-order algorithms.

Each of our two main results (Theorem 1 and Theorem 2) provides a duality relationship between a primal average sequence and a dual average sequence of the iterates generated by our first-order meta-algorithm. These kinds of average sequences underlie (sometimes implicitly) the construction and analyses of several first-order algorithms, prime examples being the analysis of the mirror descent method [25] and the dual averaging method of Nesterov [27]. It is noteworthy that Theorem 1 and Theorem 2 do not depend on any assumptions concerning Lipschitz continuity or strong convexity. Indeed, our main developments in Section 2 do not rely on any norms at all. We build on and extend ideas introduced in [18, 31]. However, unlike these predecessors that apply only to Euclidean proximal methods, the main results in this paper (Theorem 1 and Theorem 2) apply in the more general context of Bregman proximal methods and hold under minimal assumptions.

Our work is related to a variety of recent articles that aim to add intuition, explain, and unify the convergence of first-order methods. Some of these approaches are based on continuous-time dynamics [13, 33], control theory [22], online optimization [2, 1, 37], and geometric [11, 15] techniques. In contrast to these approaches, our results apply in the broad context of Bregman proximal methods as developed in the recent articles [4, 19, 24, 34, 14] and are predicated upon a perturbed Fenchel duality property.

Our developments facilitate a unified analysis of not only the Bregman gradient and subgradient schemes as discussed in [34], but also the universal proximal gradient scheme [29], the recent accelerated Bregman method of [19], and the conditional gradient method [3, 21]. The conditional gradient method corresponds to a special and particularly simple case of our first-order meta-algorithm by setting a suitable reference function identically to zero. This simple connection between the conditional gradient method and Bregman proximal methods appears to have been previously overlooked.

As a nice byproduct of our approach, we readily obtain convergence properties for the conditional gradient method that are novel and stronger along several dimensions. First, we consider a composite minimization problem as the one considered in [30] but without any boundedness assumption on the domain. The composite format includes as a special case the common format of convex minimization over a convex compact domain that is typically discussed in the literature on the conditional gradient method [3, 6, 16, 21]. Second, we provide a bound on the Fenchel duality gap between the main (primal) iterates and some canonical dual iterates. Our bound applies to a composite objective and thus is more general than the kind of duality gap bound discussed in [16, 21] that only applies to smooth convex minimization over a compact convex domain. Third, we derive a rate of convergence that relies on a curvature condition that generalizes the curvature constant introduced by Jaggi [21]. Fourth, our approach naturally suggests a line-search strategy for selecting the stepsize at each main iteration. This line-search strategy yields a novel linear convergence result on the duality gap when the components of the composite objective satisfy a type of joint smoothness and strong convexity assumption.

1.1 Main contribution: perturbed Fenchel duality

Consider the convex minimization problem

[TABLE]

and its Fenchel dual

[TABLE]

where ${\mathbb{E}}$ and ${\mathbb{F}}$ are real vector spaces, $A:{\mathbb{E}}\rightarrow{\mathbb{F}}$ is a linear mapping, and $f:{\mathbb{F}}\rightarrow{\mathbb{R}}\cup\{\infty\}$ and $\Psi:{\mathbb{E}}\rightarrow{\mathbb{R}}\cup\{\infty\}$ are closed convex functions. The spaces ${\mathbb{E}}^{*}$ and ${\mathbb{F}}^{*}$ are the continuous dual vector spaces of ${\mathbb{E}}$ and ${\mathbb{F}}$, and the functions $f^{*}:{\mathbb{F}}^{*}\rightarrow{\mathbb{R}}\cup\{\infty\}$ and $\Psi^{*}:{\mathbb{E}}^{*}\rightarrow{\mathbb{R}}\cup\{\infty\}$ are the convex conjugates of $f$ and $\Psi$ respectively, namely,

[TABLE]

It is known and easy to see, as detailed in [5, 10, 20, 32], that weak duality holds between the primal and dual problems (1) and (2), that is, for all $x\in{\mathbb{E}}$ and $u\in{\mathbb{F}}^{*}$

[TABLE]

From (3) it readily follows that $\bar{x}\in{\mathbb{E}}$ and $\bar{u}\in{\mathbb{F}}^{*}$ are optimal solutions to (1) and (2) respectively if the following Fenchel duality identity holds

[TABLE]

The central contributions of this paper (Theorem 1 and Theorem 2) show that a generic first-order meta-algorithm (Algorithm 1), which includes a wide class of first-order methods, generates average sequences $x_{k}\in{\mathbb{E}}$ and $u_{k}\in{\mathbb{F}}^{*}$ that satisfy the following perturbed version of (4) for some $\delta_{k}\geq 0$ and some $d_{k}:{\mathbb{E}}\rightarrow{\mathbb{R}}_{+}\cup\{\infty\}$

[TABLE]

Inequality (5) can be considered a type of perturbed Fenchel duality property. This property automatically yields the generic convergence results (6) and (7) below. Observe that for all $x\in{\mathbb{E}}$ we have

[TABLE]

Therefore the perturbed Fenchel duality property (5) implies that for all $x\in{\mathbb{E}}$

[TABLE]

and so $f(Ax_{k})+\Psi(x_{k})$ converges to $\min_{x\in{\mathbb{E}}}\{f(Ax)+\Psi(x)\}$ provided both $d_{k}$ and $\delta_{k}$ converge to zero.

The perturbed Fenchel duality property (5) also yields the following duality gap bound on a sequence of perturbed problems. By adding $d_{k}(x_{k})$ to both sides of (5), we can rewrite (5) as follows

[TABLE]

This gives a generic duality gap bound for the following perturbation of the original problem (1)

[TABLE]

and its dual

[TABLE]

The duality gap bound (7) automatically implies that both $f(Ax_{k})+\Psi(x_{k})$ and $-f^{*}(u_{k})-\Psi^{*}(-A^{*}u_{k})$ converge to $\min_{x\in{\mathbb{E}}}\{f(Ax)+\Psi(x)\}=\max_{u\in{\mathbb{F}}^{*}}\{-f^{*}(u)-\Psi^{*}(-A^{*}u)\}$ provided both $d_{k}$ and $\delta_{k}$ converge to zero.

Theorem 1 and Theorem 2 exhibit explicit expressions for the objects $x_{k}\in{\mathbb{E}},u_{k}\in{\mathbb{E}}^{*},\delta_{k}\in{\mathbb{R}}_{+},$ and $d_{k}:{\mathbb{E}}\rightarrow{\mathbb{R}}_{+}\cup\{\infty\}$ that appear in (5) in terms of the steps that the meta-algorithm performs. More precisely, equation (11) shows that

[TABLE]

where $D_{h}$ is the Bregman distance generated by some reference function $h$, $x_{0}$ is the initial point for the meta-algorithm, and $\{t_{i}:i=0,1,\dots\}$ are stepsizes that the meta-algorithm chooses at each main iteration. Equations (14) in Theorem 1 and (19) in Theorem 2 provide a similar expression for $\delta_{k}$ as a fraction where the denominator is also $\sum_{i=0}^{k-1}t_{i}$ and the numerator depends on both $\{t_{i}:i=0,1,\dots\}$ and $\{y_{i}:i=0,1,\dots\}.$ Therefore the following key insight is readily revealed: both $d_{k}$ and $\delta_{k}$ in (5) can be driven to zero provided the meta-algorithm makes judicious choices of $\{t_{i}:i=0,1,\dots\}$ and $\{y_{i}:i=0,1,\dots\}$ so that the denominator $\sum_{i=0}^{k-1}t_{i}$ grows sufficiently faster than the numerator in each of the fractions in (11), (14), and (19).

As detailed in Section 3, we leverage the perturbed Fenchel duality results (Theorem 1 and Theorem 2) to provide a unified derivation of the best known convergence rates of the most popular first-order algorithms, namely: the conditional gradient method [3, 21], the Bregman proximal gradient and subgradient algorithms [4, 7, 23, 24, 34], and the fast and universal Bregman proximal gradient algorithms [8, 19, 28, 29]. Each of these algorithms corresponds to a special case of our first-order meta-algorithm for some specific choice of the sequence $\{y_{k}:k=0,1,\dots\}$. Furthermore, the convergence rate of each of these algorithms follows from the above perturbed Fenchel duality approach by driving $d_{k}$ and $\delta_{k}$ to zero in some particular way.

Section 3.1 discusses a generalized conditional subgradient method. This is a special case of our first-order meta-algorithm where $h\equiv 0$ and thus $d_{k}\equiv 0$. For this method the stepsizes can be chosen so that $\delta_{k}={\mathcal{O}}(1/k^{\nu})$ and thus the duality gap converges to zero at the rate ${\mathcal{O}}(1/k^{\nu})$ provided a suitable curvature condition (29) holds for some $\nu>0$ (see Proposition 1) and a suitable choice of stepsizes. The popular convergence rate ${\mathcal{O}}(1/k)$ of the conditional gradient method discussed in [6, 16, 21] can be recovered as a special case of Proposition 1 when $\nu=1$. Most interestingly, when a suitable joint smoothness and strong convexity condition (36) holds, we show that the duality gap converges linearly to zero for properly chosen stepsizes (see Proposition 2).

Section 3.2.1 discusses the Bregman gradient method. For this method the stepsizes can be chosen so that both $1/\sum_{i=0}^{k-1}t_{i}={\mathcal{O}}(1/k)$ and $\delta_{k}=0$ provided a suitable relative smoothness condition (39) holds. The iconic ${\mathcal{O}}(1/k)$ convergence rate from [4, 24, 34] thus readily follows (see Proposition 3). Section 3.2.2 discusses the Bregman subgradient method. For this method we show that $\delta_{k}$ can be bounded above by an expression of the form ${\mathcal{O}}\left(\sum_{i=0}^{k-1}t_{i}^{2}/\sum_{i=0}^{k-1}t_{i}\right)$ provided a suitable relative continuity condition (41) holds (see Proposition 4). The iconic ${\mathcal{O}}(1/\sqrt{k})$ convergence rate of the subgradient method [23, 34] thus follows by choosing the first $k$ stepsizes $t_{0},\dots,t_{k-1}$ all equal to a constant multiple of $1/\sqrt{k}$.

Section 3.3.1 discusses the fast Bregman gradient method. In this case the stepsizes can be chosen so that both $1/\sum_{i=0}^{k-1}t_{i}={\mathcal{O}}(1/k^{\gamma})$ and $\delta_{k}=0$ provided a suitable relative smoothness condition (47) holds for some $\gamma>1$ (see Proposition 5). The iconic ${\mathcal{O}}(1/k^{2})$ convergence rate of fast proximal gradient methods [8, 26, 28] as well as the more recent related results in [19] can be obtained as a special case of Proposition 5. Section 3.3.2 discusses the universal Bregman gradient method. In this case for any $\epsilon>0$ the stepsizes can be chosen so that both $1/\sum_{i=0}^{k-1}t_{i}={\mathcal{O}}(1/\epsilon^{\frac{1-\nu}{1+\nu}}k^{\frac{1+3\nu}{1+\nu}})$ and $\delta_{k}=\epsilon$ provided a suitable relative smoothness condition (51) holds for some $\nu\in[0,1]$ (Proposition 6). The optimal ${\mathcal{O}}(1/k^{\frac{1+3\nu}{2}})$ universal convergence rate for minimization of smooth convex functions with $\nu$-Hölder continuous gradient [29] follows as a special case of Proposition 6.

1.2 Organization of the paper

The main sections of the paper are organized as follows. Section 2 presents our central developments. This section describes a first-order meta-algorithm (Algorithm 1) and its perturbed Fenchel duality properties (Theorem 1 and Theorem 2). Section 2 also states a generic convergence result for Algorithm 1 (Corollary 1). Section 3 leverages the perturbed Fenchel duality properties of Section 2 to provide succinct and unified derivations of the best known convergence rates for a variety of popular first-order algorithms. Section 4 presents the proofs of our main results, Theorem 1 and Theorem 2.

Our exposition is self-contained. We rely primarily on standard convex analysis techniques as discussed in [5, 10, 20, 32]. The seed for this paper was the technical report [17], which can be considered an initial draft of this work. The exposition and main ideas first laid out in [17] have been substantially re-organized, cleaned up, and extended.

2 First-order meta-algorithm

Suppose ${\mathbb{E}}$ and ${\mathbb{F}}$ are real vector spaces, $A:{\mathbb{E}}\rightarrow{\mathbb{F}}$ is a linear mapping, and $f:{\mathbb{F}}\rightarrow{\mathbb{R}}\cup\{\infty\}$ and $\Psi:{\mathbb{E}}\rightarrow{\mathbb{R}}\cup\{\infty\}$ are closed convex functions such that ${\mathrm{dom}}(\Psi)\subseteq{\mathrm{dom}}(\partial(f\circ A))$. Furthermore, suppose $h:{\mathbb{E}}\rightarrow{\mathbb{R}}\cup\{\infty\}$ is a convex differentiable reference function such that ${\mathrm{dom}}(\Psi)\subseteq\overline{{\mathrm{dom}}(h)}$. Algorithm 1 describes a first-order meta-algorithm for problem (1).

The key step of Algorithm 1, namely Step 5, hinges on the Bregman proximal mapping

[TABLE]

Here $D_{h}$ is the Bregman distance generated by the reference function $h$, that is,

[TABLE]

We note that for additional flexibility, we relax the common strict convexity requirement of the reference function $h$. This flexibility enables us to use linear oracles to solve (8) when $h\equiv 0$ and thus includes the conditional gradient method in our framework. Since $h$ is not required to be strictly convex, the minimizer of the problem (8) may not be unique. Step 5 of Algorithm 1 does not require uniqueness of the solution to (8). Instead it only relies on the assumption that a solution to (8) is computable. Algorithm 1 also relies on a flexibly selected sequence $\{y_{k}:k=0,1,\dots\}$. This sequence in turn yields the input $g_{k}\in\partial f(Ay_{k})$ used to generate the main sequence of iterates $\{s_{k}:k=0,1,\dots\}$ via (8). Section 3 below details how specific choices of $\{y_{k}:k=0,1,\dots\}$ turn Algorithm 1 into each of the following popular first-order methods: conditional gradient, Bregman proximal gradient and subgradient, and Bregman fast and universal proximal gradient.

Algorithm 1 naturally relies on the assumption that each instance of (8) in Step 5 is well-posed and computable. Indeed, we will make the following blanket assumption throughout the paper.

Assumption 1**.**

At each iteration $k=0,1,\dots$ of Algorithm 1 Step 5 is well-posed, that is, a minimizer $s_{k}\in\operatorname*{arg\,min}_{s}\{t_{k}(\left\langle A^{*}g_{k},s\right\rangle+\Psi(s))+D_{h}(s,s_{k-1})\}$ exists, is computable, and satisfies the optimality conditions

[TABLE]

for some $g^{\Psi}_{k}\in\partial\Psi(s_{k})$.

We should highlight that the subgradient $g^{\Psi}_{k}$ in Assumption 1 can be interpreted as a certificate of optimality for the solution $s_{k}\in\operatorname*{arg\,min}_{s}\{t_{k}(\left\langle A^{*}g_{k},s\right\rangle+\Psi(s))+D_{h}(s,s_{k-1})\}$. This certificate is used in the construction of the sequence $w_{k}\in{\mathbb{E}}^{*}$ in equation (11) below. This sequence is a key component of our main results.

Our main results (Theorem 1 and Theorem 2) show that Algorithm 1 yields average sequences of iterates that satisfy a perturbed Fenchel duality inequality of the form (5). It is worthwhile noting that these two results hold under the minimal assumptions stated above. As we subsequently detail in Section 3, several popular first-order algorithms including the conditional gradient, Bregman proximal gradient and subgradient algorithms, and fast and universal Bregman proximal gradient algorithms, can all be seen as special cases of Algorithm 1. Most interesting, their convergence properties readily follow the from the perturbed Fenchel results stated in Theorem 1 and Theorem 2.

Our perturbed Fenchel duality results concern the primal average sequences $z_{k},x_{k}\in{\mathbb{E}}$, and the dual average sequence $u_{k}\in{\mathbb{F}}^{*}$ defined by Algorithm 1 as follows

[TABLE]

For notational convenience we let $x_{0}:=z_{0}:=s_{-1}$. Our perturbed Fenchel duality results also rely on the sequences $d_{k}:{\mathbb{E}}\rightarrow{\mathbb{R}}\cup\{\infty\}$ and $w_{k}\in{\mathbb{E}}^{*}$ defined as follows

[TABLE]

Theorem 1**.**

The iterates generated by Algorithm 1 satisfy

[TABLE]

In particular, the following perturbed Fenchel duality property holds

[TABLE]

for

[TABLE]

As we detail in Section 3 below, Theorem 1 yields the known convergence rates of the proximal subgradient method [9, 23, 34]. Theorem 1 also yields our second perturbed Fenchel duality, namely Theorem 2 below. The statement of Theorem 2 relies on two more pieces of notation. First, let $\theta_{k}\in[0,1],\;k=0,1,\dots$ be defined as follows

[TABLE]

The sequence $\theta_{k}\in[0,1],\;k=0,1,2,\dots$ can be seen as a transformation of the sequence $t_{k}\in(0,\infty),\;k=0,1,\dots$. Indeed, the sequences $t_{k}>0,\;k=0,1,\dots$ and $\theta_{k}\in[0,1],\;k=0,1,2,\dots$ are in one-to-one correspondence. Observe that the output sequence $x_{k},\;k=1,2,\dots$ defined via (10) satisfies

[TABLE]

and a similar identity holds for $z_{k},\;k=0,1,2,\dots$.

Second, let ${\mathcal{D}}:{\mathrm{dom}}(\Psi)\times{\mathrm{dom}}(\partial(f\circ A))\times{\mathrm{dom}}(\Psi)\times[0,1]\rightarrow{\mathbb{R}}$ be defined as follows

[TABLE]

where $D_{f\circ A}(s,y)=f(As)-f(Ay)-\left\langle g,A(s-y)\right\rangle$ for some $g\in\partial f(Ay)$ that will always be clear from the context.

Theorem 2**.**

The iterates generated by Algorithm 1 satisfy

[TABLE]

In particular, the following perturbed Fenchel duality property holds

[TABLE]

for

[TABLE]

Theorem 2 readily yields the following generic convergence result for the iterates generated by Algorithm 1.

Corollary 1**.**

Suppose the stepsizes $t_{k}$ chosen in Step 4 of Algorithm 1 satisfy

[TABLE]

Then

[TABLE]

In particular,

[TABLE]

for all $x\in{\mathbb{E}}$.

For ease of exposition we defer the proofs of Theorem 1 and Theorem 2 to Section 4. Section 3 below shows that the convergence rates of several popular first-order algorithms can be obtained from Theorem 1, Theorem 2, and Corollary 1.

3 Popular first-order algorithms

We next detail how several popular first-order algorithms and their convergence properties can be seen as instances of Algorithm 1 and the perturbed Fenchel duality results established in Section 2. For ease of notation and to be consistent with the most common format in the literature on first-order methods, throughout this section we consider the special case when ${\mathbb{F}}={\mathbb{E}}$ and $A$ is the identity mapping. In other words, we consider the convex minimization problem

[TABLE]

All of the developments in this section extend in a straightforward fashion to the more general problem (1) albeit with a bit of notation overhead.

A critical detail in all of the convergence results discussed below is the choice of stepsize $t_{k}$ in Step 4 of Algorithm 1. To that end, we will rely on the following terminology and assumption.

Definition 1**.**

We shall say that $t>0$ is admissible for $(s_{-},g)$ if

[TABLE]

is well-posed and it has a solution $\bar{s}$ that satisfies the optimality conditions

[TABLE]

for some $g^{\Psi}\in\partial\Psi(\bar{s})$.

We will make the following assumption throughout this section.

Assumption 2**.**

The functions $f,\Psi,h$ satisfy the following three conditions:

• (a)

The function $f+\Psi$ is bounded below.

• (b)

If $t>0$ and $(s_{-},g)$ are such that the mapping

[TABLE]

is bounded below then $t$ is admissible for $(s_{-},g)$.

• (c)

There exists an oracle that takes as input a tuple $(t,s_{-},g)$ and either determines that $t$ is not admissible for $(s_{-},g)$ or computes an optimal solution of (23) that satisfies the optimality conditions (9).

We bifurcate our analysis into two main cases based on the reference function $h$. Subsection 3.1 discusses a generalized conditional subgradient algorithm, which is a special instance of Algorithm 1 where the reference function $h$ is identically zero. Subsections 3.2 and 3.3 are devoted to Bregman proximal methods which are instances of Algorithm 1 where $h$ is a general (typically nonzero) convex differentiable reference function.

3.1 Generalized conditional subgradient algorithm ($h\equiv 0$)

Algorithm 2 describes a generalized conditional subgradient algorithm for problem (22). This format is the same one developed by Nesterov [30] but without any boundedness assumption on the domain. The usual conditional gradient method [3, 6, 21], also known as Frank-Wolfe method, corresponds to the case when $\Psi$ is the indicator function $i_{C}$ of some compact convex set $C\subseteq{\mathbb{E}}$ equipped with a computable linear oracle, and $f$ is differentiable. Algorithm 2 does not require $\Psi$ to be of this form. Instead, it requires nothing more than the specialization of our blanket admissibility assumption (Assumption 1) to the case when $h\equiv 0$.

Assumption 3**.**

At each iteration $k=0,1,\dots$ of Algorithm 2 Step 4 is well-posed, that is, a minimizer $s_{k}\in\operatorname*{arg\,min}_{s}\{\left\langle g_{k},s\right\rangle+\Psi(s)\}$ exists and is computable.

We note that Assumption 3 holds if ${\mathrm{dom}}(\Psi^{*})={\mathbb{E}}^{*}$ and there exists a computable oracle for $\partial\Psi^{*}$.

We shall also make the mild assumption that $\theta_{0}=1$ and $\theta_{k}\in(0,1),\;k=1,\dots$ in Algorithm 2. Under this assumption, it is evident that Algorithm 2 can be obtained as a special case of Algorithm 1 by picking $h\equiv 0,\;s_{-1}:=x_{0},\;y_{k}=x_{k},$ and $t_{k}>0,\;k=0,1,\dots$ so that (15) holds.

Since $h\equiv 0$ it follows that $D_{h}\equiv 0$ so (9) becomes $g_{k}+g_{k}^{\Psi}=0$. Hence $w_{k}=0$ and $d_{k}^{*}(-w_{k})=0$ for $k=0,1,\dots$. Thus Theorem 2 implies that for $k=1,2,\dots$

[TABLE]

where $D:{\mathrm{dom}}(\Psi)\times{\mathrm{dom}}(\Psi)\times[0,1]\rightarrow{\mathbb{R}}$ is the simplified version of ${\mathcal{D}}$ defined as follows

[TABLE]

Observe that (3.1) can be written as

[TABLE]

where ${\mathsf{CGgap}}_{k},\;k=1,2,\dots$ is defined via ${\mathsf{CGgap}}_{1}=D(x_{0},s_{0},1)=D_{f}(s_{0},x_{0})$ and

[TABLE]

We will consider the case when $f,\Psi$ satisfy the following kind of curvature condition: there exists a constant $M>0$ and $\nu>0$ such that for $x,s\in{\mathrm{dom}}(\Psi)$

[TABLE]

The curvature condition (29) holds in particular when ${\mathrm{dom}}(\Psi)$ is bounded and $\nabla f$ satisfies the following Hölderian continuity assumption for some norm in ${\mathbb{E}}$: for all $x,y\in{\mathrm{dom}}(f)$

[TABLE]

The curvature condition (29) is inspired by and generalizes the following curvature constant defined by Jaggi [21] in the special case when $f$ is differentiable and $\Psi=i_{C}$ for some compact convex set $C\subseteq{\mathbb{E}}$:

[TABLE]

Indeed, $\mathcal{C}_{f}$ is the smallest constant such that for $x,s\in C$

[TABLE]

Observe that (31) is identical to (29) when $\nu=1$ and $\Psi=i_{C}$.

The proof of our ultimate proposition for the generalized conditional subgradient method will rely on the following weighted arithmetic-geometric mean (AM-GM) inequality: If $a,b>0$ and $\alpha,\beta\geq 0$ are such that $\alpha+\beta>0$ then

[TABLE]

This inequality is also pivotal to our analysis of the fast and universal Bregman proximal methods.

Proposition 1**.**

Suppose $f,\Psi$ satisfy (29) and Step 5 of Algorithm 2 chooses $\theta_{k}=\frac{1+\nu}{k+1+\nu}$. Then the iterates $x_{k},\;k=1,2,\dots$ generated by Algorithm 2 satisfy

[TABLE]

Proof.

By (27) it suffices to show that

[TABLE]

We proceed by induction. For $k=1$ inequality (29) implies that

[TABLE]

The second step follows from $(2+\nu)^{\nu}\leq(1+\nu)^{1+\nu}$ which in turn follows from the weighted AM-GM inequality (32) applied to $a=1,b=2+\nu,\alpha=1,\beta=\nu$.

Suppose (34) holds for $k\geq 1.$ Then (28), (29), and $\theta_{k}=\frac{1+\nu}{k+1+\nu}$ imply that

[TABLE]

The last step follows from $(k+1)(k+2+\nu)^{\nu}\leq(k+1+\nu)^{1+\nu}$ which in turn follows from the weighted AM-GM inequality (32) applied to $a=k+1,b=k+2+\nu,\alpha=1,\beta=\nu$. ∎

The above convergence rate generalizes the popular convergence rate for the conditional gradient algorithm [6, 21], which relies on the curvature condition (31) and sets $\theta_{k}=2/(k+2)$. The latter case corresponds to the special case $\nu=1$ and $\Psi=i_{C}$ in Proposition 1.

In principle Proposition 1 requires knowledge of $\nu$ since it relies on the stepsize choice $\theta_{k}=(\nu+1)/(k+\nu+1)$. However, it is easy to see that the ${\mathcal{O}}(1/k^{\nu})$ convergence rate in Proposition 1 remains attainable without knowledge of $\nu$ if the stepsize $\theta_{k}\in[0,1]$ is instead chosen via the following line-search procedure

[TABLE]

The above results are similar in spirit to the ${\mathcal{O}}(1/k^{\nu})$ convergence results developed in [30, Section 2]. However, unlike Proposition 1 which is entirely norm-independent, the results in [30] have an additional dependence on the diameter of the domain of the problem which is assumed to be bounded.

The linear convergence result stated in Proposition 2 below illustrates the power of the line-search procedure (35). To that end, we consider the case when $f,\Psi$ satisfy the following type of joint smoothness and strong convexity property: there exists a constant $M>0$ such that for all $(x,s,\theta)\in{\mathrm{dom}}(\Psi)\times{\mathrm{dom}}(\Psi)\times[0,1]$

[TABLE]

It is easy to see that (36) holds with $M=L/\mu$ when $\nabla f$ is $L$-Lipschitz continuous and $\Psi$ is $\mu$-strongly convex on ${\mathrm{dom}}(\Psi)$ for some norm in ${\mathbb{E}}$ and some $L,\mu>0$. Thus (36) can indeed be seen as a type of joint smoothness and strong convexity of $f,\Psi$. Observe that (36) is intrinsic to $f,\Psi$ and is norm-independent as is the following result.

Proposition 2**.**

Suppose $f$ and $\Psi$ satisfy (36) some constant $M>0$ and Step 5 of Algorithm 2 chooses $\theta_{k}\in[0,1]$ via the line-search procedure (35). Then the iterates generated by Algorithm 2 satisfy

[TABLE]

Proof.

From (26) and (36) it follows that for all $(x,s,\theta)\in{\mathrm{dom}}(\Psi)\times{\mathrm{dom}}(\Psi)\times[0,1]$

[TABLE]

In particular, for $\bar{\theta}=1/(M+1)\in(0,1)$ we have

[TABLE]

Therefore (28) and the choice of $\theta_{k}$ via the line-search procedure (35) imply that

[TABLE]

Consequently

[TABLE]

∎

Notice that this linear convergence result does not require any boundedness assumption as in [30]. Furthermore, the line-search procedure (35) does not require knowledge of $M$.

3.2 Bregman proximal gradient and subgradient algorithm

($h\not\equiv 0$)

Algorithm 3 describes a template that includes both a Bregman proximal gradient and a Bregman proximal subgradient algorithm for problem (22). Note that Algorithm 3 can be obtained as a special case of Algorithm 1 by picking $y_{k}=s_{k-1}$.

3.2.1 Bregman proximal gradient

Suppose Step 4 of Algorithm 3 chooses an admissible stepsize $t_{k}>0$ so that the following version of (20) holds for $y_{k}=s_{k-1}$

[TABLE]

Then Corollary 1 implies that for all $x\in{\mathbb{E}}$

[TABLE]

We will assume that Step 4 of Algorithm 3 chooses a sufficiently large admissible $t_{k}$ so that (37) holds. To that end, observe that for any fixed $r>1$ a straightforward back-tracking procedure finds $t_{k}$ within a factor of $r$ of the largest admissible $t_{k}>0$ such that (37) holds. When that is the case we will say that $t_{k}$ is $r$-large for (37). We note that such a back-tracking procedure is implementable provided Assumption 2 holds.

We next consider the case when $f$ is differentiable and satisfies the following kind of smoothness relative to $h$: there exists some constant $L>0$ such that

[TABLE]

Observe that (39) holds in particular when $\nabla f$ is $L$-Lipschitz and $h$ is 1-strongly convex for some norm in ${\mathbb{E}}$. However, the relative smoothness condition (39) holds much more broadly. The concept of relative smoothness and its close connection with first-order methods has been developed and discussed in [4, 24, 34, 35, 36].

Our analysis of the Bregman proximal convergence will rely on the following property of the function ${\mathcal{D}}$ defined via (16). The convexity of $f+\Psi$ implies that

[TABLE]

Proposition 3**.**

Suppose (39) holds and Step 4 of Algorithm 3 chooses $t_{k}$ that is $r$-large for (37) for some $r>1$. Then the sequence $x_{k},\;k=1,2,\dots$ generated by Algorithm 3 satisfies

[TABLE]

for all $x\in{\mathbb{E}}$.

Proof.

The smoothness condition (39), Assumption 2, inequality (40), and $y_{k}=s_{k-1}$ imply that $t_{k}>0$ is admissible for $(s_{k-1},g_{k}):=(s_{k-1},\nabla f(s_{k-1}))$ and makes (37) hold if $t_{k}\leq 1/L$.

Indeed, if $t_{k}\in(0,1/L]$ then (39) implies that

[TABLE]

Hence Assumption 2 implies that any $t_{k}\in(0,1/L]$ is admissible. Furthermore, inequality (40), $t_{k}\in(0,1/L]$, $y_{k}=s_{k-1}$, and (39) implies that

[TABLE]

which is precisely (37).

Thus Step 4 of Algorithm 3 chooses $t_{k}\geq 1/(rL)$ so that (37) holds and consequently (38) implies that

[TABLE]

for all $x\in{\mathbb{E}}$. ∎

Proposition 3 recovers the classic ${\mathcal{O}}(1/k)$ convergence rate of proximal gradient methods [4, 8, 24, 34].

3.2.2 Bregman proximal subgradient

Consider now the case when $f$ is not necessarily differentiable but instead the function $f$ satisfies the following kind of continuity relative to $(h,\Psi)$: there exists a constant $M>0$ such that for all $y\in{\mathrm{dom}}(h)\cap{\mathrm{dom}}(\Psi),\;g\in\partial f(y),\;s\in{\mathrm{dom}}(h)\cap{\mathrm{dom}}(\Psi)$ and $t>0$

[TABLE]

(Recall that we assume ${\mathrm{dom}}(\Psi)\subseteq{\mathrm{dom}}(\partial f)$.) The above concept of relative continuity coincides with the ones proposed by Lu [23] and by Teboulle [34] in the following special case. When $\Psi=i_{C}$ for some closed convex set $C\subseteq{\mathbb{E}}$ the relative continuity condition (41) holds if and only if for all $y\in{\mathrm{dom}}(h)\cap C,\;g\in\partial f(y),$ and $s\in{\mathrm{dom}}(h)\cap C$

[TABLE]

To motivate how the above condition is related to continuity, we next show that (41) holds for $y\in\text{relint}({\mathrm{dom}}(f))\cap{\mathrm{dom}}(\Psi)$ and $s\in{\mathrm{dom}}(\Psi)$ if both $f$ and $\Psi$ are Lipschitz continuous on ${\mathrm{dom}}(\Psi)$ and $h$ is 1-strongly convex on ${\mathrm{dom}}(\Psi)$ for some norm $\|\cdot\|$ in ${\mathbb{E}}$. Indeed, suppose $f$ and $\Psi$ are respectively $L_{f}$ and $L_{\Psi}$ Lipschitz continuous on ${\mathrm{dom}}(\Psi)$, and $y\in\text{relint}({\mathrm{dom}}(f))\cap{\mathrm{dom}}(\Psi)$. It thus follows that for $g\in\partial f(y)$ and $s\in{\mathrm{dom}}(\Psi)$

[TABLE]

and consequently

[TABLE]

for $M:=L_{\Psi}+L_{f}$. Therefore for all $t>0$

[TABLE]

Assumption 2 implies that if (41) holds then any $t>0$ is admissible for $(y,g)$ for all $y\in{\mathrm{dom}}(\Psi)$ and $g\in\partial f(y)$.

Proposition 4**.**

Suppose (41) holds. Then the sequence $z_{k},\;k=1,2,\dots$ generated by Algorithm 3 satisfies

[TABLE]

In particular, for all $x\in{\mathbb{E}}$

[TABLE]

and thus if $t_{i}:=\mathcal{C}/\sqrt{k},\;i=0,1,\dots,k-1$, then we get

[TABLE]

Proof.

Since Algorithm 3 chooses $y_{k}=s_{k-1},\;k=0,1,2,\dots$, the relative continuity condition (41) implies that for $k=0,1,2,\dots$

[TABLE]

Thus (42) follows from (13) and (14) in Theorem 1. ∎

Proposition 4 recovers the classic ${\mathcal{O}}(1/\sqrt{k})$ convergence rate of the projected gradient method, that is, the case when $\Psi=i_{C}$ for some closed convex $C\subseteq{\mathbb{R}}^{n}$ as discussed in [23, 34].

3.3 Fast and universal Bregman proximal gradient algorithm

Suppose $f$ is differentiable. Algorithm 4 describes a template that includes both a fast and a universal Bregman proximal gradient method. The fast gradient method is nearly identical to the iconic fast gradient algorithms proposed by Beck and Teboulle [8] and by Nesterov [28]. The universal gradient method is nearly identical to the method proposed by Nesterov [29]. Algorithm 4 relies on the sequence $\theta_{k},\;k=0,1,\dots$ defined via (15). Algorithm 4 can be obtained as a special case of Algorithm 1 by picking $y_{k}=(1-\theta_{k})x_{k}+\theta_{k}s_{k-1}$.

The main convergence properties of Algorithm 4 rely on some properties of the sequences $t_{k},\;k=0,1,\dots$ and $\theta_{k},\;k=0,1,\dots$. Observe that the construction (15) implies that for $k=0,1,\dots$

[TABLE]

and thus

[TABLE]

We will also rely on the following lemma. For ease of exposition, we defer the proof of Lemma 1 to Section 3.3.3.

Lemma 1**.**

Let $\theta_{0}=1,t_{0}>0$ and let $t_{k},\theta_{k}$ for $k=0,1,\dots$ be such that (43) holds for $k=0,1,\dots.$ If $\gamma\geq 1$ and $L>0$ are constants such that

[TABLE]

then

[TABLE]

Our analyses of the fast and universal Bregman proximal convergence will also rely on the following property of the function ${\mathcal{D}}$ defined via (16). The convexity of $\Psi$ and $f$ imply that

[TABLE]

3.3.1 Fast Bregman proximal gradient

As in Algorithm 3, suppose Step 4 of Algorithm 4 chooses an admissible $t_{k}$ so that (37) holds. Thus (38) holds as well. We next show that under a suitable smoothness assumption, the denominator $\sum_{i=0}^{k-1}t_{i}$ grows faster than in Algorithm 3 thereby attaining acceleration. To that end, we consider the case when $f$ satisfies the following kind of smoothness relative to $h$: there exist constants $L>0$ and $\gamma>1$ such that for $x,s,s_{-}\in{\mathrm{dom}}(f)$ and $\theta\in[0,1]$

[TABLE]

The smoothness condition (47) holds in particular for $\gamma=2$ when $h$ is 1-strongly convex and $\nabla f$ is $L$-Lipschitz continuous for some norm in ${\mathbb{E}}$. It also holds when $f$ satisfies the relative smoothness condition (39) and $D_{h}$ has triangle scaling exponent $\gamma$ as defined in [19].

The smoothness condition (47) is evidently stronger than the relative smoothness condition (39) since (47) for $\theta=1$ is identical to (39). The recent developments in [14] show that a condition stronger than (39) is inevitable for achieving a Bregman proximal gradient method with convergence rate faster than ${\mathcal{O}}(1/k)$.

Proposition 5**.**

Suppose (47) holds and Step 4 of Algorithm 4 chooses $t_{k}$ that is $r$-large for (37) for some $r>1$. Then the sequence $x_{k},\;k=1,2,\dots$ generated by Algorithm 4 satisfies

[TABLE]

for all $x\in{\mathbb{E}}$.

Proof.

The smoothness condition (47), Assumption 2, and inequality (3.3) imply that $t_{k}>0$ is admissible for $(s_{k-1},g_{k}):=(s_{k-1},\nabla f(y_{k}))$ and (37) holds if $t_{k}\theta_{k}^{\gamma-1}\leq 1/L$.

Indeed, if $t_{k}>0$ and $t_{k}\theta_{k}^{\gamma-1}\in(0,1/L]$ then (47) and the convexity of $\Psi$ imply that

[TABLE]

Hence Assumption 2 implies that any $t_{k}>0$ with $t_{k}\theta_{k}^{\gamma-1}\in(0,1/L]$ is admissible. Furthermore, inequality (3.3), $t_{k}\theta_{k}^{\gamma-1}\in(0,1/L]$, $y_{k}=(1-\theta_{k})x_{k}+\theta_{k}s_{k-1}$, and (47) implies that

[TABLE]

which is precisely (37).

Thus Step 4 of Algorithm 4 chooses $t_{k}>0$ so that (37) holds with $t_{k}\theta_{k}^{\gamma-1}\geq 1/(r^{\gamma}L)$. Therefore (38) and Lemma 1 applied to $r^{\gamma}L$ in lieu of $L$ imply (48). ∎

Proposition 5 recovers the optimal convergence rate of fast proximal gradient methods [8, 26, 28]. Indeed, all of the results discussed in [8, 26, 28] apply when $\nabla f$ is $L$-Lipchitz and $h$ is $1$-strongly convex for some norm in ${\mathbb{E}}$. In that case the relative smoothness condition (47) holds for $\gamma=2$ and thus the iconic ${\mathcal{O}}(1/k^{2})$ rate of convergence follows from Proposition 5.

3.3.2 Universal Bregman proximal gradient

Suppose $\epsilon>0$ is fixed and Algorithm 4 chooses an admissible $t_{k}$ so that the following version of (20) holds:

[TABLE]

Then Corollary 1 implies that for all $x\in{\mathbb{E}}$

[TABLE]

We next consider the case when $f$ satisfies the following kind of relative smoothness condition: there exist constants $\nu\in[0,1]$ and $M>0$ such that for $x,s,s_{-}\in{\mathrm{dom}}(f)$ and $\theta\in[0,1]$

[TABLE]

The smoothness condition (51) holds in particular when $h$ is 1-strongly convex and $\nabla f$ satisfies the Hölderian continuity condition (30) for some norm in ${\mathbb{E}}$.

Proposition 6**.**

Suppose (51) holds and Step 4 of Algorithm 4 chooses $t_{k}$ that is $r$-large for (49) for some fixed $\epsilon>0$ and some $r>1$. Then the sequence $x_{k},\;k=1,2,\dots$ generated by Algorithm 4 satisfies

[TABLE]

for all $x\in{\mathbb{E}}$.

Proof.

We prove (52) for $\nu\in[0,1)$. The case $\nu=1$ follows as a limit case. The smoothness condition (51), Assumption 2, and inequality (3.3) imply that $t_{k}>0$ is admissible and (49) holds if the following inequality holds for $s\in{\mathrm{dom}}(h)$:

[TABLE]

Indeed, if $t_{k}>0$ and (53) holds then (51) and the convexity of $\Psi$ imply that

[TABLE]

Hence Assumption 2 implies that any $t_{k}>0$ that satisfies (53) is admissible. Furthermore, $y_{k}=(1-\theta_{k})x_{k}+\theta_{k}s_{k-1}$, together with (3.3), (51), and (53) imply that

[TABLE]

which is precisely (49).

If we let $\alpha:=\frac{1-\nu}{2},\beta:=\frac{1+\nu}{2},a:=\frac{1+\nu}{1-\nu}\cdot\frac{\epsilon}{M\theta_{k}^{\nu}},$ and $b:=\frac{D_{h}(s,s_{k-1})}{a^{\frac{1-\nu}{1+\nu}}}$ then the weighted AM-GM inequality (32) implies that

[TABLE]

Therefore (53) holds if $t_{k}>0$ satisfies the following inequality

[TABLE]

Consequently, Step 4 of Algorithm 4 chooses an admissible $t_{k}>0$ so that (49) holds with

[TABLE]

Lemma 1 applied to $\gamma=\frac{1+3\nu}{1+\nu}$ and $L=r^{\frac{1+3\nu}{1+\nu}}\left(\frac{M^{2}}{\epsilon^{1-\nu}}\right)^{\frac{1}{1+\nu}}=\frac{r^{\frac{1+3\nu}{1+\nu}}M^{\frac{2}{1+\nu}}}{\epsilon^{\frac{1-\nu}{1+\nu}}}$ implies that

[TABLE]

The last step holds because $\nu\in[0,1)$ implies that $1\leq\gamma<2$. Thus (50) yields (52). ∎

Proposition 6 recovers the optimal convergence rate of the universal Bregman gradient algorithm established by Nesterov [29].

3.3.3 Proof of Lemma 1

We prove (45) by induction on $k$. For $k=0$ inequality (45) is the same as (44) for $i=0$ and thus readily follows from it. Suppose (45) holds for $k\geq 0$. Hence for some $\hat{k}\geq k$ we have

[TABLE]

Putting together (43), (54), and (44) applied to $i=k+1$ we get

[TABLE]

Hence $\theta_{k+1}\geq\hat{\theta}$ where $\hat{\theta}\in(0,1)$ is the positive root of the equation

[TABLE]

Next, observe that the weighted AM-GM inequality (32) applied to $a=\hat{k}+1,b=\hat{k}+1+\gamma,\alpha=1,\beta=\gamma-1$ yields

[TABLE]

Thus for $\theta:=\frac{\gamma}{\hat{k}+1+\gamma}$ we get

[TABLE]

Hence the monotonicity of $\theta\mapsto\frac{\theta^{\gamma}}{1-\theta}$ implies that the positive root $\hat{\theta}$ of (55) satisfies $\hat{\theta}\leq\frac{\gamma}{\hat{k}+1+\gamma}\leq\frac{\gamma}{k+1+\gamma}.$ Finally, the inequality $\theta_{k+1}\geq\hat{\theta}$ together with (43) and (54) imply that

[TABLE]

Therefore (45) also holds for $k+1$. ∎

4 Proofs of Theorem 1 and Theorem 2

Proof of Theorem 1.

By rearranging terms, identity (12) can be rewritten as follows

[TABLE]

Identity (56) in turn follows from identities (57) and (58) below. For $k=1,2,\dots$

[TABLE]

and

[TABLE]

Indeed, (57), and the construction (11) of $w_{k}$ and $d_{k}$ readily yield

[TABLE]

Thus

[TABLE]

Therefore, after dividing (58) by $\sum_{i=0}^{k-1}t_{i}$ we obtain (56).

We next prove (57) and (58). Identity (57) is an immediate consequence of (9). We prove (58) by induction. First, observe that $g_{i}\in\partial f(Ay_{i})$ and $g_{i}^{\Psi}\in\partial\Psi(s_{i})$ imply that

[TABLE]

for $i=0,1,\dots$. Thus from (9) it follows that for $i=0,1,\dots$

[TABLE]

For $k=1$ identity (58) immediately follows from (59) applied to $i=0$. Suppose (58) holds for $k\geq 1$. The three-point property of $D_{h}$ [12, Lemma 3.1] implies that

[TABLE]

Thus, after adding up (58), (59) applied to $i=k$, and (60) we get

[TABLE]

Hence (58) also holds for $k+1$.

To finish the proof of Theorem 1 we next prove (13) for $\delta_{k}$ as in (14). The convexity of $f,\Psi,f^{*},\Psi^{*}$, the constructions (10) and (11) of $z_{k},u_{k}$ and $w_{k}$, and (12) imply that

[TABLE]

Since $(\Psi+d_{k})^{*}(-A^{*}u_{k})\leq\Psi^{*}(w_{k}-A^{*}u_{k})+d_{k}^{*}(-w_{k})$, inequality (13) for $\delta_{k}$ as in (14) follows from (61).

∎

Proof of Theorem 2.

We rely on Theorem 1. Observe that identity (12) can be rewritten as follows

[TABLE]

Thus to show (17) it suffices to show

[TABLE]

We prove (62) by induction. The case $k=1$ readily follows as both sides in (62) equal zero when $k=1$. Suppose (62) holds for $k$. Since $x_{k+1}=x_{k}+\theta_{k}(s_{k}-x_{k})$, from (16) it follows that

[TABLE]

After adding up $(1-\theta_{k})$ times (62) plus (63) we get

[TABLE]

Thus (62) also holds for $k+1$.

To finish the proof of Theorem 2 we next prove (18) for $\delta_{k}$ as in (19). The convexity of $f^{*},\Psi^{*}$ and the constructions (10) and (11) of $u_{k}$ and $w_{k}$ imply that

[TABLE]

Since $(\Psi+d_{k})^{*}(-A^{*}u_{k})\leq\Psi^{*}(w_{k}-A^{*}u_{k})+d_{k}(-w_{k})$, inequality (18) for $\delta_{k}$ as in (19) follows by putting together (17) and (64). ∎