SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Sayantan Choudhury
AMS & MINDS
Nicolas Loizou
Single-call stochastic extragradient methods, like stochastic past extragradient (SPEG) and stochastic optimistic gradient (SOG), have gained a lot of interest in recent years and are one of the most efficient algorithms for solving large-scale min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. However, despite their undoubted popularity, current convergence analyses of SPEG and SOG require strong assumptions like bounded variance or growth conditions. In addition, several important questions regarding the convergence properties of these methods are still open, including mini-batching, efficient step-size selection, and convergence guarantees under different sampling strategies. In this work, we address these questions and provide convergence guarantees for two large classes of structured non-monotone VIPs: (i) quasi-strongly monotone problems (a generalization of strongly monotone problems) and (ii) weak Minty variational inequalities (a generalization of monotone and Minty VIPs). We introduce the expected residual condition, explain its benefits, and show how it allows us to obtain a strictly weaker bound than previously used growth conditions, expected co-coercivity, or bounded variance assumptions. Finally, our convergence analysis holds under the arbitrary sampling paradigm, which includes importance sampling and various mini-batching strategies as special cases.
Differentiable game formulations where several parameterized models/players compete to minimize their respective objective functions have recently gained much attention from the machine learning community. Some landmark advances in machine learning that are framed as games (or in their simplified form as min-max optimization problems) are Generative Adversarial Networks (GANs) [19, 2], adversarial training of neural networks [46, 72], reinforcement learning [9, 64], and distributionally robust learning [51, 73].
In this work, we consider a more abstract formulation of the problem and focus on solving the following unconstrained stochastic variational inequality problem (VIP):
$$
\text {F i n d} x ^ {*} \in \mathbb {R} ^ {d}: \text {s u c h t h a t} F \left(x ^ {*}\right) = \frac {1}{n} \sum_ {i = 1} ^ {n} F _ {i} \left(x ^ {*}\right) = 0 \tag {1}
$$
where each  $F_{i}:\mathbb{R}^{d}\to \mathbb{R}^{d}$  is a Lipschitz continuous operator. Problem (1) generalizes the solution of several types of stochastic smooth games [16, 44, 20, 7]. The simplest example is the unconstrained min-max optimization problem (also called a zero-sum game):
$$
\min  _ {x _ {1} \in \mathbb {R} ^ {d _ {1}}} \max  _ {x _ {2} \in \mathbb {R} ^ {d _ {2}}} \frac {1}{n} \sum_ {i = 1} ^ {n} g _ {i} \left(x _ {1}, x _ {2}\right), \tag {2}
$$
where each component function  $g_{i}:\mathbb{R}^{d_{1}}\times \mathbb{R}^{d_{2}}\to \mathbb{R}$  is assumed to be smooth. In this scenario, operator  $F_{i}$  of (1) represents the appropriate concatenation of the block-gradients of  $g_{i}\colon F_{i}(x):= (\nabla_{x_{1}}g_{i}(x_{1},x_{2}); - \nabla_{x_{2}}g_{i}(x_{1},x_{2}))$ , where  $x:= (x_{1};x_{2})$ . Solving (1) then amounts to finding a stationary point  $x^{*} = (x_{1}^{*};x_{2}^{*})$  for (2), which under a convex-concavity assumption for  $g_{i}$ , implies that it is a global solution for the min-max problem.
However, in modern machine learning applications, game-theoretical formulations that are special cases of problem (1) are rarely monotone. That is, the min-max optimization problem (2) does not satisfy the popular and well-studied convex-concave setting. For this reason, the ML community started focusing on non-monotone problems with extra structural properties. In this work, we focus on such settings (structured non-monotone operators) for which we are able to provide tight convergence guarantees and avoid the standard issues (like cycling and divergence of the methods) appearing in the more general non-monotone regime. In particular, we focus on understanding and efficiently analyze the performance of single-call extragradient methods for solving (i)  $\mu$ -quasi-strongly monotone VIPs [44, 6] and (ii) weak Minty variational inequalities [14, 33].
Classes of structured non-monotone VIPs. Throughout this work we assume that operator  $F$  in (1) is  $L$ -Lipschitz i.e.  $\forall x, y \in \mathbb{R}^d$  operator  $F$  satisfy  $\| F(x) - F(y) \| \leq L \| x - y \|$ .
As we have already mentioned, in this work, we deal with two classes of structured non-monotone problems: the  $\mu$ -quasi strongly monotone VIPs and the weak Minty variational inequalities.
Definition 1.1.  $F$  is said to be  $\mu$ -quasi strongly monotone if there is  $\mu > 0$  such that:
$$
\forall x \in \mathbb {R} ^ {d} \quad \langle F (x), x - x ^ {*} \rangle \geq \mu \| x - x ^ {*} \| ^ {2}. \tag {3}
$$
Condition (3) is a relaxation of  $\mu$ -strong monotonicity, and it includes several non-monotone games as special cases [44]. Inequality (3) can be seen as an extension of the popular quasi-strong convexity assumption from optimization literature [53, 25] to the VIPs [44]. In the literature of variational inequality problems, quasi strongly monotone problems are also known as strong coherent VIPs [66] or VIPs satisfying the strong stability condition [47], or strong Minty variational inequality [14].
One of the weakest possible assumptions on the structure of non-monotone VIPs is the weak Minty variational inequality [14].
Definition 1.2. We say weak Minty Variational Inequality (MVI) holds for  $F$  if for some  $\rho >0$ :
$$
\forall x \in \mathbb {R} ^ {d} \quad \langle F (x), x - x ^ {*} \rangle \geq - \rho \| F (x) \| ^ {2}. \tag {4}
$$
To the best of our knowledge, the weak Minty variational inequality (4) as an assumption was first introduced in [14]. The more popular and extensively studied Minty variational inequality [12, 37, 38, 48] is a particular case of (4) with  $\rho = 0$ . In addition, the weak MVI condition is implied by the negative comonotonicity [4] or, equivalently, the positive cohypomonotonicity [11]. Finally, when we focus on min-max optimization problems (2), weak MVI condition (with  $\rho = 0$ ) is satisfied for several non-convex non-concave families of min-max objectives, including quasi-convex quasi-concave or star convex-star concave [20]. Extragradient-type methods for solving VIPs satisfying the weak MVI have been proposed in [14, 54] and [8].
Our main contributions are summarized below.
- Expected Residual. We propose the expected residual (ER) condition for stochastic variational inequality problems (1). We explain the benefits of ER and show how it can be used to derive an upper bound on  $\mathbb{E}\| g(x)\|^2$  (see Lemma 3.2) that it is strictly weaker than the bounded variance assumption and "growth conditions" previously used for the analysis of stochastic algorithms for solving (1). We prove that ER holds for a large class of operators, i.e., whenever  $F_{i}$  of (1) are Lipschitz continuous.
- Novel Convergence Guarantees. We prove the first convergence guarantees for SPEG (7) in the quasi-strongly monotone (3) and weak MVI (4) cases without using the bounded variance
<sup>1</sup>The computation of approximate first-order locally optimal solutions for general non-monotone problems (without extra structure) is intractable. See [13] and [14] for more details.
Table 1: Summary of known and new convergence results for versions of SEG and SPEG with constant step-sizes applied to solve quasi-strongly monotone variational inequalities and variational inequalities with operators satisfying Weak Minty condition. Columns: "Setup" = quasi-strongly monotone or Weak MVI; "No UBV?" = is the result derived without bounded variance assumption?; "Single-call" = does the method require one oracle call per iteration?; "Convergence rate" = rate of convergence neglecting numerical factors. Notation:  $K =$  number of iterations;  ${L}_{\max } = \mathop{\max }\limits_{{i \in  \left\lbrack  n\right\rbrack  }}{L}_{i}$  ,where  ${L}_{i}$  is a Lipschitz constant of  ${F}_{i};\bar{\mu } = \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}{\mu }_{i}$  ,where  ${\mu }_{i}$  is quasi-strong monotonicity constant of  ${F}_{i}$  (see details in [20]);  ${\sigma }_{\mathrm{{US}} * }^{2} = \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}{\begin{Vmatrix}{F}_{i}\left( {x}^{ * }\right) \end{Vmatrix}}^{2};\bar{L} = \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}{L}_{i};{\sigma }_{\mathrm{{IS}} * }^{2} = \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}\frac{\bar{L}}{{L}_{i}}{\begin{Vmatrix}{F}_{i}\left( {x}^{ * }\right) \end{Vmatrix}}^{2};L =$  Lipschitz constant of  $F;\mu  =$  quasi-strong monotonicity constant of  $F;\delta ,{\sigma }_{ * }^{2} =$  parameters from (8);  $\rho  =$  parameter from Weak Minty condition;  $\tau  =$  batchsize.
Setup	Method	No UBV?	Single-call?	Convergence rate
Quasi-strong mon.	S-SEG-US [20]	✓(1)	X	Lmax/μ exp(-μ/LmaxK) + σ2/μ*K
S-SEG-IS [20]	✓(1)	X	L/μ exp(-μ/LK) + σ2/μ*K
SPEG [28]	X(2)	✓	L/μ exp(-μ/LK) + σ2/μ*K (3)
SPEG (This work)	✓	✓	max{L/μ, δ/μ2} exp(-min{μ/L, μ2/δ}K) + σ2/μ*K
Weak MVI(4)	SEG+ [14]	X(2)	X	L2||x0-x*||2/K(1-8√2Lρ) + σ2/τ(1-8√2Lρ) (5)
OGDA+ [8]	X(2)	✓	||x0-x*||2/Kac(a-ρ) + σ2/τL2ac(a-ρ) (6)
SPEG (This work)	✓	✓	(1+48ωγδ/τ(1-Lγ)2)K||x0-x*||2/Kωγ(1-L(γ+4ω)) + (1+1-Lγ/K(1+48ωγδ/τ(1-Lγ)2)K)σ2/τ(1-Lγ)(1-L(γ+4ω)) (7)
(1) Quasi-strong monotonicity of all  $F_{i}$  is assumed.
(2) It is assumed that (8) holds with  $\delta = 0$
(3) [28] do not derive this result but it can be obtained from their proof using standard choice of step-sizes.
(4) All mentioned results in this case require large batchesizes  $\tau = \mathcal{O}(K)$  to get  $\mathcal{O}(1 / K)$  rate.
(5) The result is derived for  $\rho < 1 / 8\sqrt{2} L$
(6) The result is derived for  $\rho < 3 / sL$ . Here  $a$  and  $c$  are assumed to satisfy  $aL \leq \frac{7 - \sqrt{1 + 48c^2}}{8(1 + c)}$ ,  $c > 0$  and  $a > \rho$ .
(7) The result is derived for  $\rho < 1/2L$ . Here we assume that  $\max \{2\rho, 1/(2L)\} < \gamma < 1/L$  and  $0 < \omega < \min \{\gamma - 2\rho, (4 - \gamma L)/4L\}$ .
assumption. We achieve that by using the proposed (ER) condition. In particular, for the class of quasi-strongly monotone VIPs, we show a linear convergence rate to a neighborhood of  $x^{*}$  when constant step-sizes are used. We also provide theoretically motivated step-size switching rules that guarantee exact convergence of SPEG to  $x^{*}$ . In the weak MVI case, we prove the convergence of SPEG for  $\rho < 1 / 2L$ , improving the existing restrictions on  $\rho$ . We compare our results with the existing literature in Table 1.
- Arbitrary Sampling. Via a stochastic reformulation of the variational inequality problem (1) we explain how our convergence guarantees of SPEG hold under the arbitrary sampling paradigm. This allows us to cover a wide range of samplings for SPEG that were never considered in the literature before, including mini-batching, uniform sampling, and importance sampling as special cases. In this sense, our analysis of SPEG is unified for different sampling strategies. Finally, to highlight the tightness of our analysis, we show that the best-known convergence guarantees of deterministic PEG for strongly monotone and weak MVI can be obtained as special cases of our main theorems.
2 Stochastic Reformulation of VIPs & Single-Call Extragradient Methods
In this work, we provide a theoretical analysis of single-call stochastic extragradient methods that allows us to obtain convergence guarantees of any minibatch and reasonable sampling selection. We achieve that by using the recently proposed "stochastic reformulation" of the variational inequality problem (1) from [44]. That is, to allow for any form of minibatching, we use the arbitrary sampling notation
$$
g (x) = F _ {v} (x) := \frac {1}{n} \sum_ {i = 1} ^ {n} v _ {i} F _ {i} (x), \tag {5}
$$
where  $v \in \mathbb{R}_+^n$  is a random sampling vector drawn from a user-defined distribution  $\mathcal{D}$  such that  $\mathbb{E}_{\mathcal{D}}[v_i] = 1$ , for  $i = 1, \dots, n$ . In this setting, the original problem (1) can be equivalently written as,
$$
\text {F i n d} x ^ {*} \in \mathbb {R} ^ {d}: \mathbb {E} _ {\mathcal {D}} \left[ F _ {v} \left(x ^ {*}\right) := \frac {1}{n} \sum_ {i = 1} ^ {n} v _ {i} F _ {i} \left(x ^ {*}\right) \right] = 0, \tag {6}
$$
where the equivalence trivially holds since  $\mathbb{E}_{\mathcal{D}}[F_v(x)] = \frac{1}{n}\sum_{i=1}^{n}\mathbb{E}_{\mathcal{D}}[v_i]F_i(x) = F(x)$ .
In this work, we consider Stochastic Past Extragradient Method (SPEG) applied to (6):
$$
\hat {x} _ {k} = x _ {k} - \gamma_ {k} F _ {v _ {k - 1}} (\hat {x} _ {k - 1})
$$
$$
\begin{array}{l} w _ {k} \quad w _ {k} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array}
$$
where  $\hat{x}_{-1} = x_0$  and  $v^k\sim \mathcal{D}$  is sampled i.i.d at each iteration and  $\gamma_{k} > 0$  and  $\omega_{k} > 0$  are the extrapolation step-size and update step-size respectively. We note that in our convergence analysis, we allow selecting any distribution  $\mathcal{D}$  that satisfies  $\mathbb{E}_{\mathcal{D}}[v_i] = 1\forall i$ . This means that for a different selection of  $\mathcal{D}$ , (7) yields different interpretations of SPEG for solving the original problem (1).
One example of distribution  $\mathcal{D}$  is  $\tau$ -minibatch sampling, which is defined as follows.
Definition 2.1 ( $\tau$ -Minibatch sampling). Let  $\tau \in [n]$ . We say that  $v \in \mathbb{R}^n$  is a  $\tau$ -minibatch sampling if for every subset  $S \in [n]$  with  $|S| = \tau$ , we have that  $\mathbb{P}\left[v = \frac{n}{\tau} \sum_{i \in S} e_i\right] := \frac{1}{\binom{n}{\tau}} = \frac{\tau!(n - \tau)!}{n!}$ .
By using a double counting argument, one can show that if  $v$  is a  $\tau$ -minibatch sampling, it is also a valid sampling vector  $(\mathbb{E}_{\mathcal{D}}[v_i] = 1)$  [25]. We highlight that our analysis holds for every form of minibatching and for several choices of sampling vectors  $v$ . Later in Section 5, we provide more details related to non-uniform sampling. In addition, by Definition 2.1, it is clear that if  $\tau = n$ , then  $v_i = 1$  for all  $i \in [n]$ . Later in Section 4, we prove how our analysis captures the deterministic Past Extragradient Method as a special case.
In [44], an analysis of stochastic gradient descent-ascent  $(x_{k + 1} = x_k - \omega_k F_{v_k}(x_k))$  under the arbitrary sampling paradigm was proposed for solving star-co-coercive VIPs. Later [20], extended this approach and provided general convergence guarantees for stochastic extragradian method (SEG) (a stochastic variant of the popular extragradian method [32, 30]) for solving quasi-strongly monotone and monotone VIPs. Despite its popularity, SEG requires two oracle calls per iteration which makes it prohibitively expensive in many large-scale applications and not easily applicable to the online learning problems [18]. This motivates us to explore in detail the convergence guarantees of single-call variants of extragradian methods (extragradian methods that require only a single oracle call per iteration).
On Single-Call Extragradient Methods. The seminal work of [56] is the first paper that proposes the deterministic Past Extragradient method. In the stochastic setting, [28] provides an analysis of several stochastic single-call extragradient methods for solving strongly monotone VIPs. In [28], it was also shown that in the unconstrained setting, the update rules of Past Extragradient and Optimistic Gradient are exactly equivalent (see also Proposition B.6 in appendix). Through this connection, and via our stochastic reformulation (6) our theoretical results hold also for the Stochastic Optimistic Gradient Method (SOG):  $x_{k + 1} = x_k - \omega_k F_{v_k}(x_k) - \gamma_k (F_{v_k}(x_k) - F_{v_{k - 1}}(x_{k - 1}))$ . [8] provides the convergence guarantees of SOG for weak MVI. To the best of our knowledge, our work is the first that provides convergence guarantees for SOG under the arbitrary sampling paradigm (captures sampling beyond uniform sampling) and also without using the bounded variance assumption.
3 Expected Residual
In our theoretical results, we rely on Expected Residual (ER) condition. In this section, we define ER and explain how it is connected with similar conditions used in optimization literature. We further provide sufficient conditions for ER to hold and prove how it can be used to obtain a strictly weaker upper bound of  $\mathbb{E}\| g(x)\|^2$  than previously used growth conditions, expected co-coercivity, or bounded variance assumptions.
Assumption 3.1. We say the Expected Residual (ER) condition holds if there is a parameter  $\delta >0$  such that for an unbiased estimator  $g(x)$  of the operator  $F$ , we have
$$
\mathbb {E} \left[ \| (g (x) - g (x ^ {*})) - (F (x) - F (x ^ {*})) \| ^ {2} \right] \leq \frac {\delta}{2} \| x - x ^ {*} \| ^ {2}. \tag {ER}
$$
The ER condition bounds how far the stochastic estimator  $g(x) = F_v(x)$  (5) used in SPEG is from the true operator  $F(x)$ . ER depends on both the properties of the operator  $F(x)$  and of the selection of sampling (via  $g(x)$ ). Conditions similar to ER appeared before in optimization literature but they
have never been used in operator theory and the analysis of SPEG. In particular, [24] used a similar condition for analyzing SGD in stochastic optimization problems but with the right-hand side of ER to be the function suboptimality  $f(x) - f(x^{*})$  (such concept is not available in VIPs). In [68] and [22], similar conditions appear under the name "Hessian variance" assumption for distributed minimization problems. In the context of distributed VIPs, a similar but stronger condition to ER is used by [5].
Bound on Operator Noise. A common approach for proving the convergence of stochastic algorithms for solving the VIPs is assuming uniform boundedness of the stochastic operator or uniform boundedness of the variance. However, as we explain below, these assumptions either do not hold or are true only for a restrictive set of problems. In our work, we do not assume such bounds. Instead, we use the following direct consequence of ER.
Lemma 3.2. Let  $\sigma_{*}^{2} \coloneqq \mathbb{E}\|g(x^{*})\|^{2} < \infty$  (operator noise at the optimum is finite). If ER holds, then
$$
\mathbb {E} \| g (x) \| ^ {2} \leq \delta \| x - x ^ {*} \| ^ {2} + \| F (x) \| ^ {2} + 2 \sigma_ {*} ^ {2}. \tag {8}
$$
Sufficient Conditions for ER. Let us now provide sufficient conditions which guarantee that the ER condition holds and give a closed-form expression for the expected residual parameter  $\delta$  and  $\sigma_{*}^{2} = \mathbb{E}\| g(x^{*})\|^{2}$  for the case of  $\tau$ -minibatch sampling (Def. 2.1).
Proposition 3.3. Let  $F_{i}$  of problem (1) be  $L_{i}$ -Lipschitz operators, then ER holds. If, in addition, vector  $\boldsymbol{v} \in \mathbb{R}^n$  is a  $\tau$ -minibatch sampling (Def. 2.1) then:  $\delta = \frac{2}{n\tau}\frac{n - \tau}{n - 1}\sum_{i = 1}^{n}L_{i}^{2}$ , and  $\sigma_{*}^{2} = \frac{1}{n\tau}\frac{n - \tau}{n - 1}\sum_{i = 1}^{n}\| F_{i}(x^{*})\|^{2}$ .
Similar results to Prop. 3.3 but under different sufficient conditions have been obtained for  $\tau-$  minibatch sampling under expected smoothness and a variant of expected residual for solving minimization problems in [25] and [24] respectively. In [44], a similar proposition was derived but for the much more restrictive class of co-coercive operators.
Connection to Other Assumptions. In the proofs of our convergence results, we use the bound (8), which, as we explained above, is a direct consequence of ER. In this paragraph, we place this bound in a hierarchy of common assumptions used for the analysis of stochastic algorithms for solving VIPs. In the literature on stochastic algorithms for solving the VIPs and min-max optimization problems, previous works assume either bounded operator  $(\mathbb{E}\| g(x)\| ^2\leq c)$  [1, 52], bounded variance  $(\mathbb{E}\| g(x) - F(x)\| ^2\leq c)$  [35, 69, 30] (in Appendix C we provide a simple example where bounded variance assumption does not hold) or growth condition  $(\mathbb{E}\| g(x)\| ^2\leq c_1\| F(x)\| ^2 +c_2)$  [36]. In all of these conditions, the parameters  $c$ ,  $c_{1}$ , and  $c_{2}$  are usually constants that do not have a closed-form expression. The closer works to our results are [44, 6] which assumes existence of  $l_{F} > 0$  such that the expected co-coercivity condition  $(\mathbb{E}\| g(x) - g(x^{*})\|^{2}\leq l_{F}\langle F(x),x - x^{*}\rangle)$  holds. Their convergence guarantees provide an efficient analysis for several variants of SGDA for solving cocoercive VIPs. In the proposition below, we prove how these conditions are related to the bound (8) obtained using ER.
Proposition 3.4. Suppose  $F$  is a  $L$ -Lipschitz operator. Then we have the following hierarchy of assumptions:
Let us also mention that [29] provided convergence guarantee of double-oracle stochastic extragradient (SEG) method under the variance control condition  $\mathbb{E}\| g(x) - F(x)\| ^2\leq (a\| x - x^*\| +b)^2$  where  $a,b\ge 0$ . In their work, they focus on solving VIPs satisfying the error-bound condition, and they did not provide closed-form expressions of parameters  $a$  and  $b$ . Although the analysis of [29]
can be conducted with  $a > 0$ , the authors only provide rates for the case  $a = 0$ . The main difference between their results (for SEG) and our results (for SPEG) is that our bound (8) is not really an assumption, but it holds for free when  $F_{i}$  are  $L_{i}$ -Lipschitz. In addition, the values of parameters  $\delta$  and  $\sigma_{*}^{2}$  in (8) could have different values based on the sampling used in the update rule of SPEG.
In this section, we present and discuss the main convergence results of this work. In the first part, we focus on the ones derived for  $\mu$ -quasi strongly monotone problems (3) (both for constant and decreasing step-sizes), and in the second part on the Weak Minty VIP (4).
4.1 Quasi-Strongly Monotone Problems
Constant Step-size: We start with the case of  $\mu$ -quasi strongly monotone problems and consider the convergence of SPEG with constant step-size.
Theorem 4.1. Let  $F$  be  $L$ -Lipschitz,  $\mu$ -quasi strongly monotone, and let ER hold. Choose step-sizes  $\gamma_k = \omega_k = \omega$  such that
$$
0 <   \omega \leq \min  \left\{\frac {\mu}{1 8 \delta}, \frac {1}{4 L} \right\} \tag {9}
$$
for all  $k$ . Then the iterates produced by SPEG, given by (7) satisfy
$$
R _ {k} ^ {2} \leq \left(1 - \frac {\omega \mu}{2}\right) ^ {k} R _ {0} ^ {2} + \frac {2 4 \omega \sigma_ {*} ^ {2}}{\mu}, \tag {10}
$$
where  $R_{k}^{2} \coloneqq \mathbb{E}\left[\| x_{k} - x^{*}\|^{2} + \| x_{k} - \hat{x}_{k - 1}\|^{2}\right]$ . Hence, given any  $\varepsilon >0$ , and choosing  $\omega = \min \left\{\frac{\mu}{18\delta},\frac{1}{4L},\frac{\varepsilon\mu}{48\sigma_{*}^{2}}\right\}$ , SPEG achieves  $\mathbb{E}\| x_K - x^*\| ^2\leq \varepsilon$  after  $K\geq \max \left\{\frac{8L}{\mu},\frac{36\delta}{\mu^2},\frac{96\sigma_{*}^2}{\varepsilon\mu^2}\right\} \log \left(\frac{2R_0^2}{\varepsilon}\right)$  iterations.