Abstract
Convergence analysis of accelerated firstorder methods for convex optimization problems are developed from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient (NAG) flow, is derived from the connection between acceleration mechanism and Astability of ODE solvers, and the exponential decay of a tailored Lyapunov function along with the solution trajectory is proved. Numerical discretizations of NAG flow are then considered and convergence rates are established via a discrete Lyapunov function. The proposed differential equation solver approach can not only cover existing accelerated methods, such as FISTA, Güler’s proximal algorithm and Nesterov’s accelerated gradient method, but also produce new algorithms for composite convex optimization that possess accelerated convergence rates. Both the convex and the strongly convex cases are handled in a unified way in our approach.
Introduction
We consider iterative methods for solving the unconstrained minimization problem
where V is a Hilbert space, and \(f:V\rightarrow {\mathbb {R}}\cup \{+\infty \}\) is a properly closed convex function. We shall first consider smooth f on the entire space V and later focus on the composite case \(f = h + g\) where both h (smooth) and g (nonsmooth) are convex on some (simple) closed convex set \(Q\subseteq V\). We are mainly interested in the development and analysis of accelerated firstorder methods.
Suppose V is equipped with the inner product \((\cdot ,\cdot )\) and the correspondingly induced norm \(\left\Vert {\cdot } \right\Vert \). We use \(\left\langle {\cdot ,\cdot } \right\rangle \) to denote the duality pair between \(V^*\) and V, where \(V^*\) is the continuous dual space of V and is endowed with the conventional dual norm \(\left\Vert {\cdot } \right\Vert _{*}\). For any interval \(I\subseteq {\mathbb {R}}\), denote by \(C^k(I;V)\) the space of all ktimes continuous differentiable Vvalued functions on I, and the superscript k is dropped when \(k=0\). Let \(\varOmega \subseteq V\) be some closed convex subset, we say \(f\in \mathcal S_{\mu }^{1}(\varOmega )\) if it is continuous differentiable on \(\varOmega \) and there exists \(\mu \geqslant 0\) such that
We call (2) the \(\mu \)convexity of f and when \(\mu >0\), we say f is strongly convex. We also write \(f\in \mathcal S^{1,1}_{\mu ,L}(\varOmega )\) if \(f\in {\mathcal {S}}_{\mu }^{1}(\varOmega )\) and \(\nabla f\) is Lipschitz continuous on \(\varOmega \): there exists \(0<L<\infty \) such that
By [29, Theorem 2.1.5], this implies the inequality
For \(\varOmega = V\), we shall write \({\mathcal {S}}_{\mu }^{1}(\varOmega )\) and \({\mathcal {S}}^{1,1}_{\mu ,L}(\varOmega )\) as \({\mathcal {S}}_{\mu }^{1}\) and \({\mathcal {S}}^{1,1}_{\mu ,L}\), respectively.
The above functional classes are what we work with in this paper. As for the optimization problem (1), we also care about the global minimizer(s) of f. For strongly convex case, it is wellknown that the minimizer exists uniquely. However, for convex case, to promise the existence of minimizers, additional assumption, such as the coercivity condition, is usually imposed. Throughout, we denote by \(\mathrm{argmin}\,f\) the set of global minimizers of (1) and assume it is nonempty.
One approach to derive the gradient descent (GD) method is discretizing an ordinary differential equation (ODE), i.e., the socalled gradient flow:
Here we introduce an artificial time variable t and \(x'\) is the derivative taken with respect to t. For ease of notation, in the sequel, we shall omit t when no confusion arises. The simplest forward (explicit) Euler method with step size \(\eta _k>0\) leads to the GD method
In the terminology of numerical analysis, it is wellknown that this method is conditionally Astable (cf. Sect. 2), and for \(f\in {\mathcal {S}}_{\mu ,L}^{1,1}\) with \(0\leqslant \mu \leqslant L<\infty \), the step size \(\eta _k=1/L\) is allowed to get the rate (see [29, Chapter 2])
One can also consider the backward (implicit) Euler method
which is unconditionally Astable (cf. Sect. 2) and coincides with the wellknown proximal point algorithm (PPA) [33]
Note that this method allows f to be nonsmooth and possesses linear convergence rate even for convex functions, as long as \(\eta _k\geqslant \eta >0\) for all \(k>0\).
Main results
Let us start from the quadratic objective \(f(x) = \frac{1}{2}x^{\top }Ax\) over \({{\mathbb {R}}}^d\), for which the gradient flow (5) reads simply as
where A is symmetric positive semidefinite and makes \(f\in {\mathcal {S}}_{\mu ,L}^{1,1}\). Instead of solving (9), we turn to a general linear ODE system
Briefly speaking, our main idea is to seek such a system (10) with some asymmetric block matrix G that transforms the spectrum of A from the real line to the complex plane and reduces the condition number from \(\kappa (A) = L/\mu \) to \(\kappa (G) = O(\sqrt{L/\mu })\). Afterwards, accelerated gradient methods may be constructed from Astable methods for solving (10) with a significant larger step size and consequently improve the contraction rate from \(O((1\mu /L)^k)\) to \(O((1\sqrt{\mu /L})^k)\). Furthermore, to handle the convex case \(\mu =0\), we combine the transformation idea with suitable time scaling technique; for more details, we refer to Sect. 2.
One successful and important transformation is given below
where the builtin scaling factor \(\gamma \) is positive and satisfies
Based on this, for general \(f\in {\mathcal {S}}_\mu ^1\) with \(\mu \geqslant 0\), we replace A in (11) with \(\nabla f\) and write \(y=(x,v)\) to obtain a firstorder dynamical system:
Eliminating v, we arrive at a secondorder ODE of x:
which is actually a heavy ball model (cf. (21)) with variable damping coefficients in front of \(x''\) and \(x'\). Thanks to the scaling factor \(\gamma \), we can handle both the convex case (\(\mu = 0\)) and the strongly convex case (\(\mu > 0\)) in a unified way. Moreover, we shall prove the exponential decay property
for a tailored Lyapunov function
where \(x^*\in \mathrm{argmin}\,f\) is a global minimizer of f.
Accelerated gradient methods based on numerical discretizations of the dynamical system (13) with \(f\in \mathcal S_{\mu ,L}^{1,1}\) are then considered and analyzed by means of a discrete version of the Lyapunov function (16). It will be shown that the implicit scheme (see (72)) possesses linear convergence rate as long as the time step size is uniformly bounded below. This matches the exponential decay rate (15) in the continuous level. Also, for convex case \(\mu =0\), this implicit method amounts to an accelerated PPA, that is very close to Güler’s PPA [20] and enjoys the same rate \(O(1/k^2)\) (cf. Theorem 4). In Sect. 5, for semiimplicit schemes with suitable corrections (either an extrapolation or a gradient step), we prove the following convergence rate
which is optimal in the sense of [29]. Moreover, we can recover Nesterov’s optimal method [27, 29] exactly from a semiimplicit scheme with a gradient descent correction; see Sect. 6. Therefore, instead of using estimate sequence, our ODE approach provides an alternative derivation of Nesterov’s method and hopefully more intuitive for understanding the acceleration mechanism. From this point of view, we name both (13) and (14) as Nesterov accelerated gradient (NAG) flow.
As a proof of concepts, we also generalize our NAG flow to the composite case
where \(Q\subseteq V\) is a (simple) closed convex set, \(h\in \mathcal S_{\mu ,L}^{1,1}(Q)\) with \(0\leqslant \mu \leqslant L<\infty \) and \(g:V\rightarrow {\mathbb {R}}\cup \{+\infty \}\) is proper, closed, and convex. We use \( \mathbf{dom\,}g\) to denote the effective domain of g and assume that \(Q\cap \mathbf{dom}\, g\ne \emptyset \). Treating (18) as an unconstrained minimization of \(F=f+i_Q\) where \(i_Q\) denotes the indicator function of Q, the generalized version of (14) is a secondorder differential inclusion
We shall give the existence of the solution to (19) in proper sense and then obtain the exponential decay (15) for almost all \(t>0\).
For the unconstrained case \(Q = V\), by using the tool of composite gradient mapping [29, Chapter 2], a semiimplicit scheme with correction for the generalized NAG flow (19) is presented and leads to an accelerated proximal gradient method (APGM); see Algorithm 2. We also give a simplified variant that is closely related to FISTA [12]. For the constrained problem (18), an accelerated forwardbackward method is proposed in Algorithm 4. Both two algorithms call the proximal operation of g (over Q) only once in each iteration, and they are proved to share the same accelerated convergence rate (17).
The rest of this paper is organized as follows. In the continuing of the introduction, we will review some existing works devoting to the accelerated gradient methods from the ODE point of view. Next, in Sect. 2, we shall explain the acceleration mechanism from Astability theory of ODE solvers and derive our NAG flow as well. Then in Sect. 3 we focus on the NAG flow and prove its exponential decay. After that, accelerated gradient methods based on numerical discretizations of NAG flow are proposed and analyzed in Sects. 4, 5 and 6. Finally, in Sect. 7, we extend the our NAG flow to composite optimization and propose two new accelerated methods with convergence analysis.
Related works
The wellknown momentum method can be traced back to the 1960s. In [34], Polyak studied the heavy ball (HB) method
and its continuous analogue, the heavy ball dynamical system:
Local linear convergence results for (20) via spectrum analysis were established in [34, Theorem 9]. Note that the HB method (20) adds a momentum term up to the gradient step and is sensitive to its parameters. For \(f\in \mathcal S_{\mu ,L}^{1,1}\), it shares the same theoretical convergence rate (6) as the gradient descent method; see [18, 40]. To our best knowledge, no work has established the global accelerated rate (17) for the original HB method (20). Recently, Nguyen et al. [26] developed the socalled accelerated residual method which combines (20) with an extra gradient descent step:
Numerically, they verified the efficiency and usefulness of this method with a restart strategy. We refer to [1, 3, 11, 19] for further investigations of the HB system (21).
To understand an accelerated gradient method with the rate \(O(1/k^2)\) proposed by Nesterov [27], Su, Boyd and Candès [37] derived the following secondorder ODE
where \(\alpha >0\) and \(f\in {\mathcal {S}}_{0,L}^{1,1}\). If \(\alpha \geqslant 3\) or \(1<\alpha <3\) and \((ff(x^*))^{(\alpha 1)/2}\) is convex, they proved the decay rate \(O(t^{2})\). If \(\alpha \geqslant 3\) and f is strongly convex, they also obtained a faster rate \(O(t^{2\alpha /3})\). Later on, Aujol and Dossal [10] established a generic result:
where \(\beta >0\) and \((ff(x^*))^\beta \) is convex. Almost at the same time, Attouch et al. [8] obtained the estimate (23) for \(\beta =1\) and considered numerical discretizations for (22) with the convergence rate \(O(k^{\min \{2,2\alpha /3\}})\). Also, Vassilis et al. [42] studied the nonsmooth version of (22):
They proved that the solution trajectory of (24) converges to a minimizer of f and derived the decay estimate (23) for \(\beta =1\). For more works and generalizations related to the model (22) and the corresponding algorithms, we refer to [2, 5,6,7, 14] and references therein.
Recently, Wibisono et al. [43] introduced a Lagrangian
for smooth and convex f, where \(\alpha :{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) is continuous and \(\beta :\mathbb R_+\rightarrow {\mathbb {R}}_+ \) satisfies
The Lagrangian (25) itself introduces a variational problem, the Euler–Lagrange equation to which is
They then established the convergence rate (cf. [43, Theorem 2.1])
by means of the Lyapunov function
Following this work, for \(f\in {\mathcal {S}}_{\mu }^1\) with \(\mu >0\), Wilson et al. [44] introduced another Lagrangian whose Euler–Lagrange equations reads as
with the same scaling function \(\alpha \) in (25). They proved the decay estimate (28) as well, by using the Lyapunov function
When \(\alpha =\sqrt{\mu }\), (29) gives the following model
which reduces to an HB system (cf. (21)); see also Siegel [38].
In addition, Siegel [38] and Wilson et al. [44] proposed two semiexplicit schemes for (31) individually. Both of their schemes are supplemented with an extra gradient descent step and share the same linear convergence rate \(O((1\sqrt{\mu /L})^k)\).
Recently, introducing the socalled duality gap which is the difference of appropriate upper and lower bound approximations for the objective function, Diakonikolas and Orecchia [17] presented a general framework for the construction and analysis of continuous time dynamical systems and the corresponding numerical discretizations. They recovered several existing ODE models such as the gradient flow (5), the mirror descent dynamic system and its accelerated version. We mention that the derivation of our NAG flow and analyses of discrete algorithms are fundamentally different from their duality gap technique.
Stability of ODE solvers and acceleration
In what follows, for any \(M\in \mathbb R^{d\times d}\), \(\sigma (M)\) denotes the spectrum of M, i.e., the set of all eigenvalues of M. The spectral radius is then defined by \(\rho (M) := \max _{\lambda \in \sigma (M)} \lambda \), and when M is invertible, its condition number \(\kappa (M) := \rho (M^{1})\rho (M)\). If \(\sigma (M)\subset {\mathbb {R}}\), then \(\lambda _{\min }(M)\) and \(\lambda _{\max }(M)\) stand for the minimum and maximum of \(\sigma (M)\), respectively. Moreover, \(\left\Vert {\cdot } \right\Vert _2\) is the usual 2norm for vectors and matrices.
To present our main idea as simple as possible, in this section, unless other specified, we restrict ourselves to the quadratic objective \(f(x) = \frac{1}{2}x^{\top }Ax\), where A is a symmetric matrix with the bound
For this model example, \(\nabla f(x) = Ax\) and the gradient flow (5) reads as \(x'=Ax\). The global minimal is achieved at \(x^*=0\), and when \(\mu >0\), the condition number of A is \(\kappa (A)=L/\mu \).
Astability of ODE solvers
Let \(G\in {\mathbb {R}}^{d\times d}\) and assume \({{\mathfrak {R}}}{{\mathfrak {e}}}(\lambda ) < 0\) for all \(\lambda \in \sigma (G)\). For the linear ODE system
it is not hard to derive that \(\left\Vert {y(t)} \right\Vert _2\rightarrow 0\) as \(t\rightarrow \infty \) (see [13, Theorem 7] for instance). Hence \(y^*=0\) is an equilibrium of the dynamic system (32).
We now recall the concept of Astability of ODE solves [23, 39]. A onestep method \(\phi \) for (32) with step size \(\alpha >0\) can be formally written as
As \(y^* = 0\) is an equilibrium point, (33) also gives the error equation. The scheme \(\phi \) is called absolute stable or Astable if \(\rho (E_{\phi }( \alpha , G)) < 1\) from which the asymptotic convergence \(y_{k} \rightarrow 0\) follows (cf. [16, Theorem 6.1]). If \(\rho (E_{\phi }( \alpha , G)) < 1\) holds for all \(\alpha >0\), then it is called unconditionally Astable, and if \(\rho (E_{\phi }( \alpha , G)) < 1\) for any \(\alpha \in I\), where I is an interval of the positive half line, then the scheme is called conditionally Astable.
If \(E_{\phi }(\alpha ,G)\) is normal, then \(\Vert E_{\phi }(\alpha ,G) \Vert _2=\rho (E_{\phi }(\alpha ,G))\). Therefore for Astable methods the linear convergence follows directly from the norm contraction
In general cases, however, bounding the spectral radius by one does not imply the norm contraction, i.e., (34) may not be true when \(E_{\phi }(\alpha ,G)\) is nonnormal, even if (33) is Astable. Nevertheless, we shall continue using the tool of Astability through spectral analysis and comment on its limitation in Sect. 2.6.
Implicit and explicit Euler methods
It is well known that the implicit Euler (IE) method
is unconditionally Astable. Indeed, \(E_{\mathrm{IE}} ( \alpha , G) = (I  \alpha G)^{1}\) and \(\rho (E_{\mathrm{IE}} ( \alpha , G))<1\) for all \(\alpha >0\) since all eigenvalues of \(\alpha G\) lie on the left of the complex plane and their distances to 1 are larger than one. Moreover, as it has no restriction on the step size, the implicit Euler method can achieve faster convergent rate by time rescaling which is equivalent to choosing a large step size.
In contract, the explicit Euler method
is only conditionally Astable. Let us consider the case \(G=A\) with \(\mu >0\). Then (35) is exactly the gradient descent method for minimizing \(\frac{1}{2}x^{\top }Ax\). It is not hard to obtain that
Hence \(\rho (E_{\mathrm{GD}}(\alpha , A) ) <1\) provided \(0<\alpha <2/ L\). Thanks to the symmetry of A, we have \(\Vert E_{\mathrm{GD}}(\alpha , A) \Vert _2 = \rho (E_{\mathrm{GD}}(\alpha , A) ) \) and the norm convergence with linear rate follows. Moreover, based on (36), a standard argument outputs the optimal choice \(\alpha ^* = 2/(\mu + L)\), which gives the minimal spectrum
A quasioptimal but simpler choice is \(\alpha _* = 1/ L\) which yields
We formulate the convergence rates (37) and (38) in terms of the condition number \(\kappa (A)\) as it is invariant to the rescaling of A, i.e., \(\kappa (cA) = \kappa (A)\) for any real number \(c\ne 0\). To be Astable, one has to choose \(0<\alpha <2/\lambda _{\max }(A)\). It seems that a simple rescaling to cA can reduce \(\lambda _{\max }(cA)\) and thus enlarge the range of the step size. However, the condition number \(\kappa (cA) = \kappa (A)\) is invariant. From this we see that for the GD method (35), the simple rescaling cA is in vain.
The magnitude of the step size is relative to \(\min \lambda (G)\). To fix the discussion, we chose \(G =  A/\mu \) in (35) so that \(\lambda _{\min }(A/\mu ) = 1\). Then in order for the explicit Euler method to be Astable it is equivalent to choose \(\alpha = O(1/\kappa (A))\) which leads to the contraction rate \(11/\kappa (A)\). Consequently for illconditioned problems, a tiny step size proportional to \(1/\kappa (A)\) is required.
Rather than the rescaling, our main intuition is to seek some transformation G of A, that reduces \(\kappa (A)\) to \(\kappa (G)=O(\sqrt{\kappa (A)})\). We wish to construct explicit Astable methods which can enlarge the step size from \(O(1/\kappa (A))\) to \(O(1/\sqrt{\kappa (A)})\) and consequently improve the contraction rate from \(1  1/\kappa (A)\) to \(O(1  1/\sqrt{\kappa (A)})\).
Transformation to the complex plane
Let us first consider the case \(\mu >0\) and embed A into some \(2\times 2\) block matrix G with a rotation builtin. Specifically, we construct two candidates
Due to the asymmetrical fact, \(\sigma (A)\) will be transformed from the real line to the complex plane. This may shrink the condition number; see the following result.
Proposition 1
For \(G=G_{_{\mathrm{HB}}}\) or \(G_{_{\mathrm{NAG}}}\) given in (39), it satisfies \({{\mathfrak {R}}}{{\mathfrak {e}}}(\lambda ) < 0\) for any \(\lambda \in \sigma (G)\), which promises the decay property \(\left\Vert {y(t)} \right\Vert _2\rightarrow 0\,\) for the system \(y'=Gy\). Moreover, we have \(\kappa (G_{_{\mathrm{HB}}}) = \kappa (G_{_{\mathrm{NAG}}}) = \sqrt{\kappa (A)}\).
Proof
Let us first consider \(G=G_{_{\mathrm{HB}}}\). As A is symmetric, we can write \(A = U\varLambda U^{\top }\) with unitary matrix U and diagonal matrix \(\varLambda \) consisting of eigenvalues of A. By applying the similar transform to G with the block diagonal matrix \(\mathrm{diag}(U, U)\), it suffices to consider eigenvalues of
It is clear that \(\det R_{_{\mathrm{HB}}} = \theta \) and \(\mathrm{tr}\, R_{_{\mathrm{HB}}}=2<0\). In addition, since \(\left{\mathrm{tr}\,R_{_\mathrm{HB}} } \right^2\leqslant {}4\det R_{_{\mathrm{HB}}}\), any eigenvalue \(\lambda _R\in \sigma (R_{_{\mathrm{HB}}})\) is a complex number and
As \(1 = \lambda _{\min }(A/\mu )\leqslant \theta \leqslant \lambda _{\max }(A/\mu ) = \kappa (A),\) we conclude \(\kappa (G_{_\mathrm{HB}}) = \sqrt{\kappa (A)}\).
Apply the similar transformation with \(P = \begin{pmatrix} 1&{}\; 0\\ 1&{}\; 1 \end{pmatrix},\) we observe that
So \(\sigma (R_{_{\mathrm{NAG}}} ) = \sigma (R_{_{\mathrm{HB}}})\) and consequently \(\kappa (G_{_{\mathrm{NAG}}}) = \sqrt{\kappa (A)}\). This completes the proof of this proposition. \(\square \)
We write \(y = (x, v)^{\top }\) and eliminate v in \(y'=Gy\) to get a second order ODE of x, in which we replace Ax by general form \(\nabla f(x)\). Both \(G_{_{\mathrm{HB}}}\) and \(G_{_{\mathrm{NAG}}} \) yield the same thing
which is a special case of the HB model (cf. (21)).
Note that we can find a lot of transformations G and derive corresponding ODE models. Indeed, given any G that meets our demand, both cG and \(QGQ^{1}\) are acceptable candidates, where \(c>0\) and Q is some invertible matrix. We are not going further deep beyond those two transformations given in (39) for the strongly convex case \(\mu >0\) but aim to combine the transformation with a refined time scaling to propose another one for convex case \(\mu =0\) in Sect. 2.5.
Acceleration from a Gauss–Seidel splitting
We now consider numerical discretization for (32) with \(G=G_{_{\mathrm{HB}}}\) and \(G_{_{\mathrm{NAG}}}\) given in (39). As discussed in Sect. 2.2, the implicit Euler method is unconditionally Astable. But computing \((I  \alpha G)^{1}\) needs significant effort and may not be practical.
One may hope that the explicit Euler method \(y_{k+1} = (I + \alpha G) y_k\) will be Astable with step size \(\alpha = O(1/\kappa (G))= O(1/\sqrt{\kappa (A)})\). Unfortunately, unlike the discussion for (35) with \(G=A\), where \(\sigma (I\alpha A)\) lies on the real line and \(\rho (I\alpha A)\) can be easily shrunk by choosing \(\alpha = 1/\rho (A)\) (cf. (36)), the general asymmetric G spreads the spectrum on the complex plane. For both \(G=G_{_{\mathrm{HB}}}\) and \(G=G_{_{\mathrm{NAG}}}\), we have \(\mathfrak {R}(\lambda ) = 1\) for all \(\lambda \in \sigma (G)\). Denote by \(r = \rho (G)\). Then \(\rho ^2(I + \alpha G) = (1\alpha )^2 + \alpha ^2 (r^21)\). To be Astable, requiring \(\rho (I + \alpha G) < 1\) is equivalent to letting \(0< \alpha < 2/r^2 = O(1/\kappa (A))\), where small step size \(\alpha = O(1/\kappa (A))\) is still needed. The optimal choice \(\alpha ^*=r^{2}\) only gives
where no acceleration has been obtained.
We then expect that an explicit scheme closer to the implicit Euler method will hopefully have better stability with a larger step size.
Motivated by the Gauss–Seidel (GS) method [45] for computing \((I  \alpha G)^{1}\), we consider the matrix splitting \(G = M + N\) with M being the lower triangular part of G (including the diagonal) and \(N = G  M\), and propose the following Gauss–Seidel splitting scheme
which gives the relation
Note that for \(G=G_{_{\mathrm{HB}}}\) and \(G_{_{\mathrm{NAG}}}\), the scheme (41) is still explicit as the lower triangular block matrix \(I  \alpha M\) can be inverted easily, without involving \(A^{1}\).
The spectrum bound is given below and for the algebraic proof details, we refer to “Appendix A”.
Theorem 1
For \(G = G_{_{\mathrm{HB}}}\) or \(G_{_{\mathrm{NAG}}}\) given in (39), if \( 0< \alpha \leqslant 2/\sqrt{\kappa (A)}\), then the Gauss–Seidel splitting scheme (41) is Astable and
Dynamic time rescaling for the convex case
The ODE model (40) given in Sect. 2.3 cannot treat the case \(\mu =0\) and the previous spectral analysis fails. Equivalently the condition number \(\kappa (A)\) is infinity and the spectrum bound becomes 1. To conquer this, a careful rescaling is needed. Throughout this subsection, we assume \(\mu = 0\).
For the gradient flow
one can easily establish the sublinear rate \(f(x(t))f(x^*)\leqslant C/t\); see [37]. To recover the exponential rate, we introduce a time rescaling \(t(s) =e^{s}\) and let \(y(s)=x(t(s))\). Then (43) becomes the following rescaled gradient flow
with the scaling factor \(\gamma (s)=e^s\). Besides, the previous sublinear rate \(f(x(t))f(x^*)\leqslant C/t\) turns into \(f(y(s))f(x^*) \leqslant Ce^{s}\). That is in the continuous level, we can achieve the exponential decay through suitable rescaling of time even for \(\mu =0\) .
Now let us go back to our model case \(f(x) = \frac{1}{2}x^{\top }Ax\) with \(\mu =0\) and \(\lambda _{\max }(A) = L\). Coupled with the transformation \(G_{_{\mathrm{NAG}}} \), we consider
where \(y = (x, v)^{\top }\) and
This gives a secondorder ODE in terms of x:
which is in the HB type but with variable damping coefficients.
Obviously, the implicit Euler method for solving (45) is still unconditional Astable. We now apply the GS splitting (41) to (45) and get
where \(E(\alpha _{k}, G(\gamma _{k+1}))\) is defined in (42). The equation (46) is discretized by
Eliminating \(v_{k}\) in (48) will give an HB method with variable coefficients
Instead of studying the spectrum bound \(E(\alpha _k,G(\gamma _{k+1}))\) which is 1, we apply the scaling technique to obtain a regularized matrix
which is nearly similar with \(E(\alpha _k,G(\gamma _{k+1}))\). Set \(z_k=\begin{pmatrix} I &{} \; O \\ O &{} \; \gamma _{k} I \end{pmatrix} y_{k} \), then the discrete system (48) for \(\{y_k\}\) becomes
With a careful chosen step size, the spectrum bound of \({\widetilde{E}}_{k}\) is given below and for the algebraic proof details, we refer to “Appendix A”. We note that, the step size in Theorem 2 is only to agree with the setting of Lemma B2 and for general choice \(L\alpha _k^2/\gamma _k=O(1)\) and suitable initial value \(\gamma _0\), it is possible to maintain the spectrum bound (51) together with the decay estimate (52).
Theorem 2
If \(\gamma _0=L\) and \(L\alpha _{k}^2 =\gamma _{k}(1+\alpha _k)\), then both the scheme (48) and its equivalent form (50) are Astable and we have
which further implies that
Limitation of spectral analysis
For quadratic objective f, both the ODE models (40) and (47) are linear and the spectrum bound of \(E(\alpha ,G)\) for the Gauss–Seidel splitting (42) is derived. But as pointed out in the beginning, for Astable methods, bounding the spectral radius by one is not sufficient for the norm convergence if the matrix \(E(\alpha ,G)\) is nonnormal; see convincible examples in [23, Appendix D.2] and [23, Appendix D.4].
Moving beyond quadratic f and nonlinear ODE systems, transient growth or instability of perturbed problems can easily lead to nonlinear instabilities. Particularly, for the HB system (21), it is shown in [22] that the parameters optimized for linear ODE models does not guarantee the global convergence for a nonlinear system.
To provide rigorous convergence analysis for both continuous and discrete levels, in the sequel we shall introduce the tool of Lyapunov function. Following many related works [6, 37, 43], we first analyze some proper ODEs via a Lyapunov function, then construct optimization algorithms from numerical discretizations of continuous models and use a discrete Lyapunov function to establish the convergence rates of the proposed algorithms.
Nesterov accelerated gradient flow
Continuous problem
In the previous section, we have obtained two ODE models for quadratic objective \(f(x) = \frac{1}{2}x^{\top }Ax\) with \(\mu > 0\) and \(\mu = 0\), respectively. To handle those two cases in a unified way, we combine \(G_{_{\mathrm{NAG}}}\) in (39) with \(G(\gamma )\) in (45) and consider the following transformation
where
One can solve the above equation and obtain
Since \(\gamma _0>0\), we have that \(\gamma (t)>0\) for all \(t\geqslant 0\) and \(\gamma (t)\) converges to \(\mu \) exponentially and monotonically as \(t \rightarrow +\infty \). In particular, if \(\gamma _0=\mu >0\), then \(\gamma (t)=\mu \). Therefore, when \(\mu =0\), (53) reduces to (45) and when \(\gamma _0=\mu >0\), (53) recovers (39) indeed. Correspondingly, the transform (53) gives the system
Heuristically, for general \(f\in {\mathcal {S}}_\mu ^1\) with \(\mu \geqslant 0\), we just replace Ax in (55) with \(\nabla f(x)\) and obtain our NAG flow
with initial conditions \(x(0)=x_0\) and \(v(0)=v_0\). The equivalent secondorder ODE (will also be abbreviated as NAG flow) reads as follows
with initial conditions \(x(0)=x_0\) and \(x'(0)=v_0x_0\). Clearly, if \(\gamma _0 = \mu >0\), then (57) becomes (40), and if \( \mu =0\), then (57) coincides with (47).
Motivated by (30), we introduce a Lyapunov function for (56):
In addition, we need the following lemma, which is trivial but very useful for the convergence analysis in both of the continuous and discrete levels.
Lemma 1
For any \(u,v,w\in V\), we have
We first present the wellposedness of (57) and prove the exponential decay property of the Lyapunov function (58).
Lemma 2
If \(f\in {\mathcal {S}}_{\mu ,L}^{1,1}\) with \(\mu \geqslant 0\), then the NAG flow (57) admits a unique solution \(x\in C^2([0,\infty ); V)\) and moreover
which implies that
Proof
Basically, as \(\nabla f\) is Lipschitz continuous, applying the standard existence and uniqueness results of ODE (see [9, Theorem 4.1.4]) yields the fact that the system (56) admits a unique classical solution \((x,v)\in C^1([0,\infty ); V)\times C^1([0,\infty ); V)\). This implies that \(x' =vx\in C^1([0,\infty ); V)\), and therefore \(x\in C^2([0,\infty ); V)\) is also the unique solution to our NAG flow (57).
It remains to prove (59), which yields the exponential decay (60) immediately. A straightforward calculation yields that
and by (54) and (56), we replace \(\gamma '\) and \(v'\) by their right hand side terms and obtain
Let us focus on the last term. Thanks to Lemma 1,
and the gradient term is split as follows
By the relation \(x' = v x\), the first term in (62) becomes \(\left\langle {\nabla f(x), x'} \right\rangle \) which cancels the first term in (61). Combining all identities together gives
As f is \(\mu \)strongly convex (cf.(2)), there holds
and plugging this into (63) implies that
which proves (59) and thus completes the proof of this lemma. \(\square \)
Remark 1
According to the proof of Lemma 2, the Eq. (54) for \(\gamma \) can be relaxed to \(\gamma ' \leqslant \mu \gamma \). This makes (61) and (63) become inequality but leaves the final estimate (59) invariant. \(\square \)
Rescaling property
Based on our NAG flow (56) (or (57)), it is possible to use time scaling technique to construct more ODE systems with any desirable convergence rate. It is worth distinguishing the connection and difference with existing dynamical models.
Specifically, let \(\alpha \) be any continuous nonnegative function on \({\mathbb {R}}_+\), and consider the time rescaling
Set \(y(\tau ) = x(t(\tau )),w(\tau ) = v(t(\tau ))\) and \(\beta (\tau ) = \gamma (t(\tau ))\), then it is clear that
Similarly, \(w'(\tau ) = \alpha (\tau ) v'(t(\tau ))\) and plugging those facts into (56) gives the scaled NAG flow
with initial conditions \(y(0)=x_0\) and \(y'(0)=\alpha (0)x'(0)\). By Remark 1, the Eq. (54) can be replaced by \(\gamma '\leqslant \mu \gamma \), which becomes
Correspondingly, the Lyapunov function (58) reads as follows
Analogously to (59), we can prove
which implies that
Therefore, larger scaling factor \(\alpha \) promises faster decay rate.
We note that the scaled NAG flow (65) is very close to the two models (27) and (29), which are derived in [43] and [44] respectively, via the variational perspective. Indeed, they differs mainly from the coefficient of \(w'\). By (66), an elementary calculation gives
Therefore, (65) chooses variable coefficient \(\beta (\tau )\) for \(\mu \geqslant 0\), while (27) considers dynamically changing coefficient (26) only for \(\mu =0\) and (29) adopts fixed parameter \(\mu >0\). For strongly convex case \(\mu >0\), if we take \(\beta =\mu \), which satisfies (66), then the scaled system (65) coincides with (29). For convex case \(\mu =0\), if both (27) and (66) are equalities, then (65) agrees with (27). Hence, we conclude that our NAG flow system is more tight and provides a unified way to handle \(\mu =0\) and \(\mu >0\).
Now, let us look at a concrete rescaling example. Let the scaling factor \(\alpha \) satisfy
For instance, the following choice is allowed:
For the equality case of (68), we have a closedform solution
where
We now set \(\beta = \alpha ^2\) which fulfills (66) by our assumption (68), then the scaled NAG flow (65) gives a new HB system
According to (67), we have the estimate
Particularly, if \(\mu >0\) and \(\alpha \) satisfies (70) with \(\gamma _0=\sqrt{\mu }\), then \(\alpha (\tau )=\sqrt{\mu }\) and (71) recovers (31) with the same rate \(O(e^{\sqrt{\mu }\tau })\). Moreover, if \(\mu =0\) and \(\alpha \) satisfies (69) with \(\gamma _0=4\) and \(b=2\), then \(\alpha (\tau )=2/(\tau +1)\) and (71) becomes
which gives the decay rate \(O(\tau ^{2})\) and coincides with the prevailing ODE model (22) derived in [37].
An implicit scheme
Exponential decay of an implicit discretization for solving (56) can be established, which is more or less straightforward since one can easily follow the proof from the continuous problem. However, the implicit scheme requires efficient solver or proximal calculation and may not be practical sometimes. It is presented here to bridge the analysis from the continuous level to semiimplicit and explicit schemes.
Consider the following implicit scheme
where \(\alpha _k>0\) denotes the time step size to discretize the time derivative and the parameter Eq. (54) is also discretized implicitly
We shall present the convergence result for the implicit scheme (72)–(73). To do so, we introduce a suitable Lyapunov function
which is clearly a discrete analogue to the continuous one (58).
Theorem 3
If \( f\in {\mathcal {S}}_{\mu }^{1}\) with \(\mu \geqslant 0\), then for the scheme (72) with \(\alpha _k>0\), we have
Proof
It suffices to prove
Let us mimic the proof of Lemma 2. Instead of the derivative, we compute the difference as follows
Analogously to the continuous level, we focus on the last term
By (72), it follows that
and we use Lemma 1 to split the cross term into squares:
For the gradient term, we have \(v_{k+1}x^{*} = v_{k+1}x_{k+1} + x_{k+1}  x^{*}\) and use (72) to obtain
Consequently, using the \(\mu \)strongly convex property (cf.(2)) of f and dropping surplus negative square terms, we see
This proves (75) and concludes the proof of this theorem. \(\square \)
We observe from Theorem 3 that the fully implicit scheme (72) achieves linear convergence rate as long as \(\alpha _k\geqslant \alpha >0\) for all \(k>0\) and larger \(\alpha _k\) yields faster convergence rate. We also mention that (72) can be rewritten as
where the proximal operator \(\mathbf{prox}_{\eta _k f}\) has been introduced in (8) and
Therefore, it allows f to be nonsmooth and we claim that Theorem 3 still holds true in this case. One just replaces the gradient \(\nabla f(x_{k+1})\) with the subgradient \((y_kx_{k+1})/\eta _k\in \partial f(x_{k+1})\); see (105) and (112).
For convex case, i.e., \(\mu =0\), our method (76) is very close to Güler’s proximal point algorithm [20]
where \(\gamma _{k+1}\gamma _{k}=\alpha _{k}\gamma _k\) and \( y_k ={} \alpha _kv_k+(1\alpha _k)x_k\). Indeed, with suitable step size, they share the similar rate; see [20, Theorem 2.3] and Theorem 4 below.
Theorem 4
If f is proper, closed and convex and we choose \(\alpha _k^2=\eta _k\gamma _k(1+\alpha _k)\) with \(\eta _k>0\), then for the proximal point algorithm (76) with \(\mu =0\), we have
which means if \(\sum _{k=0}^{\infty }\sqrt{\eta _k}=\infty \) then \({\mathcal {L}}_k\rightarrow 0\) as \(k\rightarrow \infty \). Moreover, it holds that
Proof
For convenience and later use, define a sequence \(\{\rho _{k}\}\) by that
As mentioned above, Theorem 3 holds true for such a nonsmooth f and thus it is evident that \(\mathcal L_k\leqslant \rho _k{\mathcal {L}}_0\). Invoking Lemma B2 proves (77) and it is trivial to obtain (78) from (77). This finishes the proof. \(\square \)
Remark 2
Note that the sequence \(\{\gamma _k\}\) in (73) is bounded: \(0<\gamma _{k}\leqslant \max \{\mu ,\gamma _{0}\}\) and \(\gamma _k\rightarrow \mu \) as \(k\rightarrow \infty \). Hence, even for large \(\gamma _0\), the Lyapunov function \({\mathcal {L}}_k\) is asymptotically bounded as \(k\rightarrow \infty \). In addition, from (77) and (78), we see that, for small \(\gamma _0\), the convergence rate depends on \(\gamma _0\) but large \(\gamma _0\) does not pollute the final rate. This fact also holds true for all the forthcoming convergence bounds. \(\square \)
Gauss–Seidel splitting with corrections
This section considers the Gauss–Seidel splitting (41), which is a semiimplicit discretization. In Sect. 2.4, we have established the spectrum bound \(O(1\sqrt{\mu /L})\) with step size \(\alpha _k=O(\sqrt{\mu /L})\) for quadratic objectives. However, as we summarized in Sect. 2.6, spectral analysis is not sufficient for (norm) convergence.
Indeed, in the sequel, we further show that, for the discrete Lyapunov function (74), with any step size \(\alpha _k>0\), the naive discretization (41), reformulated as (80), does not lead to the contraction property like (75). This motivates us to add some proper correction steps.
The Gauss–Seidel splitting
Recall the Gauss–Seidel splitting (41): given step size \(\alpha _k>0\) and previous result \((x_k,v_k)\), compute \((x_{k+1},v_{k+1})\) from
In addition, the parameter Eq. (54) of \(\gamma \) is still discretized implicitly via (73).
Lemma 3
If \(f\in {\mathcal {S}}_{\mu }^{1}\) with \(\mu \geqslant 0\), then for (80) with any step size \(\alpha _k>0\), we have
and
Proof
Following the proof of Theorem 3, we start from the difference
Using the update for \(x_{k+1}\) in (80), we split the gradient term as below
As \(f\in {\mathcal {S}}_{\mu }^{1}\), we obtain that
Ignoring all the negative terms of the second line, the above estimate implies (81).
As we see, different from (75), the estimate (81) contains a combination of a negative term and another cross term. Obviously, an easy application of CauchySchwarz inequality yields
This proves another bound (82) that only involves a positive gradient norm. \(\square \)
A predictor–corrector method
To conquer the cross term \( \alpha _k\left\langle {\nabla f(x_{k+1}),v_{k+1}  v_k} \right\rangle \) in (81), we add an extra extrapolation step to (80) which can be thought as an semiimplicit discretization of \(x'= v  x\) with the newest update \(v_{k+1}\). More precisely, consider
This is in line with the spirit of the predictorcorrector method for ODE solvers [39, Section 3.8]. The variable \(y_k\) is the predictor produced by an explicit scheme and \(x_{k+1}\) is the corrector by an implicit scheme. It can be also thought of as a symmetric Gauss–Seidel iteration for approximating the implicit Euler method. Again, the parameter Eq. (54) of \(\gamma \) is still discretized via (73).
As the first two steps of (83) agree with (80), with \(x_{k+1}\) being \(y_k\), recalling the estimate (81), we have
where
Therefore, it follows that
From the update for \(y_k\) and \(x_{k+1}\) in (83), we find the relation
and if \(f\in {\mathcal {S}}^{1,1}_{\mu ,L}\), then there comes the estimate (cf. (4))
As a result, we obtain
The second term vanishes if we choose suitable step size; see the theorem below.
Theorem 5
Assume that \(f\in \mathcal {S}_{\mu ,L}^{1,1}\) with \(0\leqslant \mu \leqslant L<\infty \) and \(L\alpha _k^2 =\gamma _k(1+\alpha _k)\), then for the predictorcorrector scheme (83) together with (73), we have
where \({\mathcal {L}}_k\) is defined by (74). Consequently, for all \(k\geqslant 0\),
and moreover, for all \(k\geqslant 1\),
where
Proof
The inequality (85) suggests the choice \(L\alpha _k^2 =\gamma _k(1+\alpha _k)\) and promises (86). Recalling the sequence \(\{\rho _{k}\}\) defined by (79), we have \( {\mathcal {L}}_k\leqslant \rho _k{\mathcal {L}}_0\). Hence, using Lemma B2 gives the decay estimate of \(\rho _k\) and proves (87).
It remains to check (88) for all \(k\geqslant 1\).
From Lemma B2 we easily get
On the other hand, by the relation \(L\alpha _0^2=\gamma _0(1+\alpha _0)\), it is evident that
which implies
The above estimate also indicates that
Applying Lemma B2 shows \(\alpha _{k}\geqslant \sqrt{\min \{\gamma _0,\mu \}/L}\) and it follows that
Collecting this estimate and (90) establishes the final rate (88) and thus completes the proof of this theorem. \(\square \)
Remark 3
We mention that the estimate (88) verifies the claim made previously in Remark 2. That is, the convergence rate given in Theorem 5 depends on small \(\gamma _0\) but is robust when \(\gamma _0\geqslant L\). \(\square \)
Correction via a gradient step
Motivated by the estimate (82), we can also aim to cancel the squared gradient norm. One preferable choice is the gradient descent step and according to our discussion below, any other correction step satisfying the decay property (94) is acceptable. Note that the two numerical schemes proposed in [38, 44] for the HB Eq. (31) also have additional gradient steps.
As what we did before, replace \(x_{k+1}\) by \(y_k\) in (80) and consider the following corrected scheme: given \(\alpha _k>0\) and \((x_k,v_k)\), compute \((x_{k+1},v_{k+1})\) from
The implicit discretization (73) for the parameter Eq. (54) keeps unchanged here. In the first equation \(y_k\) can be solved in terms of the known data \((x_k, v_k)\). After that, we evaluate the gradient \(\nabla f(y_k)\) once and use it to update \((x_{k+1}, v_{k+1})\).
Theorem 6
Assume that \(f\in \mathcal {S}_{\mu ,L}^{1,1}\) with \(0\leqslant \mu \leqslant L<\infty \) and \(L\alpha _k^2 =\gamma _k(1+\alpha _k)\), then for the corrected scheme (91) together with (73), we have
where \({\mathcal {L}}_k\) is defined by (74), and both the two estimates (87) and (88) hold true here.
Proof
According to (82) in Lemma 3, we have established that
where \(\widehat{{\mathcal {L}}}_k\) is defined by (84). Thanks to the additional gradient step in (91), we have the basic gradient descent inequality:
which comes from (4) since \(f\in \mathcal S_{\mu ,L}^{1,1}\) and implies that
Plugging this into (93) gives
This together with the condition \(L\alpha _k^2 =\gamma _k(1+\alpha _k)\) yields (92).
As we choose the same step size as Theorem 5, based on the contraction (92), it is trivial to conclude that the two estimates (87) and (88) hold true here indeed. This completes the proof of this theorem. \(\square \)
A corrected semiimplicit scheme from NAG method
In this section, we consider another semiimplicit scheme which comes exactly from Nesterov accelerated gradient method.
NAG method
In [29, Chapter 2, General scheme of optimal method], by using the estimate sequence, Nesterov presented an accelerated gradient method for solving (1) with \(f\in {\mathcal {S}}_{\mu ,L}^{1,1}\), \(0\leqslant \mu \leqslant L<\infty \); see Algorithm 1 below.
Note that we have many choices for \(x_{k+1}\) in step 5 of Algorithm 1. One noticeable example is the gradient descent step (see [29, Chapter 2, Constant Step Scheme, I]):
With this choice, the sequence \(\{v_k\}\) in Algorithm 1 can be eliminated and \(y_{k+1}\) is updated by that (see [29, Chapter 2, Constant Step Scheme, II])
where \(\alpha _{k+1}\in (0,1)\) is calculated from the quadratic equation
If \(\mu >0\) and \(\alpha _0=\sqrt{\mu /L}\), then \(\alpha _k = \sqrt{\mu /L}\); see [29, Chapter 2, Constant Step Scheme, III]. In particular, if \(\mu =0\), then Algorithm 1 (with \(x_{k+1}\) updated by (95)) coincides with the accelerated scheme proposed by Nesterov early in the 1980s [27].
NAG method as a corrected semiimplicit scheme
After simple calculations, we can rewrite Algorithm 1 as an equivalent form
where in addition we update \(x_{k+1}\) satisfying
Surprisingly, (96) formulates a semiimplicit discretization for our NAG flow (56) with a correction step (97) and an explicit discretization for the Eq. (54) of \(\gamma \). Similarly to (91), we can adopt the gradient descent step which promises (97).
Based on subtle algebraic calculations of the estimate sequence, Nesterov [29, Chapter 2] proved the convergence rate of Algorithm 1. In the following, we give an alternative proof by using the Lyapunov function (74).
Theorem 7
Assume that \(f\in \mathcal {S}_{\mu ,L}^{1,1}\) with \(0\leqslant \mu \leqslant L<\infty \). If \(L\alpha _k^2= \gamma _{k+1}\), then for Algorithm 1, i.e., the scheme (96) together with (97), we have \(0<\alpha _k\leqslant 1\) and
where \({\mathcal {L}}_k\) is defined by (74). Consequently for all \(k\geqslant 0\),
Moreover, for all \(k\geqslant 1\),
where \(C_{\gamma _0,L}\) has been defined in (89).
Proof
Let us first prove (98). By (96), we find
and a direct computation gives
Dropping the negative term \(\left\Vert {y_kx_k} \right\Vert ^2\) and using the \(\mu \)convexity of f imply that
and we get the inequality
Consequently, by (97) and the relation \(L\alpha _k^2=\gamma _{k+1}\), the right hand side of the above inequality is negative, which proves (98).
In this case, we modify (79) as follows
then by (98) it is clear that \( \mathcal L_k\leqslant \rho _k{\mathcal {L}}_0\), and invoking Lemma B1 proves (99). As the proof of (100) is very similar with that of (88), we omit the details here and conclude the proof of this theorem. \(\square \)
Remark 4
Similar to our corrected schemes (83) and (91), NAG method (i.e., Algorithm 1) generates a threeterm sequence \(\{(x_k,y_{k},v_{k})\}\) as well. If \(\mu =0\), then they share the same convergence rate bound
and when \(\gamma _0=\mu >0\), we have
In view of the trivial fact
we see the rates in (102) are asymptotically the same and NAG method can achieve a slightly better convergence rate. However, we note that they share the same computational complexity
which is optimal, in the sense that [29] it achieves the complexity lower bound of firstorder algorithms for the function class \({\mathcal {S}}_{\mu ,L}^{1,1}\) with \(0\leqslant \mu \leqslant L<\infty \). \(\square \)
Remark 5
Unlike the gradient descent method, the function value \(f(x_k)\) of accelerated gradient methods may not decrease in each step. It is the discrete Lyapunov function \({\mathcal {L}}_k\) that is always decreasing; see (86), (92) and (98). \(\square \)
Remark 6
To reduce the function value, one can adopt the restating strategy [31]. Specifically, given \((\gamma _0,v_0,x_0)\), if \(f(x_k)\) is increasing after kiteration, then set \(k=0\) and restart the iteration process with another initial guess \(({\tilde{\gamma }}_0,{\tilde{v}}_0,{\tilde{x}}_0)\). By Theorems 5, 6 and 7, when \(f\in {\mathcal {S}}_{0,L}^{1,1}\) and \(\gamma _0=L,v_0 = x_0\), we only have the sublinear convergence rate
where we used (4), which promises
Additionally, assume f satisfies the quadratic growth condition with \(\sigma >0\):
where \(\mathrm{dist}(x,\mathrm{argmin}\,f) = \inf _{x^*\in \mathrm{argmin}\,f}\left\Vert {xx^*} \right\Vert \). As (103) holds for all \(x^*\in \mathrm{argmin}\,f\), we have immediately that
Therefore, as analyzed in [30], if we consider fixed restart technique [31] every k steps, then after \(N=nk\) steps we will get
Evidently, the optimal choice \( k_{\#} =e \sqrt{4L/\sigma }\) yields the linear rate
If the parameter \(\sigma \) is unknown, one can use the adaptive restart technique [31].
When f is quadratic and convex, changing \(\gamma _k\) from L to \(\mu \) periodically will smoothing out error in different frequencies and can further optimize the constant in front of the accelerated rate. That is, the dynamically changing parameter \(\{\gamma _k\}\) hopefully outperforms the fixed one \(\gamma _k=\mu \). For general nonlinear convex functions, a rigorous justification of the restart strategy is under investigation. \(\square \)
Composite convex optimization
In this part we mainly focus on the composite optimization
where \(Q\subseteq V\) is a simple closed convex set, \(h\in \mathcal S_{\mu ,L}^{1,1}(Q)\) with \(0\leqslant \mu \leqslant L<\infty \) and \(g:V\rightarrow {\mathbb {R}}\cup \{+\infty \}\) is proper, closed and convex, and \(Q\cap \mathbf{dom}\, g\ne \emptyset \). In general g is not differentiable but its subdifferential \(\partial g\) exists as a setvalued function. More precisely, the subdifferential \(\partial g(x)\) of g at x is defined by that
Remark 7
For the case that \(h\in {\mathcal {S}}_{0,L}^{1,1}(Q)\) and g is \(\mu \)strongly convex with \(\mu \geqslant 0\), we can split \(h+g\) as \((h(x) + \frac{\mu }{2}\Vert x\Vert ^2) + (g(x)  \frac{\mu }{2}\Vert x\Vert ^2)\), which reduces to our current assumption for (104). \(\square \)
We shall apply our ODE solver approach to the problem (104). The first step is to generalize the dynamical system (56) to the current nonsmooth setting. Basically, we set \(F = f+i_Q\) with \(i_Q\) being the indicator function of Q and obtain a differential inclusion for minimizing F on V, which is equivalent to minimize f over Q. After that, optimization methods (see Algorithms 2 and 4) for solving the original problem (104) with the accelerated convergence rate
are proposed from numerical discretizations of the continuous model (106). This is a proof of the effective and usefulness of our NAG flow model (106) and the ODE solver approach, by which we can construct new accelerated methods easily.
Continuous model
For minimizing a nonsmooth function F over V, our NAG flow (56) becomes a differential inclusion
To ensure solution existence, suitable initial conditions shall be imposed later. Correspondingly, the secondorder ODE (57) reads as a secondorder differential inclusion
As the subdifferential \(\partial F\) is a setvalued maximal monotone operator, classical \(C^2\) solution to (107) may not exist because discontinuity can occur in \(x'\). Therefore, the concept of energyconserving solution has been introduced in [15, 32, 36].
Let us assume the initial data
where \({\mathcal {T}}_{\mathbf{dom} F}(x_0)\) denotes the tangent cone of \(\mathbf{dom} F\) at \(x_0\):
In addition, we shall introduce some vectorvalued functional spaces. Given any interval \(I\subset {\mathbb {R}}\), let M(I; V) be the space of Vvalued Radon measures on I; for any \(m\in {\mathbb {N}}\) and \(1\leqslant p\leqslant \infty \), \(W^{m,p}(I;V)\) denotes the standard Vvalued Sobolev space [21]; the space of all Vvalued functions with bounded variation is defined by BV(I; V) [4]. Also, \(W_\mathrm{loc}^{m,p}(I;V)\) and \(BV_{\mathrm{loc}}(I;V)\) consist of all the sets \(W^{m,p}(\omega ;V)\) and \(BV(\omega ;V)\) respectively, where \(\omega \subset I\) is any compact subset.
Definition 1
We call \(x:[0,\infty )\rightarrow V\) an energyconserving solution to (107) with initial data (108) if it satisfies the following.

1.
\(x\in W^{1,\infty }_{\mathrm{loc}}(0,\infty ;V),\,x(0) = x_0\) and \(x(t)\in \mathbf{dom} F\) for all \(t>0\).

2.
\(x'\in BV_{\mathrm{loc}}([0,\infty );V),\,x'(0+) = x_1\).

3.
For almost all \(t>0\), there holds the energy equality:
$$\begin{aligned} \begin{aligned} {}&F(x(t)) + \frac{\gamma (t)}{2}\left\Vert {x'(t)} \right\Vert ^2 + \int _{0}^{t}\frac{\mu +3\gamma (s)}{2}\left\Vert {x'(s)} \right\Vert ^2 {{\mathrm{d}}}s = {} F(x_0) + \frac{\gamma _0}{2}\left\Vert {x_1} \right\Vert ^2. \end{aligned} \end{aligned}$$ 
4.
There exists some \(\nu \in M(0,\infty ;V)\) such that
$$\begin{aligned} \gamma x'' + (\mu +\gamma )x'+\nu = 0 \end{aligned}$$holds in the sense of distributions, and for any \(T>0\), we have
$$\begin{aligned} \int _{0}^{T}\big (F(y(t))F(x(t))\big ){{\mathrm{d}}}t \geqslant \left\langle { \nu ,yx} \right\rangle _{C([0,T];V)} \quad \text {for all } y\in C([0,T];V). \end{aligned}$$
In [25], the problem (107) has been extended to a general case
where \(\xi \) stands for small perturbation. Therefore, according to [25, Theorem 2.1], we have the existence of an energyconserving solution to (107) and by [25, Theorems 2.2 and 2.3], we obtain the exponential decay, which is a nonsmooth version of (60).
Theorem 8
Assume V is a finite dimensional Hilbert space. In the sense of Definition 1, the differential inclusion (107) admits an energyconserving solution \(x:[0,\infty )\rightarrow V\) satisfying
for almost all \(t>0\), where \( {\mathcal {L}}_0 := F(x_0)F(x^*) +\frac{\gamma _0}{2}\left\Vert {x_0+x_1x^*} \right\Vert ^2\).
Remark 8
If additionally \(\mathbf{dom} F = V\), then \(x\in W^{2,\infty }_\mathrm{loc}(0,\infty ;V) \cap C^1([0,\infty );V)\) and (109) holds for all \(t>0\). \(\square \)
An APGM for unconstrained optimization
Let us first consider the unconstrained case \(Q=V\), i.e.,
where \(f\in {\mathcal {S}}_{\mu ,L}^{1,1}\) with \(0\leqslant \mu \leqslant L<\infty \) and \(g:V\rightarrow {\mathbb {R}}\cup \{+\infty \}\) is a properly closed and convex function and possibly nonsmooth.
Gradient mapping
To treat the nonsmooth part g, we introduce the tool of gradient mapping. Following [29, Chapter 2], given any \(\eta >0\), the composite gradient mapping \({\mathcal {G}}_f(x,\eta )\) of f at x is defined by that
where \(S_f(x,\eta ):=\mathbf{prox}_{\eta g}(x\eta \nabla h(x))\) and the proximal operator \(\mathbf{prox}_{\eta g}\) has been defined by (8). Note that \(S_f(x,\eta )\) is clearly welldefined and so is \({\mathcal {G}}_f(x,\eta )\).
It is well known [33, 35] that
which yields the fact
From this we conclude that the fixedpoint set of \(S_f(\cdot ,\eta )\) is \(\mathrm{argmin}\,f\). Indeed, \(x = S_f(x,\eta )\) if and only if \(0\in \partial f(x) \). We also observe from (113) that the gradient mapping (111) is defined reversely from the proximalgradient step for minimizing \(f = h+g\), i.e.,
Hence it plays the role of the gradient \(\nabla f\) in the smooth case. Particularly, if \(g = 0\), then \({\mathcal {G}}_f(x,\eta ) = \nabla h(x)\) and \(S_f(x,\eta )=x\eta \nabla h(x)\) is nothing but a gradient step.
To move on, we present an auxiliary lemma, which is a key ingredient for our convergence analysis. As we will fix \(\eta =1/L\), for simplicity, we set \({\mathcal {G}}_f(x):= {\mathcal {G}}_f(x,1/L)\) and \(S_f(x):=S_f(x,1/L)\).
Lemma 4
Assume \(f = h+g\), where \(h\in {\mathcal {S}}_{\mu ,L}^{1,1}\) with \(0\leqslant \mu \leqslant L<\infty \) and \(g:V\rightarrow \mathbb R\cup \{+\infty \}\) is properly closed and convex. Then for any \(x,y\in V\),
Proof
Since \(h\in {\mathcal {S}}_{\mu ,L}^{1,1}\), applying (2) and (4) gives
which implies that
Observing (113), we get
Summing the above two inequalities and using the split
we finally arrive at (114) and end the proof of this lemma. \(\square \)
Remark 9
For a fixed x, the right hand side of (114) defines a quadratic approximation of f at x, and it is strongly reminiscent of the quadratic lower bound approximation (2) for the smooth case. However, compared to (2), the constant is shifted from f(x) to a lower value \(f(S_f(x)) + \frac{1}{2L} \left\Vert {{\mathcal {G}}_f(x)} \right\Vert ^2\). The first order part is \({\mathcal {G}}_f(x)\) instead of the subgradient at x. The quadratic part \(\frac{\mu }{2}\left\Vert {yx} \right\Vert ^2\) is due to the \(\mu \)convexity. \(\square \)
The proposed method
Based on the corrected semiimplicit scheme (91) for NAG flow (56), it is possible to generalize it to solve the differential inclusion (106). We simply replace the gradient \(\nabla f(y_k)\) with the gradient mapping \({\mathcal {G}}_f(y_k)\) and set the correction as \( x_{k+1} ={}S_f(y_k)\). More precisely, consider
Once \(x_{k+1}=S_f(y_k)=\mathbf{prox}_{\eta g}(y_k\eta \nabla h(y_k))\) is obtained, we can update \(v_{k+1}\) with known datum \(x_k,y_k,v_k\) and \(x_{k+1}\). Thus in each iteration, (115) only calls the proximal operation \(\mathbf{prox}_{\eta g}\) once.
We still use the step size \(L\alpha _k^2=\gamma _k(1+\alpha _k)\) and summarize the semiimplicit scheme (115) in Algorithm 2, which is called semiimplicit APGM (SemiAPGM for short). Also, the convergence rate is derived via the discrete Lyapunov function (74).
Theorem 9
For Algorithm 2, we have
where \({\mathcal {L}}_k = f(x_k)f(x^*) + \frac{\gamma _k}{2} \left\Vert {v_kx^*} \right\Vert ^2 \), and both (87) and (88) hold true here.
Proof
The proof of (116) is very similar to that of (92). Replacing \(x_{k+1}\) and its gradient \(\nabla f(x_{k+1})\) in (80) respectively with \(y_k\) and \({\mathcal {G}}_f(y_k)\), we can proceed as the proof of Lemma 3 and use Lemma 4 to obtain
where \(\widehat{ {\mathcal {L}}}_{k}\) is defined by (84). Thanks to the relation \(L\alpha _k^2=\gamma _k(1+\alpha _k)\), the second line of (117) vanishes, and inserting the identity \(f(y_k)f(x_{k+1})= \widehat{ {\mathcal {L}}}_{k} {\mathcal {L}}_{k+1}\) into (117) gives (116). Based on this, it is not hard to see that both (87) and (88) hold true. This finishes the proof of this theorem. \(\square \)
We mention that with another choice
we can drop the sequence \(\{v_{k}\}\) from (115). The procedure is not straightforward but very similar to that of Nesterov’s optimal method in [29, page 80]. We omit the details and only list the following algorithm.
This can be viewed as a generalization of [29, Chapter 2, Constant Step Scheme, II] to problem (110). Particularly, for convex case \(\mu =0\), it is very close to FISTA [12]. Both of them share the same spirit: applying one proximal gradient step first and then using some extrapolation formulae. The difference comes only from the use of the two sequences \(\{\alpha _k\}\) and \(\{\beta _k\}\). We also claim that Algorithm 3 has the same accelerated convergence rate as Algorithm 2, i.e., \(O(\min (L/k^2,(1+\sqrt{\mu /L})^{k}))\). In contrast FISTA is designed for \(\mu =0\) and has only the sublinear rate \(O(L/k^2)\).
We mention that, accelerated proximal gradient methods for solving (110) with only one evaluation of \(\mathbf{prox}_{\eta g}\) in each iteration can be found in [38] (only for strongly convex case) and [24, Chapter 2, Algorithm 2.2] (for both convex and strongly convex cases).
Both Algorithms 2 and 3 cannot be applied directly to the general constraint case (104). The main issue comes from the definition (111) of the gradient mapping \({\mathcal {G}}_f(x,\eta )\), where we impose the restriction \(x\in Q\) and calculate the proximal operator \(\mathbf{prox}_{\eta g}\) over Q to obtain \(S_f(x)\in Q\). For both two algorithms, we shall compute \(x_{k+1}=S_f(y_k)=\mathbf{prox}_{\eta g}(y_k\eta \nabla h(y_k))\). But the sequence \(\{y_k\}\) in Algorithms 2 and 3 may be outside the constraint set. This is not acceptable because \(\nabla h(y_k)\) might not exist: for instance, \(Q = [0,\infty )\) and h is the entropy function.
The original FISTA [12] and the methods in [38] and [24, Chapter 2, Algorithm 2.2] mentioned above, cannot be applied to the constrained problem (104) either. This stimulates us to propose a new operator splitting scheme to conquer this problem.
An accelerated forward–backward method for constrained optimization
We now go back to the constrained problem (104). As mentioned above, the tool of gradient mapping is not convenient for us to handle this case. To avoid using it, we utilize the separable structure of \(f=h+g\) and apply explicit and implicit schemes for h and g, respectively. This is the socalled operator splitting technique in ODE solvers and is also known as the forwardbackward method.
Let us start from the predictorcorrector scheme (83) and rewrite it as follows
For minimizing \(f=h+g\) over Q, we modify the above method as follows
where \(x_0,\,v_0\in Q\) and the parameter sequence \(\{\gamma _k\}\) comes from the implicit discretization (73) of the Eq. (54). Clearly, as convex combinations are used, the method (119) preserves the threeterm sequence \(\{(x_k,y_k,v_k)\}\) in Q and it requires the proximal computation of g over Q only once in each iteration.
We choose \(L\alpha _k^2=\gamma _{k}(1+\alpha _k)\) as before and rewrite (119) in Algorithm 4, which is called semiimplicit accelerated forwardbackward (SemiAFB for short) method.
In [41], Tseng considered problem (104) only with convex assumption, i.e., \(\mu =0\), and proposed an APGM that possesses the rate \(O(L/k^2)\). By using the technique of estimate sequence, Nesterov [28] presented an accelerated method for solving (104) with the assumption that h is Lsmooth over Q and g is \(\mu \)strongly convex with \(\mu \geqslant 0\). Both our Algorithm 4 and Nesterov’s method generate a threeterm sequence \(\{(x_k,y_k,v_k)\}\) and have the same accelerated rate \(O(\min (L/k^2,(1+\sqrt{\mu /L})^{k}))\); see [28, Theorem 6] and our Theorem 10. However, as mentioned in [12], the later used an accumulated history of the past iterations to build recursively a sequence of estimate functions, and in each iteration, to update \(x_{k+1}\) and \(v_{k+1}\), Nesterov’s method in [28] calls \(\mathbf{prox}_{ g}\) over Q twice.
Below, we shall establish the convergence rate of Algorithm 4 via the analysis of a Lyapunov function. It is well known [28, Eq (2.9)] that the firstorder optimality condition for \(v_{k+1}\) in (119) is the variational inequality
where \(p_{k+1}\in \partial g(v_{k+1})\). Expanding \(w_k\), we observe the relation
where \(x\in Q\) is arbitrary.
Theorem 10
For Algorithm 4, we have
where \({\mathcal {L}}_k= f(x_k)f(x^*) + \frac{\gamma _k}{2} \left\Vert {v_kx^*} \right\Vert ^2\), and both (87) and (88) hold true here.
Proof
As before, we calculate the difference
Thanks to (120), we have
where \(p_{k+1} \in \partial g(v_{k+1})\).
By Lemma 1, the first term in (122) is split as follows
The gradient term in (122) is more subtle. Firstly, by convexity of g, we have
and secondly, according to the update for \(y_{k}\) (see step 4 in Algorithm 4), we find
As h is \(\mu \)strongly convex on Q, by the fact \(\{(x_k,y_k,v_k)\}\subset Q\), it follows that
Therefore, collecting all the estimates and dropping surplus negative terms related to \(\left\Vert {x_ky_k} \right\Vert ^2\) and \(\Vert y_{k}v_{k+1}\Vert ^{2}\), we get
Let us consider the additional terms in (123). In view of (4), we have
Thanks to the extrapolation step for \(x_{k+ 1}\) (see step 6 in Algorithm 4), we find a crucial relation
which gives that
as \(L\alpha _k^2=\gamma _k(1+\alpha _k)\). Moreover, since \(x_{k+1}\) is a convex combination of \(x_k\) and \(v_{k+1}\), the estimate follows
Plugging this and the previous inequality into (123) gives
which establishes (121).
By the relation \(L\alpha _k^2=\gamma _{k}(1+\alpha _k)\) and the contraction (121), it is clear that the two estimates (87) and (88) hold true. This completes the proof of this theorem. \(\square \)
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Acknowledgements
The authors would like to thank the anonymous reviewers for valuable suggestions and careful comments, which significantly improved the qualify of an early version of the paper.
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Long Chen was supported in part by the National Science Foundation under grants DMS1913080 and DMS2012465. Hao Luo was supported by the China Scholarship Council (CSC) joint Ph.D. student scholarship (Grant 201806240132).
Appendices
A Spectral analysis
Proof of Theorem 1
Let us start from the scalar case
where \(a,b,c,d\geqslant 0\) and \(\mathrm{tr}\,R<0<\det R\). Set
By direct computations we have
where \(\delta :=(1+a\alpha )(1+d\alpha )\). Since \(\mathrm{tr}\,R<0\), we see that
Note that any eigenvalue \(\theta \) of \(E(\alpha ,R)\) satisfies
We now arrive at the following lemma, which says the spectrum of \(E(\alpha ,R)\) can be transformed to the circle \( \left{\theta } \right = \sqrt{\det E(\alpha ,R)}< 1\), with proper \(\alpha \). \(\square \)
Lemma 5
Assume
with \(a,b,c,d\geqslant 0\) such that \(\mathrm{tr}\, R<{} 0<\det R \). Let \(E(\alpha ,R)\) be defined by (124). If \(\alpha >0\) satisfies
then we have
Proof
If \(\varDelta =\left{\mathrm{tr}\,E(\alpha ,R)} \right^24\det E(\alpha ,R)\leqslant 0\), then any solution to (125) satisfies that \(\left{\theta } \right = \sqrt{\det E(\alpha ,R)}\) and the conclusion follows. By direct calculation, \(\varDelta \leqslant 0\) is equivalent to
Square the inequality \(\alpha \sqrt{\det R}  1 \leqslant \sqrt{\delta }\) and cancel one \(\alpha \) to get the upper bound in (126). The lower bound can be proved similarly. \(\square \)
We now in the position of establishing Theorem 1. We first consider \( G= G_{_{\mathrm{HB}}}\), for which we have
It is clear that \(\theta \in \sigma (E(\alpha ,G))\Leftrightarrow \theta \in \sigma (E(\alpha ,R(\lambda )))\), where \(E(\alpha ,R(\lambda ))\) is defined by (124) with
As \(\mathrm{tr}\, R(\lambda ) \leqslant 2\sqrt{\det R(\lambda )}\), by Lemma 5, if
then we can obtain
Similarly, for \(G = G_{_{\mathrm{NAG}}}\) with condition (127), we can establish
Consequently, for both two cases, taking \(\alpha = 2/\sqrt{\kappa (A)}\) yields the spectrum bound
This concludes the proof of Theorem 1. \(\square \)
Proof of Theorem 2
Observe that \({{\widetilde{E}}}_{k}\) is similar with
where
To prove (51), it is sufficient to verify \(\rho \left( H_{k}\right) =1\).
Given any eigenvalue \(\theta \in \sigma (H_{k})\), it solves
with some \(\lambda \in \sigma (A)\subset [0,L]\). By (49), \(\{\gamma _k\}\) is decreasing and thus \(\gamma _{k}\leqslant \gamma _0=L\). According to our choice \(L\alpha _k^2=\gamma _k(1+\alpha _k)\), we have \(0<\alpha _k\leqslant 2\) and moreover \(0<\lambda \alpha _k^2/\gamma _k\leqslant L\alpha _k^2/\gamma _k = 1+\alpha _k\leqslant 3\). This implies \(\varDelta = (\lambda \alpha _{k}^2/\gamma _{k}2)^24\leqslant 0\) for all \(\lambda \in \sigma (A)\). Therefore, we conclude that \(\left{\theta } \right = 1\) for all \(\theta \in \sigma (H_{k})\), which proves \(\rho \left( H_{k}\right) =1\) and thus establishes (51).
Thanks to Lemma B2, there holds
This proves (52) and completes the proof of Theorem 2. \(\square \)
B Decay rates
Lemma B1
Let \(\gamma _0>0\) and \( \mu \geqslant 0\) be given and assume there is a real positive sequence \(\{L_k\}\) such that \( L_k\geqslant \mu \). Define \(\{(\alpha _k,\gamma _k)\}\) by that
Then we have \(\gamma _k>0,0<\alpha _k\leqslant 1\) and \(\alpha _k \geqslant \sqrt{\min \{\gamma _1,\mu \}/L}\), where \(L: = \sup _{k\in {\mathbb {N}}}L_k\). Moreover, for all \(k\geqslant 1\),
and if \(\mu =0\), then we have the lower bound
Proof
Let us first check that \(0<\alpha _k\leqslant 1\) and \(\gamma _k>0\). Since \(\gamma _0>0\), by (128) we have
from which we claim that \(0< \alpha _0\leqslant 1\). Thus by the second step in (128) we have \(\gamma _1>0\). A sequential argument implies that \(0< \alpha _k\leqslant 1\) and \(\gamma _k>0 \) for all \(k\geqslant 0\).
It is not hard to find the fact: if \(\gamma _0>\mu \), then \(\mu<\gamma _{k+1}<\gamma _k\) and if \(\gamma _0<\mu \), then \(\gamma _{k}<\gamma _{k+1}<\mu \). Particularly, if \(\gamma _0=\mu \), then \(\gamma _k=\mu \). Based on this observation and the fact \(L_k\leqslant L\), we conclude that \(\alpha _k\geqslant \sqrt{\min \{\gamma _1,\mu \}/L}\) and thus
Next, let us prove the estimate
where \(\rho _k\) is defined by (101). We start from the trivial equality
where we used the relation \(\rho _{k+1}=\rho _{k}(1\alpha _k)\). By (128), for any \(i\geqslant 0\), it holds that
and multiplying the above inequality from \(i=0\) to \(i=k1\) gives \(\rho _{k}\leqslant \gamma _{k}/\gamma _0\). Plugging this into (132) and using the relation \(L_k\alpha _k^2=\gamma _{k+1}\) and the fact \(0<\alpha _k\leqslant 1\) imply
which further indicates that
Therefore, a simple calculation proves (131) and concludes the proof of this lemma.
For \(\mu =0\), we have the relation \(\rho _{k}=\gamma _{k}/\gamma _0\), and proceeding as the above derivation, it is not hard to establish the lower bound (130). This concludes the proof of this lemma. \(\square \)
Similarly, we can establish the following result, the proof of which is omitted for simplicity.
Lemma B2
Let \(\gamma _0>0\) and \( \mu \geqslant 0\) be given and assume there is a real positive sequence \(\{L_k\}\) such that \( L_k\geqslant \mu \). Define \(\{(\alpha _k,\gamma _k)\}\) by that
Then we have \(\gamma _k>0\) and \(\alpha _k\geqslant \sqrt{\min \{\gamma _0,\mu \}/L}\), where \(L: = \sup _{k\in \mathbb N}L_k\). Moreover, for all \(k\geqslant 1\),
and if \(\mu =0\), then we have the lower bound
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Luo, H., Chen, L. From differential equation solvers to accelerated firstorder methods for convex optimization. Math. Program. (2021). https://doi.org/10.1007/s10107021017133
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Keywords
 Accelerated firstorder methods
 Ordinary differential equation
 Convergence analysis
 Convex optimization
 Lyapunov function
 Exponential decay
Mathematics Subject Classification
 37N40
 65L20
 65B99
 90C25