SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof outline of Theorem 5.1. The general strategy is to find discretized counterparts of corresponding continuous-time analyses. However, directly discretizing the conservation law of Proposition 3.2 was difficult due to technical reasons. Instead, we obtain differently scaled but equivalent conservation laws using dilated coordinates and then performed the discretization. The specific dilated coordinates, inspired by [67], are  $W_{1}(t) = X(t) - X_{0}$  for  $p > 1$ ,  $W_{2}(t) = t^{p}\left(X(t) - X_{0}\right)$  for  $0 < p < 1$ ,  $W_{3}(t) = t\left(X(t) - X_{0}\right)$  for  $p = 1$ ,  $\gamma \geq 1$  and  $W_{4}(t) = t^{\gamma}\left(X(t) - X_{0}\right)$  for  $p = 1$ ,  $0 < \gamma < 1$ .
In the discrete-time analyses, the behavior of the leading-order terms is predictable as they match the continuous-time counterpart. The difficult part is, however, controlling the higher-order terms that were not present in the continuous-time analyses. Through our detailed analyses, we bound such higher-order terms and show that they do not affect the convergence rate in the end. We provide the details in Appendix G.3.
6 Convergence analysis under strong monotonicity
In this section, we analyze the dynamics of the anchor ODE (3) for  $\mu$ -strongly monotone  $\mathbb{A}$ . When  $\beta(t) = \frac{1}{t}$  and  $\mathbb{A} = \left( \begin{array}{cc} \mu & 0 \\ 0 & \mu \end{array} \right)$ , Lemma 4.1 tells us that  $\mathbb{A}(X(t)) = \frac{1}{t} \left( I - e^{-tA} \right) X_0$  and therefore that  $\| \mathbb{A}(X(t)) \|^2 = \Theta \left( \frac{1}{t^2} \right)$ , which is a slow rate for the strongly monotone setup. On the other hand, we will see that  $\beta(t) = \frac{2\mu}{e^{2\mu t} - 1}$  is a better choice leading to a faster rate in this setup.
Our analysis of this section is also based on a conservation law, but we use a slightly modified version to exploit strong monotonicity.
Proposition 6.1. Suppose  $\tilde{\mathbb{A}}$  is monotone and Lipschitz continuous. Let  $X$  be the solution of (3) and let  $R:[0,\infty)\to (0,\infty)$  be a differentiable function. For  $t_0 > 0$ , define  $E:(0,\infty)\rightarrow \mathbb{R}$  as
$$
\begin{array}{l} E = \frac {C (t) ^ {2} R (t) ^ {2}}{2} \Bigg (\left\| \tilde {\mathbf {A}} (X (t)) \right\| ^ {2} + 2 \beta (t) \left\langle \tilde {\mathbf {A}} (X (t)), X (t) - X _ {0} \right\rangle + \left(\beta (t) ^ {2} + \dot {\beta} (t)\right) \left\| X (t) - X _ {0} \right\| ^ {2} \Bigg) \\ \left. \right. - \int_ {t _ {0}} ^ {t} \frac {d}{d s} \left(\frac {C (s) ^ {2} R (s) ^ {2} \dot {\beta} (s)}{2}\right) \| X (t) - X _ {0} \| ^ {2} d s + \int_ {t _ {0}} ^ {t} C (s) ^ {2} R (s) ^ {2} \left(\left\langle \frac {d}{d s} \tilde {\mathbf {A}} (X (s)), \dot {X} (s) \right\rangle - \frac {\dot {R} (s)}{R (s)} \| \dot {X} (s) \| ^ {2}\right) d s. \\ \end{array}
$$
Then  $E$  is a constant function for  $t \in [0, \infty)$ .
Proposition 6.1 generalizes Proposition 3.2, since it corresponds to the special case with  $R(t) \equiv 1$ .
Recall from (2), when  $\mathbb{A}$  is  $\mu$ -strongly monotone we have
$$
\left\langle \frac {d}{d s} \mathbb {A} (X (t)), \dot {X} (t) \right\rangle - \mu \left\| \dot {X} (t) \right\| ^ {2} \geq 0.
$$
This motivates the choice  $R(t) = e^{\mu t}$ , since  $\frac{R(s)}{R(s)} = \mu$ . From calculation provided in Appendix H.2, the choice  $\beta(t) = \frac{2\mu}{e^{2\mu t} - 1}$  makes  $\frac{d}{ds}\left(\frac{C(s)^2R(s)^2\dot{\beta}(s)}{2}\right) = 0$ . Plugging these choices into Proposition 6.1 and following arguments of Section 3, we arrive at the following theorem.
Theorem 6.2. Let  $\mathbb{A}$  be a  $\mu$ -strongly maximal monotone operator with  $\mu > 0$  and assume  $\mathrm{Zer}\mathbb{A} \neq \emptyset$ . Let  $X$  be a solution of the differential inclusion (3) with  $\beta(t) = \frac{2\mu}{e^{2\mu t} - 1}$ , i.e. for almost all  $t$ ,
$$
\dot {X} \in - \mathbb {A} (X) - \frac {2 \mu}{e ^ {2 \mu t} - 1} (X - X _ {0}). \tag {10}
$$
Let  $\tilde{\mathbb{A}}(X(t))$  be the selection of  $\mathbb{A}(X(t))$  as in Section 2.1. Define  $V:[0,\infty)\to \mathbb{R}$  as
$$
V (t) = \frac {\left(e ^ {\mu t} - e ^ {- \mu t}\right) ^ {2}}{2} \left\| \tilde {\mathbf {A}} (X (t)) \right\| ^ {2} + 2 \mu \left(1 - e ^ {- 2 \mu t}\right) \left(\left\langle \tilde {\mathbf {A}} (X (t)), X (t) - X _ {0} \right\rangle - \mu \left\| X (t) - X _ {0} \right\| ^ {2}\right).
$$
Then  $V(t)\leq V(0)$  holds for  $t\geq 0$  . Furthermore for  $X_{\star}\in \mathrm{Zer}\mathbb{A}$
$$
\left\| \tilde {\mathbf {A}} (X (t)) \right\| ^ {2} \leq \left(\frac {2 \mu}{e ^ {\mu t} - 1}\right) ^ {2} \| X _ {0} - X _ {\star} \| ^ {2} = \mathcal {O} \left(\frac {1}{e ^ {2 \mu t}}\right).
$$
In Appendix H.2.3, we show that (10) is a continuous-time model for OS-PPM of Park and Ryu [54]. In Appendix C, we show the existence and uniqueness of the solution.
Since  $\beta(t) = \frac{2\mu}{e^{2\mu t} - 1} \in L^1[t_0, \infty)$  for any  $t_0 > 0$ , Theorem 4.3 implies that  $\tilde{\mathbb{A}}(X(t)) \to 0$  when  $\mathbb{A}$  is merely monotone. This tells us that the optimal choice of  $\beta(t)$  for should depend on the properties of  $\mathbb{A}$ . In the following section, we describe how  $\beta(t)$  can be chosen to adapt to the operator's properties.
7 Adaptive anchor acceleration and experiments
In this section, we present an adaptive method for choosing the anchor coefficient  $\beta$ , and we theoretically and experimentally show that this choice allows the dynamics to adapt to the operator's properties. It achieves the optimal  $\mathcal{O}(1 / k^2)$ -convergence rate when  $\mathbb{A}$  is monotone and an exponential convergence rate when  $\mathbb{A}$  is furthermore  $\mu$ -strongly monotone and Lipschitz continuous.
Theorem 7.1. Suppose  $\tilde{\mathbb{A}}$  is Lipschitz continuous and monotone. Consider the anchor ODE
$$
\dot {X} = - \tilde {\mathbf {A}} (X) + \underbrace {\frac {\left\| \tilde {\mathbf {A}} (X) \right\| ^ {2}}{2 \langle \tilde {\mathbf {A}} (X) , X - X _ {0} \rangle}} _ {= - \beta (t)} (X - X _ {0}) \tag {11}
$$
with initial condition  $X(0) = X_0$  and  $\|\tilde{\mathbf{A}}(X_0)\| \neq 0$ . Suppose the solution exists and  $\dot{X}$  is continuous at  $t = 0$ . Moreover, suppose  $\beta: (0, \infty) \to \mathbb{R}$  is well-defined, i.e., no division by zero occurs in the definition of  $\beta(t)$ . Then for  $t > 0$  and  $X_\star \in \mathrm{Zer}\tilde{\mathbf{A}}$ , we have  $\beta(t) > 0$  and
$$
\left\| \tilde {\mathbf {A}} (X (t)) \right\| ^ {2} \leq 4 \beta (t) ^ {2} \| X _ {0} - X _ {\star} \| ^ {2}
$$
$$
\beta (t) ^ {2} \leq \frac {1}{t ^ {2}}.
$$
If  $\tilde{\mathbb{A}}$  is furthermore  $\mu$ -strongly monotone, then for  $t > 0$ ,
$$
\beta (t) ^ {2} \leq \left(\frac {\mu / 2}{e ^ {\mu t / 2} - 1}\right) ^ {2}.
$$
We provide the proof in Appendix I.1. Note that anchor coefficient (11) is chosen so that
$$
\Phi (t) = \left\| \tilde {\mathbf {A}} (X (t)) \right\| ^ {2} + 2 \beta (t) \left\langle \tilde {\mathbf {A}} (X (t)), X (t) - X _ {0} \right\rangle = 0.
$$
So left-hand side of (8) is zero and a  $\mathcal{O}\left(\beta(t)^2\right)$  convergence rate is immediate. An analogous discrete-time result is shown in the following theorem.
Theorem 7.2. Let  $\mathbb{A}$  be a maximal monotone operator. Let  $x^0 = y^0\in \mathbb{R}^n$ . Consider
$$
x ^ {k} = \mathbb {J} _ {\mathbb {A}} y ^ {k - 1}
$$
$$
y ^ {k} = \left(1 - \beta_ {k}\right) \left(2 x ^ {k} - y ^ {k - 1}\right) + \beta_ {k} x ^ {0}
$$
$$
\beta_ {k} = \left\{ \begin{array}{l l} \frac {\| \tilde {\mathbf {A}} x ^ {k} \| ^ {2}}{- \langle \tilde {\mathbf {A}} x ^ {k} ,   x ^ {k} - x ^ {0} \rangle + \| \tilde {\mathbf {A}} x ^ {k} \| ^ {2}} & i f \| \tilde {\mathbf {A}} x ^ {k} \| ^ {2} \neq 0 \\ 0 & i f \| \tilde {\mathbf {A}} x ^ {k} \| ^ {2} = 0, \end{array} \right.
$$
for  $k = 1,2,\ldots$  , where  $\tilde{\mathbf{A}} x^{k} = y^{k - 1} - x^{k}$
Then  $\beta_{k}\geq 0$  and
$$
\left\| \tilde {\mathbf {A}} x ^ {k + 1} \right\| ^ {2} \leq \beta_ {k} ^ {2} \left\| x ^ {0} - x ^ {\star} \right\| ^ {2}
$$
$$
\beta_ {k} ^ {2} \leq \frac {1}{(k + 1) ^ {2}}
$$
for  $k = 1,2,\ldots$  and  $x^{\star}\in \mathrm{ZerA}$
If  $\mathbb{A}$  is furthermore  $\mu$ -strongly monotone and  $L$ -Lipschitz continuous, then for  $k = 1,2,\ldots$
$$
\beta_ {k} ^ {2} \leq \left(\frac {\mu / (1 + L ^ {2})}{\left(1 + \mu / (1 + L ^ {2})\right) ^ {k} - 1 + \mu / (1 + L ^ {2})}\right) ^ {2}.
$$
Figure 2: (Left) Network graph. (Right) Squared  $M$ -norm  $\| \tilde{\mathbb{A}} x^k\| _M^2$  vs.  $k$ . Halpern corresponds to the method in Theorem 5.1, we use  $p = 1.5$  and  $\gamma = 2.0$ .
For the monotone setup, the rate  $\left\| \tilde{\mathbf{A}} x^{k + 1}\right\| ^2\leq \frac{1}{(k + 1)^2}\left\| x^0 -x^\star \right\|^2$  matches the exact optimal rate of APPM [54]. In the limit  $\mu \to 0$ , the result for the  $\mu$ -strongly monotone case reduces to the result for the monotone case. We provide the proof and details of Theorem 7.2 in Appendix I.3.
The method of Theorem 7.2 is a discrete-time counterpart of the ODE of (11). The extra term  $\| \tilde{\mathbb{A}} x^k\| ^2$  of the denominator in the definition of  $\beta_{k}$  vanishes in the continuous-time limit. We provide further details in Appendix I.2. Analogous to the continuous-time case, a key property of the discrete-time adaptive method is that the counterpart of  $\Phi (t)$  is kept nonpositive. In the proof of Lemma I.3, the fact  $\beta_{k} < 1$  plays the key role in proving this property. The extra term  $\| \tilde{\mathbb{A}} x^{k}\|^{2}$  in the denominator and the fact that  $\langle \tilde{\mathbb{A}} x^k,x^k -x^0\rangle < 0$  when  $\| \tilde{\mathbb{A}} x^k\| ^2\neq 0$  leads to  $\beta_{k} < 1$
7.1 Experiment details
We now show an experiment with the method of Theorem 7.2 applied to a decentralized compressed sensing problem Shi et al. [63]. We assume that we have the measurement  $b_{i} = A_{(i)}x + e_{i}$ , where  $A_{(i)}$  is a measurement matrix available for each local agent  $i$ ,  $x$  is an unknown shared signal we hope to recover, and  $e_{i}$  is an error in measurement. We solve this problem in a decentralized manner in which the local agents keep their measurements private and only communicate with their neighbors.
As in Shi et al. [63], we formulate the problem into an unconstrained  $\ell_1$ -regularized least squares problem
$$
\underset {x \in \mathbb {R} ^ {d}} {\text {m i n i m i z e}} \quad \frac {1}{n} \sum_ {i = 1} ^ {n} \left\{\frac {1}{2} \| A _ {(i)} x - b _ {i} \| ^ {2} + \rho \| x \| _ {1} \right\},
$$
and apply PG-EXTRA. We compare vanilla PG-EXTRA with the various anchored versions of PG-EXTRA with  $\beta_{k}$  as in Theorem 5.1 and Theorem 7.2. We show the results in Figure 2. Further details of the experiment are provided in Appendix J.
This work introduces a continuous-time model of anchor acceleration, the anchor ODE  $\dot{X} \in -\mathbb{A}(X) - \beta(t)(X - X_0)$ . We characterize the convergence rate as a function of  $\beta(t)$  and thereby obtain insight into the anchor acceleration mechanism. Finally, inspired by the continuous-time analyses, we present an adaptive method and establish its effectiveness through theoretical analyses and experiments.
Prior work analyzing continuous-time models of Nesterov acceleration had inspired various follow-up research, such as analyses based on Lagrangian and Hamiltonian mechanics [71, 72, 26], high-resolution ODE model [62], and continued framework [29]. Carrying out similar analyses for the anchor ODE are interesting directions of future work.
This work was supported by the Samsung Science and Technology Foundation (Project Number SSTF-BA2101-02). We thank Jaeyeon Kim for providing valuable feedback. We thank Hangjun Cho for the helpful discussions on well-posedness of ODEs. We also thank anonymous reviewers for the constructive comments.
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A Prior work
Acceleration for smooth convex function in discrete setting. There had been rich amount of research on acceleration about smooth convex functions. Nesterov [51] introduced accelerated gradient method (AGM), which has a faster  $\mathcal{O}(1 / k^2)$  rate than  $\mathcal{O}(1 / k)$  rate of gradient descent [22] in reducing the function value. Optimized gradient method (OGM) [36] improved AGM's rate by a constant factor, and is proven to be optimal [27]. For smooth strongly convex setup, strongly convex AGM [52] achieves an accelerated rate, and further improvements were studied [70, 53, 68, 61]. Recently, OGM-G [37] was introduced as an accelerated method reducing squared gradient magnitude for smooth convex minimization.
Acceleration for smooth convex function in continuous setting. Continuous-time analysis of Nesterov acceleration has been thoroughly studied as well. Su et al. [65, 66] introduced an ODE model of AGM  $\dot{X}(t) + \frac{r}{t} X + \nabla f(X) = 0$ , providing  $f(X(t)) - f_{\star} \in \mathcal{O}\left(1 / t^{2}\right)$  rate for  $r \geq 3$ . Attouch et al. [7] improved the constant of bound for  $r > 3$  and proved convergence of the trajectories. Attouch et al. [9] achieved  $\mathcal{O}\left(1 / t^{-2r / 3}\right)$  rate for  $0 < r < 3$ . Apidopoulos et al. [2] generalized their results to differential inclusion with non-differentiable convex function. Furthermore, wide range of variations of AGM ODE has been studied [3, 8, 12, 16, 18, 5, 10, 19]. Also, applications to monotone inclusion problem were studied by Attouch and Peypouquet [6], Attouch and László [4], Boş and Hulett [17].
Motivated from above continuous-time analysis for accelerated methods, tools analyzing ODEs have further developed. Wibisono et al. [71], Wilson et al. [72] and Kim and Yang [38] adopted Lagrangian mechanics and introduced first, second, unified Bregman Lagrangian to provide unified analysis for generalized family of ODE, where the latter provided analysis for strongly convex AGM. Systemical approach to obtain Lyapunov functions exploiting Hamiltonian mechanics [26] and dilated coordinate system [67] were proposed, and analysis of OGM-G was provided by dilated coordinate framework. Different forms of continuous-time models such as high-resolution ODE [62] and continued framework [29] were developed.
On the other hand, another type of acceleration called anchor acceleration recently gained attention. As Yoon and Ryu [76] focused, many recently discovered accelerated methods for both minimax optimization and fixed-point problems are based on anchor acceleration. More recently, practical applications of anchor acceleration to detecting infeasibility for constrained optimization problems [55], and accelerating value iteration for dynamic programming and reinforcement learning [41] were introduced as well.
Fixed-point problem. The history of studies on fixed-point problem dates back to the work of Banach [14], which established that the Picard iteration with contractive operator is convergent. Kransnosel'skii-Mann iteration (KM) [40, 47] was introduced, which is a generalization of Picard iteration. Convergence of KM iteration with general nonexpansive operators was proven by Martinet [49]. For iteration of Halpern [34], convergence with wide choice of parameter were shown by Wittmann [73].
The squared norm  $\| y_{k} - \mathbb{T}y_{k}\|^{2}$  of fixed-point residual is a common error measure for fixed-point problems. KM iteration was shown to exhibit  $\mathcal{O}(1 / k)$  rate [23, 44, 21] and  $o(1 / k)$ -rate [13, 50]. For Halpern iteration,  $\mathcal{O}(1 / (\log k)^2)$ -rate was established by Leustean [43], then improve to  $\mathcal{O}(1 / k)$  rate by Kohlenbach [39]. First accelerated  $\mathcal{O}(1 / k^2)$  rate was achieved by Sabach and Shtern [60] and the constant was improved by Lieder [45] by a factor of 16.
It is known that there is an equivalence between solving fixed-point problem and solving monotone inclusion problem [30, 28, 54]. Proximal point method (PPM) [48] achieves  $\mathcal{O}(1/k)$ -rate in terms of  $\left\| \tilde{\mathbb{A}} x_k \right\|^2$  [33]. Accelerated proximal point method (APPM) [35] improved the rate to accelerated  $\mathcal{O}\left(1/k^2\right)$ -rate. Park and Ryu [54] showed APPM is exactly optimal method for this problem and provided exactly optimal method for  $\mu$ -strongly monotone operator named OS-PPM, which achieved  $\mathcal{O}\left(1/e^{4\mu k}\right)$  rate. The optimal methods APPM and OS-PPM are based on anchor acceleration [76].
Minimax problems. Minimax optimization problem of the form  $\min_x\max_y\mathbf{L}(x,y)$  have recently gained attention in machine learning society. One of the commonly considered theoretical setting is smooth convex-concave setup, with squared gradient norm as error measure. In terms of  $\| \partial \mathbf{L}(x,y)\| ^2$  , classical EG [64] and OG [56, 57, 24] was shown to achieved  $\mathcal{O}\left(1 / k\right)$  -rate [32]. SGDA [59] achieved  $\mathcal{O}\left(1 / k^{2 - 2p}\right)$  rate for  $p > 1 / 2$  with introducing the term anchor. With introducing a parameter-free Halpern type method, Diakonikolas [25] achieved  $\mathcal{O}(\log k / k^2)$  . Recently, EAG [75] first achieved accelerated rate  $\mathcal{O}(1 / k^2)$  with anchor acceleration, followed by FEG [42], anchored Popov's scheme [69] and moving anchor methods [1]. Fast ODGA [20] also achieved accelerated  $o(1 / k^2)$  rate. For  $\partial \mathbf{L}$  is furthermore strongly monotone with condition number  $\kappa$  SM-EAG+ [76] achieved accelerated  $\mathcal{O}\left(1 / e^{2k\kappa}\right)$  rate.
However, continuous-time analysis for anchor acceleration is, to the best of our knowledge, insufficient. Continuous-time analyses of acceleration for monotone inclusion problem were studied by Bot et al. [20], Lin and Jordan [46], but they did not consider anchor acceleration. Ryu et al. [59] considered continuous-time analysis of anchor acceleration, but only with limited cases  $\dot{X}(t) = -\mathbb{A}(X(t)) - \frac{\gamma}{t} (X - X_0)$  for  $\gamma \geq 1$ . In this paper, we provide a unified continuous-time analysis for anchor acceleration with generalized anchor coefficient.
B Proof of Theorem 2.2
B.1 Proof of uniqueness
Proof of uniqueness is immediate from Lemma 2.3, we first prove the lemma.
Proof of Lemma 2.3. It is trivial for  $t = 0$ , so we may assume  $t > 0$ .
By monotonicity of  $\mathbb{A}$  and Young's inequality, we get the following inequality.