SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\theta \simeq \frac {p \eta^ {- 1} - 1}{\beta p \eta^ {- 1} - 1}, \quad M \simeq \max \{2; \sqrt {p ^ {- 1} (1 + p / \beta)} \}, \quad B = \lceil b \log_ {2} M \rceil ,
$$
$$
\begin{array}{l} \mathbb {E} \left[ \| x ^ {N} - x ^ {*} \| ^ {2} + \frac {6}{\mu} (f (x _ {f} ^ {N}) - f (x ^ {*})) \right] \lesssim \exp \left(- N \sqrt {\frac {p ^ {2} \mu \gamma}{3}}\right) \left[ \| x ^ {0} - x ^ {*} \| ^ {2} + \frac {6}{\mu} (f (x ^ {0}) - f (x ^ {*})) \right] \\ + \frac {p \sqrt {\gamma}}{\mu^ {3 / 2}} \left(\sigma^ {2} \tau b ^ {- 1} + \sigma^ {2} \tau^ {2} b ^ {- 2}\right). (7) \\ \end{array}
$$
The proof is provided in Appendix B.3. The result of Theorem 1 can be rewritten as an upper complexity bound under an appropriate choice of the remaining free parameter  $b$ :
Corollary 1. Under the conditions of Theorem  $\square$  choosing  $b = \tau$  and  $\gamma$  as
$$
\gamma \simeq \min  \left\{\frac {1}{L}; \frac {1}{p ^ {2} \mu N ^ {2}} \ln \left(\max  \left\{2; \frac {\mu^ {2} N [ \| x ^ {0} - x ^ {*} \| ^ {2} + 6 \mu^ {- 1} (f (x _ {f} ^ {0}) - f (x ^ {*})) ]}{\sigma^ {2}} \right\}\right) \right\},
$$
in order to achieve  $\varepsilon$ -approximate solution (in terms of  $\mathbb{E}[\|x - x^{*}\|^{2}] \lesssim \varepsilon$ ) it takes
$$
\tilde {\mathcal {O}} \left(\tau \left[ (1 + \delta^ {2}) \sqrt {\frac {L}{\mu}} \log \frac {1}{\varepsilon} + \frac {\sigma^ {2}}{\mu^ {2} \varepsilon} \right]\right) \quad o r a c l e \quad c a l l s. \tag {8}
$$
The results of Corollary  $\square$  are obtained with fixed parameters of the method. In Corollary  $\square$  these parameters are selected a bit artificially, e.g., the stepsize  $\gamma$  depends on the iteration horizon  $N$ . In Appendix B.4 we show how one can similar results, but with a decreasing stepsize.
Comparison. Running the procedure above requires to know the mixing time  $\tau$ . Estimating the mixing time from a single trajectory of the running Markov chain is known to be computationally hard problem, see e.g. [101] and references therein. At the same time, methods, which share the same (optimal) linear scaling of the sample complexity w.r.t. the mixing time also share the same drawback as our method. In particular, it holds true for the EMD algorithm [23], SGD-DD algorithm [71], and usual SGD with Markovian data [28]. At the same time, in the non-convex scenario the paper [21] is truly oblivious to mixing time, allowing to obtain sample complexity rates for non-convex problems, which are homogeneous w.r.t.  $\tau$  with AdaGrad-type learning rate. An interesting direction for the future work to suggest a procedure that would allow to generalize the results of [21] to accelerated SGD setting.
It is possible that the sample complexity bound (8) is worse than the respective bounds for non-accelerated SGD with Markov data, provided that  $\delta^2$  grows quickly with  $L / \mu$ . At the same time, this drawback is shared by the classical results on learning under the strong growth condition, see e.g. [97]. As it is shown in [62], the respective rates can be worse than the ones obtained by usual SGD even under the i.i.d. noise setting, see Appendix F.3 in [62]. Making the analysis of accelerated SGD 'backward compatible' w.r.t. the rates of usual SGD requires to perform analysis in terms of additional problem-specific quantities, see [47, 62].
The closest equivalent of the result Corollary  $\square$  is given by [20 Theorem 3]. However, the corresponding bound of [20 Theorem 3] is incomplete, since the factor  $\tau^2$  is lost in the proof (see equations (64 - 66)). With this completion, the bound of [20 Theorem 3] yields a variance term of order  $\tilde{\mathcal{O}}\left(\frac{\sigma^2\tau^2}{\mu^2\varepsilon}\right)$ , which is suboptimal with respect to  $\tau$ . Moreover, the corresponding analysis relies heavily on the assumption of a bounded domain. In [28], the author considers Markovian noise with a finite number of states and manages to obtain a rather interesting result of the form  $O\left(\frac{L}{\mu}\log \frac{1}{\varepsilon} +\frac{L\tau\sigma_{*}^{2}}{\mu^{3}\varepsilon}\right)$ . Here the first term does not depend on  $\tau$ , and the second consists only  $\sigma^{*}$  (stochasticity in  $x^{*}$ ), but the price for this is an additional factor  $L / \mu$  in the second term and more strict assumption that all realizations  $F(\cdot ,z)$  are smooth and strongly convex. In the context of overparameterized learning, our results are almost consistent with the bound of [97, Theorem 1] under i.i.d. sampling. The difference is that the term  $\delta^2$  in A can be more pessimistic than the expectation bound in [97].
2.2 Lower bounds
We start with a lower bound for the complexity of Markovian stochastic optimization under the assumptions A1-A4. Below we provide a result that highlights that the bound of Theorem 1 is tight provided that  $\delta$  does not scale with the instance-dependent quantities, e.g., condition number  $L / \mu$ .
Theorem 2. There exists an instance of the optimization problem satisfying assumptions  $A \square -A \square$  with  $\delta = 1$  and arbitrary  $\sigma \geq 0, L, \mu > 0, \tau \in \mathbb{N}^*$ , such that for any first-order gradient method it takes at least
$$
N = \Omega \left(\tau \sqrt {\frac {L}{\mu}} \log \frac {1}{\varepsilon} + \frac {\tau \sigma^ {2}}{\mu^ {2} \varepsilon}\right)
$$
oracle calls in order to achieve  $\mathbb{E}[\| x^N -x^*\| ^2 ]\leq \varepsilon$
The proof is provided in Appendix B.5 The idea of the constructed lower for deterministic part bound  $\Omega\left(\tau \sqrt{\frac{L}{\mu}} \log \frac{1}{\varepsilon}\right)$  goes back to [76, Theorem 2.1.13]. The stochastic part lower bound goes back to the classical statistical reasoning, and is well explained for i.i.d. noise in [59, Chapter 4.1]. Our adaptation for Markovian setting is based on Le Cam's theory, see [105]. For the case of Markov noise this lower bound is, to the best of our knowledge, original. The closest result to ours is the stochastic term lower bound in [28, Proposition 1], but it is valid only for the vanilla stochastic gradient methods. Below we provide another lower bound showing that the dependence of the sample complexity Corollary 1 on  $\delta$  is not an artefact of the proof.
Proposition 1. There exists an instance of the optimization problem satisfying assumptions A[7]–A[4] with arbitrary  $L, \mu > 0, \tau \in \mathbb{N}^*$ ,  $\delta = \frac{L}{\mu}$ , and  $\sigma = 0$ , such that for any first-order gradient method it takes at least
$$
N = \Omega \left(\tau \frac {L}{\mu} \log \frac {1}{\varepsilon}\right)
$$
gradient calls in order to achieve  $\mathbb{E}[\| x^N -x^*\| ^2 ]\leq \varepsilon$
This lower bound is adapted from the information-theoretic lower bound [71]. The detailed proof can be found in Appendix B.5. Recent studies [54, 71, 13] have revealed the impossibility of accelerating stochastic gradient descent (SGD) for online linear regression problems with specific noise structures. To address this issue, researchers have proposed various solutions, such as the MaSS algorithm [62] and the approach presented in [48]. However, these methods rely heavily on the particular structure of the online regression setup. Another question that naturally arises is whether one can get rid of the dependence on  $\tau$  in the deterministic part of (8) if  $\delta = 0$ . The following counterexample shows that this is not the case in general.
Proposition 2. There exists an instance of the optimisation problem satisfying assumptions  $A[7] - A[4]$  with with arbitrary  $L, \mu > 0, \tau \in \mathbb{N}^*$ ,  $\sigma = 1, \delta = 0$ , such that for any first-order gradient method it takes at least
$$
N = \Omega \left(\left(\tau + \sqrt {\frac {L}{\mu}}\right) \log \frac {1}{\varepsilon}\right)
$$
The proof is provided in Appendix B.5.
2.3 Non-convex problems
Now we proceed with a randomized batch size version of the simple SGD algorithm. It is summarized in Algorithm 2 and can be shown to achieve optimal rates of convergence for smooth non-convex problems. For the case of non-convex problems with Markov noise similar analysis appeared in [21, Theorem 4].
Algorithm 2 Randomized GD
1: Parameters: stepsize  $\gamma > 0$ , number of iterations  $K$ , bound on batchsize  $B$ , mixing time  $\tau$ ;
2: Initialization: choose  $x^0\in \mathcal{X}$
3: for  $k = 0,1,2,\dots ,N - 1$  do
4: Sample  $J_{k} \sim \mathrm{Geom}\left(\frac{1}{2}\right)$
5:  $g^{k} = g_{0}^{k} + \left\{ \begin{array}{ll}2^{J_{k}}\left(g_{J_{k}}^{k} - g_{J_{k} - 1}^{k}\right), & \text{if } 2^{J_{k}} \leq M \\ 0, & \text{otherwise} \end{array} \right.$  with  $g_{j}^{k} = 2^{-j}B^{-1}\sum_{i=1}^{2^{j}B}\nabla f(x^{k},Z_{T^{k+i}})$ .
6:  $x^{k + 1} = x^k -\gamma g^k$
7:  $T^{k + 1} = T^k +2^{J_k}B$
By balancing the values of  $B$  and  $M$  with Lemma 2, we establish the following result:
Theorem 3. Assume  $A \square A \square A \square$  Let problem (1) be solved by Algorithm 2. Let  $f^{*}$  be a global (maybe not unique) minimum of  $f$ . Then for any  $b \in \mathbb{N}^*$ , and  $\gamma, M$  satisfying
$$
\gamma \lesssim (L [ 1 + \delta^ {2} \tau b ^ {- 1} + \delta^ {2} \tau^ {2} b ^ {- 2} ]) ^ {- 1}, \quad M \simeq \max  \{2; \sqrt {\gamma^ {- 1} L ^ {- 1}} \}, \quad B = \lceil b \log_ {2} M \rceil ,
$$
$$
\mathbb {E} \left[ \frac {1}{N} \sum_ {k = 0} ^ {N - 1} \| \nabla f (x ^ {k}) \| ^ {2} \right] \lesssim \frac {f (x ^ {0}) - f ^ {*}}{\gamma N} + L \gamma \cdot \left[ \sigma^ {2} \tau b ^ {- 1} + \sigma^ {2} \tau^ {2} b ^ {- 2} \right].
$$
The proof is provided in Appendix B.6. The next corollary immediately follows from the theorem.
Corollary 2. Under the conditions of Theorem 3 if we choose  $b = \tau$  and  $\gamma$  given by
$$
\gamma \simeq \min \left\{\frac {1}{L (1 + \delta^ {2})}; \sqrt {\frac {f (x ^ {0}) - f ^ {*}}{L N \sigma^ {2}}} \right\},
$$
then to achieve  $\varepsilon$ -solution (in terms of  $\mathbb{E}[\|\nabla f(x)\|^2] \lesssim \varepsilon^2$ ) we need
$$
\tilde {\mathcal {O}} \left(\tau \cdot \left[ \frac {(1 + \delta^ {2}) L (f (x ^ {0}) - f ^ {*})}{\varepsilon^ {2}} + \frac {L (f (x ^ {0}) - f ^ {*}) \sigma^ {2}}{\varepsilon^ {4}} \right]\right) \quad o r a c l e \text {c a l l s}.
$$
Comparison. The respective bound for the non-convex setting provided in [20, Theorem 1] yields the sample complexity of order  $\tilde{\mathcal{O}}\left(\frac{\tau^2L(f(x^0) - f(x^*))\sigma^2}{\varepsilon^4}\right)$ . Also we can note the results of [28, Theorem 2] with the following estimate  $O\left(\frac{\tau(L(f(x^0) - f(x^*)) + \sigma^2)}{\varepsilon^2} + \frac{\tau(L(f(x^0) - f(x^*)) + \sigma^2)\sigma^2}{\varepsilon^4}\right)$ .
To achieve linear convergence rates in the non-convex setting we can use the Polyak-Lojasiewicz (PL) condition [83]. The respective result is provided in Appendix B.7.
2.4 Variational inequalities
In this section, we are interested in the following problem:
$$
\text {F i n d} x ^ {*} \in \mathcal {X} \text {s u c h t h a t} \langle F (x ^ {*}), x - x ^ {*} \rangle + r (x) - r (x ^ {*}) \geq 0 \text {f o r a l l} x \in \mathcal {X}. \tag {9}
$$
Here  $F: \mathbb{R}^d \to \mathbb{R}^d$  is an operator,  $\mathcal{X}$  a convex set, and  $r: \mathbb{R}^d \to \mathbb{R}$  is a regularization term (a suitable lower semicontinuous convex function) which is assumed to have a simple structure. As mentioned earlier, this problem is quite general and covers a wide range of possible problem formulations. For example, if the operator  $F$  is the gradient of a convex function  $f$ , then the problem (9) is equivalent to the composite minimization problem [6], i.e., minimization of  $f(x) + r(x)$ . In the meantime, (9) is also a reformulation of the min-max problem
$$
\min  _ {x _ {1} \in \mathcal {X} _ {1}} \max  _ {x _ {2} \in \mathcal {X} _ {2}} r _ {1} \left(x _ {1}\right) + g \left(x _ {1}, x _ {2}\right) - r _ {2} \left(x _ {2}\right), \tag {10}
$$
with convex-concave continuously differentiable  $g$ , convex sets  $\mathcal{X}_1, \mathcal{X}_2$  and convex functions  $r_1, r_2$ . Using the first-order optimality conditions, it is easy to verify that (10) is equivalent to (9) with  $x = (x_1^T, x_2^T)^T$ ,  $F(x) = (\nabla_{x_1}f(x_1, x_2)^T, -\nabla_{x_2}f(x_1, x_2)^T)^T$ , and  $r(x) = r_1(x_1) + r_2(x_2)$ .
A 5. The operator  $F$  is  $L$ -Lipschitz continuous on  $\mathcal{X}$  with  $L > 0$ , i.e., the following inequality holds for all  $x, y \in \mathcal{X}$ :
$$
\| F (x) - F (y) \| \leq L \| x - y \|, \quad \forall x, y \in \mathcal {X}.
$$
A 6. The operator  $F$  is  $\mu_F$ -strongly monotone on  $\mathcal{X}$ , i.e., the following inequality holds for all  $x, y \in \mathcal{X}$ :
$$
\langle F (x) - F (y), x - y \rangle \geq \mu_ {F} \| x - y \| ^ {2}. \tag {11}
$$
The function  $r$  is  $\mu_r$ -strongly convex on  $\mathcal{X}$ , i.e. for all  $x, y \in \mathcal{X}$  and any  $r'(x) \in \partial r(x)$  we have
$$
r (y) \geq r (x) + \left\langle r ^ {\prime} (x), y - x \right\rangle + \left(\mu_ {r} / 2\right) \| x - y \| ^ {2}. \tag {12}
$$
These two assumptions are more than standard for the study of variational inequalities and are found in all the papers from Table 2. We consider two cases: strongly monotone/convex with  $\mu_F + \mu_r > 0$  and monotone/convex with  $\mu_F + \mu_r = 0$ .
A 7. For all  $x \in \mathbb{R}^d$  it holds that  $\mathbb{E}_{\pi}[F(x,Z)] = F(x)$ . Moreover, for all  $z \in \mathbb{Z}$  and  $x \in \mathcal{X}$  it holds that
$$
\left\| F (x, z) - F (x) \right\| ^ {2} \leq \sigma^ {2} + \Delta^ {2} \| x - x ^ {*} \| ^ {2}, \tag {13}
$$
where  $x^{*}$  is some point from the solution set.
A 7 is found in the literature on variational inequalities [43, 45, 38] and is considered to be analog to A 4 on overparametrized learning.
Just as the Nesterov accelerated method is optimal for smooth convex minimization problems, the ExtraGradient method [57, 72, 52] is optimal for monotone variational inequalities. Therefore, we take it as a base. On the extrapolation step (Line 4) of Algorithm 3, we simply collect a batch of size  $B$ , but on the main step (Line 8) we use the randomization as in Algorithm 1. The next theorem gives the convergence of our method.
Theorem 4. Assume  $A \boxplus A \boxminus$  with  $\mu_F + \mu_r > 0$ ,  $A \boxplus A \boxminus$  Let problem (9) be solved by Algorithm 3. Then for any  $b \in \mathbb{N}^*$ , and  $\gamma, M$  satisfying
$$
\begin{array}{l} \gamma \lesssim \min  \left\{(\mu_ {F} + \mu_ {r}) ^ {- 1}; L ^ {- 1}; (\mu_ {F} + \mu_ {r}) (\Delta^ {2} \tau b ^ {- 1} + \Delta^ {2} \tau^ {2} b ^ {- 2}) ^ {- 1}; \sqrt {\Delta^ {- 2} \tau^ {- 1} b} \right\}, \\ M \simeq \max  \{2; \sqrt {\gamma^ {- 1} \left(\mu_ {F} + \mu_ {r}\right) ^ {- 1}} \}, \quad B = \lceil b \log_ {2} M \rceil , \\ \end{array}
$$
$$
\mathbb {E} \left[ \| x ^ {N} - x ^ {*} \| ^ {2} \right] \lesssim \exp \left(- \frac {N (\mu_ {F} + \mu_ {r}) \gamma}{2}\right) \| x ^ {0} - x ^ {*} \| ^ {2} + \frac {\gamma}{\mu} \left(\sigma^ {2} \tau b ^ {- 1} + \sigma^ {2} \tau^ {2} b ^ {- 2}\right).
$$
The proof is postponed to Appendix B.8. One can get an estimate on oracle complexity.
Algorithm 3 Randomized ExtraGradient
1: Parameters: stepsize  $\gamma > 0$ , number of iterations  $N$
2: Initialization: choose  $x^0 \in \mathcal{X}$
3: for  $k = 0,1,2,\ldots ,N - 1$  do
4:  $x^{k + 1 / 2} = \mathrm{prox}_{\gamma r}\big(x^k -\gamma B^{-1}\sum_{i = 1}^B F(x^k,Z_{T^{k + i}})\big)$
5:  $T^{k + 1 / 2} = T^k + B$
6: Sample  $J_{k} \sim \mathrm{Geom}\left(\frac{1}{2}\right)$
7:  $g^{k} = g_{0}^{k} + \left\{ \begin{array}{ll}2^{J_{k}}\left(g_{J_{k}}^{k} - g_{J_{k} - 1}^{k}\right), & \text{if } 2^{J_{k}} \leq M \\ 0, & \text{otherwise} \end{array} \right.$  with  $g_{j}^{k} = 2^{-j}B^{-1}\sum_{i=1}^{2^{j}\cdot B}F(x^{k+1/2}, Z_{T^{k+1/2}+i})$
8:  $x^{k + 1} = \mathrm{prox}_{\gamma r}\big(x^k -\gamma g^k\big)$
9:  $T^{k + 1} = T^{k + 1 / 2} + 2^{J_k}B$
Corollary 3. Under the conditions of Theorem 4 if we choose  $b = \tau$  and  $\gamma$  as follows
$$
\gamma \simeq \min  \left\{\frac {1}{\mu_ {F} + \mu_ {r}}; \frac {1}{L}; \frac {\mu_ {F} + \mu_ {r}}{\Delta^ {2}}; \frac {1}{\Delta}; \frac {1}{N (\mu_ {F} + \mu_ {r})} \ln \left(\max  \left\{2; \frac {\mu N \| x ^ {0} - x ^ {*} \| ^ {2}}{\sigma^ {2}} \right\}\right) \right\},
$$
then to achieve  $\varepsilon$ -solution (in terms of  $\mathbb{E}[\|x - x^{*}\|^{2}] \lesssim \varepsilon$ ) we need
$$
\tilde {\mathcal {O}} \left(\tau \cdot \left[ \left(1 + \frac {L}{\mu_ {F} + \mu_ {r}} + \frac {\Delta}{\mu_ {F} + \mu_ {r}} + \frac {\Delta^ {2}}{(\mu_ {F} + \mu_ {r}) ^ {2}}\right) \log \frac {1}{\varepsilon} + \frac {\sigma^ {2}}{\mu^ {2} \varepsilon} \right]\right) o r a c l e c a l l s.
$$
Note that one provide an (almost) matching lower complexity bounds for variational inequalities via lower bounds for saddle point problems, which are a special case of variational inequalities. The method for obtaining lower bounds for saddle point problems is reduced to obtaining estimates for the strongly convex minimization problem (see [106, 40] for respective deterministic lower bounds), which we provide in Section 2.2. Similarly, the question of constructing a lower bound which is tight w.r.t.  $\Delta$  remains open.
For the monotone case, we use the gap function as a convergence criterion:
$$
\operatorname {G a p} (x) = \sup  _ {y \in \mathcal {X}} [ \langle F (y), x - y \rangle + r (x) - r (y) ]. \tag {14}
$$
Such a criterion is standard and classical for monotone variational inequalities [72, 52]. An important assumption for the gap function is the boundedness of the set  $\mathcal{X}$ .
A 8. The set  $\mathcal{X}$  is bounded and has a diameter  $D$ , i.e., for all  $x, y \in \mathcal{X}$ :  $\| x - y\|^2 \leq D^2$ .
A 8 can be slightly relaxed. We need to use a simple trick from [77]. In particular, we need to consider  $\mathcal{C}$  - a compact subset of  $\mathcal{X}$  and change  $\mathcal{X}$  to  $\mathcal{C}$  in (14). But such a technique is rather technical and does not change the essence. Finally, the following result holds.
Theorem 5. Assume  $A \Box, A \Box$  with  $\mu_F + \mu_r = 0$ ,  $A \Box, A \Box, A \Box$ . Let problem (9) be solved by Algorithm 3. Then for any  $B \in \mathbb{N}^*$ , and  $\gamma, M$  satisfying  $\gamma \lesssim L^{-1}$ ,  $M = \sqrt{N}$ , it holds that
$$
\mathbb {E} \left[ G a p (\bar {x} ^ {N}) \right] \lesssim \frac {D ^ {2}}{\gamma N} + \gamma (\tau B ^ {- 1} \log_ {2} N + \tau^ {2} B ^ {- 2}) (\sigma^ {2} + \Delta^ {2} D ^ {2}) w h e r e \bar {x} ^ {N} = \frac {1}{N} \sum_ {k = 0} ^ {N - 1} x ^ {k + 1 / 2}.
$$
The proof is postponed to Appendix B.9. The following corollary holds.
Corollary 4. Under the conditions of Theorem 5 if we choose  $B = \tau$  and  $\gamma$  as follows
$$
\gamma \simeq \min \left\{\frac {1}{L}; \sqrt {\frac {D ^ {2}}{(\sigma^ {2} + \Delta^ {2} D ^ {2}) N}} \right\},
$$
then to achieve  $\varepsilon$ -solution (in terms of  $\mathbb{E}[Gap(x)] \lesssim \varepsilon$ ) we need
$$
\tilde {\mathcal {O}} \left(\tau \left[ \frac {L D ^ {2}}{\varepsilon} + \frac {\sigma^ {2} D ^ {2} + \Delta^ {2} D ^ {4}}{\varepsilon^ {2}} \right]\right) \quad o r a c l e \text {c a l l s}.
$$
Comparison. These results is the first for variational inequalities with Markovian stochasticity, either in the strongly monotone or monotone cases. The only close work is [99]. The authors work with convex-concave saddle point problems and provide the following estimate on the oracle complexity  $\mathcal{O}\left(\tau^2\cdot \frac{G^4}{\varepsilon^2} +\frac{D^2}{\varepsilon^2}\right)$  (with  $G$  - the uniform bound of the operator), which is worse than ours at least in terms of  $\tau$ . Moreover, the authors consider the case of a finite Markov chain, which is a special case of our setup.
In this paper, we present a unified random batch size framework that achieves optimal finite-time performance for non-convex and strongly convex optimization problems with Markov noise, as well as for variational inequalities. Unlike existing methods, our framework relaxes the assumptions typically imposed on the domain and stochastic gradient oracle. We also provide a variety of lower bounds, which are to the best of our knowledge original in the Markov setting.
This research of A. Beznosikov has been supported by The Analytical Center for the Government of the Russian Federation (Agreement No. 70-2021-00143 dd. 01.11.2021, IGK 000000D730321P5Q0002). E. Moulines received support from the grant ANR-19-CHIA-002 SCAI and parts of his work has been done under the auspices of Lagrange Center for maths and computing.
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