| Algorithm | General convex | Adaptive |
| SVRG (Johnson & Zhang, 2013) | - | No |
| SVRG++ (Allen-Zhu & Yuan, 2016) | O(n log β/ε + β/ε) | No |
| Katyusha (Allen-Zhu, 2017) | O(n log β/ε + √nβ/ε) | No |
| VARAG (Lan et al., 2019) | O(n min{log β/ε, log n} + √nβ/ε) | No |
| VRADA (Song et al., 2020) | O(n min{log log β/ε, log log n} + √nβ/ε) | No |
| AdaSVRG (Dubois-Taine et al., 2021) | O(nβ/ε)(fixed sized inner loop, only if ε = Ω(β/n)) | Yes |
| O(n log β/ε + β/ε)(multi-stage) |
| AdaVRAE (unknown β) (This Paper) | O(n min{log log β/ε, log log n} + √nβ/ε) | Yes |
| VRAE (known β) (This Paper) | O(n min{log log β/ε, log log n} + √nβ/ε) | No |
| AdaVRAG (unknown β) (This Paper) | O(n min{log log β/ε, log log n} + √nβ log β/ε) | Yes |
| VRAG (known β) (This Paper) | O(n min{log log β/ε, log log n} + √nβ/ε) | No |
| Lower Bound (Woodworth & Srebro, 2016) | Ω(n + √nβ/ε) | - |
2021; Ene & Nguyen, 2021). Remarkably, these works have shown that, in the setting where we have access to the exact full gradient in each iteration, it is possible to match the convergence rates of both unaccelerated and accelerated gradient descent methods without any prior knowledge of the smoothness parameter. These methods have also been analyzed in the stochastic setting under a bounded variance assumption, and they achieve a convergence rate that is comparable to that of SGD.
Given the theoretical and practical success of adaptive methods, it is natural to ask whether one can design VR methods that achieve state of the art convergence guarantees without any prior knowledge of the smoothness parameter. The recent work of (Dubois-Taine et al., 2021) gives the first adaptive VR method — AdaSVRG — with the gradient complexity of $\mathcal{O}\left(n\log \frac{\beta}{\epsilon} +\frac{\beta}{\epsilon}\right)$ . AdaSVRG builds on the AdaGrad (Duchi et al., 2011) and SVRG algorithms (Johnson & Zhang, 2013), both of which are not accelerated.
Our contributions: In this work, we take this line of work further and design the first accelerated VR methods that do not require any prior knowledge of the smoothness parameter. Our algorithms, Adaptive Variance Reduced Accelerated Extra-Gradient (AdaVRAE) and Adaptive Variance Reduced Accelerated Gradi
ent (AdaVRAG), only use $\mathcal{O}\left(n\log \log n + \sqrt{\frac{n\beta}{\epsilon}}\right)$ and $\mathcal{O}\left(n\log \log n + \sqrt{\frac{n\beta\log\beta}{\epsilon}}\right)$ gradient evaluations respectively to attain an $\mathcal{O}(\epsilon)$ -suboptimal solution when $\beta$ is unknown, both of which significantly improve the convergence rate of AdaSVRG. Table 1 compares our algorithms and prior VR methods and Section 2 discusses our algorithmic approaches and techniques. The convergence rate of AdaVRAE matches up to constant factors the best-known convergence rate of non-adaptive VR methods (Song et al., 2020; Joulani et al., 2020). Both of our algorithms follow a different approach from these methods and are based on extra-gradient and mirror descent, instead of dual averaging.
We demonstrate the efficiency of our algorithms in practice on multiple real-world datasets. We show that AdaVRAG and AdaVRAE are competitive with existing standard and adaptive VR methods while having the advantage of not requiring hyperparameter tuning, and in many cases AdaVRAG outperforms these benchmarks.
# 1.1. Related work
Variance reduced gradient methods: Variance reduction technique (Roux et al., 2012; Schmidt et al., 2017; Shalev-
Shwartz & Zhang, 2013; Mairal, 2013; Johnson & Zhang, 2013; Defazio et al., 2014) has been proposed to improve the convergence rate of stochastic gradient descent algorithms in the finite sum problem and has since become widely-used in many successful algorithms. Notable improvements can be seen in strongly convex optimization problems where earliest algorithms such as SVRG (Johnson & Zhang, 2013) or SAGA (Defazio et al., 2014) obtain $\mathcal{O}\left((n + \kappa)\log \frac{1}{\epsilon}\right)$ convergence rate compared with $\mathcal{O}\left(\frac{\sigma^2\kappa}{\beta\epsilon}\right)$ of plain SGD, with the latter requiring an additional assumption on the $\sigma^2$ -boundedness of the variance term, i.e., $\mathbb{E}_i\left[\| \nabla f_i(x) - \nabla f(x)\|^2\right] \leq \sigma^2$ . However, these non-accelerated methods do not achieve the optimal convergence rate. Recent works such as (Lin et al., 2015; Allen-Zhu, 2017; Lan et al., 2019) focus on designing accelerated methods and successfully match the optimal lower bound for strongly convex optimization of $\Omega\left((n + \sqrt{n\kappa})\log \frac{1}{\epsilon}\right)$ given by (Lan & Zhou, 2018).
In non-strongly convex problems, however, existing works do not yet match the lower bound of $\Omega\left(n + \sqrt{\frac{\beta n}{\epsilon}}\right)$ shown in (Woodworth & Srebro, 2016). The best effort so far can be found in the line of accelerated methods started by (Allen-Zhu, 2017) and followed by (Allen-Zhu, 2018; Lan et al., 2019; Li, 2021) that rely on incorporating the checkpoint in each update. AdaVRAG follows the same idea but offers simpler update and more efficient choice of coefficients that results in a better convergence rate, equivalent to VRADA (Song et al., 2020). By comparison, while VRADA is a dual-averaging scheme, AdaVRAG is a mirror descent method and AdaVRAE is an extra-gradient algorithm.
In a different line of research (Allen-Zhu & Hazan, 2016; Fang et al., 2018; Zhou et al., 2018), variance reduction has been applied to non-convex optimization to find critical points with much better convergence rate.
Adaptive methods with variance reduction: There has been extensive research on adaptive methods (Duchi et al., 2011; Kingma & Ba, 2014; Reddi et al., 2018; Tieleman et al., 2012; Dozat, 2016) in the setting where we compute a full gradient in each iteration. However, there are only few works combining adaptive methods with VR techniques in the finite sum setup. Most relevant for our work is AdaSVRG (Dubois-Taine et al., 2021). This algorithm is built upon SVRG which as mentioned earlier is a non-accelerated method and has a slower convergence rate. AdaSVRG uses the gradient norm to update the step size, similar to (Duchi et al., 2011) and the step is reset in every epoch, which could lead to step sizes that are too large in later stages. In contrast, both AdaVRAG and AdaVRAE are accelerated VR methods and use a cumulative step size. AdaVRAG uses the iterate movement to update the step size,
as in (Bach & Levy, 2019; Ene et al., 2021). AdaVRAE improves the convergence rate by a $\sqrt{\log\beta}$ factor by using the gradient difference similarly to (Mohri & Yang, 2016; Joulani et al., 2020; Ene & Nguyen, 2021). In a different direction, (Xu et al., 2017) propose an adaptive VR algorithm adaptive to the unknown growth parameter instead of the smoothness parameter.
A different line of work considers VR methods that set the step size using stochastic line search (Schmidt et al., 2017; Mairal, 2013) or Barzilai-Borwein step size (Tan et al., 2016; Li et al., 2020). The former methods do not have theoretical guarantees, and the latter methods require knowledge of the smoothness parameter in order to obtain theoretical bounds.
Recent works design variance-reduced methods for nonconvex optimization. STORM (Cutkosky & Orabona, 2019) and $\mathrm{STORM}^{+}$ (Levy et al., 2021) design an adaptive step size, though the former still requires the smoothness parameter in the step size. Super-Adam (Huang et al., 2021) also requires their parameters to satisfy some inequality involving the smoothness parameter like STORM.
# 1.2. Notation and problem setup
Let $[n]$ denote the set $\{1,2,\dots ,n\}$ . For simplicity, we only consider the Euclidean norm $\| \cdot \| \coloneqq \| \cdot \| _2$ (Our work can be extended to $\| x\| _A\coloneqq \sqrt{x^\top Ax}$ for any $A\succ 0$ with almost no change). $x^{+}$ represents max $\{x,0\}$ .
We are interested in solving the following problem
$$
\min _ {x \in \mathcal {X}} \left\{F (x) = f (x) + h (x) \right\}
$$
where $f(x) \coloneqq \frac{1}{n} \sum_{i=1}^{n} f_i(x)$ and for $i \in [n]$ , $f_i: \mathbb{R}^d \to \mathbb{R}$ and $h: \mathcal{X} \to \mathbb{R}$ are convex functions with a closed convex set $\mathcal{X} \subseteq \mathbb{R}^d$ . Let $x^* = \arg \min_{x \in \mathcal{X}} F(x)$ . We say a function $G$ is $\beta$ -smooth if $\| \nabla G(x) - \nabla G(y) \| \leq \beta \| x - y \|$ for all $x, y \in \mathbb{R}^d$ . Equivalently, we have $G(y) \leq G(x) + \langle \nabla G(x), y - x \rangle + \frac{\beta}{2} \| y - x \|^2$ . In this paper we always assume that each $f_i$ is $\beta$ -smooth, which implies that $f$ is also $\beta$ -smooth. We assume that we can efficiently solve optimization problems of the form $\arg \min_{x \in \mathcal{X}} \left( \gamma h(x) + \frac{1}{2} \| x - v \|^2 \right)$ where $\gamma \geq 0$ and $v \in \mathbb{R}^d$ . When the smoothness parameter $\beta$ is unknown, we additionally assume that $\mathcal{X}$ is compact with diameter $D$ , i.e., $\sup_{x, y \in \mathcal{X}} \| x - y \| \leq D$ .
# 2. Our algorithms and convergence guarantees
In this section, we describe our algorithms and state their convergence guarantees. Our algorithm AdaVRAE shown in Algorithm 1 is a novel accelerated scheme that uses past extra-gradient update steps in the inner loop and novel averaging to achieve acceleration. In each inner iteration, the
# Algorithm 1 AdaVRAE
Input: initial point $u^{(0)}$ , domain diameter $D$ .
Parameters: $\{a^{(s)}\} ,\{T_s\}$ $A_{T_0}^{(0)} > 0$ $\eta >0$
$\overline{x}_0^{(1)} = z_0^{(1)} = u^{(0)}$ , compute $\nabla f(u^{(0)})$
Initialize $\gamma_0^{(1)} = \gamma$ , where $\gamma$ is any small constant
for $s = 1$ to $S$ :
$$
A _ {0} ^ {(s)} = A _ {T _ {s - 1}} ^ {(s - 1)} - T _ {s} \left(a ^ {(s)}\right) ^ {2}
$$
for $t = 1$ to $\bar{T}_s$ :
$$
\begin{array}{l} x _ {t} ^ {(s)} = \arg \min _ {x \in \mathcal {X}} \left\{a ^ {(s)} \left\langle g _ {t - 1} ^ {(s)}, x \right\rangle + a ^ {(s)} h (x) \right. \\ \left. + \frac {\gamma_ {t - 1} ^ {(s)}}{2} \left\| x - z _ {t - 1} ^ {(s)} \right\| ^ {2} \right\} \\ \end{array}
$$
Let $A_{t}^{(s)} = A_{t - 1}^{(s)} + a^{(s)} + \left(a^{(s)}\right)^{2}$
$$
\bar {x} _ {t} ^ {(s)} = \frac {1}{A _ {t} ^ {(s)}} \left(A _ {t - 1} ^ {(s)} \bar {x} _ {t - 1} ^ {(s)} + a ^ {(s)} x _ {t} ^ {(s)} + \left(a ^ {(s)}\right) ^ {2} u ^ {(s - 1)}\right)
$$
if $t\neq T_s$
$$
\text {P i c k} i _ {t} ^ {(s)} \sim \text {U n i f o r m} ([ n ])
$$
$$
g _ {t} ^ {(s)} = \nabla f _ {i _ {t} ^ {(s)}} (\bar {x} _ {t} ^ {(s)}) - \nabla f _ {i _ {t} ^ {(s)}} (u ^ {(s - 1)}) + \nabla f (u ^ {(s - 1)})
$$
else:
$$
g _ {t} ^ {(s)} = \nabla f (\bar {x} _ {t} ^ {(s)})
$$
$$
\gamma_ {t} ^ {(s)} = \frac {1}{\eta} \sqrt {\eta^ {2} \left(\gamma_ {t - 1} ^ {(s)}\right) ^ {2} + \left(a ^ {(s)}\right) ^ {2} \left\| g _ {t} ^ {(s)} - g _ {t - 1} ^ {(s)} \right\| ^ {2}}
$$
$$
\begin{array}{l} z _ {t} ^ {(s)} = \arg \min _ {z \in \mathcal {X}} \left\{a ^ {(s)} \left\langle g _ {t} ^ {(s)}, z \right\rangle + a ^ {(s)} h (z) \right. \\ \left. + \frac {\gamma_ {t - 1} ^ {(s)}}{2} \left\| z - z _ {t - 1} ^ {(s)} \right\| ^ {2} + \frac {\gamma_ {t} ^ {(s)} - \gamma_ {t - 1} ^ {(s)}}{2} \left\| z - x _ {t} ^ {(s)} \right\| ^ {2} \right\} \\ \end{array}
$$
$$
u ^ {(s)} = \bar {x} _ {0} ^ {(s + 1)} = \bar {x} _ {T _ {s}} ^ {(s)}, z _ {0} ^ {(s + 1)} = z _ {T _ {s}} ^ {(s)}, g _ {0} ^ {(s + 1)} = g _ {T _ {s}} ^ {(s)},
$$
$$
\begin{array}{l} \gamma_ {0} ^ {(s + 1)} = \gamma_ {T _ {s}} ^ {(s)} \\ \text {r e t u r n} u ^ {(S)} \\ \end{array}
$$
# Algorithm 2 AdaVRAG
Input: initial point $u^{(0)}$ , domain diameter $D$ .
Parameters: $\{a^{(s)}\} ,a^{(s)}\in (0,1),\{q^{(s)}\} ,\{T_s\} ,\eta >0.$
$$
x _ {0} ^ {(1)} = u ^ {(0)}
$$
Initialize $\gamma_0^{(1)} = \gamma$ , where $\gamma$ is any small constant
for $s = 1$ to $S$ :
$$
\bar {x} _ {0} ^ {(s)} = a ^ {(s)} x _ {0} ^ {(s)} + (1 - a ^ {(s)}) u ^ {(s - 1)}, \text {c o m p u t e} \nabla f \left(u ^ {(s - 1)}\right)
$$
for $t = 1$ to $T_{s}$ :
Pick $i_t^{(s)} \sim \text{Uniform}([n])$
$$
g _ {t} ^ {(s)} = \nabla f _ {i _ {t} ^ {(s)}} (\bar {x} _ {t - 1} ^ {(s)}) - \nabla f _ {i _ {t} ^ {(s)}} (u ^ {(s - 1)}) + \nabla f (u ^ {(s - 1)})
$$
$$
\begin{array}{l} x _ {t} ^ {(s)} = \arg \min _ {x \in \mathcal {X}} \left\{\left\langle g _ {t} ^ {(s)}, x \right\rangle + h (x) \right. \\ \left. + \frac {\gamma_ {t - 1} ^ {(s)} q ^ {(s)}}{2} \left\| x - x _ {t - 1} ^ {(s)} \right\| ^ {2} \right\} \\ \end{array}
$$
$$
\bar {x} _ {t} ^ {(s)} = a ^ {(s)} x _ {t} ^ {(s)} + (1 - a ^ {(s)}) u ^ {(s - 1)}
$$
Option I: $\gamma_{t}^{(s)} = \gamma_{t - 1}^{(s)}\sqrt{1 + \frac{\left\|x_{t}^{(s)} - x_{t - 1}^{(s)}\right\|^{2}}{\eta^{2}}}$
Option II: $\gamma_t^{(s)} = \gamma_{t - 1}^{(s)} + \frac{\left\|x_t^{(s)} - x_{t - 1}^{(s)}\right\|^2}{\eta^2}$
$$
\begin{array}{l} u ^ {(s)} = \frac {1}{T _ {s}} \sum_ {t = 1} ^ {T _ {s}} \bar {x} _ {t} ^ {(s)}, x _ {0} ^ {(s + 1)} = x _ {T _ {s}} ^ {(s)}, \gamma_ {0} ^ {(s + 1)} = \gamma_ {T _ {s}} ^ {(s)} \\ \text {r e t u r n} u ^ {(S)} \\ \end{array}
$$
new average iterate $\overline{x}_t^{(s)}$ is obtained by combining the old average iterate $\overline{x}_{t - 1}^{(s)}$ , the new iterate $x_{t}^{(s)}$ and the checkpoint $u^{(s - 1)}$ with coefficients $A_{t - 1}^{(s)}$ , $a^{(s)}$ and $(a^{(s)})^2$ normalized by the sum of them, i.e by $A_{t}^{(s)} = A_{t - 1}^{(s)} + a^{(s)} + (a^{(s)})^2$ . We will explain the intuition behind this choice of coefficients in the analysis outline. At the beginning of each epoch, we set $A_0^{(s)} = A_{Ts - 1}^{(s - 1)} - T_s(a^{(s)})^2$ so that at the end, we only accumulate the coefficients of the new iterates $x_{t}^{(s)}$ . AdaVRAE adaptively sets the step sizes based on the stochastic gradient difference. Our choice of step sizes is a novel adaptation to the VR setting of the step sizes used by the works (Mohri & Yang, 2016; Kavis et al., 2019; Joulani et al., 2020; Ene & Nguyen, 2021) in the batch/full-gradient setting. Our algorithm builds on the work (Ene & Nguyen, 2021), which provides an unaccelerated past extra-gradient algorithm in the batch/full-gradient setting.
Theorem 2.1 states the parameter choices and the convergence guarantee for AdaVRAE, and we give its proof in Section A in the appendix. The convergence rate of AdaVRAE matches up to constant factors the rate of the state of the art non-adaptive VR methods (Joulani et al., 2020; Song et al., 2020). The initial step size $\gamma_0^{(1)}$ can be set to any small constant $\gamma$ , which in practice we choose $\gamma = 0.01$ . Similarly to AdaGrad, setting $\eta = \Theta(D)$ gives us the optimal dependence of the convergence rate in the domain diameter. For simplicity, we state the convergence in Theorem 2.1 and 2.2 when $\eta = \Theta(D)$ . We refer the reader to Theorems A.1 and B.1 in the appendix for the precise choice of parameters as well as the full dependence of the convergence rate on arbitrary choices of $\gamma$ and $\eta$ . In both Theorem 2.1 and 2.2, we measure convergence using the number of individual gradient evaluations $\nabla f_i$ , assuming that the exact computation of $\nabla f$ takes $n$ gradient evaluations.
Theorem 2.1. (Convergence of AdaVRAE) Define $s_0 = \lceil \log_2\log_24n\rceil$ , $c = \frac{3}{2}$ . Suppose we set the parameters of Algorithm 1 as follows:
$$
\begin{array}{l} a ^ {(s)} = \left\{ \begin{array}{l l} (4 n) ^ {- 0. 5 ^ {s}} & 1 \leq s \leq s _ {0} \\ \frac {s - s _ {0} - 1 + c}{2 c} & s _ {0} < s \end{array} \right., \\ T _ {s} = n, \\ A _ {T _ {0}} ^ {(0)} = \frac {5}{4}. \\ \end{array}
$$
Suppose that $\mathcal{X}$ is a compact convex set with diameter $D$ and we set $\eta = \Theta(D)$ . The number of individual gradient evaluations to achieve a solution $u^{(S)}$ such that $\mathbb{E}\left[F(u^{(S)}) - F(x^*)\right] \leq \epsilon$ for Algorithm 1 is
$$
\# \text {g r a d s} = \left\{ \begin{array}{l l} \mathcal {O} \left(n \log \log \frac {V _ {1}}{\epsilon}\right) & i f \epsilon \geq \frac {V _ {1}}{n} \\ \mathcal {O} \left(n \log \log n + \sqrt {\frac {n V _ {1}}{\epsilon}}\right) & i f \epsilon < \frac {V _ {1}}{n} \end{array} \right.
$$
where $V_{1} = \mathcal{O}\left(F(u^{(0)}) - F(x^{*}) + (\gamma +\beta)D^{2}\right)$ .
Our algorithm AdaVRAG is shown in Algorithm 2. Compared with AdaVRAE, AdaVRAG has a worse dependence on the smoothness parameter $\beta$ but it performs only one projection onto $\mathcal{X}$ in each inner iteration. Additionally, as we discuss in more detail below, it uses adaptive step sizes based on the iterate movement.
AdaVRAG follows a similar framework to existing VR methods such as VARAG (Lan et al., 2019) and VRADA (Song et al., 2020). Similarly to VRADA, the algorithm achieves acceleration at the epoch level, where an epoch is an iteration of the outer loop. The iterations in an epoch update the main iterates via mirror descent with novel choices of step sizes and coefficients. The stochastic gradient is computed at a point that is a convex combination between the current iterate and the checkpoint; the coefficients of this combination remain fixed throughout the epoch. The step sizes are adaptively set based on the iterate movement.
The structure of the inner iterations of our algorithm differs from both VARAG and VRADA in several notable aspects. VARAG also uses mirror descent to update the main iterates and it computes the stochastic gradient at suitable combinations of the iterates and the checkpoint. AdaVARAG uses a different averaging of the iterates to compute the snapshots. Moreover, it uses a very different and simpler choice for the coefficient used to combine the main iterates and the checkpoint in order to obtain the points at which the stochastic gradients are evaluated. In VARAG, this coefficient is set to a constant (namely, $1/2$ ) in the initial iterations, whereas in AdaVRAG, it starts from a small number and is increased gradually. This choice is critical for improving the first term in the convergence from $\mathcal{O}(n\log n)$ to $\mathcal{O}(n\log \log n)$ . In a similar manner, VRADA attains the same convergence by a new choice of coefficient. However, this is achieved via a very different approach based on dual-averaging.
The step sizes used by AdaVRAG have two components: the step $\gamma_t^{(s)}$ that is updated based on the iterate movement and the per-epoch coefficient $q^{(s)}$ to achieve acceleration at the epoch level. Our analysis is flexible and allows the use of several approaches for updating the steps $\gamma_t^{(s)}$ . One approach, shown as option I in Algorithm 2, is based on the multiplicative update rule of AdaGrad+ (Ene et al., 2021) which generalizes the AdaGrad update to the constrained setting. We also propose a different variant, shown as option II, that updates the steps in an additive manner. Our analysis shows a similar convergence guarantee for both options, with the main difference being in the dependence on the smoothness: option I incurs a dependence of $\sqrt{\beta\log\beta}$ , whereas option II has a worse dependence of $\beta$ . Option II achieved improved performance in our experiments.
Theorem 2.2 states the parameter choices and the convergence guarantee for AdaVRAG, and we give its proof in Section B in the appendix. Analogously to AdaVRAE, the
initial step size $\gamma$ can be set to any small constant.
Theorem 2.2. (Convergence of AdaVRAG) Define $s_0 = \lceil \log_2\log_24n\rceil$ , $c = \frac{3 + \sqrt{33}}{4}$ . Suppose we set the parameters of Algorithm 2 as follows:
$$
a ^ {(s)} = \left\{ \begin{array}{l l} 1 - (4 n) ^ {- 0. 5 ^ {s}} & 1 \leq s \leq s _ {0} \\ \frac {c}{s - s _ {0} + 2 c} & s _ {0} < s \end{array} \right.,
$$
$$
q ^ {(s)} = \left\{ \begin{array}{l l} \frac {1}{(1 - a ^ {(s)}) a ^ {(s)}} & 1 \leq s \leq s _ {0} \\ \frac {8 (2 - a ^ {(s)}) a ^ {(s)}}{3 (1 - a ^ {(s)})} & s _ {0} < s \end{array} \right.,
$$
$$
T _ {s} = n.
$$
Suppose that $\mathcal{X}$ is a compact convex set with diameter $D$ and we set $\eta = \Theta(D)$ . Additionally, we assume that $2\eta^2 > D^2$ if Option I is used for setting the step size. The number of individual gradient evaluations to achieve a solution $u^{(S)}$ such that $\mathbb{E}\left[F(u^{(S)}) - F(x^*)\right] \leq \epsilon$ for Algorithm 2 is
$$
\# \text {g r a d s} = \left\{ \begin{array}{l l} \mathcal {O} \left(n \log \log \frac {V _ {2}}{\epsilon}\right) & \epsilon \geq \frac {V _ {2}}{n} \\ \mathcal {O} \left(n \log \log n + \sqrt {\frac {n V _ {2}}{\epsilon}}\right) & \epsilon < \frac {V _ {2}}{n} \end{array} , \right.
$$
where
$$
V _ {2} = \left\{ \begin{array}{l l} \mathcal {O} \left(F (u ^ {(0)}) - F (x ^ {*}) + \left(\gamma + \beta \log \left(\frac {\beta}{\gamma}\right)\right) D ^ {2}\right) & \\ & f o r O p t i o n I \\ \mathcal {O} \left(F (u ^ {(0)}) - F (x ^ {*}) + \left(\gamma + \beta^ {2}\right) D ^ {2}\right) & \\ & f o r O p t i o n I I \end{array} \right.
$$
Comparison to AdaSVRG: As noted in the introduction, the state of the art adaptive VR method is the AdaSVRG algorithm (Dubois-Taine et al., 2021), which is a non-accelerated method. Both of our algorithms achieve a faster convergence using different approaches and step sizes. AdaSVRG resets the step sizes in each epoch, whereas our algorithms use a cumulative update approach for the step sizes. In our experimental evaluation, the resetting of the step sizes led to slower convergence. AdaSVRG (multi-stage variant) uses varying epoch lengths similarly to $\mathrm{SVRG}^{++}$ (Allen-Zhu & Yuan, 2016), whereas our algorithms use epoch lengths that are set to $n$ . Using an epoch of length $n$ allows for implementing the random sampling via a random permutation of $[n]$ and is the preferred approach in practice.
Both our algorithms and AdaSVRG require that the domain $\mathcal{X}$ has bounded diameter. This is a restriction that is shared by almost all existing adaptive methods. Recent work (Antonakopoulos et al., 2021; Ene & Nguyen, 2021) in the batch/full-gradient setting have proposed unaccelerated methods that are suitable for unbounded domains, at a loss of additional factors in the convergence. All of the existing accelerated methods require that the domain is bounded,
even in the batch/full-gradient setting. We note that our analysis holds for arbitrary compact domains, whereas the analysis of AdaSVRG only applies to domains that contain the global optimum. Similarly to AdaGrad, both our algorithms and AdaSVRG can be used in the unconstrained setting under the promise that the iterates do not move too far from the optimum.
Non-adaptive variants of our algorithms: In the setting where the smoothness parameter is known, we can set the step sizes of our algorithms based on the smoothness, as shown in Algorithms 3 and 4 (Sections C and D in the appendix). Both algorithms match the convergence rates of the state of the art VR methods (Joulani et al., 2020; Song et al., 2020) using different algorithmic approaches based on mirror descent and extra-gradient instead of dual-averaging. We experimentally compare the non-adaptive algorithms to existing methods in Section E of the appendix.
# 2.1. Analysis outline
We outline some of the key steps in the analysis of AdaVRAE. For the purpose of simplicity, we assume $h = 0$ and $\eta = D$ . Starting from the observation $\overline{x}_t^{(s)} = \frac{1}{A_t^{(s)}}\left(A_{t - 1}^{(s)}\overline{x}_{t - 1}^{(s)} + a^{(s)}x_t^{(s)} + \left(a^{(s)}\right)^2 u^{(s - 1)}\right)$ and $A_{t}^{(s)} = A_{t - 1}^{(s)} + a^{(s)} + \left(a^{(s)}\right)^2$ , we have
$$
\overline {{x}} _ {t} ^ {(s)} - \overline {{x}} _ {t - 1} ^ {(s)} = \frac {a ^ {(s)}}{A _ {t - 1} ^ {(s)}} \left(x _ {t} ^ {(s)} - \overline {{x}} _ {t} ^ {(s)}\right) + \frac {\left(a ^ {(s)}\right) ^ {2}}{A _ {t - 1} ^ {(s)}} \left(u ^ {(s - 1)} - \overline {{x}} _ {t} ^ {(s)}\right)
$$
which allows us to carry out the analysis for the function progress in one iteration, i.e, $f(\overline{x}_t^{(s)}) - f(\overline{x}_{t - 1}^{(s)})$ and obtain
$$
\begin{array}{l} \mathbb {E} \left[ \left(A _ {t} ^ {(s)} - \left(a ^ {(s)}\right) ^ {2}\right) \left(f \left(\bar {x} _ {t} ^ {(s)}\right) - f \left(x ^ {*}\right)\right) \right. \\ \left. - A _ {t - 1} ^ {(s)} \left(f (\bar {x} _ {t - 1} ^ {(s)}) - f (x ^ {*})\right) \right] \\ \leq \mathbb {E} \left[ a ^ {(s)} \underbrace {\left\langle g _ {t} ^ {(s)} , x _ {t} ^ {(s)} - x ^ {*} \right\rangle} _ {\text {s t o c h a s t i c r e g r e t}} \right] \\ + \mathbb {E} \left[ \left(a ^ {(s)}\right) ^ {2} \left\langle \nabla f (\bar {x} _ {t} ^ {(s)}), u ^ {(s - 1)} - \bar {x} _ {t} ^ {(s)} \right\rangle \right] \\ - \mathbb {E} \left[ \frac {A _ {t - 1} ^ {(s)}}{2 \beta} \left\| \nabla f (\overline {{x}} _ {t} ^ {(s)}) - \nabla f (\overline {{x}} _ {t - 1} ^ {(s)}) \right\| ^ {2} \right]. \\ \end{array}
$$
By building on the standard analysis of the stochastic regret for extra-gradient methods, we obtain the following result
for the progress of one iteration:
$$
\begin{array}{l} \mathbb {E} \left[ \left(A _ {t} ^ {(s)} - \left(a ^ {(s)}\right) ^ {2}\right) \left(f \left(\bar {x} _ {t} ^ {(s)}\right) - f \left(x ^ {*}\right)\right) \right. \\ \left. - A _ {t - 1} ^ {(s)} \left(f (\bar {x} _ {t - 1} ^ {(s)}) - f (x ^ {*})\right) \right] \\ \leq \mathbb {E} \left[ \frac {\gamma_ {t - 1} ^ {(s)}}{2} \left\| z _ {t - 1} ^ {(s)} - x ^ {*} \right\| ^ {2} - \frac {\gamma_ {t} ^ {(s)}}{2} \left\| z _ {t} ^ {(s)} - x ^ {*} \right\| ^ {2} \right] \\ + \mathbb {E} \left[ \frac {\gamma_ {t} ^ {(s)} - \gamma_ {t - 1} ^ {(s)}}{2} \left\| x _ {t} ^ {(s)} - x ^ {*} \right\| ^ {2} \right] \\ + \mathbb {E} \left[ \left(a ^ {(s)}\right) ^ {2} \left\langle \nabla f (\bar {x} _ {t} ^ {(s)}), u ^ {(s - 1)} - \bar {x} _ {t} ^ {(s)} \right\rangle \right] \\ + \mathbb {E} \left[ \frac {\left(a ^ {(s)}\right) ^ {2}}{2 \gamma_ {t} ^ {(s)}} \left\| g _ {t} ^ {(s)} - g _ {t - 1} ^ {(s)} \right\| ^ {2} \right] \\ - \underbrace {\mathbb {E} \left[ \frac {A _ {t - 1} ^ {(s)}}{2 \beta} \left\| \nabla f \left(\bar {x} _ {t} ^ {(s)}\right) - \nabla f \left(\bar {x} _ {t - 1} ^ {(s)}\right) \right\| ^ {2} \right]} _ {\text {g a i n}}. \tag {2} \\ \end{array}
$$
In comparison to the standard analysis, the coefficient for the checkpoint appears in the coefficient of $f(\overline{x}_t^{(s)}) - f(x^*)$ , which becomes $\left(A_t^{(s)} - (a^{(s)})^2\right)$ instead of the usual $A_{t}^{(s)}$ , making the sum not telescope immediately. To resolve this, we first turn our attention to the analysis of the stochastic gradient difference $\left\| g_t^{(s)} - g_{t - 1}^{(s)}\right\|^2$ .
The key idea is to split $\frac{\left(a^{(s)}\right)^2}{2\gamma_t^{(s)}}\left\| g_t^{(s)} - g_{t - 1}^{(s)}\right\|^2$
into $\left(\frac{1}{2\gamma_t^{(s)}} -\frac{1}{16\beta}\right)\left(a^{(s)}\right)^2\left\| g_t^{(s)} - g_{t - 1}^{(s)}\right\| ^2 +$
$\frac{\left(a^{(s)}\right)^2}{16\beta}\left\| g_t^{(s)} - g_{t - 1}^{(s)}\right\|^2$ and bound each term in turn. For the first term, we build on the techniques from prior work in the batch/full-gradient setting (Ene & Nguyen, 2021) when taking the sum over the iterations and epochs. For intuition, part of the gain $-\frac{1}{16\beta}\left(a^{(s)}\right)^2\left\| g_t^{(s)} - g_{t - 1}^{(s)}\right\|^2$ will be used to cancel out the term $\mathbb{E}\left[\frac{\gamma_{t}^{(s)} - \gamma_{t - 1}^{(s)}}{2}\left\| x_{t}^{(s)} - x^{*}\right\|^2\right]$ . We then only need to consider the time before $\gamma_{t}^{(s)}$ goes above $\mathcal{O}(\beta)$ , and notice that $\left(a^{(s)}\right)^2\left\| g_t^{(s)} - g_{t - 1}^{(s)}\right\|^2 = \left(\gamma_t^{(s)}\right)^2 -\left(\gamma_{t - 1}^{(s)}\right)^2$ , thus we can upperbound the first term via the last $\gamma_{t}^{(s)}$ that is still small than $\mathcal{O}(\beta)$ . We provide more details below.
For the second term, we use Young's inequality to write $\mathbb{E}\left[\left\| g_t^{(s)} - g_{t - 1}^{(s)}\right\|^2\right] \leq$
$$
\mathbb {E} \left[ 4 \left\| \nabla f (\overline {{x}} _ {t} ^ {(s)}) - g _ {t} ^ {(s)} \right\| ^ {2} + 4 \left\| \overline {{\nabla f (\overline {{x}} _ {t - 1} ^ {(s)}) - g _ {t - 1} ^ {(s)}} \right\| ^ {2} \right] \quad +
$$
$\mathbb{E}\left[2\left\| \nabla f(\overline{x}_t^{(s)}) - \nabla f(\overline{x}_{t - 1}^{(s)})\right\| ^2\right]$ . The gradient difference loss term is cancelled by the gain term in (2), and thus we can focus on the first two variance terms. We apply the usual variance reduction technique put forward by (Lan et al., 2019) (see Lemma A.2) to bound the two variance terms, as follows:
$$
\begin{array}{l} \mathbb {E} \left[ \left\| g _ {t} ^ {(s)} - \nabla f (\bar {x} _ {t} ^ {(s)}) \right\| ^ {2} \right] \leq \mathbb {E} \left[ 2 \beta (f (u ^ {(s - 1)}) - f (\bar {x} _ {t} ^ {(s)}) \right. \\ \left. \left. - \left\langle \nabla f \left(\bar {x} _ {t} ^ {(s)}\right), u ^ {(s - 1)} - \bar {x} _ {t} ^ {(s)} \right\rangle\right) \right]. \\ \end{array}
$$
Thus we obtain an upper bound on $\frac{\left(a^{(s)}\right)^2}{16\beta}\left\|g_t^{(s)}-g_{t-1}^{(s)}\right\|^2$ in terms of $\left(a^{(s)}\right)^2\left(f(u^{(s-1)})-f(\overline{x}_t^{(s)})\right)$ . This is the reason for setting the coefficient for the checkpoint to $\left(a^{(s)}\right)^2$ , so that the LHS of (2) can become the usual telescoping sum $A_t^{(s)}\left(f(\overline{x}_t^{(s)})-f(x^*)\right)-A_{t-1}^{(s)}\left(f(\overline{x}_{t-1}^{(s)})-f(x^*)\right)$ . Using the convexity of $f$ , we obtain the following key result for the progress of each epoch:
$$
\begin{array}{l} \mathbb {E} \left[ A _ {T _ {s}} ^ {(s)} \left(f (\bar {x} _ {T _ {s}} ^ {(s)}) - f (x ^ {*})\right) - A _ {0} ^ {(s)} \left(f (\bar {x} _ {0} ^ {(s)}) - f (x ^ {*})\right) \right] \\ \leq \mathbb {E} \left[ \frac {\gamma_ {0} ^ {(s)}}{2} \left\| z _ {0} ^ {(s)} - x ^ {*} \right\| ^ {2} - \frac {\gamma_ {T _ {s}} ^ {(s)}}{2} \left\| z _ {T _ {s}} ^ {(s)} - x ^ {*} \right\| ^ {2} \right] \\ + \mathbb {E} \left[ \sum_ {t = 1} ^ {T _ {s}} \frac {\gamma_ {t} ^ {(s)} - \gamma_ {t - 1} ^ {(s)}}{2} \left\| x _ {t} ^ {(s)} - x ^ {*} \right\| ^ {2} \right] \\ + \mathbb {E} \left[ T _ {s} \left(a ^ {(s)}\right) ^ {2} \left(f \left(u ^ {(s - 1)}\right) - f \left(x ^ {*}\right)\right) \right] \\ + \mathbb {E} \left[ \sum_ {t = 1} ^ {T _ {s}} \left(\frac {1}{2 \gamma_ {t} ^ {(s)}} - \frac {1}{1 6 \beta}\right) \left(a ^ {(s)}\right) ^ {2} \left\| g _ {t} ^ {(s)} - g _ {t - 1} ^ {(s)} \right\| ^ {2} \right]. \\ \end{array}
$$
Intuitively, we want to have another telescoping sum when summing up the above inequality across all epochs $s$ . To do so, we can set the starting points of the next epoch to be the ending points of the previous one, i.e., $\overline{x}_{T_s}^{(s)} = \overline{x}_0^{(s + 1)} = u^{(s)}, \gamma_{T_s}^{(s)} = \gamma_0^{(s + 1)}, z_{T_s}^{(s)} = z_0^{(s + 1)}$ . However, an extra term $T_s(a^{(s)})^2(f(u^{(s - 1)}) - f(x^*))$ appears on the RHS. We need to reset the new starting coefficient in the new epoch $A_0^{(s)}$ to $A_{T_{s - 1}}^{(s - 1)} - T_s(a^{(s)})^2$ so that we can telescope the LHS.
To bound the term $\begin{array}{l} \sum_{s = 1}^{S}\sum_{t = 1}^{T_s}\frac{\gamma_t^{(s)} - \gamma_{t - 1}^{(s)}}{2}\left\| x_t^{(s)} - x^*\right\| ^2 +\\ \left(\frac{1}{2\gamma_t^{(s)}} -\frac{1}{16\beta}\right)\left(a^{(s)}\right)^2\left\| g_t^{(s)} - g_{t - 1}^{(s)}\right\| ^2, \end{array}$ since
$\gamma_{T_s}^{(s)} = \gamma_0^{(s + 1)}$ and, the sequence $\left(\gamma_t^{(s)}\right)$ is not decreasing, we can make the first part of the sum telescope: $\sum_{s = 1}^{S}\sum_{t = 1}^{T_s}\frac{\gamma_t^{(s)} - \gamma_{t - 1}^{(s)}}{2}\left\| x_t^{(s)} - x^*\right\|^2\leq$
$$
\sum_ {s = 1} ^ {S} \sum_ {t = 1} ^ {T _ {s}} \frac {\gamma_ {t} ^ {(s)} - \gamma_ {t - 1} ^ {(s)}}{2} D ^ {2} = \frac {D ^ {2}}{2} \left(\gamma_ {T _ {S}} ^ {(S)} - \gamma_ {0} ^ {(1)}\right). \quad \text {M o r e -}
$$
over, $g_{T_s}^{(s)} = g_0^{(s + 1)}$ , we can consider the doubly indexed sequences $\left(\gamma_t^{(s)}\right)$ and $\left(g_t^{(s)}\right)$ as two singly indexed sequences $(\gamma_k)$ and $(g_k)$ and the coefficient $a^{(s)}$ to be another sequence $(a_k)$ . Then we can employ the following two inequalities:
$$
\begin{array}{l} \frac {D ^ {2}}{2} \left(\gamma_ {K} - \gamma_ {0}\right) - \frac {1}{4 8 \beta} \sum_ {k = 1} ^ {K} a _ {k} ^ {2} \| g _ {k} - g _ {k - 1} \| ^ {2} \\ \leq 1 2 \beta D ^ {2} \\ \sum_ {k = 1} ^ {K} \left(\frac {1}{2 \gamma_ {k}} - \frac {1}{2 4 \beta}\right) a _ {k} ^ {2} \| g _ {k} - g _ {k - 1} \| ^ {2} \\ \leq 1 2 \beta D ^ {2} \\ \end{array}
$$
Finally, we need to choose the parameters $a^{(s)}$ so that the conditions needed for our analysis are satisfied and $A_{T_s}^{(s)}$ is sufficiently large, so that we attain a fast convergence. We have to choose $a^{(s)}$ such that $\left(a^{(s)}\right)^2 \leq 4A_{t-1}^{(s)}$ for all $s, t \geq 1$ and that $A_0^{(s)} = A_{T_{s-1}}^{(s-1)} - T_s\left(a^{(s)}\right)^2 \geq 0$ . The main idea is to divide the epochs into two phases: in the first phase, $A_{T_s}^{(s)}$ quickly rises to $\Omega(n)$ and in the second phase, to achieve the optimal $\sqrt{\frac{n\beta}{\epsilon}}$ rate, $A_{T_s}^{(s)} = \Omega(n^2)$ . The nearly-optimal choice of $a^{(s)}$ in the first phase is $(4n)^{-0.5^s}$ , stopping at $s = s_0 = \lceil \log_2 \log_2 4n \rceil$ , while in the second phase, we have to be more conservative and choose $a^{(s)} = \frac{s - s_0 + \frac{1}{2}}{3}$ . With this we can obtain the convergence rate of $\mathcal{O}\left(n \min \left\{\log \log \frac{\beta}{\epsilon}, \log \log n\right\} + \sqrt{\frac{n\beta}{\epsilon}}\right)$ .
# 3. Experiments
In this section we demonstrate the performances of AdaVRAG and AdaVRAE in comparison with the existing standard and adaptive VR methods. We use the experimental setup and the code base of (Dubois-Taine et al., 2021)