amortizedimplicitdifferentiationforstochasticbileveloptimization/full.md

AMORTIZED IMPLICIT DIFFERENTIATION FOR STOCHASTIC BILEVEL OPTIMIZATION

Michael Arbel & Julien Mairal

Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France.

ABSTRACT

We study a class of algorithms for solving bilevel optimization problems in both stochastic and deterministic settings when the inner-level objective is strongly convex. Specifically, we consider algorithms based on inexact implicit differentiation and we exploit a warm-start strategy to amortize the estimation of the exact gradient. We then introduce a unified theoretical framework inspired by the study of singularly perturbed systems (Habets, 1974) to analyze such amortized algorithms. By using this framework, our analysis shows these algorithms to match the computational complexity of oracle methods that have access to an unbiased estimate of the gradient, thus outperforming many existing results for bilevel optimization. We illustrate these findings on synthetic experiments and demonstrate the efficiency of these algorithms on hyper-parameter optimization experiments involving several thousands of variables.

1 INTRODUCTION

Bilevel optimization refers to a class of algorithms for solving problems with a hierarchical structure involving two levels: an inner and an outer level. The inner-level problem seeks a solution $y^{\star}(x)$ minimizing a cost $g(x, y)$ over a set $\mathcal{V}$ given a fixed outer variable $x$ in a set $\mathcal{X}$ . The outer-level problem minimizes an objective of the form $\mathcal{L}(x) = f(x, y^{\star}(x))$ over $\mathcal{X}$ for some upper-level cost $f$ . When the solution $y^{\star}(x)$ is unique, the bilevel optimization problem takes the following form:

$$\begin{array}{cccc}& \underset{x\in \mathcal{X}}{\min }\mathcal{L}(x):=f(x,y^{\star }(x)),\text{s u c h t h a t}{y}^{\star }(x)=g(x,y)\mathrm{.}& & \text{(1)}\end{array}\backslash min\; \_\; \{x\; \backslash in\; \backslash mathcal\; \{X\}\}\; \backslash mathcal\; \{L\}\; (x)\; :=\; f\; (x,\; y\; ^\; \{\backslash star\}\; (x)),\; \backslash quad\; \backslash text\; \{s\; u\; c\; h\; t\; h\; a\; t\}\; y\; ^\; \{\backslash star\}\; (x)\; =\; \backslash underset\; \{y\; \backslash in\; \backslash mathcal\; \{Y\}\}\; \{\backslash arg\; \backslash min\; \}\; g\; (x,\; y).\; \backslash tag\; \{1\}$$

First introduced in the field of economic game theory by Stackelberg (1934) and long studied in optimization (Ye and Zhu, 1995; Ye and Ye, 1997; Ye et al., 1997), this problem has recently received increasing attention in the machine learning community (Domke, 2012; Gould et al., 2016; Liao et al., 2018; Blondel et al., 2021; Liu et al., 2021; Shaban et al., 2019; Ablin et al., 2020). Indeed, many machine learning applications can be reduced to (1) including hyper-parameter optimization (Feurer and Hutter, 2019), meta-learning (Bertinetto et al., 2018), reinforcement learning (Hong et al., 2020b; Liu et al., 2021) or dictionary learning (Mairal et al., 2011; Lecouat et al., 2020a;b).

The hierarchical nature of (1) introduces additional challenges compared to standard optimization problems, such as finding a suitable trade-off between the computational budget for approximating the inner and outer level problems (Ghadimi and Wang, 2018; Dempe and Zemkoho, 2020). These considerations are exacerbated in machine learning applications, where the costs $f$ and $g$ often come as an average of functions over a large or infinite number of data points (Franceschi et al., 2018). All these challenges highlight the need for methods that are able to control the computational costs inherent to (1) while dealing with the large-scale setting encountered in machine learning.

Gradient-based bilevel optimization methods appear to be viable approaches for solving (1) in large-scale settings (Lorraine et al., 2020). They can be divided into two categories: Iterative differentiation (ITD) and Approximate implicit differentiation (AID). ITD approaches approximate the map $y^{\star}(x)$ by a differentiable optimization algorithm $\mathcal{A}(x)$ viewed as a function of $x$ . The resulting surrogate loss $\tilde{\mathcal{L}}(x) = f(x, \mathcal{A}(x))$ is optimized instead of $\mathcal{L}(x)$ using reverse-mode automatic differentiation (see Baydin et al., 2018). AID approaches (Pedregosa, 2016) rely on an expression of the gradient $\nabla \mathcal{L}$ resulting from the implicit function theorem (Lang, 2012, Theorem 5.9). Unlike ITD, AID avoids differentiating the algorithm approximating $y^{\star}(x)$ and, instead, approximately

solves a linear system using only Hessian and Jacobian-vector products to estimate the gradient $\nabla \mathcal{L}$ (Rajeswaran et al., 2019). These methods can also rely on stochastic approximation to increase scalability (Franceschi et al., 2018; Grazzi et al., 2020; 2021).

In the context of machine-learning, Ghadimi and Wang (2018) provided one of the first comprehensive studies of the computational complexity for a class of bilevel algorithms based on AID approaches. Subsequently, Hong et al. (2020b); Ji et al. (2021); Ji and Liang (2021); Yang et al. (2021) proposed different algorithms for solving (1) and obtained improved overall complexity by achieving a better trade-off between the cost of the inner and outer level problems. Still, the question of whether these complexities can be improved by better exploiting the structure of (1) through heuristics such as warm-start remains open (Grazzi et al., 2020). Moreover, these studies proposed separate analysis of their algorithms depending on the convexity of the loss $\mathcal{L}$ and whether a stochastic or deterministic setting is considered. This points out to a lack of unified and systematic theoretical framework for analyzing bilevel problems, which is what the present work addresses.

We consider the Amortized Implicit Gradient Optimization (AmIGO) algorithm, a bilevel optimization algorithm based on Approximate Implicit Differentiation (AID) approaches that exploits a warm-start strategy when estimating the gradient of $\mathcal{L}$ . We then propose a unified theoretical framework for analyzing the convergence of AmIGO when the inner-level problem is strongly convex in both stochastic and deterministic settings. The proposed framework is inspired from the early work of Habets (1974) on singularly perturbed systems and analyzes the effect of warm start by viewing the iterates of AmIGO algorithm as a dynamical system. The evolution of such system is described by a total energy function which allows to recover the convergence rates of unbiased oracle methods which have access to an unbiased estimate of $\nabla \mathcal{L}$ (c.f. Table 1). To the best of our knowledge, this is the first time a bilevel optimization algorithm based on a warm-start strategy provably recovers the rates of unbiased oracle methods across a wide range of settings including the stochastic ones.

2 RELATED WORK

Singularly perturbed systems (SPS) are continuous-time deterministic dynamical systems of coupled variables $(x(t), y(t))$ with two time-scales where $y(t)$ evolves much faster than $x(t)$ . As such, they exhibit a hierarchical structure similar to (1). The early work of Habets (1974); Saberi and Khalil (1984) provided convergence rates for SPS towards equilibria by studying the evolution of a single scalar energy function summarizing these systems. The present work takes inspiration from these works to analyze the convergence of AmIGO which involves three time-scales.

Two time-scale Stochastic Approximation (TTSA) can be viewed as a discrete-time stochastic version of SPS. (Kaledin et al., 2020) showed that TTSA achieves a finite-time complexity of $O\left(\epsilon^{-1}\right)$ for linear systems while Doan (2020) obtained a complexity of $O\left(\epsilon^{-3/2}\right)$ for general non-linear systems by extending the analysis for SPS. Hong et al. (2020b) further adapted the non-linear TTSA for solving (1). In the present work, we obtain faster rates by taking into account the dynamics of a third variable $z_k$ appearing in AmIGO, thus resulting in a three time-scale dynamics.

Warm-start in bilevel optimization. Ji et al. (2021); Ji and Liang (2021) used a warm-start for the inner-level algorithm to obtain an improved computational complexity over algorithms without warm-start. In the deterministic non-convex setting, Ji et al. (2021) used a warm-start strategy when solving the linear system appearing in AID approaches to obtain improved convergence rates. However, it remained open whether using a warm-start when solving both inner-level problem and linear system arising in AID approaches can yield faster algorithms in the more challenging stochastic setting (Grazzi et al., 2020). In the present work, we provide a positive answer to this question.

3 AMORTIZED IMPLICIT GRADIENT OPTIMIZATION

3.1 GENERAL SETTING AND MAIN ASSUMPTIONS

Notations. In all what follows, $\mathcal{X}$ and $\mathcal{Y}$ are Euclidean spaces. For a differentiable function $h(x,y)$ : $\mathcal{X} \times \mathcal{Y} \to \mathbb{R}$ , we denote by $\nabla h$ its gradient w.r.t. $(x,y)$ , by $\partial_x h$ and $\partial_y h$ its partial derivatives w.r.t. $x$ and $y$ and by $\partial_{xy} h$ and $\partial_{yy} h$ the partial derivatives of $\partial_y h$ w.r.t $x$ and $y$ , respectively.

Geometries	Setting	Algorithms	Complexity
(SC)	(D)	BA (Ghadimi and Wang, 2018)	O(κL2 ∨ κg2 log2 ε-1)
		AccBio (Ji and Liang, 2021)	O(κL1/2 κg1/2 log2 ε-1)
		AmIGO (Corollary 1)	O(κLκg log ε-1)
	(S)	BSA (Ghadimi and Wang, 2018)	O(κL4 ε-2)
		TTSA (Hong et al., 2020b)	O(κL0.5(κg8.5 + κg3)ε-3/2 log ε-1)
		AmIGO (Corollary 2)	O(κL2 κg3 ε-1 log ε-1)
(NC)	(D)	BA (Ghadimi and Wang, 2018)	O(κg5 ε-5/4)
		AID-BiO (Ji et al., 2021)	O(κg4 ε-1)
		AmIGO (Corollary 3)	O(κg4 ε-1)
	(S)	BSA (Ghadimi and Wang, 2018)	O(κg9 ε-3 + κg6 ε-2)
		TTSA (Hong et al., 2020b)	O(κg16 ε-5/2 log ε-1)
		stocBiO (Ji et al., 2021)	O(κg9 ε-2 + κg6 ε-2 log ε-1)
		MRBO/VRBO* (Yang et al., 2021)	O(poly(κg) ε-3/2 log ε-1)
		AmIGO (Corollary 4)	O(κg9 ε-2)

Table 1: Cost of finding an $\epsilon$ -accurate solution as measured by $\mathbb{E}[\mathcal{L}(x_k) - \mathcal{L}^\star] \wedge 2^{-1}\mu \mathbb{E}\left[| x_k - x^\star|^2\right]$ when $\mathcal{L}$ is $\mu$ -strongly-convex (SC) and $\frac{1}{k}\sum_{i=1}^{k}\mathbb{E}\left[|\nabla\mathcal{L}(x_i)|^2\right]$ when $\mathcal{L}$ is non-convex (NC). The settings (D) and (S) stand for the deterministic and stochastic settings. The cost corresponds to the total number of gradients, Jacobian and Hessian-vector products used by the algorithm. $\kappa_L$ and $\kappa_g$ are the conditioning numbers of $\mathcal{L}$ and $g$ whenever applicable. The dependence on $\kappa_L$ and $\kappa_g$ for TTSA and AccBio are derived in Proposition 11 of Appendix A.4. The rate of MRBO/VRBO is obtained under the additional mean-squared smoothness assumption (Arjevani et al., 2019).

To ensure that (1) is well-defined, we consider the setting where the inner-level problem is strongly convex so that the solution $y^{\star}(x)$ is unique as stated by the following assumption:

Assumption 1. For any $x \in \mathcal{X}$ , the function $y \mapsto g(x, y)$ is $L_g$ -smooth and $\mu_g$ -strongly convex.

Assumption 1 holds in the context of hyper-parameter selection when the inner-level is a kernel regression problem (Franceschi et al., 2018), or when the variable $y$ represents the last linear layer of a neural network as in many meta-learning tasks (Ji et al., 2021). Under Assumption 1 and additional smoothness assumptions on $f$ and $g$ , the next proposition shows that $\mathcal{L}$ is differentiable:

Proposition 1. Let $g$ be a twice differentiable function satisfying Assumption 1. Assume that $f$ is differentiable and consider the quadratic problem:

$$\begin{array}{cccc}& \underset{z\in {\mathbb{R}}^{d_y}}{\min }Q(x,y,z):=\frac{1}{2}{z}^{\mathrm{\top }}({\mathrm{\partial }}_{yy}g(x,y))z+z^{\mathrm{\top }}{\mathrm{\partial }}_{y}f(x,y)\mathrm{.}& & \text{(2)}\end{array}\backslash min\; \_\; \{z\; \backslash in\; \backslash mathbb\; \{R\}\; ^\; \{d\; \_\; \{y\}\}\}\; Q\; (x,\; y,\; z)\; :=\; \backslash frac\; \{1\}\{2\}\; z\; ^\; \{\backslash top\}\; \backslash left(\backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\backslash right)\; z\; +\; z\; ^\; \{\backslash top\}\; \backslash partial\_\; \{y\}\; f\; (x,\; y).\; \backslash tag\; \{2\}$$

Then, (2) admits a unique minimizer $z^{\star}(x,y)$ for any $(x,y)$ in $\mathcal{X}\times \mathcal{Y}$ . Moreover, $y^{\star}(x)$ is unique and well-defined for any $x$ in $\mathcal{X}$ and $\mathcal{L}$ is differentiable with gradient given by:

$$\begin{array}{cccc}& \mathrm{\nabla }\mathcal{L}(x)={\mathrm{\partial }}_{x}f(x,y^{\star }(x))+{\mathrm{\partial }}_{xy}g(x,y^{\star }(x))z^{\star }(x,y^{\star }(x))\mathrm{.}& & \text{(3)}\end{array}\backslash nabla\; \backslash mathcal\; \{L\}\; (x)\; =\; \backslash partial\_\; \{x\}\; f\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; +\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; z\; ^\; \{\backslash star\}\; (x,\; y\; ^\; \{\backslash star\}\; (x)).\; \backslash tag\; \{3\}$$

Proposition 1 follows by application of the implicit function theorem (Lang, 2012, Theorem 5.9) and provides an expression for $\nabla \mathcal{L}$ solely in terms of partial derivatives of $f$ and $g$ evaluated at $(x,y^{\star}(x))$ . Following Ghadimi and Wang (2018), we further make two smoothness assumptions on $f$ and $g$ :

Assumption 2. There exist positive constants $L_{f}$ and $B$ such that for all $x, x' \in \mathcal{X}$ and $y, y' \in \mathcal{Y}$ :

$$\mathrm{\parallel }\mathrm{\nabla }f(x,y)-\mathrm{\nabla }f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\parallel }\le L_f\mathrm{\parallel }(x,y)-(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\parallel },\mathrm{\parallel }{\mathrm{\partial }}_{y}f(x,y)\mathrm{\parallel }\le B\mathrm{.}\backslash |\; \backslash nabla\; f\; (x,\; y)\; -\; \backslash nabla\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash |\; \backslash leq\; L\; \_\; \{f\}\; \backslash |\; (x,\; y)\; -\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash |,\; \backslash quad\; \backslash |\; \backslash partial\_\; \{y\}\; f\; (x,\; y)\; \backslash |\; \backslash leq\; B.$$

Assumption 3. There exit positive constants $L_{g}^{\prime}, M_{g}$ such that for any $x, x^{\prime} \in \mathcal{X}$ and $y, y^{\prime} \in \mathcal{Y}$ :

$$\begin{array}{c}\max \{\mathrm{\parallel }{\mathrm{\partial }}_{xy}g(x,y)-{\mathrm{\partial }}_{xy}g(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\parallel },\mathrm{\parallel }{\mathrm{\partial }}_{yy}g(x,y)-{\mathrm{\partial }}_{yy}g(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\parallel }\}\le M_g\mathrm{\parallel }(x,y)-(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\parallel }\\ \mathrm{\parallel }{\mathrm{\partial }}_{y}g(x,y)-{\mathrm{\partial }}_{y}g(x^{\mathrm{\prime }},y)\mathrm{\parallel }\le L_g^{\mathrm{\prime }}\mathrm{\parallel }x-x^{\mathrm{\prime }}\mathrm{\parallel }\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash max\; \backslash left\backslash \{\backslash |\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y)\; -\; \backslash partial\_\; \{x\; y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash |,\; \backslash |\; \backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\; -\; \backslash partial\_\; \{y\; y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash |\; \backslash right\backslash \}\; \backslash leq\; M\; \_\; \{g\}\; \backslash |\; (x,\; y)\; -\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash |\; \backslash \backslash \; \backslash |\; \backslash partial\_\; \{y\}\; g\; (x,\; y)\; -\; \backslash partial\_\; \{y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\backslash right)\; \backslash |\; \backslash leq\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash |.\; \backslash \backslash \; \backslash end\{array\}$$

Assumptions 1 to 3 allow a control of the variations of $y^{\star}$ and $z^{\star}$ and ensure $\mathcal{L}$ is $L$ -smooth for some positive constant $L$ as shown in Proposition 6 of Appendix B.2. As an $L$ -smooth function, $\mathcal{L}$ is necessarily weakly convex (Davis et al., 2018), meaning that $\mathcal{L}$ satisfies the inequality $\mathcal{L}(x) - \mathcal{L}(y) \leq \nabla \mathcal{L}(x)^{\top}(x - y) - \frac{\mu}{2}| x - y|^{2}$ for some fixed $\mu \in \mathbb{R}$ with $|\mu| \leq L$ . In particular, $\mathcal{L}$ is convex when $\mu \geq 0$ , strongly convex when $\mu > 0$ and generally non-convex when $\mu < 0$ . We thus consider two cases for $\mathcal{L}$ , the strongly convex case $(\mu > 0)$ and the non-convex case $(\mu < 0)$ . When $\mathcal{L}$ is convex, we denote by $\mathcal{L}^{\star}$ its minimum value achieved at a point $x^{\star}$ and define $\kappa_{\mathcal{L}} = L / \mu$ when $\mu > 0$ .

Stochastic/deterministic settings. We consider the general setting where $f(x,y)$ and $g(x,y)$ are expressed as an expectation of stochastic functions $\hat{f}(x,y,\xi)$ and $\hat{g}(x,y,\xi)$ over a noise variable $\xi$ . We recover the deterministic setting as a particular case when the variable $\xi$ has zero variance, thus allowing us to treat both stochastic (S) and deterministic (D) settings in a unified framework. As often in machine-learning, we assume we can always draw a new batch $\mathcal{D}$ of i.i.d. samples of the noise variable $\xi$ with size $|\mathcal{D}| \geq 1$ and use it to compute stochastic approximations of $f$ and $g$ defined by abuse of notation as $\hat{f}(x,y,\mathcal{D}) \coloneqq \frac{1}{|\mathcal{D}|}\sum_{\xi \in \mathcal{D}}\hat{f}(x,y,\xi)$ and $\hat{g}(x,y,\mathcal{D}) \coloneqq \frac{1}{|\mathcal{D}|}\sum_{\xi \in \mathcal{D}}\hat{g}(x,y,\xi)$ . We make the following noise assumptions which are implied by those in Ghadimi and Wang (2018):

Assumption 4. For any batch $\mathcal{D}$ , $\nabla \hat{f}(x, y, \mathcal{D})$ and $\partial_y \hat{g}(x, y, \mathcal{D})$ are unbiased estimator of $\nabla f(x, y)$ and $\partial_y g(x, y)$ with a uniformly bounded variance, i.e. for all $x, y \in \mathcal{X} \times \mathcal{Y}$ :

$$\mathbb{E}[\parallel \mathrm{\nabla }\widehat{f}(x,y,\mathcal{D})-\mathrm{\nabla }f(x,y){\parallel }^2]\le {\tilde{\sigma }}_f^2\mathrm{\mid }\mathcal{D}{\mathrm{\mid }}^{-1},\mathbb{E}[\mathrm{\parallel }{\mathrm{\partial }}_y\widehat{g}(x,y,\mathcal{D})-{\mathrm{\partial }}_{y}g(x,y){\mathrm{\parallel }}^2]\le {\tilde{\sigma }}_g^2\mathrm{\mid }\mathcal{D}{\mathrm{\mid }}^{-1}\mathrm{.}\backslash mathbb\; \{E\}\; \backslash bigg\; [\; \backslash bigg\; \backslash |\; \backslash nabla\; \backslash hat\; \{f\}\; (x,\; y,\; \backslash mathcal\; \{D\})\; -\; \backslash nabla\; f\; (x,\; y)\; \backslash bigg\; \backslash |\; ^\; \{2\}\; \backslash bigg\; ]\; \backslash leq\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\; |\; \backslash mathcal\; \{D\}\; |\; ^\; \{-\; 1\},\; \backslash qquad\; \backslash mathbb\; \{E\}\; \backslash bigg\; [\; \backslash |\; \backslash partial\_\; \{y\}\; \backslash hat\; \{g\}\; (x,\; y,\; \backslash mathcal\; \{D\})\; -\; \backslash partial\_\; \{y\}\; g\; (x,\; y)\; \backslash |\; ^\; \{2\}\; \backslash bigg\; ]\; \backslash leq\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\; |\; \backslash mathcal\; \{D\}\; |\; ^\; \{-\; 1\}.$$

Assumption 5. For any batch $\mathcal{D}$ , the matrices $F_{1}(x,y,\mathcal{D})\coloneqq \partial_{xy}\hat{g} (x,y,\mathcal{D}) - \partial_{xy}g(x,y)$ and $F_{2}(x,y,\mathcal{D}):= \partial_{yy}\hat{g} (x,y,\mathcal{D}) - \partial_{yy}g(x,y)$ have zero mean and satisfy for all $x,y\in \mathcal{X}\times \mathcal{Y}$ :

$${\parallel \mathbb{E}[F_1(x,y,\mathcal{D}{)}^{\mathrm{\top }}{F}_1(x,y,\mathcal{D})]\parallel }_{op}\le {\tilde{\sigma }}_{g_{xy}}^2\mathrm{\mid }\mathcal{D}{\mathrm{\mid }}^{-1},{\parallel \mathbb{E}[F_2(x,y,\mathcal{D}{)}^{\mathrm{\top }}{F}_2(x,y,\mathcal{D})]\parallel }_{op}\le {\tilde{\sigma }}_{g_{yy}}^2\mathrm{\mid }\mathcal{D}{\mathrm{\mid }}^{-1}\mathrm{.}\backslash left\backslash |\; \backslash mathbb\; \{E\}\; \backslash big\; [\; F\; \_\; \{1\}\; (x,\; y,\; \backslash mathcal\; \{D\})\; ^\; \{\backslash top\}\; F\; \_\; \{1\}\; (x,\; y,\; \backslash mathcal\; \{D\})\; \backslash big\; ]\; \backslash right\backslash |\; \_\; \{o\; p\}\; \backslash leq\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; |\; \backslash mathcal\; \{D\}\; |\; ^\; \{-\; 1\},\; \backslash quad\; \backslash left\backslash |\; \backslash mathbb\; \{E\}\; \backslash big\; [\; F\; \_\; \{2\}\; (x,\; y,\; \backslash mathcal\; \{D\})\; ^\; \{\backslash top\}\; F\; \_\; \{2\}\; (x,\; y,\; \backslash mathcal\; \{D\})\; \backslash big\; ]\; \backslash right\backslash |\; \_\; \{o\; p\}\; \backslash leq\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; |\; \backslash mathcal\; \{D\}\; |\; ^\; \{-\; 1\}.$$

For conciseness, we will use the notations $\sigma_f^2 \coloneqq \tilde{\sigma}f^2 |\mathcal{D}|^{-1}$ , $\sigma_g^2 \coloneqq \tilde{\sigma}g^2 |\mathcal{D}|^{-1}$ , $\sigma{g{xy}}^2 \coloneqq \tilde{\sigma}{g{xy}}^2 |\mathcal{D}|^{-1}$ and $\sigma_{g_{yy}}^2 \coloneqq \tilde{\sigma}{g{yy}}^2 |\mathcal{D}|^{-1}$ , without explicit reference to the batch $\mathcal{D}$ . Next, we describe the algorithm.

3.2 ALGORITHMS

Amortized Implicit Gradient Optimization (AmIGO) is an iterative algorithm for solving (1). It constructs iterates $x_{k}$ , $y_{k}$ and $z_{k}$ such that $x_{k}$ approaches a stationary point of $\mathcal{L}$ while $y_{k}$ and $z_{k}$ track the quantities $y^{\star}(x_k)$ and $z^{\star}(x_k,y_k)$ . AmIGO computes the iterate $x_{k + 1}$ using an update equation $x_{k + 1} = x_{k} - \gamma_{k}\hat{\psi}{k}$ for some given step-size $\gamma{k}$ and a stochastic estimate $\hat{\psi}k$ of $\nabla \mathcal{L}(x_k)$ based on (3) and defined according to (4) below for some new batches of samples $\mathcal{D}f$ and $\mathcal{D}{g{xy}}$ .

$$\begin{array}{cccc}& {\widehat{\psi }}_k:={\mathrm{\partial }}_x\widehat{f}(x_k,y_k,{\mathcal{D}}_f)+{\mathrm{\partial }}_{x,y}\widehat{g}{(x_k,y_k,{\mathcal{D}}_{g_{xy}})}^{\mathrm{\top }}{z}_k\mathrm{.}& & \text{(4)}\end{array}\backslash hat\; \{\backslash psi\}\; \_\; \{k\}\; :=\; \backslash partial\_\; \{x\}\; \backslash hat\; \{f\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\}\backslash right)\; +\; \backslash partial\_\; \{x,\; y\}\; \backslash hat\; \{g\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{x\; y\}\}\backslash right)\; ^\; \{\backslash top\}\; z\; \_\; \{k\}.\; \backslash tag\; \{4\}$$

Algorithm 1 AmIGO

1: Inputs: $x_0, y_{-1}, z_{-1}$ .
2: Parameters: $\gamma_{k}, K$ .
3: for $k \in {0, \dots, K}$ do
4: $y_{k} \gets \mathcal{A}{k}(x{k},y_{k - 1})$
5: Sample batches $\mathcal{D}f, \mathcal{D}g$ .
6: $(u{k},v{k})\gets \nabla f(x_{k},y_{k},\mathcal{D}{f}).$
7: $z{k}\gets \mathcal{B}{k}(x{k},y_{k},v_{k},z_{k - 1})$
8: $w_{k}\gets \partial_{xy}\hat{g} (x_{k},y_{k},\mathcal{D}{g{xy}})z_{k}$
9: $\psi_{k - 1}\gets u_k + w_k$
10: $x_{k}\gets x_{k - 1} - \gamma_{k}\psi_{k - 1}$
11: end for
12: Return $x_{K}$ .

AmIGO computes $\hat{\psi}k$ in 4 steps given iterates $x_k, y{k-1}$ and $z_{k-1}$ . A first step computes an approximation $y_k$ to $y^*(x_k)$ using a stochastic algorithm $\mathcal{A}k$ initialized at $y{k-1}$ . A second step computes unbiased estimates $u_k = \partial_x \hat{f}(x_k, y_k, \mathcal{D}_f)$ and $v_k = \partial_y \hat{f}(x_k, y_k, \mathcal{D}f)$ of the partial derivatives of $f$ w.r.t. $x$ and $y$ . A third step computes an approximation $z_k$ to $z^*(x_k, y_k)$ using a second stochastic algorithm $\mathcal{B}k$ for solving (2) initialized at $z{k-1}$ . To increase efficiency, algorithm $\mathcal{B}k$ uses the pre-computed vector $v_k$ for approximating the partial derivative $\partial_y f$ in (2). Finally, the stochastic estimate $\hat{\psi}k$ is computed using (4) by summing the pre-computed vector $u_k$ with the jacobian-vector product $w_k = \partial{xy} \hat{g}(x_k, y_k, \mathcal{D}{g{xy}}) z_k$ . AmIGO is summarized in Algorithm 1.

Algorithms $\mathcal{A}_k$ and $\mathcal{B}_k$ . While various choices for $\mathcal{A}_k$ and $\mathcal{B}_k$ are possible, such as adaptive algorithms (Kingma and Ba, 2015), or accelerated stochastic algorithms (Ghadimi and Lan, 2012), we

focus on simple stochastic gradient descent algorithms with a pre-defined number of iterations $T$ and $N$ . These algorithms compute intermediate iterates $y^{t}$ and $z^n$ optimizing the functions $y \mapsto g(x_k, y)$ and $z \mapsto Q(x_k, y_k, z)$ starting from some initial values $y^0$ and $z^0$ and returning the last iterates $y^T$ and $z^N$ as described in Algorithms 2 and 3. Algorithm $\mathcal{A}k$ updates the current iterate $y^{t-1}$ using a stochastic gradient $\partial_y \hat{g}(x_k, y^{t-1}, \mathcal{D}g)$ for some new batch of samples $\mathcal{D}g$ and a fixed step-size $\alpha_k$ . Algorithm $\mathcal{B}k$ updates the current iterate $z^{t-1}$ using a stochastic estimate of $\partial_z Q(x_k, y_k, z^{t-1})$ with step-size $\beta_k$ . The stochastic gradient is computed by evaluating the Hessian-vector product $\partial{yy} \hat{g}(x_k, y_k, \mathcal{D}{g{yy}}) z^{t-1}$ for some new batch of samples $\mathcal{D}{g_{yy}}$ and summing it with a vector $v_k$ approximating $\partial_y f(x_k, y_k)$ provided as input to algorithm $\mathcal{B}_k$ .

Warm-start for $y^0$ and $z^0$ . Following the intuition that $y^\star(x_k)$ remains close to $y^\star(x_{k-1})$ when $x_k \simeq x_{k-1}$ , and assuming that $y_{k-1}$ is an accurate approximation to $y^\star(x_{k-1})$ , it is natural to initialize $\mathcal{A}k$ with the iterate $y{k-1}$ . The same intuition applies when initializing $\mathcal{B}k$ with $z{k-1}$ . Next, we introduce a framework for analyzing the effect of warm-start on the convergence speed of AmIGO.

Algorithm 2 Ak(x,y0)	Algorithm 3 Bk(x,y,v,z0)
1: Parameters: αk,T	1: Parameters: βk,N.
2: for t ∈ {1,...,T} do	2: for n ∈ {1,...,N} do
3: Sample batch Dg,t,k.	3: Sample batch Dgyy.n,k.
4: yt← yt-1- αk∂yhat(x,yt-1, Dgt,k).	4: zn← zn-1- βk(∂yyhat(x,y, Dgyy.n,k)zt-1+v).
5: end for	5: end for
6: Return yT.	6: Return zN.

4 ANALYSIS OF AMORTIZED IMPLICIT GRADIENT OPTIMIZATION

4.1 GENERAL APPROACH AND MAIN RESULT

The proposed approach consists in three main steps: (1) Analysis of the outer-level problem, (2) Analysis of the inner-level problem and (3) Analysis of the joint dynamics of both levels.

Outer-level problem. We consider a quantity $E_{k}^{x}$ describing the evolution of $x_{k}$ defined as follows:

where $u \in {0,1}$ is set to 1 in the stochastic setting and to 0 in the deterministic one and $\delta_{k}$ is a positive sequence that determines the convergence rate of the outer-level problem and is defined by:

with $\eta_0$ such that $\gamma_0^{-1} \geq \eta_0 \geq \mu$ if $\mu \geq 0$ and $\eta_0 = L$ if $\mu < 0$ and where we choose the step-size $\gamma_k$ to be a non-increasing sequence with $\gamma_0 \leq \frac{1}{L}$ . With this choice for $\delta_k$ and by setting $u = 1$ in (5), $E_k^x$ recovers the quantity considered in the stochastic estimate sequences framework of Kulunchakov and Mairal (2020) to analyze the convergence of stochastic optimization algorithms when $\mathcal{L}$ is convex. When $\mathcal{L}$ is non-convex, $E_k^x$ recovers a standard measure of stationarity (Davis and Drusvyatskiy, 2018). In Section 4.3, we control $E_k^x$ using bias and variance error $E_{k - 1}^{\psi}$ and $V_{k - 1}^{\psi}$ of $\hat{\psi}_k$ given by (6) below where $\mathbb{E}k$ denotes expectation conditioned on $(x_k, y_k, z{k - 1})$ .

$$\begin{array}{cccc}& E_k^{\psi }:=\mathbb{E}\left[{\parallel {\mathbb{E}}_k\left[{\widehat{\psi }}_k\right]-\mathrm{\nabla }\mathcal{L}(x_k)\parallel }^2\right],V_k^{\psi }:=\mathbb{E}\left[{\parallel {\widehat{\psi }}_k-{\mathbb{E}}_k\left[{\widehat{\psi }}_k\right]\parallel }^2\right]\mathrm{.}& & \text{(6)}\end{array}E\; \_\; \{k\}\; ^\; \{\backslash psi\}\; :=\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash mathbb\; \{E\}\; \_\; \{k\}\; \backslash left[\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\}\; \backslash right]\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right],\; \backslash quad\; V\; \_\; \{k\}\; ^\; \{\backslash psi\}\; :=\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\}\; -\; \backslash mathbb\; \{E\}\; \_\; \{k\}\; \backslash left[\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\}\; \backslash right]\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right].\; \backslash tag\; \{6\}$$

Inner-level problems. We consider the mean-squared errors $E_{k}^{y}$ and $E_{k}^{z}$ between initializations $(y^{0} = y_{k - 1}$ and $z^0 = z_{k - 1}$ ) and stationary values $(y^{\star}(x_k)$ and $z^{\star}(x_k,y_k))$ of algorithms $\mathcal{A}_k$ and $\mathcal{B}_k$ :

$$E_k^y:=\mathbb{E}\left[{\parallel y_k^0-y^{\star }(x_k)\parallel }^2\right],E_k^z:=\mathbb{E}\left[{\parallel z_k^0-z^{\star }(x_k,y_k)\parallel }^2\right]\mathrm{.}E\; \_\; \{k\}\; ^\; \{y\}\; :=\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; y\; \_\; \{k\}\; ^\; \{0\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right],\; \backslash qquad\; E\; \_\; \{k\}\; ^\; \{z\}\; :=\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; \_\; \{k\}\; ^\; \{0\}\; -\; z\; ^\; \{\backslash star\}\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right].$$

In Section 4.3, we show that the warm-start strategy allows to control $E_{k}^{y}$ and $E_{k}^{z}$ in terms of previous iterates $E_{k-1}^{y}$ and $E_{k-1}^{z}$ as well as the bias and variance errors in (6). We further prove that such bias and variance errors are, in turn, controlled by $E_{k}^{y}$ and $E_{k}^{z}$ .

Joint dynamics. Following Habets (1974), we consider an aggregate error $E_{k}^{tot}$ defined as a linear combination of $E_{k}^{x}$ , $E_{k}^{y}$ and $E_{k}^{z}$ with carefully selected coefficients $a_{k}$ and $b_{k}$ :

$$\begin{array}{cccc}& E_k^{tot}=E_k^x+a_k{E}_k^y+b_k{E}_k^z\mathrm{.}& & \text{(7)}\end{array}E\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; =\; E\; \_\; \{k\}\; ^\; \{x\}\; +\; a\; \_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{y\}\; +\; b\; \_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{z\}.\; \backslash tag\; \{7\}$$

As such $E_k^{tot}$ represents the dynamics of the whole system. The following theorem provides an error bound for $E_k^{tot}$ in both convex and non-convex settings for a suitable choice of the coefficients $a_k$ and $b_k$ provided that $T$ and $N$ are large enough:

Theorem 1. Choose a batch-size $\left|\mathcal{D}{g{yy}}\right| \geq 1 \vee \frac{\tilde{\sigma}{g{yy}}^2}{\mugL_g}$ and the step-sizes $\alpha_k = L_g^{-1}$ , $\beta_k = (2L_g)^{-1}$ , $\gamma_k = L^{-1}$ . Set the coefficients $a_k$ and $b_k$ to be $a_k := \delta_0(1 - \alpha_k\mu_g)^{1/2}$ and $b_k := \delta_0\big(1 - \frac{1}{2}\beta_k\mu_g\big)^{1/2}$ and set the number of iterations $T$ and $N$ of Algorithms 2 and 3 to be of order $T = O(\kappa_g)$ and $N = O(\kappa_g)$ up to a logarithmic dependence on $\kappa_g$ . Let $\hat{x}k = u(1 - \delta_k)\hat{x}{k-1} + (1 - u(1 - \delta_k))x_k$ , with $\hat{x}_0 = x_0$ . Then, under Assumptions 1 to 5, $E_k^{tot}$ satisfies:

where $\mathcal{W}^2$ , defined in (20) of Appendix A.2, is the effective variance of the problem with $\mathcal{W}^2 = 0$ in the deterministic setting and, in the stochastic setting, $\mathcal{W}^2 > 0$ is of the following order:

$${\mathcal{W}}^2=O({\delta }_0^{-1}{\kappa }_g^5\mathrm{\mid }{\mathcal{D}}_g{\mathrm{\mid }}^{-1}{\tilde{\sigma }}_g^2+{\kappa }_g^3\mathrm{\mid }{\mathcal{D}}_{g_{yy}}{\mathrm{\mid }}^{-1}{\tilde{\sigma }}_{g_{yy}}^2+{\kappa }_g^2\mathrm{\mid }{\mathcal{D}}_{g_{xy}}{\mathrm{\mid }}^{-1}{\tilde{\sigma }}_{g_{xy}}^2+{\kappa }_g^2\mathrm{\mid }{\mathcal{D}}_f{\mathrm{\mid }}^{-1}{\tilde{\sigma }}_f^2),\backslash mathcal\; \{W\}\; ^\; \{2\}\; =\; O\; \backslash Big\; (\backslash delta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash kappa\_\; \{g\}\; ^\; \{5\}\; |\; \backslash mathcal\; \{D\}\; \_\; \{g\}\; |\; ^\; \{-\; 1\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{3\}\; |\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{y\; y\}\}\; |\; ^\; \{-\; 1\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; |\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{x\; y\}\}\; |\; ^\; \{-\; 1\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; |\; \backslash mathcal\; \{D\}\; \_\; \{f\}\; |\; ^\; \{-\; 1\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\; \backslash Big),$$

We describe the strategy of the proof in Section 4.3 and provide a proof outline in Appendix A.1 with exact expressions for all variables including the expressions of $T$ , $N$ and $\mathcal{W}^2$ . The full proof is provided in Appendix A.2. The choice of $a_{k}$ and $b_{k}$ ensures that $E_k^y$ and $E_k^z$ contribute less to $E_k^{tot}$ as the algorithms $\mathcal{A}_k$ and $\mathcal{B}_k$ become more accurate. The effective variance $\mathcal{W}^2$ accounts for interactions between both levels in the presence of noise and becomes proportional to the outer-level variance $\sigma_f^2$ when the inner-level problem is solved exactly. In the deterministic setting, all variances $\tilde{\sigma}f^2$ , $\tilde{\sigma}g^2$ , $\tilde{\sigma}{gxy}^2$ and $\tilde{\sigma}{gyy}^2$ vanish so that $\mathcal{W}^2 = 0$ . Hence, we characterize such setting by $\mathcal{W}^2 = 0$ and the stochastic one by $\mathcal{W}^2 > 0$ . Next, we apply Theorem 1 to obtain the complexity of AmIGO.

4.2 COMPLEXITY ANALYSIS

We define the complexity $\mathcal{C}(\epsilon)$ of a bilevel algorithm to be the total number of queries to the gradients of $f$ and $g$ , Jacobian/hessian-vector products needed by the algorithm to achieve an error $\epsilon$ according to some pre-defined criterion. Let the number of iterations $k$ , $T$ and $N$ and sizes of the batches $|\mathcal{D}g|$ , $|\mathcal{D}f|$ , $|\mathcal{D}{g{xy}}|$ and $|\mathcal{D}{g{yy}}|$ , be such that AmIGO achieves a precision $\epsilon$ . Then $\mathcal{C}(\epsilon)$ is given by:

$$\begin{array}{cccc}& \mathcal{C}(ϵ)=k(T\mid {\mathcal{D}}_g\mid +N\mid {\mathcal{D}}_{g_{yy}}\mid +\mid {\mathcal{D}}_{g_{xy}}\mid +\mid {\mathcal{D}}_f\mid ),& & \text{(8)}\end{array}\backslash mathcal\; \{C\}\; (\backslash epsilon)\; =\; k\; \backslash left(T\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\}\; \backslash right|\; +\; N\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{y\; y\}\}\; \backslash right|\; +\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{x\; y\}\}\; \backslash right|\; +\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{f\}\; \backslash right|\backslash right),\; \backslash tag\; \{8\}$$

We provide the complexity of AmIGO in the 4 settings of Table 1 in the form of Corrolaries 1 to 4.

Corollary 1 (Case $\mu >0$ and $\mathcal{W}^2 = 0$ ). Use batches of size 1. Achieving $\mathcal{L}(x_k) - \mathcal{L}^\star +\frac{\mu}{2}| x_k - x^\star | ^2\leq \epsilon$ requires $\mathcal{C}(\epsilon) = O\left(\kappa_{\mathcal{L}}\kappa_g\log \left(\frac{E_0^{tot}}{\epsilon}\right)\right)$ .

Corollary 1 outperforms the complexities in Table 1 in terms of the dependence on $\epsilon$ . It is possible to improve the dependence on $\kappa_{g}$ to $\kappa_{g}^{1/2}$ using acceleration in $\mathcal{A}{k}$ and $\mathcal{B}{k}$ as discussed in Appendix A.5.1, or using generic acceleration methods such as Catalyst (Lin et al., 2018).

Corollary 2 (Case $\mu \mathcal{W}^2 >0$ ). Choose $|\mathcal{D}g| = \Theta (\epsilon^{-1}\kappa{\mathcal{L}}\kappa_g^2\tilde{\sigma}g^2)$ , $|\mathcal{D}{g_{xy}}| = \Theta (\epsilon^{-1}\tilde{\sigma}{g{xy}}^2)$ , $|\mathcal{D}f| = \Theta \left(\frac{\tilde{\sigma}f^2}{\epsilon}\right)$ and $|\mathcal{D}{g{yy}}| = \Theta \left(\tilde{\sigma}{g{yy}}^2\left(\frac{1}{\epsilon}\vee \kappa_g\right)\right)$ . Achieving $\mathbb{E}[\mathcal{L}(\hat{x}_k) - \mathcal{L}^\star ] + \frac{\mu}{2}\mathbb{E}\big[| x_k - x^\star | ^2\big]\leq \epsilon$ requires:

$$\mathcal{C}(ϵ)=O\left({\kappa }_{\mathcal{L}}\right({\kappa }_{\mathcal{L}}{\kappa }_g^3{\tilde{\sigma }}_g^2+{\kappa }_g(1\vee ϵ{\kappa }_g){\tilde{\sigma }}_{g_{yy}}^2+{\tilde{\sigma }}_{g_{xy}}^2+{\tilde{\sigma }}_f^2)\frac{1}{ϵ}\log \left(\frac{E_0^{tot}+\mathbb{E}[\mathcal{L}(x_0)-{\mathcal{L}}^{\star }]}{ϵ}\right))\mathrm{.}\backslash mathcal\; \{C\}\; (\backslash epsilon)\; =\; O\; \backslash Bigg\; (\backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; \backslash Big\; (\backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; \backslash kappa\_\; \{g\}\; ^\; \{3\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; (1\; \backslash vee\; \backslash epsilon\; \backslash kappa\_\; \{g\})\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\; \backslash Big)\; \backslash frac\; \{1\}\{\backslash epsilon\}\; \backslash log\; \backslash left(\backslash frac\; \{E\; \_\; \{0\}\; ^\; \{t\; o\; t\}\; +\; \backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{0\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\}\{\backslash epsilon\}\backslash right)\; \backslash Bigg).$$

Corollary 2 improves over the results in Table 1 in the stochastic strongly-convex setting and recovers the dependence on $\epsilon$ of stochastic gradient descent for smooth and strongly convex functions up to a logarithmic factor.

Corollary 3 (Case $\mu < 0$ and $\mathcal{W}^2 = 0$ ). Choose batches of size 1. Achieving $\frac{1}{k}\sum_{i=1}^{k}|\nabla\mathcal{L}(x_i)|^2 \leq \epsilon$ requires $\mathcal{C}(\epsilon) = \mathcal{O}\Big(\frac{\kappa_g^4}{\epsilon}((\mathcal{L}(x_0) - \mathcal{L}^\star) + E_0^y + E_0^z)\Big)$ .

Corollary 3 recovers the complexity of AID-BiO (Ji et al., 2021) in the deterministic non-convex setting. This is expected since AID-BiO also exploits warm-start for both $\mathcal{A}_k$ and $\mathcal{B}_k$ .

Corollary 4 (Case $\mu < 0$ and $\mathcal{W} > 0$ ). Choose $|\mathcal{D}{g{xy}}| = \Theta \left(\frac{\kappa_g^2}{\epsilon}\tilde{\sigma}{g{xy}}^2\right)$ , $|\mathcal{D}{g{yy}}| = \Theta \left(\frac{\kappa_g^3}{\epsilon}\tilde{\sigma}{g{yy}}^2\left(1\vee \epsilon \mu_g^2\right)\right)$ , $|\mathcal{D}_f| = \Theta \left(\frac{\kappa_g^2\tilde{\sigma}_f^2}{\epsilon}\right)$ and $|\mathcal{D}_g| = \Theta \left(\frac{\kappa_g^5\tilde{\sigma}g^2}{\epsilon}\right)$ . Achieving an error $\frac{1}{k}\sum{i=1}^{k}\mathbb{E}\left[|\nabla\mathcal{L}(x_i)|^2\right] \leq \epsilon$ requires:

$$\mathcal{C}(ϵ)=O\left(\frac{{\kappa }_g^5}{{ϵ}^2}\right({\kappa }_g^4{\tilde{\sigma }}_g^2+{\kappa }_g^2(1\vee ϵ{\mu }_g^2){\tilde{\sigma }}_{g_{yy}}^2+{\tilde{\sigma }}_{g_{xy}}^2+{\tilde{\sigma }}_f^2)(\mathbb{E}[\mathcal{L}(x_k)-{\mathcal{L}}^{\star }]+E_0^y+E_0^z))\backslash mathcal\; \{C\}\; (\backslash epsilon)\; =\; O\; \backslash left(\backslash frac\; \{\backslash kappa\_\; \{g\}\; ^\; \{5\}\}\{\backslash epsilon^\; \{2\}\}\; \backslash Big\; (\backslash kappa\_\; \{g\}\; ^\; \{4\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; \backslash big\; (1\; \backslash vee\; \backslash epsilon\; \backslash mu\_\; \{g\}\; ^\; \{2\}\; \backslash big)\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\; \backslash Big)\; (\backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\; +\; E\; \_\; \{0\}\; ^\; \{y\}\; +\; E\; \_\; \{0\}\; ^\; \{z\})\backslash right)$$

Corollary 4 recovers the optimal dependence on $\epsilon$ of $O\left(\frac{1}{\epsilon^2}\right)$ achieved by stochastic gradient descent in the smooth non-convex case (Arjevani et al., 2019, Theorem 1). It also improves over the results in (Ji et al., 2021) which involve an additional logarithmic factor $\log (\epsilon^{-1})$ as $N$ is required to be $O(\kappa_g\log (\epsilon^{-1}))$ . In our case, $N$ remains constant since $\mathcal{B}_k$ benefits from warm-start. The faster rates of MRBO/VRBO* (Yang et al., 2021) are obtained under the additional mean-squared smoothness assumption (Arjevani et al., 2019), which we do not investigate in the present work. Such assumption allows to achieve the improved complexity of $O(\epsilon^{-3 / 2}\log (\epsilon^{-1}))$ . However, these algorithms still require $N = O(\log (\epsilon^{-1}))$ , indicating that the use of warm-start in $\mathcal{B}_k$ could further reduce the complexity to $O(\epsilon^{-3 / 2})$ which would be an interesting direction for future work.

4.3 OUTLINE OF THE PROOF

The proof of Theorem 1 proceeds by deriving a recursion for both outer-level error $E_{k}^{x}$ and inner-level errors $E_{k}^{y}$ and $E_{k}^{z}$ and then combining those to obtain an error bound on the total error $E_{k}^{tot}$ .

Outer-level recursion. To allow a unified analysis of the behavior of $E_k^x$ in both convex and nonconvex settings, we define $F_k$ as follows:

The following proposition, with a proof in Appendix C.1, provides a recursive inequality on $E_k^x$ involving the errors in (6) due to the inexact gradient $\hat{\psi}_k$ :

Proposition 2. Let $\rho_{k}$ be a non-increasing sequence with $0 < \rho_{k} < 2$ . Assumptions 1 to 3 ensure that:

$$\begin{array}{cccc}& F_k+E_k^x\le (1-(1-2^{-1}{\rho }_k){\delta }_k)E_{k-1}^x+{\gamma }_k{s}_k{V}_{k-1}^{\psi }+{\gamma }_k(s_k+{\rho }_k^{-1})E_{k-1}^{\psi },& & \text{(10)}\end{array}F\; \_\; \{k\}\; +\; E\; \_\; \{k\}\; ^\; \{x\}\; \backslash leq\; \backslash left(1\; -\; \backslash left(1\; -\; 2\; ^\; \{-\; 1\}\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\backslash right)\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; +\; \backslash gamma\_\; \{k\}\; s\; \_\; \{k\}\; V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; \backslash gamma\_\; \{k\}\; \backslash left(s\; \_\; \{k\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\},\; \backslash tag\; \{10\}$$

with $s_k$ defined as $s_k \coloneqq \frac{1}{2}\delta_k + \left(\frac{u}{2}\delta_k + (1 - u)\right)\mathbb{1}_{\mu > 0}$ .

In the ideal case where $y_{k} = y^{\star}(x_{k})$ and $z_{k} = z^{\star}(x_{k},y_{k})$ , the bias $E_k^\psi$ vanishes and (10) simplifies to (Kulunchakov and Mairal, 2019, Proposition 1) which recovers the convergence rates for stochastic gradient methods in the convex case. However, $y_{k}$ and $z_{k}$ are generally inexact solutions and introduce a positive bias $E_k^\psi$ . Therefore, controlling the inner-level iterates is required to control the bias $E_k^\psi$ which, in turn, impacts the convergence of the outer-level as we discuss next.

Controlling the inner-level iterates $y_{k}$ and $z_{k}$ . Proposition 3 below controls the expected mean squared errors between iterates $y_{k}$ and $z_{k}$ and their limiting values $y^{\star}(x_k)$ and $z^{\star}(x_k, y_k)$ :

Proposition 3. Let the step-sizes $\alpha_{k}$ and $\beta_{k}$ be such that $\alpha_{k} \leq L_{g}^{-1}$ and $\beta_{k} \leq \frac{1}{2L_{g}} \wedge \frac{\mu_{g}}{\mu_{g}^{2} + \sigma_{gyy}^{2}}$ . Let $\Lambda_{k} := (1 - \alpha_{k}\mu_{g})^{T}$ and $\Pi_{k} := \left(1 - \frac{\beta_{k}\mu_{g}}{2}\right)^{N}$ . Under Assumptions 1, 4 and 5, it holds that:

$$\begin{array}{cccc}& \mathbb{E}[\mathrm{\parallel }{y}_k-y^{\star }\left(x_k\right){\mathrm{\parallel }}^2]\le {\mathrm{\Lambda }}_k{E}_k^y+R_k^y,\mathbb{E}[\mathrm{\parallel }{z}_k-z^{\star }(x_k,y_k){\mathrm{\parallel }}^2]\le {\mathrm{\Pi }}_k{E}_k^z+R_k^z,& & \text{(11)}\end{array}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash Lambda\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{y\}\; +\; R\; \_\; \{k\}\; ^\; \{y\},\; \backslash quad\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; z\; \_\; \{k\}\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash Pi\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{z\}\; +\; R\; \_\; \{k\}\; ^\; \{z\},\; \backslash tag\; \{11\}$$

where $R_{k}^{y} = O\left(\kappa_{g}\sigma_{g}^{2}\right)$ and $R_{k}^{z} = O\left(\kappa_{g}^{3}\sigma_{g_{yy}}^{2} + \kappa_{g}^{2}\sigma_{f}^{2}\right)$ are defined in (14) of Appendix A.2.

While Proposition 3 is specific to the choice of the algorithms $\mathcal{A}_k$ and $\mathcal{B}_k$ in Algorithms 2 and 3, our analysis directly extends to other algorithms satisfying inequalities similar to (11) such as accelerated or variance reduced algorithms discussed in Appendices A.5.1 and A.5.2. Proposition 4 below controls the bias and variance terms $V_k^\psi$ and $E_k^\psi$ in terms of the warm-start error $E_k^y$ and $E_k^z$ .

Proposition 4. Under Assumptions 1 to 5, the following inequalities hold:

$$E_k^{\psi }\le 2L_{\psi }^2({\mathrm{\Lambda }}_k{E}_k^y+{\mathrm{\Pi }}_k{E}_k^z+R_k^y),V_k^{\psi }\le w_x^2+{\sigma }_x^2{\mathrm{\Pi }}_k{E}_k^z,E\; \_\; \{k\}\; ^\; \{\backslash psi\}\; \backslash leq\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash left(\backslash Lambda\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{y\}\; +\; \backslash Pi\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{z\}\; +\; R\; \_\; \{k\}\; ^\; \{y\}\backslash right),\; \backslash quad\; V\; \_\; \{k\}\; ^\; \{\backslash psi\}\; \backslash leq\; w\; \_\; \{x\}\; ^\; \{2\}\; +\; \backslash sigma\_\; \{x\}\; ^\; \{2\}\; \backslash Pi\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{z\},$$

where $w_{x}^{2} = O\left(\kappa_{g}^{2}\left(\sigma_{f}^{2} + \sigma_{g_{xy}}^{2}\right) + \kappa_{g}^{3}\sigma_{g_{yy}}^{2}\right)$ , $\sigma_{x}^{2} = O\left(\sigma_{g_{xy}}^{2} + \kappa_{g}^{2}\sigma_{g_{yy}}^{2}\right)$ and $L_{\psi} = O(\kappa_g^2)$ are positive constants defined in (13) and (16) of Appendix A.2 with $L_{\psi}$ controlling the variations of $\mathbb{E}_k[\hat{\psi}_k]$ .

Proposition 4 highlights the dependence of $E_{k}^{\psi}$ and $V_{k}^{\psi}$ on the inner-level errors. It suggests analyzing the evolution of $E_{k}^{y}$ and $E_{k}^{z}$ to quantify how large the bias and variances can get:

Proposition 5. Let $\zeta_k > 0$ , a $2 \times 2$ matrix $\pmb{P}_k$ , two vectors $\pmb{U}_k$ and $\pmb{V}_k$ in $\mathbb{R}^2$ all independent of $x_k$ , $y_k$ and $z_k$ be as defined in Proposition 8 of Appendix A.2. Under Assumptions 1 to 5, it holds that:

$$\begin{array}{cccc}& \left(\genfrac{}{}{0px}{}{E_k^y}{E_k^z}\right)\le P_{\mathit{k}}\left(\genfrac{}{}{0px}{}{{\mathrm{\Lambda }}_{k-1}{E}_{k-1}^y+R_{k-1}^y}{{\mathrm{\Pi }}_{k-1}{E}_{k-1}^z+R_{k-1}^z}\right)+2{\gamma }_k(E_{k-1}^{\psi }+V_{k-1}^{\psi }+{\zeta }_k{E}_{k-1}^x){\mathit{U}}_{\mathit{k}}+{\mathit{V}}_{\mathit{k}}\mathrm{.}& & \text{(12)}\end{array}\backslash binom\; \{E\; \_\; \{k\}\; ^\; \{y\}\}\; \{E\; \_\; \{k\}\; ^\; \{z\}\}\; \backslash leq\; P\; \_\; \{\backslash boldsymbol\; \{k\}\}\; \backslash binom\; \{\backslash Lambda\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\}\; \{\backslash Pi\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\}\; +\; 2\; \backslash gamma\_\; \{k\}\; \backslash left(E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; \backslash zeta\_\; \{k\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\backslash right)\; \backslash boldsymbol\; \{U\}\; \_\; \{\backslash boldsymbol\; \{k\}\}\; +\; \backslash boldsymbol\; \{V\}\; \_\; \{\backslash boldsymbol\; \{k\}\}.\; \backslash tag\; \{12\}$$

Proposition 5 describes the evolution of the inner-level errors as the number of iterations $k$ increases. The matrix $P_{k}$ and vectors $U_{k}$ and $V_{k}$ arise from discretization errors and depend on the step-sizes and constants of the problem. The second term of (12) represents interactions with the outer-level through $E_{k-1}^{x}, V_{k-1}^{\psi}$ and $E_{k-1}^{\psi}$ . Propositions 2, 4 and 5 describe the joint dynamics of $(E_{k}^{x}, E_{k}^{y}, E_{k}^{z})$ from which the evolution of $E_{k}^{tot}$ can be deduced as shown in Appendices A.1 and A.2.

5 EXPERIMENTS

We run three sets of experiments described in Sections 5.1 to 5.3. In all cases, we consider AmIGO with either gradient descent (AmIGO-GD) or conjugate gradient (AmIGO-CG) for algorithm $\mathcal{B}_k$ . We AmIGO with AID methods without warm-start for $\mathcal{B}_k$ which we refer to as (AID-GD) and (AID-CG) and with (AID-CG-WS) which uses warm-start for $\mathcal{B}_k$ but not for $\mathcal{A}_k$ . We also consider other variants using either a fixed-point algorithm (AID-FP) (Grazzi et al., 2020) or Neumann series expansion (AID-N) (Lorraine et al., 2020) for $\mathcal{B}_k$ . Finally, we consider two algorithms based on iterative differentiation which we refer to as (ITD) (Grazzi et al., 2020) and (Reverse) (Franceschi et al., 2017). For all methods except (AID-CG-WS), we use warm-start in algorithm $\mathcal{A}_k$ , however only AmIGO, AmIGO-CG and AID-CG-WS exploits warm-start in $\mathcal{B}_k$ the other AID based methods initializing $\mathcal{B}_k$ with $z^0 = 0$ . In Sections 5.2 and 5.3, we also compare with BSA algorithm (Ghadimi and Wang, 2018), TTSA algorithm (Hong et al., 2020a) and stocBio (Ji et al., 2021). An implementation of AmIGO is available in https://github.com/MichaelArbel/AmIGO.

5.1 SYNTHETIC PROBLEM

To study the behavior of AmIGO in a controlled setting, we consider a synthetic problem where both inner and outer level losses are quadratic functions with thousands of variables as described in details in Appendix F.1. Figure 1(a) shows the complexity $\mathcal{C}(\epsilon)$ needed to reach $10^{-6}$ relative error amongst the best choice for $T$ and $M$ over a grid as the conditioning number $\kappa_{g}$ increases. AmIGO-CG achieves the lowest time and is followed by AID-CG thus showing a favorable effect of warm-start for $\mathcal{B}k$ . The same conclusion holds for AmIGO-GD compared to AID-GD. Note that AID-CG is still faster than AmIGO-CG for larger values of $\kappa{g}$ highlighting the advantage of using algorithms $\mathcal{B}k$ with $O(\sqrt{\kappa_g})$ complexity such as (CG) instead non-accelerated ones with $O(\kappa_g^{-1})$ such as (GD). Figure 1(b) shows the relative error after 10s and maintains the same conclusions. For moderate values of $\kappa{g}$ , only AmIGO and AID-CG reach an error of $10^{-20}$ as shown in Figure 1(c). We refer to Figures 2 and 3 of Appendix F for additional results on the effect of the choice of $T$ and $M$ showing that AmIGO consistently performs well for a wide range of values of $T$ and $M$ .

5.2 HYPER-PARAMETER OPTIMIZATION

We consider a classification task on the 20Newsgroup dataset using a logistic loss and a linear model. Each dimension of the linear model is regularized using a different hyper-parameter. The

Figure 1: Top row: performance on the synthetic task. The relative error is defined as a ratio between current and initial errors $(\mathcal{L}(x_k) - \mathcal{L}^\star) / (\mathcal{L}(x_0) - \mathcal{L}^\star)$ . The complexity $\mathcal{C}(\epsilon)$ as defined in (8). Bottom row: performance on the hyper-parameter optimization task.

collection of those hyper-parameters form a vector $x$ of dimension $d = 101631$ optimized using an unregularized regression loss over the validation set while the model is learned using the training set. We consider two evaluations settings: A default setting based on Grazzi et al. (2020); Ji et al. (2021) and a grid-search search setting near the default values of $\beta_{k}$ , $T$ and $N$ as detailed in Appendix F.2. We also vary the batch-size from $10^{3} * {0.1, 1, 2, 4}$ and report the best performing choices for each method. Figure 1(d,e,f) show AmIGO-CG to be the fastest, achieving the lowest error and highest validation and test accuracies. The test accuracy of AmIGO-CG decreases after exceeding $80%$ indicating a potential overfitting as also observed in Franceschi et al. (2018). Similarly, AmIGO-GD outperformed all other methods that uses an algorithm $\mathcal{B}_k$ with $O(\kappa_g)$ complexity. Moreover, all remaining methods achieved comparable performance matching those reported in Ji et al. (2021), thus indicating that the warm-start in $\mathcal{B}_k$ and acceleration in $\mathcal{B}_k$ were the determining factors for obtaining an improved performance. Additionally, Figure 4 of Appendix F report similar results for each choice of the batch-size indicating robustness to the choice of the batch-size.

5.3 DATASET DISTILLATION

Dataset distillation (Wang et al., 2018) consists in learning a synthetic dataset so that a model trained on this dataset achieves a small error on the training set. Figure 5 of Appendix F.3 shows the training loss (outer loss), the training and test accuracies of a model trained on MNIST by dataset distillation. Similarly to Figure 1, AmIGO-CG achieves the best performance followed by AID-CG. AmIGO obtains the best performance by far among methods without acceleration for $\mathcal{B}k$ while all the remaining ones fail to improve. This finding is indicative of an ill-conditioned inner-level problem as confirmed when computing the conditioning number of the hessian $\partial{yy}g(x,y)$ which we found to be of order $7\times 10^{4}$ . Indeed, when compared to the synthetic example for $\kappa_g = 10^4$ as shown in Figure 2, we also observe that only AmIGO-CG, AmIGO and AID-CG could successfully optimize the loss. Hence, these results confirm the importance of warm-start for an improved performance.

6 CONCLUSION

We studied AmIGO, an algorithm for bilevel optimization based on amortized implicit differentiation and introduced a unified framework for analyzing its convergence. Our analysis showed that AmIGO achieves the same complexity as unbiased oracle methods, thus achieving improved rates compared to methods without warm-start in various settings. We then illustrated empirically such improved convergence in both synthetic and a hyper-optimization experiments. A future research direction consists in extending the proposed framework to non-smooth objectives and analyzing acceleration in both inner and outer level problems as well as variance reduction techniques.

7 ACKNOWLEDGMENTS AND FUNDING

This project was supported by the ERC grant number 714381 (SOLARIS project) and by ANR 3IA MIAI@Grenoble Alpes, (ANR19-P3IA-0003).

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A CONVERGENCE OF AMIGO ALGORITHM

In this section, we provide a proof of Theorem 1 as well as its corollaries Corrolaries 1 to 4. In Appendix A.1, we provide an outline of the proof Theorem 1 that states the main intermediary results needed for the proof and provide explicit expressions for the quantities needed throughout the rest of the paper. Appendices A.2 and A.3 provide the proofs of Theorem 1 and Corrolaries 1 to 4. The proofs of the intermediary results are deferred to Appendices B and C.

A.1 PROOF OUTLINE OF THEOREM 1

The proof of Theorem 1 proceeds in 8 steps as discussed below.

Step 1: Smoothness properties. This step consists in characterizing the smoothness of $\nabla \mathcal{L}$ , $y^{\star}$ , $z^{\star}$ as well as the conditional expectation $\mathbb{E}[\hat{\psi}k|x_k,y_k,z_k]$ knowing $x{k},y_{k}$ and $z_{k}$ . For this purpose, we consider the function $\Psi :\mathcal{X}\times \mathcal{Y}\times \mathcal{Y}\to \mathcal{X}$ defined as follows:

$$\mathrm{\Psi }(x,y,z):={\mathrm{\partial }}_{x}f(x,y)+{\mathrm{\partial }}_{xy}g(x,y)z\mathrm{.}\backslash Psi\; (x,\; y,\; z)\; :=\; \backslash partial\_\; \{x\}\; f\; (x,\; y)\; +\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y)\; z.$$

Hence, by definition of $\hat{\psi}_k$ , it is easy to see that $\mathbb{E}[\hat{\psi}_k|x_k,y_k,z_k] = \Psi (x_k,y_k,z_k)$ . The following proposition controls the smoothness of $\nabla \mathcal{L}$ , $y^{\star}$ , $z^{\star}$ and $\Psi$ and is adapted from (Ghadimi and Wang, 2018, Lemma 2.2). We provide a proof in Appendix B.2 for completeness.

Proposition 6. Under Assumptions 1 to 3, $\mathcal{L}$ , $\Psi$ , $y^{\star}$ and $z^{\star}$ satisfy:

$$\begin{array}{c}\mathrm{\parallel }{y}^{\star }(x)-y^{\star }\left(x^{\mathrm{\prime }}\right)\mathrm{\parallel }\le L_y\mathrm{\parallel }x-x^{\mathrm{\prime }}\mathrm{\parallel },\mathrm{\parallel }{z}^{\star }(x,y)-z^{\star }(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\parallel }\le L_z(\mathrm{\parallel }x-x^{\mathrm{\prime }}\mathrm{\parallel }+\mathrm{\parallel }y-y^{\mathrm{\prime }}\mathrm{\parallel })\\ \mathrm{\parallel }\mathrm{\nabla }\mathcal{L}(x)-\mathrm{\nabla }\mathcal{L}\left(x^{\mathrm{\prime }}\right)\mathrm{\parallel }\le L\mathrm{\parallel }x-x^{\mathrm{\prime }}\mathrm{\parallel },\mathrm{\parallel }\mathrm{\Psi }(x,y,z)-\mathrm{\nabla }\mathcal{L}(x)\mathrm{\parallel }\le L_{\psi }[\mathrm{\parallel }y-y^{\star }(x)\mathrm{\parallel }+\mathrm{\parallel }z-z^{\star }(x,y)\mathrm{\parallel }]\\ \parallel z^{\star }(x,y^{\star }(x))\parallel \le {\mu }_g^{-1}B,\end{array}\backslash begin\{array\}\{l\}\; \backslash |\; y\; ^\; \{\backslash star\}\; (x)\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\}\backslash right)\; \backslash |\; \backslash leq\; L\; \_\; \{y\}\; \backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash |,\; \backslash quad\; \backslash |\; z\; ^\; \{\backslash star\}\; (x,\; y)\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash |\; \backslash leq\; L\; \_\; \{z\}\; (\backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash |\; +\; \backslash |\; y\; -\; y\; ^\; \{\backslash prime\}\; \backslash |)\; \backslash \backslash \; \backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x)\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; ^\; \{\backslash prime\}\backslash right)\; \backslash |\; \backslash leq\; L\; \backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash |,\; \backslash quad\; \backslash |\; \backslash Psi\; (x,\; y,\; z)\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x)\; \backslash |\; \backslash leq\; L\; \_\; \{\backslash psi\}\; [\; \backslash |\; y\; -\; y\; ^\; \{\backslash star\}\; (x)\; \backslash |\; +\; \backslash |\; z\; -\; z\; ^\; \{\backslash star\}\; (x,\; y)\; \backslash |\; ]\; \backslash \backslash \; \backslash left\backslash |\; z\; ^\; \{\backslash star\}\; \backslash left(x,\; y\; ^\; \{\backslash star\}\; (x)\backslash right)\; \backslash right\backslash |\; \backslash leq\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; B,\; \backslash \backslash \; \backslash end\{array\}$$

where $L_y, L_z, L_\psi$ and $L$ are given by:

$$\begin{array}{cccc}& L_y:={\mu }_g^{-1}{L}_g^{\mathrm{\prime }},L_z:={\mu }_g^{-2}{M}_{g}B+{\mu }_g^{-1}{L}_f,& & \text{(13)}\end{array}L\; \_\; \{y\}\; :=\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; L\; \_\; \{g\}\; ^\; \{\backslash prime\},\; \backslash quad\; L\; \_\; \{z\}\; :=\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; M\; \_\; \{g\}\; B\; +\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; L\; \_\; \{f\},\; \backslash tag\; \{13\}$$

$$L_{\psi }:=\max ((L_f+M_g{\mu }_g^{-1}B+L_g^{\mathrm{\prime }}{L}_z),L_g^{\mathrm{\prime }})L\; \_\; \{\backslash psi\}\; :=\; \backslash max\; \backslash left(\backslash left(L\; \_\; \{f\}\; +\; M\; \_\; \{g\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; B\; +\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; L\; \_\; \{z\}\backslash right),\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\backslash right)$$

$$L:=(L_f+{\mu }_g^{-2}{L}_g^{\mathrm{\prime }}{M}_{g}B+{\mu }_g^{-1}(L_g^{\mathrm{\prime }}{L}_f+M_{g}B))(1+{\mu }_g^{-1}{L}_g^{\mathrm{\prime }})L\; :=\; \backslash left(L\; \_\; \{f\}\; +\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; M\; \_\; \{g\}\; B\; +\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash left(L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; L\; \_\; \{f\}\; +\; M\; \_\; \{g\}\; B\backslash right)\backslash right)\; \backslash left(1\; +\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\backslash right)$$

The expressions of $L_y$ , $L_z$ , $L_\psi$ and $L$ suggests the following dependence on the conditioning $\kappa_g$ of the inner-level problem which will be useful for the complexity analysis: $L_y = O(\kappa_g)$ , $L_\psi = O(\kappa_g^2)$ , $L_z = O(\kappa_g^2)$ and $L = O(\kappa_g^3)$ .

Step 2: Convergence of the inner-level iterates. In this step, we control the mean squared errors $\mathbb{E}\left[| y_k - y^\star (x_k)|^2\right]$ and $\mathbb{E}\left[| z_k - z^\star (x_k,y_k)|^2\right]$ as stated in Proposition 3. In fact we prove a slightly stronger version stated below:

Proposition 7. Let $\alpha_{k}$ and $\beta_{k}$ be two positive sequences with $\alpha_{k} \leq L_{g}^{-1}$ and $\beta_{k} \leq \frac{1}{2L_{g}}\min \left(1, \frac{2L_{g}}{\mu_{g}(1 + \mu_{g}^{-2}\sigma_{gy}^{2})}\right)$ and define $\Lambda_{k} := (1 - \alpha_{k}\mu_{g})^{T}$ and $\Pi_{k} := \left(1 - \frac{\beta_{k}\mu_{g}}{2}\right)^{N}$ . Denote by $\bar{z}{k}$ the conditional expectation of $z{k}$ knowing $x_{k}, y_{k}$ and $z_{k}^{0}$ . Let $R_{k}^{y}$ and $R_{k}^{z}$ be defined as:

$$\begin{array}{cccc}& R_k^y:=2{\alpha }_k{\mu }_g^{-1}{\sigma }_g^2,R_k^z:={\beta }_k{B}^2{\mu }_g^{-3}{\sigma }_{g_{yy}}^2+3{\mu }_g^{-2}{\sigma }_f^2\mathrm{.}& & \text{(14)}\end{array}R\; \_\; \{k\}\; ^\; \{y\}\; :=\; 2\; \backslash alpha\_\; \{k\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\},\; \backslash quad\; R\; \_\; \{k\}\; ^\; \{z\}\; :=\; \backslash beta\_\; \{k\}\; B\; ^\; \{2\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 3\}\; \backslash sigma\_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; 3\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{f\}\; ^\; \{2\}.\; \backslash tag\; \{14\}$$

Then, under Assumptions 1, 4 and 5 the iterates $z_{k}$ and $\bar{z}_k$ satisfy:

$$\mathbb{E}[\mathrm{\parallel }{y}_k-y^{\star }(x_k){\mathrm{\parallel }}^2]\le {\mathrm{\Lambda }}_k{E}_k^y+R_k^y\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash Lambda\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{y\}\; +\; R\; \_\; \{k\}\; ^\; \{y\}$$

$$\mathbb{E}\left[{\parallel z_k-z^{\star }(x_k,y_k)\parallel }^2\right]\le {\mathrm{\Pi }}_k\mathbb{E}\left[{\parallel z_k^0-z^{\star }(x_k,y_k)\parallel }^2\right]+R_k^z,\backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; \_\; \{k\}\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash Pi\_\; \{k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; \_\; \{k\}\; ^\; \{0\}\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; +\; R\; \_\; \{k\}\; ^\; \{z\},$$

$$\mathbb{E}[\mathrm{\parallel }{\bar{z}}_k-z^{\star }(x_k,y_k){\mathrm{\parallel }}^2]\le {\mathrm{\Pi }}_k\mathbb{E}\left[{\parallel z_k^0-z^{\star }(x_k,y_k)\parallel }^2\right]\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; \backslash bar\; \{z\}\; \_\; \{k\}\; -\; z\; ^\; \{\backslash star\}\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash Pi\_\; \{k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; \_\; \{k\}\; ^\; \{0\}\; -\; z\; ^\; \{\backslash star\}\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]$$

$$\begin{array}{cccc}& \mathbb{E}\left[{\parallel z_k-{\bar{z}}_k\parallel }^2\right]\le 4{\sigma }_g^2{\mu }_g^{-2}{\mathrm{\Pi }}_k\mathbb{E}\left[{\parallel z_k^0-z^{\star }(x_k,y_k)\parallel }^2\right]+2R_k^z\mathrm{.}& & \text{(15a)}\end{array}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; \_\; \{k\}\; -\; \backslash bar\; \{z\}\; \_\; \{k\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; 4\; \backslash sigma\_\; \{g\}\; ^\; \{2\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash Pi\_\; \{k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; \_\; \{k\}\; ^\; \{0\}\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; +\; 2\; R\; \_\; \{k\}\; ^\; \{z\}.\; \backslash tag\; \{15a\}$$

It is easy to see from the above expressions that $R_{k}^{y} = O\big(\kappa_{g}\sigma_{g}^{2}\big)$ while $R_{k}^{z} = O\Big(\kappa_{g}^{3}\sigma_{g_{y}y}^{2} + \kappa_{g}^{2}\sigma_{f}^{2}\Big)$ as stated in Proposition 3. Controlling $\mathbb{E}[| y_k - y^\star (x_k)| ]^2$ follows by standard results on SGD (Kulunchakov and Mairal, 2020, Corollary 31) since the iterates of Algorithm 2 uses i.i.d. samples. The error terms $\mathbb{E}\Big[| z_k - z^\star (x_k,y_k)| ^2\Big]$ is more delicate since Algorithm 3 uses the same sample $\partial_y\hat{f} (x_k,y_k)$ for updating the iterates, therefore introducing additional correlations between these iterates. We defer the proof of Proposition 7 to Appendix B.3 which relies on a general result for stochastic linear systems with correlated noise provided in Appendix E.

Step 3: Controlling the bias and variance errors $V_{k}^{\psi}$ and $E_{k}^{\psi}$ is achieved by Proposition 4. The bias $E_{k}^{\psi}$ is controlled simply by using the smoothness of the potential $\Psi$ near the point $(x_{k},y^{\star}(x_{k}),z^{\star}(x_{k},y^{\star}(x_{k}))$ as shown in Proposition 6. The variance term $V_{k}^{\psi}$ is more delicate to control due to the multiplicative noise resulting from the Jacobian-vector product between $\partial_{xy}\hat{g} (x_k,y_k,\mathcal{D}{g{xy}})z_k$ . We defer the proof to Appendix B.4 and provide below explicit expressions for the constants $\sigma_x^2$ and $w_{x}^{2}$ :

$$\begin{array}{cccc}& \begin{array}{c}{w}_x^2:=(1+2L_g^{\mathrm{\prime }}{\mu }_g^{-1}+6({\sigma }_{g_{xy}}^2+{\left(L_g^{\mathrm{\prime }}\right)}^2){\mu }_g^{-2}){\sigma }_f^2\\ +2({\sigma }_{g_{xy}}^2+{\left(L_g^{\mathrm{\prime }}\right)}^2)B^2{L}_g^{-1}{\mu }_g^{-3}{\sigma }_{g_{yy}}^2+2B^2{\mu }_g^{-2}{\sigma }_{g_{xy}}^2\mathrm{.}\\ {\sigma }_x^2:=2{\sigma }_{g_{xy}}^2+2{\left(L_g^{\mathrm{\prime }}\right)}^2{\mu }_g^{-2}{\sigma }_{g_{yy}}^2\mathrm{.}\end{array}& & \text{(16a)}\end{array}\backslash begin\{array\}\{l\}\; w\; \_\; \{x\}\; ^\; \{2\}\; :=\; \backslash left(1\; +\; 2\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; +\; 6\; \backslash left(\backslash sigma\_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash left(L\; \_\; \{g\}\; ^\; \{\backslash prime\}\backslash right)\; ^\; \{2\}\backslash right)\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\backslash right)\; \backslash sigma\_\; \{f\}\; ^\; \{2\}\; \backslash tag\; \{16a\}\; \backslash \backslash \; +\; 2\; \backslash left(\backslash sigma\_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash left(L\; \_\; \{g\}\; ^\; \{\backslash prime\}\backslash right)\; ^\; \{2\}\backslash right)\; B\; ^\; \{2\}\; L\; \_\; \{g\}\; ^\; \{-\; 1\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 3\}\; \backslash sigma\_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; 2\; B\; ^\; \{2\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}.\; \backslash \backslash \; \backslash sigma\_\; \{x\}\; ^\; \{2\}\; :=\; 2\; \backslash sigma\_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; 2\; \backslash left(L\; \_\; \{g\}\; ^\; \{\backslash prime\}\backslash right)\; ^\; \{2\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}.\; \backslash \backslash \; \backslash end\{array\}$$

Note that $w_{x}^{2} = O\left(\kappa_{g}^{2}\left(\sigma_{f}^{2} + \sigma_{g_{xy}}^{2}\right) + \kappa_{g}^{3}\sigma_{g_{yy}}^{2}\right)$ and $\sigma_{x}^{2} = O\left(\sigma_{g_{xy}}^{2} + \kappa_{g}^{2}\sigma_{g_{yy}}^{2}\right)$ , as stated in Proposition 4.

Step 4: Outer-level error bound. This step consists in obtaining the inequality in Proposition 2 which extends the result of (Kulunchakov and Mairal, 2020, Proposition 1) to biased gradients and to the non-convex case. We defer the proof of such result to Appendix C.1.

Step 5: Inner-level error bound. This step consists in proving Proposition 5. For clarity, we provide a second statement with explicit expressions for the quantities of interest:

Proposition 8. Let $r_k$ and $\theta_k$ be two positive non-increasing sequences no greater than 1. For any $0 \leq v \leq 1$ , denote by $\phi_k$ and $\tilde{R}_k$ the following non-negative scalars:

Finally, consider the following matrices and vectors:

Then, under Assumptions 1 to 5, the following holds:

$$\left(\begin{array}{c}{E}_k^y\\ E_k^z\end{array}\right)\le \left(\begin{array}{c}{\mathrm{\Lambda }}_{k-1}{E}_{k-1}^y+R_{k-1}^y\\ {\mathrm{\Pi }}_{k-1}{E}_{k-1}^z+R_{k-1}^z\end{array}\right)+2{\gamma }_k(E_{k-1}^{\psi }+V_{k-1}^{\psi }+{\zeta }_k{E}_{k-1}^x)+\mathrm{.}\backslash left(\; \backslash begin\{array\}\{c\}\; E\; \_\; \{k\}\; ^\; \{y\}\; \backslash \backslash \; E\; \_\; \{k\}\; ^\; \{z\}\; \backslash end\{array\}\; \backslash right)\; \backslash leq\; \backslash boldsymbol\; \{P\; \_\; \{k\}\}\; \backslash left(\; \backslash begin\{array\}\{c\}\; \backslash Lambda\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; \backslash \backslash \; \backslash Pi\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; \backslash end\{array\}\; \backslash right)\; +\; 2\; \backslash gamma\_\; \{k\}\; \backslash Big\; (E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; \backslash zeta\_\; \{k\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; \backslash Big)\; \backslash boldsymbol\; \{U\; \_\; \{k\}\}\; +\; \backslash boldsymbol\; \{V\; \_\; \{k\}\}.$$

We defer the proof of the above result to Appendix C.2.

Step 6: General error bound. By combining the inequalities in Propositions 2 and 8 resulting from the analysis of both outer and inner levels, we obtain a general error bound on $E_k^{tot}$ in Proposition 9 with a proof in Appendix C.4.

Proposition 9. Choose the step-sizes $\alpha_{k}$ and $\beta_{k}$ such that they are non-increasing in $k$ and choose $r_k$ and $\theta_{k}$ such that $\delta_{k}r_{k}^{-1}$ and $\delta_{k}\theta_{k}^{-1}$ are non-increasing sequences. Choose the coefficients $a_{k}$ and $b_{k}$ defining $E_{k}^{tot}$ in (7) to be of the form $a_{k} = \delta_{k}r_{k}^{-1}\Lambda_{k}^{s}$ and $b_{k} = \delta_{k}\theta_{k}^{-1}\Pi_{k}^{s}$ for some $0 < s < 1$ , and fix a non-increasing sequence $0 < \rho_{k} < 1$ . Then, under Assumptions 1 to 5 $E_{k}^{tot}$ satisfies:

$$F_k+E_k^{tot}\le \mathrm{\parallel }{A}_k{\mathrm{\parallel }}_{\mathrm{\infty }}{E}_{k-1}^{tot}+V_k^{tot}\mathrm{.}F\; \_\; \{k\}\; +\; E\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; \backslash leq\; \backslash |\; A\; \_\; \{k\}\; \backslash |\; \_\; \{\backslash infty\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{t\; o\; t\}\; +\; V\; \_\; \{k\}\; ^\; \{t\; o\; t\}.$$

where $V_{k}$ and $\pmb{A}_{\pmb{k}}$ are given by:

$$\begin{array}{cccc}& A_k:=\left(\begin{array}{c}1-(1-\frac{1}{2}{\rho }_k){\delta }_k+2{\zeta }_k{\lambda }_k{u}_k^I\\ {\mathrm{\Lambda }}_k^{1-s}(1+r_k(1+16L_z^2{\mathrm{\Pi }}_k^s{\theta }_k^{-2}{\varphi }_k+2L_{\psi }^2{\eta }_k^{-1}(2u_k^I+s_k+{\rho }_k^{-1})))\\ {\mathrm{\Pi }}_k^{1-s}(1+{\theta }_k(1+{\eta }_k^{-1}(2L_{\psi }^2+{\sigma }_x^2)(2u_k^I+s_k+{\rho }_k^{-1})))\end{array}\right)& & \text{(18)}\end{array}A\; \_\; \{k\}\; :=\; \backslash left(\; \backslash begin\{array\}\{c\}\; 1\; -\; \backslash left(1\; -\; \backslash frac\; \{1\}\{2\}\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\; +\; 2\; \backslash zeta\_\; \{k\}\; \backslash lambda\_\; \{k\}\; u\; \_\; \{k\}\; ^\; \{I\}\; \backslash \backslash \; \backslash Lambda\_\; \{k\}\; ^\; \{1\; -\; s\}\; \backslash left(1\; +\; r\; \_\; \{k\}\; \backslash left(1\; +\; 1\; 6\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; \backslash phi\_\; \{k\}\; +\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash eta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left(2\; u\; \_\; \{k\}\; ^\; \{I\}\; +\; s\; \_\; \{k\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\backslash right)\backslash right)\; \backslash \backslash \; \backslash Pi\_\; \{k\}\; ^\; \{1\; -\; s\}\; \backslash left(1\; +\; \backslash theta\_\; \{k\}\; \backslash left(1\; +\; \backslash eta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left(2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; +\; \backslash sigma\_\; \{x\}\; ^\; \{2\}\backslash right)\; \backslash left(2\; u\; \_\; \{k\}\; ^\; \{I\}\; +\; s\; \_\; \{k\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\backslash right)\backslash right)\; \backslash end\{array\}\; \backslash right)\; \backslash tag\; \{18\}$$

$$\begin{array}{c}{V}_k^{tot}:={\delta }_k({\mathrm{\Lambda }}_k^s(1+r_k^{-1})+16L_z^2{\varphi }_k{\theta }_k^{-2}{\mathrm{\Pi }}_k^s+2L_{\psi }^2{\eta }_k^{-1}(2u_k^I+s_k+{\rho }_k^{-1}))R_{k-1}^y\\ +{\delta }_k((1+{\theta }_k^{-1}){\mathrm{\Pi }}_k^s{R}_{k-1}^z+8L_z^2{\theta }_k^{-2}{\mathrm{\Pi }}_k^s{\tilde{R}}_k^y)+{\gamma }_k(s_k+u_k^I)w_x^2\end{array}\backslash begin\{array\}\{l\}\; V\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; :=\; \backslash delta\_\; \{k\}\; \backslash left(\backslash Lambda\_\; \{k\}\; ^\; \{s\}\; \backslash left(1\; +\; r\; \_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\; +\; 1\; 6\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash phi\_\; \{k\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; +\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash eta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left(2\; u\; \_\; \{k\}\; ^\; \{I\}\; +\; s\; \_\; \{k\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\backslash right)\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; \backslash \backslash \; +\; \backslash delta\_\; \{k\}\; \backslash Big\; ((1\; +\; \backslash theta\_\; \{k\}\; ^\; \{-\; 1\})\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; +\; 8\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; \backslash tilde\; \{R\}\; \_\; \{k\}\; ^\; \{y\}\; \backslash Big)\; +\; \backslash gamma\_\; \{k\}\; \backslash big\; (s\; \_\; \{k\}\; +\; u\; \_\; \{k\}\; ^\; \{I\}\; \backslash big)\; w\; \_\; \{x\}\; ^\; \{2\}\; \backslash \backslash \; \backslash end\{array\}$$

where we introduced $u_{k}^{I} = a_{k}U_{k}^{(1)} + b_{k}U_{k}^{(2)}$ for conciseness with $U_{k}^{(1)}$ and $U_{k}^{(2)}$ being the components of the vector $\mathbf{U}_{\mathbf{k}}$ defined in Proposition 8.

Proposition 9 holds without conditions on the error made by Algorithms 2 and 3. The general form of $a_{k}$ and $b_{k}$ allows to account for potentially decreasing step-sizes $\gamma_{k}$ , $\alpha_{k}$ and $\beta_{k}$ . However, in the present work, we will restrict to the constant step-size for ease of presentation as we discuss next.

Step 7: Controlling the precision of the inner-level algorithms. In this step, we provide conditions on $T$ and $N$ in Proposition 10 below so that $| A_k|_\infty \leq 1 - (1 - \rho_k)\delta_k$ in the constant step-size case. These conditions are expressed in terms of the following constants:

$$\begin{array}{cccc}& C_1:=1+2\log (6+24L_{\psi }^2{\eta }_0^{-1})& & \text{(19a)}\end{array}C\; \_\; \{1\}\; :=\; 1\; +\; 2\; \backslash log\; \backslash left(6\; +\; 2\; 4\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash eta\_\; \{0\}\; ^\; \{-\; 1\}\backslash right)\; \backslash tag\; \{19a\}$$

$$\begin{array}{cccc}& C_2:=2\log (1+4L_y^2{L}^{-2}\max ({\eta }_0,8{\zeta }_0))& & \text{(19b)}\end{array}C\; \_\; \{2\}\; :=\; 2\; \backslash log\; \backslash left(1\; +\; 4\; L\; \_\; \{y\}\; ^\; \{2\}\; L\; ^\; \{-\; 2\}\; \backslash max\; \backslash left(\backslash eta\_\; \{0\},\; 8\; \backslash zeta\_\; \{0\}\backslash right)\backslash right)\; \backslash tag\; \{19b\}$$

$$\begin{array}{cccc}& C_3:=\max (0,-2\log \left(5L_{\psi }^2{\eta }_0^{-1}\right),-2\log \left(\frac{L}{4L_y^2}\right))& & \text{(19c)}\end{array}C\; \_\; \{3\}\; :=\; \backslash max\; \backslash left(0,\; -\; 2\; \backslash log\; \backslash left(5\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash eta\_\; \{0\}\; ^\; \{-\; 1\}\backslash right),\; -\; 2\; \backslash log\; \backslash left(\backslash frac\; \{L\}\{4\; L\; \_\; \{y\}\; ^\; \{2\}\}\backslash right)\backslash right)\; \backslash tag\; \{19c\}$$

$$\begin{array}{cccc}& C_1^{\mathrm{\prime }}:=1+2\log (4+12{\eta }_0^{-1}(2L_{\psi }^2+{\sigma }_x^2))& & \text{(19d)}\end{array}C\; \_\; \{1\}\; ^\; \{\backslash prime\}\; :=\; 1\; +\; 2\; \backslash log\; \backslash left(4\; +\; 1\; 2\; \backslash eta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash left(2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; +\; \backslash sigma\_\; \{x\}\; ^\; \{2\}\backslash right)\backslash right)\; \backslash tag\; \{19d\}$$

$$\begin{array}{cccc}& C_2^{\mathrm{\prime }}:=2\log (1+2L_z^2{L}_y^{-2}(1+16L_y^2)),& & \text{(19e)}\end{array}C\; \_\; \{2\}\; ^\; \{\backslash prime\}\; :=\; 2\; \backslash log\; \backslash left(1\; +\; 2\; L\; \_\; \{z\}\; ^\; \{2\}\; L\; \_\; \{y\}\; ^\; \{-\; 2\}\; \backslash left(1\; +\; 1\; 6\; L\; \_\; \{y\}\; ^\; \{2\}\backslash right)\backslash right),\; \backslash tag\; \{19e\}$$

$$\begin{array}{cccc}& C_3^{\mathrm{\prime }}:=\max (0,-2\log \left(\frac{L_{\psi }^2}{4L_z^2{\eta }_0}\right),-2\log \left(\gamma ({\sigma }_{g_{xy}}^2+{\left(L_g^{\mathrm{\prime }}\right)}^2)\right))& & \text{(19f)}\end{array}C\; \_\; \{3\}\; ^\; \{\backslash prime\}\; :=\; \backslash max\; \backslash left(0,\; -\; 2\; \backslash log\; \backslash left(\backslash frac\; \{L\; \_\; \{\backslash psi\}\; ^\; \{2\}\}\{4\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash eta\_\; \{0\}\}\backslash right),\; -\; 2\; \backslash log\; \backslash left(\backslash gamma\; \backslash left(\backslash sigma\_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash left(L\; \_\; \{g\}\; ^\; \{\backslash prime\}\backslash right)\; ^\; \{2\}\backslash right)\backslash right)\backslash right)\; \backslash tag\; \{19f\}$$

Proposition 10. Let Assumptions 1 to 5 hold. Choose $\rho_{k} = \frac{1}{2}$ , the step-sizes to be constant: $\alpha_{k} = \alpha \leq \frac{1}{L_{g}}$ , $\beta_{k} = \beta \leq \frac{1}{2L_{g}}$ and $\gamma_{k} = \gamma \leq \frac{1}{L}$ and choose the batch-size $|\mathcal{D}{g{yy}}| = \Theta \left(\frac{\tilde{\sigma}{g{yy}}^2}{\mu_g L_g}\right)$ . Moreover, set $s = \frac{1}{2}$ , $r_k = \theta_k = 1$ . Finally, choose $T$ and $N$ as follows:

$$T=\lfloor {\alpha }^{-1}{\mu }_g^{-1}\max (C_1,C_2,C_3)\rfloor +1,T\; =\; \backslash left\backslash lfloor\; \backslash alpha^\; \{-\; 1\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash max\; \backslash left(C\; \_\; \{1\},\; C\; \_\; \{2\},\; C\; \_\; \{3\}\backslash right)\; \backslash right\backslash rfloor\; +\; 1,$$

$$N=\lfloor 2{\beta }^{-1}{\mu }_g^{-1}(\max (C_1,C_2,C_3)+\max (C_1^{\mathrm{\prime }},C_2^{\mathrm{\prime }},C_3^{\mathrm{\prime }}))\rfloor +1N\; =\; \backslash left\backslash lfloor\; 2\; \backslash beta^\; \{-\; 1\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash left(\backslash max\; \backslash left(C\; \_\; \{1\},\; C\; \_\; \{2\},\; C\; \_\; \{3\}\backslash right)\; +\; \backslash max\; \backslash left(C\; \_\; \{1\}\; ^\; \{\backslash prime\},\; C\; \_\; \{2\}\; ^\; \{\backslash prime\},\; C\; \_\; \{3\}\; ^\; \{\backslash prime\}\backslash right)\backslash right)\; \backslash right\backslash rfloor\; +\; 1$$

with $C_1, C_2, C_3, C_1', C_2'$ and $C_3'$ defined in (19a) to (19f). Then, $|A_k|_{\infty} \leq 1 - (1 - \rho_k)\delta_k$ and $V_k^{tot} \leq \gamma \delta_0 \mathcal{W}^2$ , with $\mathcal{W}^2$ given by:

$$\begin{array}{cccc}& {\mathcal{W}}^2:=({\delta }_0^{-1}\frac{1-u}{2}{\mathbb{1}}_{\mu >0}+3)w_x^2+\frac{60L_{\psi }^2}{{\delta }_0{\mu }_g{L}_g}{\sigma }_g^2& & \text{(20)}\end{array}\backslash mathcal\; \{W\}\; ^\; \{2\}\; :=\; \backslash left(\backslash delta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash frac\; \{1\; -\; u\}\{2\}\; \backslash mathbb\; \{1\}\; \_\; \{\backslash mu\; >\; 0\}\; +\; 3\backslash right)\; w\; \_\; \{x\}\; ^\; \{2\}\; +\; \backslash frac\; \{6\; 0\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\}\{\backslash delta\_\; \{0\}\; \backslash mu\_\; \{g\}\; L\; \_\; \{g\}\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\}\; \backslash tag\; \{20\}$$

We provide a proof of Proposition 10 in Appendix D. It is easy to see from Proposition 10 that that $T = O(\kappa_g)$ and $N = O(\kappa_g)$ when $\alpha = \frac{1}{L_g}$ and $\beta = \frac{1}{2L_g}$ , where the big- $O$ notation hides a logarithmic dependence in $\kappa_g$ coming from the constants ${C_i, C_i' | i \in {1, 2, 3}}$ .

Step 8: Proving the main inequalities. The final step combines Propositions 9 and 10 to get the desired inequality. We provide a full proof in Appendix A.2 assuming Propositions 9 and 10 hold.

A.2 PROOF OF THEOREM 1

In order to prove Theorem 1 in the convex case, we need the following averaging strategy lemma, a generalization of (Kulunchakov and Mairal, 2020, Lemma 30):

Lemma 1. Let $\mathcal{L}$ be a convex function on $\mathcal{X}$ . Let $x_{k}$ be a (potentially stochastic) sequence of iterates in $\mathcal{X}$ . Let $(E_k){k\geq 0}$ , $(V{k}){k\geq 0}$ and $(\delta_k){k\geq 0}$ be non-negative sequences such that $\delta_{k}\in (0,1)$ . Fix some non-negative number $u\in [0,1]$ and define the following averaged iterates $\hat{x}k$ recursively by $\hat{x}k = u(1 - \delta_k)\hat{x}{k - 1} + (1 - (1 - \delta_k)u)x_k$ and starting from any initial point $\hat{x}0$ . Assume the iterates $(x{k}){k\geq 1}$ satisfy the following relation for all $k\geq 1$ :

$$\begin{array}{cccc}& (1-u(1-{\delta }_k))\mathbb{E}[\mathcal{L}(x_k)-{\mathcal{L}}^{\star }]+E_k\le (1-{\delta }_k)(E_{k-1}+(1-u)\mathbb{E}[\mathcal{L}(x_{k-1})-{\mathcal{L}}^{\star }])+V_k\mathrm{.}& & \text{(21)}\end{array}(1\; -\; u\; (1\; -\; \backslash delta\_\; \{k\}))\; \backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\; +\; E\; \_\; \{k\}\; \backslash leq\; (1\; -\; \backslash delta\_\; \{k\})\; \backslash left(E\; \_\; \{k\; -\; 1\}\; +\; (1\; -\; u)\; \backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\backslash right)\; +\; V\; \_\; \{k\}.\; \backslash tag\; \{21\}$$

Let $\Gamma_{k} := \prod_{t=1}^{k}(1 - \delta_{k})$ . Then the averaged iterates $(\hat{x}{k}){k \geq 1}$ satisfy the following:

$$\mathbb{E}[\mathcal{L}\left({\widehat{x}}_k\right)-{\mathcal{L}}^{\star }]+E_k\le {\mathrm{\Gamma }}_k(E_0+\mathbb{E}[\mathcal{L}\left(x_0\right)-{\mathcal{L}}^{\star }+u^k(\mathcal{L}\left({\widehat{x}}_0\right)-{\mathcal{L}}^{\star })])+{\mathrm{\Gamma }}_k\sum _{1\le t\le k}{\mathrm{\Gamma }}_t^{-1}{V}_t\mathrm{.}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash mathcal\; \{L\}\; \backslash left(\backslash hat\; \{x\}\; \_\; \{k\}\backslash right)\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; \backslash right]\; +\; E\; \_\; \{k\}\; \backslash leq\; \backslash Gamma\_\; \{k\}\; \backslash left(E\; \_\; \{0\}\; +\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{0\}\backslash right)\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; +\; u\; ^\; \{k\}\; \backslash left(\backslash mathcal\; \{L\}\; \backslash left(\backslash hat\; \{x\}\; \_\; \{0\}\backslash right)\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\backslash right)\; \backslash right]\backslash right)\; +\; \backslash Gamma\_\; \{k\}\; \backslash sum\_\; \{1\; \backslash leq\; t\; \backslash leq\; k\}\; \backslash Gamma\_\; \{t\}\; ^\; \{-\; 1\}\; V\; \_\; \{t\}.$$

Proof. For simplicity, we write $F_{k} = \mathbb{E}[\mathcal{L}(x_{k}) - \mathcal{L}^{\star}]$ and $\hat{F}_k = \mathbb{E}[\mathcal{L}(\hat{x}_k) - \mathcal{L}^\star ]$ . We first multiply (21) by $\Gamma_k^{-1}$ and sum the resulting inequalities for all $1\leq k\leq K$ to get:

$$\sum _{k=1}^K{\mathrm{\Gamma }}_k^{-1}(1-u(1-{\delta }_k))F_k+{\mathrm{\Gamma }}_k^{-1}{T}_k\le \sum _{k=1}^K{\mathrm{\Gamma }}_k^{-1}(1-{\delta }_k)(T_{k-1}+(1-u)F_{k-1})+\sum _{k=1}^K{\mathrm{\Gamma }}_k^{-1}{V}_k\mathrm{.}\backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; (1\; -\; u\; (1\; -\; \backslash delta\_\; \{k\}))\; F\; \_\; \{k\}\; +\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; T\; \_\; \{k\}\; \backslash leq\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; (1\; -\; \backslash delta\_\; \{k\})\; (T\; \_\; \{k\; -\; 1\}\; +\; (1\; -\; u)\; F\; \_\; \{k\; -\; 1\})\; +\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; V\; \_\; \{k\}.$$

Grouping the terms in $F_{k}$ together and recalling that $\Gamma_k^{-1}(1 - \delta_k) = \Gamma_{k - 1}^{-1}$ yields:

$$\sum _{k=1}^K{\mathrm{\Gamma }}_k^{-1}{F}_k-{\mathrm{\Gamma }}_{k-1}^{-1}{F}_{k-1}+u\sum _{k=1}^K{\mathrm{\Gamma }}_{k-1}^{-1}(F_{k-1}-F_k)\le \sum _{k=1}^K({\mathrm{\Gamma }}_{k-1}^{-1}{T}_{k-1}-{\mathrm{\Gamma }}_k^{-1}{T}_k)+\sum _{k=1}^K{\mathrm{\Gamma }}_k^{-1}{V}_k\mathrm{.}\backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; F\; \_\; \{k\}\; -\; \backslash Gamma\_\; \{k\; -\; 1\}\; ^\; \{-\; 1\}\; F\; \_\; \{k\; -\; 1\}\; +\; u\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash Gamma\_\; \{k\; -\; 1\}\; ^\; \{-\; 1\}\; \backslash left(F\; \_\; \{k\; -\; 1\}\; -\; F\; \_\; \{k\}\backslash right)\; \backslash leq\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash left(\backslash Gamma\_\; \{k\; -\; 1\}\; ^\; \{-\; 1\}\; T\; \_\; \{k\; -\; 1\}\; -\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; T\; \_\; \{k\}\backslash right)\; +\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; V\; \_\; \{k\}.$$

Simplifying the telescoping sums and multiplying by $\Gamma_{k}$ , we get:

$$\begin{array}{cccc}& F_K+u{\mathrm{\Gamma }}_K\sum _{k=1}^K{\mathrm{\Gamma }}_{k-1}^{-1}(F_{k-1}-F_k)\le -T_K+{\mathrm{\Gamma }}_K(F_0+T_0+\sum _{k=1}^K{\mathrm{\Gamma }}_k^{-1}{V}_k)\mathrm{.}& & \text{(22)}\end{array}F\; \_\; \{K\}\; +\; u\; \backslash Gamma\_\; \{K\}\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash Gamma\_\; \{k\; -\; 1\}\; ^\; \{-\; 1\}\; \backslash left(F\; \_\; \{k\; -\; 1\}\; -\; F\; \_\; \{k\}\backslash right)\; \backslash leq\; -\; T\; \_\; \{K\}\; +\; \backslash Gamma\_\; \{K\}\; \backslash left(F\; \_\; \{0\}\; +\; T\; \_\; \{0\}\; +\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; V\; \_\; \{k\}\backslash right).\; \backslash tag\; \{22\}$$

Consider now the quantity $\hat{F}_k$ . Recalling that $\mathcal{L}$ is convex and by definition of the iterates $\hat{x}_k$ we apply Jensen's inequality to write:

$${\widehat{F}}_k\le u(1-{\delta }_k){\widehat{F}}_{k-1}+(1-u(1-{\delta }_k))F_k\mathrm{.}\backslash hat\; \{F\}\; \_\; \{k\}\; \backslash leq\; u\; (1\; -\; \backslash delta\_\; \{k\})\; \backslash hat\; \{F\}\; \_\; \{k\; -\; 1\}\; +\; (1\; -\; u\; (1\; -\; \backslash delta\_\; \{k\}))\; F\; \_\; \{k\}.$$

By iteratively applying the above inequality, we get that:

$$\begin{array}{c}{\widehat{F}}_K\le u^K{\mathrm{\Gamma }}_K{\widehat{F}}_0+{\mathrm{\Gamma }}_K{\sum }_{k=1}^K{u}^{K-k}{\mathrm{\Gamma }}_k^{-1}(1-u(1-{\delta }_k))F_k\\ =u^K{\mathrm{\Gamma }}_K{\widehat{F}}_0+{\mathrm{\Gamma }}_K{\sum }_{k=1}^K{u}^{K-k}{\mathrm{\Gamma }}_k^{-1}{F}_k-{\sum }_{k=1}^K{u}^{K-k+1}{\mathrm{\Gamma }}_{k-1}{F}_k\\ =u^K{\mathrm{\Gamma }}_K{\widehat{F}}_0+F_K+{\mathrm{\Gamma }}_K{\sum }_{k=1}^{K-1}{u}^{K-k}{\mathrm{\Gamma }}_k^{-1}(F_k-F_{k+1})\end{array}\backslash begin\{array\}\{l\}\; \backslash hat\; \{F\}\; \_\; \{K\}\; \backslash leq\; u\; ^\; \{K\}\; \backslash Gamma\_\; \{K\}\; \backslash hat\; \{F\}\; \_\; \{0\}\; +\; \backslash Gamma\_\; \{K\}\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; u\; ^\; \{K\; -\; k\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; (1\; -\; u\; (1\; -\; \backslash delta\_\; \{k\}))\; F\; \_\; \{k\}\; \backslash \backslash \; =\; u\; ^\; \{K\}\; \backslash Gamma\_\; \{K\}\; \backslash hat\; \{F\}\; \_\; \{0\}\; +\; \backslash Gamma\_\; \{K\}\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; u\; ^\; \{K\; -\; k\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; F\; \_\; \{k\}\; -\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\}\; u\; ^\; \{K\; -\; k\; +\; 1\}\; \backslash Gamma\_\; \{k\; -\; 1\}\; F\; \_\; \{k\}\; \backslash \backslash \; =\; u\; ^\; \{K\}\; \backslash Gamma\_\; \{K\}\; \backslash hat\; \{F\}\; \_\; \{0\}\; +\; F\; \_\; \{K\}\; +\; \backslash Gamma\_\; \{K\}\; \backslash sum\_\; \{k\; =\; 1\}\; ^\; \{K\; -\; 1\}\; u\; ^\; \{K\; -\; k\}\; \backslash Gamma\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left(F\; \_\; \{k\}\; -\; F\; \_\; \{k\; +\; 1\}\backslash right)\; \backslash \backslash \; \backslash end\{array\}$$

We can therefore apply (22) to the above inequality to get the desired result.

We now proceed to prove Theorem 1.

Proof of Theorem 1. By application of Proposition 9 and using the choice of $T$ and $N$ given by Proposition 10, the following inequality holds:

$$\begin{array}{cccc}& F_k+E_k^{tot}\le (1-(1-{\rho }_k){\delta }_k)E_{k-1}^{tot}+\gamma {\delta }_0{\mathcal{W}}^2\mathrm{.}& & \text{(23)}\end{array}F\; \_\; \{k\}\; +\; E\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; \backslash leq\; (1\; -\; (1\; -\; \backslash rho\_\; \{k\})\; \backslash delta\_\; \{k\})\; E\; \_\; \{k\; -\; 1\}\; ^\; \{t\; o\; t\}\; +\; \backslash gamma\; \backslash delta\_\; \{0\}\; \backslash mathcal\; \{W\}\; ^\; \{2\}.\; \backslash tag\; \{23\}$$

with $\mathcal{W}$ defined in Proposition 10. We then distinguish two cases depending on the sign of $\mu$ :

Case $\mu \geq 0$ . Recall that $F_{k}$ and $E_k^x$ are given by:

$$F_k=u{\delta }_k\mathbb{E}[\mathcal{L}(x_k)-{\mathcal{L}}^{\star }],E_k^x=\frac{{\delta }_k}{2{\gamma }_k}\mathrm{\parallel }{x}_k-x^{\star }{\mathrm{\parallel }}^2+(1-u)\mathbb{E}[\mathcal{L}(x_k)-{\mathcal{L}}^{\star }]F\; \_\; \{k\}\; =\; u\; \backslash delta\_\; \{k\}\; \backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ],\; \backslash qquad\; E\; \_\; \{k\}\; ^\; \{x\}\; =\; \backslash frac\; \{\backslash delta\_\; \{k\}\}\{2\; \backslash gamma\_\; \{k\}\}\; \backslash |\; x\; \_\; \{k\}\; -\; x\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; (1\; -\; u)\; \backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]$$

Since $\mu > 0$ , $\mathcal{L}$ is a convex function and we can apply Lemma 1 with $V_{k} = \gamma \delta_{0}\mathcal{W}^{2}$ and $E_{k} = \frac{\delta_{k}}{2\gamma_{k}} | x_{k} - x^{\star} |^{2} + a_{k}E_{k}^{y} + b_{k}E_{k}^{z}$ . The result follows by noting that $\Gamma_{k}\sum_{t=1}^{k}\Gamma_{t}^{-1} \leq \delta_{0}^{-1}$ .

Case $\mu < 0$ . In this case, we recall that $F_{k}$ and $E_k^x$ are given by:

$$F_k=\mathbb{E}[\mathcal{L}\left(x_k\right)-\mathcal{L}\left(x_{k-1}\right)]+E_{k-1}^x-E_k^x,E_k^x=L^{-1}\mathbb{E}\left[{\parallel \mathrm{\nabla }\mathcal{L}\left(x_k\right)\parallel }^2\right]\mathrm{.}F\; \_\; \{k\}\; =\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; -\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\; \backslash right]\; +\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; -\; E\; \_\; \{k\}\; ^\; \{x\},\; \backslash quad\; E\; \_\; \{k\}\; ^\; \{x\}\; =\; L\; ^\; \{-\; 1\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right].$$

We then sum (23) for all iterations $0 \leq t \leq k$ which, by telescoping, yields:

$$\mathbb{E}[\mathcal{L}\left(x_k\right)-\mathcal{L}\left(x_0\right)]+E_0^x-E_k^x+E_k^{tot}-E_0^{tot}+\sum _{1\le t\le k}(1-{\rho }_t){\delta }_t{E}_t^{tot}\le k\gamma {\delta }_0{\mathcal{W}}^2\mathrm{.}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; -\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{0\}\backslash right)\; \backslash right]\; +\; E\; \_\; \{0\}\; ^\; \{x\}\; -\; E\; \_\; \{k\}\; ^\; \{x\}\; +\; E\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; -\; E\; \_\; \{0\}\; ^\; \{t\; o\; t\}\; +\; \backslash sum\_\; \{1\; \backslash leq\; t\; \backslash leq\; k\}\; \backslash left(1\; -\; \backslash rho\_\; \{t\}\backslash right)\; \backslash delta\_\; \{t\}\; E\; \_\; \{t\}\; ^\; \{t\; o\; t\}\; \backslash leq\; k\; \backslash gamma\; \backslash delta\_\; \{0\}\; \backslash mathcal\; \{W\}\; ^\; \{2\}.$$

Using that $\mathbb{E}[\mathcal{L}(x_k) - \mathcal{L}^\star ] + E_k^{tot} - E_k^x$ is non-negative since $E_{k}^{tot} - E_{k}^{x} = a_{k}E_{k}^{y} + b_{k}E_{k}^{z}$ , we get:

$$\sum _{1\le t\le k}(1-{\rho }_t){\delta }_t{E}_t^x\le (\mathbb{E}[\mathcal{L}(x_0)-{\mathcal{L}}^{\star }]+a_0{E}_0^y+b_0{E}_0^z)+k\gamma {\delta }_0{\mathcal{W}}^2\mathrm{.}\backslash sum\_\; \{1\; \backslash leq\; t\; \backslash leq\; k\}\; (1\; -\; \backslash rho\_\; \{t\})\; \backslash delta\_\; \{t\}\; E\; \_\; \{t\}\; ^\; \{x\}\; \backslash leq\; (\backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{0\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\; +\; a\; \_\; \{0\}\; E\; \_\; \{0\}\; ^\; \{y\}\; +\; b\; \_\; \{0\}\; E\; \_\; \{0\}\; ^\; \{z\})\; +\; k\; \backslash gamma\; \backslash delta\_\; \{0\}\; \backslash mathcal\; \{W\}\; ^\; \{2\}.$$

Finally, since $\rho_t = \frac{1}{2}$ , $\delta_t = L\gamma$ , the result follows after dividing both sides by $\frac{1}{2} kL\gamma$ .

A.3 PROOF OF CORROLARIES 1 TO 4

Proof of Corollary 1. Choosing $u = 0$ implies that $E_k^x = \frac{\mu}{2} | x_k - x^\star |^2 + \mathcal{L}(x_k) - \mathcal{L}^\star \leq E_k^{tot}$ . We can then apply Theorem 1 for $\mu > 0$ which yields the following:

$$\begin{array}{cccc}& \frac{\mu }{2}\mathrm{\parallel }{x}_k-x^{\star }\mathrm{\parallel }+\mathcal{L}(x_k)-{\mathcal{L}}^{\star }\le {(1-{\left(2{\kappa }_{\mathcal{L}}\right)}^{-1})}^k{E}_0^{tot}+\frac{2{\mathcal{W}}^2}{L}\mathrm{.}& & \text{(24)}\end{array}\backslash frac\; \{\backslash mu\}\{2\}\; \backslash |\; x\; \_\; \{k\}\; -\; x\; ^\; \{\backslash star\}\; \backslash |\; +\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; \backslash leq\; \backslash left(1\; -\; \backslash left(2\; \backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\backslash right)\; ^\; \{-\; 1\}\backslash right)\; ^\; \{k\}\; E\; \_\; \{0\}\; ^\; \{t\; o\; t\}\; +\; \backslash frac\; \{2\; \backslash mathcal\; \{W\}\; ^\; \{2\}\}\{L\}.\; \backslash tag\; \{24\}$$

In the deterministic setting, it holds all variances vanish: $\sigma_f^2 = \sigma_g^2 = \sigma_{g_{yy}}^2 = \sigma_{g_{xy}}^2 = 0$ . Hence, $\mathcal{W}^2 = 0$ by definition of $\mathcal{W}^2$ . Therefore, to achieve an error $\mathcal{L}(x_k) - \mathcal{L}^\star \leq \epsilon$ for some $\epsilon > 0$ , (24) suggests choosing $k = O\left(\kappa_{\mathcal{L}}\log \left(\frac{E_0^{tot}}{\epsilon}\right)\right)$ . Additionally, $T = \Theta (\kappa_g)$ and $N = \Theta (\kappa_g)$ as required by Theorem 1 and since $\sigma_{g_{yy}}^2 = 0$ , it holds that $N = O(\kappa_g)$ . Using batches of size 1, yields the desired complexity.

Proof of Corollary 2. Here we choose $u = 1$ and apply Theorem 1 for $\mu > 0$ which yields:

$$\mathbb{E}[\mathcal{L}({\widehat{x}}_k)-{\mathcal{L}}^{\star }]+E_k^{tot}\le {(1-(2{\kappa }_{\mathcal{L}}{)}^{-1})}^k(E_0^{tot}+\mathbb{E}[\mathcal{L}(x_0)-{\mathcal{L}}^{\star }])+2L^{-1}{\mathcal{W}}^2\backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (\backslash hat\; \{x\}\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\; +\; E\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; \backslash leq\; \backslash left(1\; -\; (2\; \backslash kappa\_\; \{\backslash mathcal\; \{L\}\})\; ^\; \{-\; 1\}\backslash right)\; ^\; \{k\}\; \backslash big\; (E\; \_\; \{0\}\; ^\; \{t\; o\; t\}\; +\; \backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{0\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\; \backslash big)\; +\; 2\; L\; ^\; \{-\; 1\}\; \backslash mathcal\; \{W\}\; ^\; \{2\}$$

Hence, to achieve an error $\mathbb{E}[\mathcal{L}(\hat{x}k) - \mathcal{L}^\star ]\leq \epsilon$ , we need $k = O\Big(\kappa{\mathcal{L}}\log \left(\frac{E_0^{tot} + \mathbb{E}[\mathcal{L}(x_0) - \mathcal{L}^\star]}{\epsilon}\right)\Big)$ to guarantee that the first term in the l.h.s. of the above inequality is $O(\epsilon)$ . Moreover, we recall that $L^{-1} = O(\kappa_g^{-3})$ from Proposition 6 and that Theorem 1 ensures the variance $\mathcal{W}$ satisfies:

$${\mathcal{W}}^2=O({\kappa }_g^5{\sigma }_g^2+{\kappa }_g^3{\sigma }_{g_{yy}}^2+{\kappa }_g^2{\sigma }_{g_{xy}}^2+{\kappa }_g^2{\sigma }_f^2)\mathrm{.}\backslash mathcal\; \{W\}\; ^\; \{2\}\; =\; O\; \backslash left(\backslash kappa\_\; \{g\}\; ^\; \{5\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{3\}\; \backslash sigma\_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; \backslash sigma\_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; \backslash sigma\_\; \{f\}\; ^\; \{2\}\backslash right).$$

Hence, ensuring the variance term $2L^{-1}\mathcal{W}^2$ is of order $\epsilon$ is achieved by choosing the size of the batches as follows:

$$\begin{array}{c}\mid {\mathcal{D}}_f\mid =\mathrm{\Theta }\left(\frac{{\tilde{\sigma }}_f^2}{ϵ}\right),\mid {\mathcal{D}}_g\mid =\mathrm{\Theta }\left(\frac{{\kappa }_{\mathcal{L}}{\kappa }_g^2{\tilde{\sigma }}_g^2}{ϵ}\right),\mid {\mathcal{D}}_{g_{xy}}\mid =\mathrm{\Theta }\left(\frac{{\tilde{\sigma }}_{g_{xy}}^2}{ϵ}\right),\\ \mid {\mathcal{D}}_{g_{yy}}\mid =\mathrm{\Theta }\left({\tilde{\sigma }}_{g_{yy}}^2(\frac{1}{ϵ}\vee {\kappa }_g)\right)\end{array}\backslash begin\{array\}\{l\}\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{f\}\; \backslash right|\; =\; \backslash Theta\; \backslash left(\backslash frac\; \{\backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\}\{\backslash epsilon\}\backslash right),\; \backslash quad\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\}\; \backslash right|\; =\; \backslash Theta\; \backslash left(\backslash frac\; \{\backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\}\{\backslash epsilon\}\backslash right),\; \backslash quad\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{x\; y\}\}\; \backslash right|\; =\; \backslash Theta\; \backslash left(\backslash frac\; \{\backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\}\{\backslash epsilon\}\backslash right),\; \backslash \backslash \; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{y\; y\}\}\; \backslash right|\; =\; \backslash Theta\; \backslash left(\backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\}\; \backslash vee\; \backslash kappa\_\; \{g\}\backslash right)\backslash right)\; \backslash \backslash \; \backslash end\{array\}$$

Recall that $T = \Theta(\kappa_g)$ and $N = \Theta(\kappa_g)$ as required by, Theorem 1, thus yielding the desired result.

Proof of Corollary 3. In the non-convex deterministic case, recall that $E_{k}^{x} = \frac{1}{L}| \nabla \mathcal{L}(x_{k})|^{2}\leq E_{k}^{tot}$ . We thus apply Theorem 1 for $\mu < 0$ , multiply by $L$ to get:

$$\frac{1}{k}\sum _{t=1}^k\mathrm{\parallel }\mathrm{\nabla }\mathcal{L}(x_t){\mathrm{\parallel }}^2\le \frac{2L}{k}(\mathcal{L}(x_0)-{\mathcal{L}}^{\star }+(E_0^y+E_0^z))+2{\mathcal{W}}^2\mathrm{.}\backslash frac\; \{1\}\{k\}\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{k\}\; \backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{t\})\; \backslash |\; ^\; \{2\}\; \backslash leq\; \backslash frac\; \{2\; L\}\{k\}\; (\backslash mathcal\; \{L\}\; (x\; \_\; \{0\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; +\; (E\; \_\; \{0\}\; ^\; \{y\}\; +\; E\; \_\; \{0\}\; ^\; \{z\}))\; +\; 2\; \backslash mathcal\; \{W\}\; ^\; \{2\}.$$

The setting being deterministic, it holds that $\mathcal{W}^2 = 0$ . Moreover, recall that $L = O(\kappa_g^3)$ from Proposition 6. Hence, to achieve an error of order $\min_{1\leq t\leq k}| \nabla \mathcal{L}(x_t)|^2\leq \epsilon$ , it suffices to choose $k = O\left(\frac{\kappa_g^3}{\epsilon} (\mathcal{L}(x_0) - \mathcal{L}^\star +(E_0^y +E_0^z))\right)$ . Thus using batches of size 1 and $T$ and $N$ of order $\kappa_{g}$ .

Proof of Corollary 4. In the non-convex stochastic case, $E_{k}^{x} = \frac{1}{L}\mathbb{E}\Big[| \nabla \mathcal{L}(x_{k})|^{2}\Big]\leq E_{k}^{tot}$ . We thus apply Theorem 1 for $\mu < 0$ , multiply by $L$ to get:

$$\frac{1}{k}\sum _{t=1}^k\mathbb{E}[\mathrm{\parallel }\mathrm{\nabla }\mathcal{L}(x_t){\mathrm{\parallel }}^2]\le \frac{2L}{k}(\mathbb{E}[\mathcal{L}(x_0)-{\mathcal{L}}^{\star }]+(E_0^y+E_0^z))+2{\mathcal{W}}^2\mathrm{.}\backslash frac\; \{1\}\{k\}\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{k\}\; \backslash mathbb\; \{E\}\; \backslash Big\; [\; \backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{t\})\; \backslash |\; ^\; \{2\}\; \backslash Big\; ]\; \backslash leq\; \backslash frac\; \{2\; L\}\{k\}\; (\backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{0\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\; +\; (E\; \_\; \{0\}\; ^\; \{y\}\; +\; E\; \_\; \{0\}\; ^\; \{z\}))\; +\; 2\; \backslash mathcal\; \{W\}\; ^\; \{2\}.$$

to achieve an error of order $\epsilon$ , we need to ensure each term in the l.h.s. of the above inequality is of order $\epsilon$ . For the first term, similarly to the deterministic setting Corollary 3, we simply need $k = O\left(\frac{\kappa_g^3}{\epsilon} (\mathcal{L}(x_0) - \mathcal{L}^\star + (E_0^y + E_0^z))\right)$ . For the second term, we need to have $\mathcal{W}^2 = O(\epsilon)$ , which is achieved using the following choice for the sizes of the batches:

$$\mid {\mathcal{D}}_f\mid =O\left(\frac{{\kappa }_g^2{\tilde{\sigma }}_f^2}{ϵ}\right),\mid {\mathcal{D}}_g\mid =O\left(\frac{{\kappa }_g^5{\tilde{\sigma }}_g^2}{ϵ}\right),\mid {\mathcal{D}}_{g_{xy}}\mid =O\left(\frac{{\kappa }_g^2{\tilde{\sigma }}_{g_{xy}}^2}{ϵ}\right),\backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{f\}\; \backslash right|\; =\; O\; \backslash left(\backslash frac\; \{\backslash kappa\_\; \{g\}\; ^\; \{2\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\}\{\backslash epsilon\}\backslash right),\; \backslash quad\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\}\; \backslash right|\; =\; O\; \backslash left(\backslash frac\; \{\backslash kappa\_\; \{g\}\; ^\; \{5\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\}\{\backslash epsilon\}\backslash right),\; \backslash quad\; \backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{x\; y\}\}\; \backslash right|\; =\; O\; \backslash left(\backslash frac\; \{\backslash kappa\_\; \{g\}\; ^\; \{2\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\}\{\backslash epsilon\}\backslash right),$$

$$\mid {\mathcal{D}}_{g_{yy}}\mid =O\left(\frac{{\kappa }_g^3{\tilde{\sigma }}_{g_{yy}}^2}{ϵ}(1\vee ϵ{\mu }_g^2)\right)\mathrm{.}\backslash left|\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{y\; y\}\}\; \backslash right|\; =\; O\; \backslash left(\backslash frac\; \{\backslash kappa\_\; \{g\}\; ^\; \{3\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\}\{\backslash epsilon\}\; \backslash left(1\; \backslash vee\; \backslash epsilon\; \backslash mu\_\; \{g\}\; ^\; \{2\}\backslash right)\backslash right).$$

Finally, as required by Theorem 1, we set $T = \Theta(\kappa_g)$ and $N = \Theta(\kappa_g)$ thus yielding the desired complexity.

A.4 COMPARAISONS WITH OTHER METHODS

In this subsection we derive and discuss the complexities of methods presented in Table 1.

A.4.1 COMPARAISON WITH TTSA (HONG ET AL., 2020B)

Proposition 11. strongly-convex case $\mu >0$ . The complexity of the TTSA algorithm in Hong et al. (2020b) to achieve an error $\frac{\mu}{2}\mathbb{E}\left[| x_k - x^\star | ^2\right]\leq \epsilon$ is given by:

$$\mathcal{C}(ϵ):=O(({\kappa }_g^{\frac{17}{2}}{\kappa }_{\mathcal{L}}^{1\mathrm{/}2}+{\kappa }_{\mathcal{L}}^{7\mathrm{/}2})\frac{1}{{ϵ}^{\frac{3}{2}}}\log \frac{1}{ϵ})\backslash mathcal\; \{C\}\; (\backslash epsilon)\; :=\; O\; \backslash left(\backslash left(\backslash kappa\_\; \{g\}\; ^\; \{\backslash frac\; \{1\; 7\}\{2\}\}\; \backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; ^\; \{1\; /\; 2\}\; +\; \backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; ^\; \{7\; /\; 2\}\backslash right)\; \backslash frac\; \{1\}\{\backslash epsilon^\; \{\backslash frac\; \{3\}\{2\}\}\}\; \backslash log\; \backslash frac\; \{1\}\{\backslash epsilon\}\backslash right)$$

non-convex case $\mu < 0$ . The complexity of the TTSA algorithm in Hong et al. (2020b) to achieve an error $\frac{1}{k}\sum_{1\leq i\leq k}\mathbb{E}\Big[| \nabla \mathcal{L}(x_i)| ^2\Big]\leq \epsilon$ is given by:

$$\mathcal{C}(ϵ):=k(1+N)=O(({\kappa }_g^{11}+{\kappa }_g^{16})\frac{1}{{ϵ}^{\frac{5}{2}}}\log \left(\frac{1}{ϵ}\right))\mathrm{.}\backslash mathcal\; \{C\}\; (\backslash epsilon)\; :=\; k\; (1\; +\; N)\; =\; O\; \backslash left(\backslash left(\backslash kappa\_\; \{g\}\; ^\; \{1\; 1\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{1\; 6\}\backslash right)\; \backslash frac\; \{1\}\{\backslash epsilon^\; \{\backslash frac\; \{5\}\{2\}\}\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\}\backslash right)\backslash right).$$

Proof. strongly-convex case $\mu > 0$ Using the choice of step-sizes in Hong et al. (2020b), the following bound holds:

$$\begin{array}{c}\mathbb{E}[\mathrm{\parallel }{x}_k-x^{\star }{\mathrm{\parallel }}^2]\lesssim {\prod }_{i=0}^k(1-\frac{8}{3(k+k_{\alpha })})({\mathrm{\Delta }}_x^0+L_{\psi }^2{\mu }_g^{-2}{\mathrm{\Delta }}_0^y)\\ +\frac{L_{\psi }^2}{{\mu }_g{\mu }^{\frac{4}{3}}}({\mu }_g^{-1}+\frac{{\mu }_g}{{\mu }^2}{L}_y^2){\left(\frac{1}{k+k_{\alpha }}\right)}^{\frac{2}{3}}\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash Big\; [\; \backslash |\; x\; \_\; \{k\}\; -\; x\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; \backslash Big\; ]\; \backslash lesssim\; \backslash prod\_\; \{i\; =\; 0\}\; ^\; \{k\}\; \backslash left(1\; -\; \backslash frac\; \{8\}\{3\; (k\; +\; k\; \_\; \{\backslash alpha\})\}\backslash right)\; \backslash big\; (\backslash Delta\_\; \{x\}\; ^\; \{0\}\; +\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash Delta\_\; \{0\}\; ^\; \{y\}\; \backslash big)\; \backslash \backslash \; +\; \backslash frac\; \{L\; \_\; \{\backslash psi\}\; ^\; \{2\}\}\{\backslash mu\_\; \{g\}\; \backslash mu^\; \{\backslash frac\; \{4\}\{3\}\}\}\; \backslash left(\backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; +\; \backslash frac\; \{\backslash mu\_\; \{g\}\}\{\backslash mu^\; \{2\}\}\; L\; \_\; \{y\}\; ^\; \{2\}\backslash right)\; \backslash left(\backslash frac\; \{1\}\{k\; +\; k\; \_\; \{\backslash alpha\}\}\backslash right)\; ^\; \{\backslash frac\; \{2\}\{3\}\}\; \backslash \backslash \; \backslash end\{array\}$$

where $\Delta_x^0 = \mathbb{E}\left[| x_0 - x^\star | ^2\right],\mathbb{E}\left[| y_1 - y^\star (x_0)| ^2\right]$ and $k_{\alpha}$ given by:

$$k_{\alpha }=\max (35(\frac{L_g^3}{{\mu }_g^3}(1+{\sigma }_g^2{)}^{\frac{3}{2}}),\frac{(512{)}^{\frac{3}{2}}{L}_{\psi }^2{L}_y^2}{{\mu }^2})\mathrm{.}k\; \_\; \{\backslash alpha\}\; =\; \backslash max\; \backslash left(3\; 5\; \backslash left(\backslash frac\; \{L\; \_\; \{g\}\; ^\; \{3\}\}\{\backslash mu\_\; \{g\}\; ^\; \{3\}\}\; (1\; +\; \backslash sigma\_\; \{g\}\; ^\; \{2\})\; ^\; \{\backslash frac\; \{3\}\{2\}\}\backslash right),\; \backslash frac\; \{(5\; 1\; 2)\; ^\; \{\backslash frac\; \{3\}\{2\}\}\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; L\; \_\; \{y\}\; ^\; \{2\}\}\{\backslash mu^\; \{2\}\}\backslash right).$$

By a simple calculation, it is easy to see that $\prod_{i=0}^{k}\left(1 - \frac{8}{3(k + k_{\alpha})}\right) \leq \frac{(k_{\alpha} - 1)^2}{(k - 1 + k_{\alpha})^2}$ . Moreover, using that $L_{\psi} = O(\mu_g^{-2})$ , $L_y = O(\mu_g^{-1})$ , we get that

$$\frac{\mu }{2}\mathbb{E}[\mathrm{\parallel }{x}_k-x^{\star }{\mathrm{\parallel }}^2]\lesssim \frac{(k_{\alpha }-1{)}^2}{(k+k_{\alpha }-1{)}^2}(\mu {\mathrm{\Delta }}_x^0+\mu {\mu }_g^{-6}{\mathrm{\Delta }}_0^y)+{\mu }_g^{-6}{\mu }^{-\frac{1}{3}}(1+{\mu }^{-2}){\left(\frac{1}{k+k_{\alpha }}\right)}^{\frac{2}{3}}\backslash frac\; \{\backslash mu\}\{2\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; x\; \_\; \{k\}\; -\; x\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash lesssim\; \backslash frac\; \{(k\; \_\; \{\backslash alpha\}\; -\; 1)\; ^\; \{2\}\}\{(k\; +\; k\; \_\; \{\backslash alpha\}\; -\; 1)\; ^\; \{2\}\}\; \backslash left(\backslash mu\; \backslash Delta\_\; \{x\}\; ^\; \{0\}\; +\; \backslash mu\; \backslash mu\_\; \{g\}\; ^\; \{-\; 6\}\; \backslash Delta\_\; \{0\}\; ^\; \{y\}\backslash right)\; +\; \backslash mu\_\; \{g\}\; ^\; \{-\; 6\}\; \backslash mu^\; \{-\; \backslash frac\; \{1\}\{3\}\}\; \backslash left(1\; +\; \backslash mu^\; \{-\; 2\}\backslash right)\; \backslash left(\backslash frac\; \{1\}\{k\; +\; k\; \_\; \{\backslash alpha\}\}\backslash right)\; ^\; \{\backslash frac\; \{2\}\{3\}\}$$

Using that $L = O(\mu_g^{-3})$ , we get $\mu_g^{-6}\mu^{-\frac{1}{3}}(1 + \mu^{-2}) = O\left(\mu_g^{-5}\kappa_{\mathcal{L}}^{\frac{1}{3}} + \mu_g\kappa_{\mathcal{L}}^{\frac{7}{3}}\right)$ . Hence, to reach an error $\epsilon$ , we need to control both terms in the above inequality. This suggests the following condition on $k$ to control the second term which dominates the error:

$$k\ge ({\mu }_g^{\frac{3}{2}}{\kappa }_{\mathcal{L}}^{7\mathrm{/}2}+{\kappa }_{\mathcal{L}}^{1\mathrm{/}2}{\mu }_g^{-\frac{15}{2}})\frac{1}{{ϵ}^{\frac{3}{2}}}\mathrm{.}k\; \backslash geq\; \backslash left(\backslash mu\_\; \{g\}\; ^\; \{\backslash frac\; \{3\}\{2\}\}\; \backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; ^\; \{7\; /\; 2\}\; +\; \backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; ^\; \{1\; /\; 2\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; \backslash frac\; \{1\; 5\}\{2\}\}\backslash right)\; \backslash frac\; \{1\}\{\backslash epsilon^\; \{\backslash frac\; \{3\}\{2\}\}\}.$$

Moreover, the result in (Hong et al., 2020b, Theorem 1) requires $N = \Theta \left(\kappa_{g}\log \frac{1}{\epsilon}\right)$ , where $N$ is the number of terms in the Neumann series used to approximate the hessian inverse $(\partial_{yy}g(x,y))^{-1}$ in the expression of the gradient $\nabla \mathcal{L}$ . Hence, the total complexity is given by the following expression:

$$\mathcal{C}(ϵ):=k(1+N)=O(({\kappa }_g^{\frac{17}{2}}{\kappa }_{\mathcal{L}}^{1\mathrm{/}2}+{\kappa }_{\mathcal{L}}^{7\mathrm{/}2})\frac{1}{{ϵ}^{\frac{3}{2}}}\log \frac{1}{ϵ})\backslash mathcal\; \{C\}\; (\backslash epsilon)\; :=\; k\; (1\; +\; N)\; =\; O\; \backslash left(\backslash left(\backslash kappa\_\; \{g\}\; ^\; \{\backslash frac\; \{1\; 7\}\{2\}\}\; \backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; ^\; \{1\; /\; 2\}\; +\; \backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; ^\; \{7\; /\; 2\}\backslash right)\; \backslash frac\; \{1\}\{\backslash epsilon^\; \{\backslash frac\; \{3\}\{2\}\}\}\; \backslash log\; \backslash frac\; \{1\}\{\backslash epsilon\}\backslash right)$$

Smooth Non-convex case $\mu < 0$ . Following Hong et al. (2020b), consider the proximal map of $\mathcal{L}$ for a fixed $\rho > 0$ :

$$\widehat{x}(z):=\arg \underset{x\in \mathcal{X}}{\min }\{\mathcal{L}(x)+\frac{\rho }{2}\mathrm{\parallel }x-z{\mathrm{\parallel }}^2\}\backslash hat\; \{x\}\; (z)\; :=\; \backslash arg\; \backslash min\; \_\; \{x\; \backslash in\; \backslash mathcal\; \{X\}\}\; \backslash left\backslash \{\backslash mathcal\; \{L\}\; (x)\; +\; \backslash frac\; \{\backslash rho\}\{2\}\; \backslash |\; x\; -\; z\; \backslash |\; ^\; \{2\}\; \backslash right\backslash \}$$

and define the quantity $\tilde{\Delta}_x^k \coloneqq \mathbb{E}\left[| \hat{x}(x_k) - x_k|^2\right]$ , where $x_k$ are the iterates produced by the TTSA algorithm. Let $K$ be a random variable uniformly distributed on ${0, \dots, K-1}$ and independent from the remaining r.v. used in the TTSA algorithm. The result in (Hong et al., 2020b, Theorem 2) provides the following error bound on $\tilde{\Delta}_x^k$

$$\frac{1}{k}\sum _{1\le i\le k}{\tilde{\mathrm{\Delta }}}_x^i\lesssim (L_{\psi }^2({\mathrm{\Delta }}^0+\frac{{\sigma }_g^2}{{\mu }_g^2})+{\mu }_g)\frac{k^{-\frac{2}{5}}}{L^2}\mathrm{.}\backslash frac\; \{1\}\{k\}\; \backslash sum\_\; \{1\; \backslash leq\; i\; \backslash leq\; k\}\; \backslash tilde\; \{\backslash Delta\}\; \_\; \{x\}\; ^\; \{i\}\; \backslash lesssim\; \backslash left(L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash left(\backslash Delta^\; \{0\}\; +\; \backslash frac\; \{\backslash sigma\_\; \{g\}\; ^\; \{2\}\}\{\backslash mu\_\; \{g\}\; ^\; \{2\}\}\backslash right)\; +\; \backslash mu\_\; \{g\}\backslash right)\; \backslash frac\; \{k\; ^\; \{-\; \backslash frac\; \{2\}\{5\}\}\}\{L\; ^\; \{2\}\}.$$

where $\rho$ is set to $2L$ and $\Delta^0\leq \max \left(\mathbb{E}[\mathcal{L}(x_0) - \mathcal{L}^\star ],\mathbb{E}\Big[||y_1 - y^\star (x_0)||^2\Big]\right)$ . Now, recall that by definition of the proximal map, we have the following identity:

$$\frac{1}{k}\sum _{1\le i\le k}\mathbb{E}[\mathrm{\parallel }\mathrm{\nabla }\mathcal{L}(x_i){\mathrm{\parallel }}^2]\le 2({\rho }^2+L^2)\frac{1}{k}\sum _{1\le i\le k}{\tilde{\mathrm{\Delta }}}_x^i\lesssim L^2\frac{1}{k}\sum _{1\le i\le k}{\tilde{\mathrm{\Delta }}}_x^i\mathrm{.}\backslash frac\; \{1\}\{k\}\; \backslash sum\_\; \{1\; \backslash leq\; i\; \backslash leq\; k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{i\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; 2\; (\backslash rho^\; \{2\}\; +\; L\; ^\; \{2\})\; \backslash frac\; \{1\}\{k\}\; \backslash sum\_\; \{1\; \backslash leq\; i\; \backslash leq\; k\}\; \backslash tilde\; \{\backslash Delta\}\; \_\; \{x\}\; ^\; \{i\}\; \backslash lesssim\; L\; ^\; \{2\}\; \backslash frac\; \{1\}\{k\}\; \backslash sum\_\; \{1\; \backslash leq\; i\; \backslash leq\; k\}\; \backslash tilde\; \{\backslash Delta\}\; \_\; \{x\}\; ^\; \{i\}.$$

Hence, we obtain the following error bound:

$$\frac{1}{k}\sum _{1\le i\le k}\mathbb{E}[\mathrm{\parallel }\mathrm{\nabla }\mathcal{L}(x_i){\mathrm{\parallel }}^2]\le ({\kappa }_g^4({\mathrm{\Delta }}^0+{\kappa }_g^2{\sigma }_g^2)+{\mu }_g)k^{-\frac{2}{5}}\backslash frac\; \{1\}\{k\}\; \backslash sum\_\; \{1\; \backslash leq\; i\; \backslash leq\; k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{i\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash left(\backslash kappa\_\; \{g\}\; ^\; \{4\}\; \backslash left(\backslash Delta^\; \{0\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\}\backslash right)\; +\; \backslash mu\_\; \{g\}\backslash right)\; k\; ^\; \{-\; \backslash frac\; \{2\}\{5\}\}$$

where we used that $L_{\psi} = O(\kappa_g^2)$ . Therefore, to reach an error of order $\epsilon$ , TTSA requires:

$$k\ge \frac{1}{{ϵ}^{\frac{5}{2}}}({\kappa }_g^{10}{\mathrm{\Delta }}_0^{\frac{5}{2}}+{\kappa }_g^{15}{\sigma }_g^5)k\; \backslash geq\; \backslash frac\; \{1\}\{\backslash epsilon^\; \{\backslash frac\; \{5\}\{2\}\}\}\; \backslash Big\; (\backslash kappa\_\; \{g\}\; ^\; \{1\; 0\}\; \backslash Delta\_\; \{0\}\; ^\; \{\backslash frac\; \{5\}\{2\}\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{1\; 5\}\; \backslash sigma\_\; \{g\}\; ^\; \{5\}\; \backslash Big)$$

Moreover, controlling the bias in the estimation of the gradient requires $N = O\left(\kappa_{g}\log \frac{1}{\epsilon}\right)$ terms in the Neumann series approximating the hessian. Hence, the total complexity of the algorithm is:

$$\mathcal{C}(ϵ):=k(1+N)=O(({\kappa }_g^{11}+{\kappa }_g^{16})\frac{1}{{ϵ}^{\frac{5}{2}}}\log \left(\frac{1}{ϵ}\right))\mathrm{.}\backslash mathcal\; \{C\}\; (\backslash epsilon)\; :=\; k\; (1\; +\; N)\; =\; O\; \backslash bigg\; (\backslash left(\backslash kappa\_\; \{g\}\; ^\; \{1\; 1\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{1\; 6\}\backslash right)\; \backslash frac\; \{1\}\{\backslash epsilon^\; \{\backslash frac\; \{5\}\{2\}\}\}\; \backslash log\; \backslash left(\backslash frac\; \{1\}\{\backslash epsilon\}\backslash right)\; \backslash bigg).$$

A.4.2 COMPARaison WITH ACCBIO (JI AND LIANG, 2021)

Complexity of AccBio. The bilevel algorithm AccBio introduced in Ji and Liang (2021) uses acceleration for both the inner and outer loops. This allows to obtain the following conditions on $k$ , $T$ and $N$ to achieve an $\epsilon$ accuracy:

$$k=O({\kappa }_{\mathcal{L}}^{\frac{1}{2}}\log \frac{1}{ϵ}),T=O\left({\kappa }_g^{\frac{1}{2}}\right),N=O({\kappa }_g^{\frac{1}{2}}\log \frac{1}{ϵ})\mathrm{.}k\; =\; O\; \backslash left(\backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash log\; \backslash frac\; \{1\}\{\backslash epsilon\}\backslash right),\; T\; =\; O\; \backslash left(\backslash kappa\_\; \{g\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\backslash right),\; N\; =\; O\; \backslash left(\backslash kappa\_\; \{g\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash log\; \backslash frac\; \{1\}\{\backslash epsilon\}\backslash right).$$

Note that, since AccBio do not use a warm-start strategy when solving the linear system, $N$ is required to grow as $\log \frac{1}{\epsilon}$ in order to achieve an $\epsilon$ accuracy. This contributes an additional logarithmic factor to the total complexity so that $\mathcal{C}(\epsilon) = O\left(\kappa_{\mathcal{L}}^{\frac{1}{2}}\kappa_{g}^{\frac{1}{2}}\left(\log \frac{1}{\epsilon}\right)^{2}\right)$ . This is by contrast with AmIGO which exploits warm start when solving the linear system and thus only needs a constant number of iterations $N = O(\kappa_g)$ although the dependence on $\kappa_{g}$ is worse compared to AccBio. However, it is possible to improve such dependence by using acceleration in the inner-level algorithms $\mathcal{A}_k$ and $\mathcal{B}_k$ as we discuss in Appendix A.5.1.

Complexity of AccBio as a function of $\mu$ and $\mu_{g}$ . The authors choose to report the complexity as a function of $\mu$ and $\mu_{g}$ instead of the conditioning numbers $\kappa_{\mathcal{L}}$ and $\kappa_{g}$ . To achieve this, they observe that, under the additional assumption that the hessian $\partial_{yy}g(x,y)$ is constant w.r.t. $y$ , the Lipschitz constant $L$ has an improved dependence on $\mu_{g}$ : $L = O(\mu_g^{-2})$ instead of $L = O(\mu_g^{-3})$ in the general case where $\partial_{yy}g(x,y)$ is only Lipschitz in $y$ . This allows them to express $\kappa_{\mathcal{L}} = \frac{L}{\mu} = O(\mu_g^{-2}\mu^{-1})$ and to report the following complexities in terms of $\mu$ and $\mu_{g}$ :

$$\mathcal{C}(ϵ)=O\left({\mu }^{-\frac{1}{2}}{\mu }_g^{-\frac{3}{2}}{(\log \frac{1}{ϵ})}^2\right)\mathrm{.}\backslash mathcal\; \{C\}\; (\backslash epsilon)\; =\; O\; \backslash left(\backslash mu^\; \{-\; \backslash frac\; \{1\}\{2\}\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; \backslash frac\; \{3\}\{2\}\}\; \backslash left(\backslash log\; \backslash frac\; \{1\}\{\backslash epsilon\}\backslash right)\; ^\; \{2\}\backslash right).$$

Note that, in the general case where $L = O(\mu_g^{-3})$ , the complexity as a function of $\mu$ and $\mu_g$ becomes $O\left(\mu^{-\frac{1}{2}}\mu_g^{-2}\left(\log \frac{1}{\epsilon}\right)^2\right)$ , while still maintaining the same expression in terms of $\kappa_{\mathcal{L}}$ and $\kappa_g$ . Hence, using the expression in terms of conditioning allows a more general expression for the complexity that is less sensitive to the specific assumptions on $g$ and is therefore more suitable for comparison with other results in the literature.

A.5 CHOICE OF THE INNER-LEVEL ALGORITHMS $\mathcal{A}_k$ AND $\mathcal{B}_k$ .

The choice of $\mathcal{A}_k$ and $\mathcal{B}_k$ has an impact on the total complexity of the algorithm. We discuss two choices for $\mathcal{A}_k$ and $\mathcal{B}_k$ which improve the total complexity of AmIGO: Accelerated algorithms (in Appendix A.5.1) and variance reduced algorithms (in Appendix A.5.2).

A.5.1 ACCELERATION OF THE INNER-LEVEL FOR AMIGO

AmIGO could benefit from acceleration in the inner-loop by using standard acceleration schemes Nesterov (2003) for $\mathcal{A}_k$ and $\mathcal{B}_k$ . As a consequence, and using analysis of accelerated algorithms (Nesterov, 2003) in the deterministic setting, the error of the inner-level iterates would satisfy:

$$\mathbb{E}[\mathrm{\parallel }{y}_k-y^{\star }(x_k){\mathrm{\parallel }}^2]\le {\tilde{\mathrm{\Lambda }}}_k{E}_k^y,\mathbb{E}[\mathrm{\parallel }{y}_k-y^{\star }(x_k){\mathrm{\parallel }}^2]\le {\tilde{\mathrm{\Pi }}}_k{E}_k^z\backslash mathbb\; \{E\}\; \backslash Big\; [\; \backslash |\; y\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash Big\; ]\; \backslash leq\; \backslash tilde\; \{\backslash Lambda\}\; \_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{y\},\; \backslash qquad\; \backslash mathbb\; \{E\}\; \backslash Big\; [\; \backslash |\; y\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash Big\; ]\; \backslash leq\; \backslash tilde\; \{\backslash Pi\}\; \_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{z\}$$

where $\tilde{\Lambda}_k$ and $\tilde{\Pi}_k$ are accelerated rates of the form $\tilde{\Lambda}_k = O((1 - \sqrt{\kappa_g})^T)$ and $\tilde{\Pi}_k = O((1 - \sqrt{\kappa_g})^N)$ . The rest of the proofs are similar provided that $\Lambda_k$ and $\Pi_k$ are replaced by their accelerated rates $\tilde{\Lambda}_k$ and $\tilde{\Pi}_k$ . This implies that $T$ and $N$ need to be only of order $T = O(\sqrt{\kappa_g})$ and $N = O(\sqrt{\kappa_g})$ so that the final complexity becomes:

$$\mathcal{C}(ϵ):=O({\kappa }_{\mathcal{L}}{\kappa }_g^{1\mathrm{/}2}\log \frac{1}{ϵ})\mathrm{.}\backslash mathcal\; \{C\}\; (\backslash epsilon)\; :=\; O\; \backslash left(\backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; \backslash kappa\_\; \{g\}\; ^\; \{1\; /\; 2\}\; \backslash log\; \backslash frac\; \{1\}\{\backslash epsilon\}\backslash right).$$

Note that using conjugate gradient for $\mathcal{B}_k$ also enjoys an accelerated convergence rate Shewchuk et al. (1994). This is confirmed in our experiments of Figure 1 where AmIGO-CG enjoys the fastest convergence.

In order to further improve the dependence on $\kappa_{\mathcal{L}}$ to $\kappa_{\mathcal{L}}^{1/2}$ , one would need to use an accelerated scheme when updating the iterates $x_{k}$ . The analysis of such schemes along with warm-start would be an interesting direction for future work.

A.5.2 VARIANCE REDUCED ALGORITHMS FOR $\mathcal{A}_k$ AND $\mathcal{B}_k$

When the inner-level cost function $g$ is a finite average of functions $g(x,y) = \frac{1}{n}\sum_{1\leq i\leq n}g_i(x,y)$ empirical average, it is possible to use variance reduced algorithms such as SAG (Schmidt et al.,

2017). If every function $g_{i}$ is $L_{g}$ -smooth, then by (Schmidt et al., 2017, Proposition 1), the inner level error becomes:

$$\mathbb{E}[\mathrm{\parallel }{y}_k-y^{\star }(x_k){\mathrm{\parallel }}^2]\lesssim {\tilde{\mathrm{\Lambda }}}_k(3E_k^y+\frac{9}{4}{L}_g^{-2}{\sigma }_g^2),\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash lesssim\; \backslash tilde\; \{\backslash Lambda\}\; \_\; \{k\}\; (3\; E\; \_\; \{k\}\; ^\; \{y\}\; +\; \backslash frac\; \{9\}\{4\}\; L\; \_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\}),$$

with $\tilde{\Lambda}k = \left(1 - \frac{\kappa_g}{8n}\right)^T$ . This has the advantage that the error due to the variance decays exponentially with the number of iterations $T$ . As a consequence, the dependence of the effective variance $\mathcal{W}^2$ on the conditioning numbers $\kappa{\mathcal{L}}$ and $\kappa_g$ can be improved to:

$${\mathcal{W}}^2=O(\mathrm{\mid }{\mathcal{D}}_g{\mathrm{\mid }}^{-1}{\tilde{\sigma }}_g^2+{\kappa }_g^3\mathrm{\mid }{\mathcal{D}}_{g_{yy}}{\mathrm{\mid }}^{-1}{\tilde{\sigma }}_{g_{yy}}^2+{\kappa }_g^2\mathrm{\mid }{\mathcal{D}}_{g_{xy}}{\mathrm{\mid }}^{-1}{\tilde{\sigma }}_{g_{xy}}^2+{\kappa }_g^2\mathrm{\mid }{\mathcal{D}}_f{\mathrm{\mid }}^{-1}{\tilde{\sigma }}_f^2)\mathrm{.}\backslash mathcal\; \{W\}\; ^\; \{2\}\; =\; O\; \backslash Big\; (|\; \backslash mathcal\; \{D\}\; \_\; \{g\}\; |\; ^\; \{-\; 1\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{3\}\; |\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{y\; y\}\}\; |\; ^\; \{-\; 1\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; |\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{x\; y\}\}\; |\; ^\; \{-\; 1\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{2\}\; |\; \backslash mathcal\; \{D\}\; \_\; \{f\}\; |\; ^\; \{-\; 1\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\; \backslash Big).$$

This can be achieved by taking $T = O(n\kappa_g)$ up to a logarithmic dependence on the condition numbers. As a consequence, the complexity in the strongly convex stochastic setting becomes:

$$\mathcal{C}(ϵ)=O\left({\kappa }_{\mathcal{L}}\right(n{\tilde{\sigma }}_g^2+{\kappa }_g(1\vee ϵ{\kappa }_g){\tilde{\sigma }}_{g_{yy}}^2+{\tilde{\sigma }}_{g_{xy}}^2+{\tilde{\sigma }}_f^2)\frac{1}{ϵ}\log \left(\frac{E_0^{tot}+\mathbb{E}[\mathcal{L}(x_0)-{\mathcal{L}}^{\star }]}{ϵ}\right))\mathrm{.}\backslash mathcal\; \{C\}\; (\backslash epsilon)\; =\; O\; \backslash Bigg\; (\backslash kappa\_\; \{\backslash mathcal\; \{L\}\}\; \backslash Big\; (n\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; (1\; \backslash vee\; \backslash epsilon\; \backslash kappa\_\; \{g\})\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\; \backslash Big)\; \backslash frac\; \{1\}\{\backslash epsilon\}\; \backslash log\; \backslash left(\backslash frac\; \{E\; \_\; \{0\}\; ^\; \{t\; o\; t\}\; +\; \backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{0\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\}\{\backslash epsilon\}\backslash right)\; \backslash Bigg).$$

In the non-convex setting, the complexity becomes:

$$\mathcal{C}(ϵ)=O\left(\frac{{\kappa }_g^4}{{ϵ}^2}\right(n{\tilde{\sigma }}_g^2+{\kappa }_g^3(1\vee ϵ{\mu }_g^2){\tilde{\sigma }}_{g_{yy}}^2+{\kappa }_g{\tilde{\sigma }}_{g_{xy}}^2+{\kappa }_g{\tilde{\sigma }}_f^2)(\mathbb{E}[\mathcal{L}(x_k)-{\mathcal{L}}^{\star }]+E_0^y+E_0^z))\mathrm{.}\backslash mathcal\; \{C\}\; (\backslash epsilon)\; =\; O\; \backslash left(\backslash frac\; \{\backslash kappa\_\; \{g\}\; ^\; \{4\}\}\{\backslash epsilon^\; \{2\}\}\; \backslash Big\; (n\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; ^\; \{3\}\; \backslash big\; (1\; \backslash vee\; \backslash epsilon\; \backslash mu\_\; \{g\}\; ^\; \{2\}\; \backslash big)\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash kappa\_\; \{g\}\; \backslash tilde\; \{\backslash sigma\}\; \_\; \{f\}\; ^\; \{2\}\; \backslash Big)\; (\backslash mathbb\; \{E\}\; [\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; ]\; +\; E\; \_\; \{0\}\; ^\; \{y\}\; +\; E\; \_\; \{0\}\; ^\; \{z\})\backslash right).$$

The downside of this approach is the dependence on the number $n$ of functions $g_{i}$ in the total complexity.

B PRELIMINARY RESULTS

B.1 EXPRESSION OF THE GRADIENT

We provide a proof of Proposition 1 which shows that $\mathcal{L}$ is differentiable and provides an expression of its gradient.

Proof. Assumption 1 ensures that $y \mapsto g(x, y)$ admits a unique minimizer $y^{\star}(x)$ defined as the unique solution to the implicit equation $\partial_y g(x, y^{\star}(x)) = 0$ . Moreover, since $g$ is twice continuously differentiable and strongly convex, it follows that $\partial_{yy}g(x,y^{\star}(x))$ is invertible for any $x \in \mathcal{X}$ . Therefore the implicit function theorem (Lang, 2012, Theorem 5.9), ensures that $x \mapsto y^{\star}(x)$ is continuously differentiable with Jacobian given by $\nabla y^{\star}(x) = -\partial_{xy}g(x,y^{\star}(x))\partial_{yy}g(x,y^{\star}(x))^{-1}$ . Hence, by composition of differentiable functions, $\mathcal{L}$ is also differentiable with gradient given by:

$$\mathrm{\nabla }\mathcal{L}(x)={\mathrm{\partial }}_{x}f(x,y^{\star }(x))-{\mathrm{\partial }}_{xy}g(x,y^{\star }(x)){\mathrm{\partial }}_{yy}g(x,y^{\star }(x){)}^{-1}{\mathrm{\partial }}_{y}f(x,y^{\star }(x))\mathrm{.}\backslash nabla\; \backslash mathcal\; \{L\}\; (x)\; =\; \backslash partial\_\; \{x\}\; f\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; -\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; \backslash partial\_\; \{y\; y\}\; g\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; ^\; \{-\; 1\}\; \backslash partial\_\; \{y\}\; f\; (x,\; y\; ^\; \{\backslash star\}\; (x)).$$

We can thus define $z^{\star}(x,y) = -\partial_{yy}g(x,y)^{-1}\partial_{y}f(x,y)$ to get the desired expression for $\nabla \mathcal{L}(x)$ and note that $z^{\star}$ is the solution to (2).

B.2 SMOOTHNESS PROPERTIES OF $\mathcal{L}$ , $\mathcal{V}^{\star}$ , $z^{\star}$ AND $\Psi$

Proof of Proposition 6. Lipschitz continuity of $x \mapsto y^{\star}(x)$ . By Assumptions 1 and 3, the implicit function theorem (Lang, 2012, Theorem 5.9) ensures $y^{\star}(x)$ is differentiable with Jacobian given by:

$$\mathrm{\nabla }{y}^{\star }(x)=-{\mathrm{\partial }}_{xy}g(x,y^{\star }(x))({\mathrm{\partial }}_{yy}g(x,y^{\star }(x)){)}^{-1}\mathrm{.}\backslash nabla\; y\; ^\; \{\backslash star\}\; (x)\; =\; -\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; (\backslash partial\_\; \{y\; y\}\; g\; (x,\; y\; ^\; \{\backslash star\}\; (x)))\; ^\; \{-\; 1\}.$$

Moreover, by Assumption 3, we know that $\partial_y g(x,y)$ is $L_{g}^{\prime}$ -Lipschitz in $x$ for any $y\in \mathcal{V}$ , hence, $| \partial_{xy}g(x,y^{\star}(x))|{op}$ is upper-bounded by $L{g}^{\prime}$ . Moreover, by Assumption 1, $g$ is $\mu_g$ -strongly convex in $y$ uniformly on $\mathcal{X}$ . Therefore, it holds that $\left| \partial_{yy}g(x,y^{\star}(x))^{-1}\right|{op}\leq \mu_g^{-1}$ . This allows to deduce that $| \nabla y^{\star}(x)|{op}\leq \mu_g^{-1}L_g'$ , and by application of the fundamental theorem of calculus that:

$$\mathrm{\parallel }{y}^{\star }(x)-y^{\star }(x^{\mathrm{\prime }})\mathrm{\parallel }\le {\mu }_g^{-1}{L}_g^{\mathrm{\prime }}\mathrm{\parallel }x-x^{\mathrm{\prime }}\mathrm{\parallel }\mathrm{.}\backslash |\; y\; ^\; \{\backslash star\}\; (x)\; -\; y\; ^\; \{\backslash star\}\; (x\; ^\; \{\backslash prime\})\; \backslash |\; \backslash leq\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash |.$$

This shows that $y^{\star}$ is $L_{y}$ -Lipschitz continuous with $L_{y} := \mu_{g}^{-1} L_{g}'$ .

Lipschitz continuity of $x \mapsto z^{\star}(x, y)$ . Let $(x, y)$ and $(x', y')$ be two points in $\mathcal{X} \times \mathcal{Y}$ . Recalling the definition of $z^{\star}(x, y)$ in Proposition 1, it is easy to see that $z^{\star}(x, y)$ admits the following expression:

$$\begin{array}{cccc}& z^{\star }(x,y)=-{\mathrm{\partial }}_{yy}g(x,y{)}^{-1}{\mathrm{\partial }}_{y}f(x,y)\mathrm{.}& & \text{(25)}\end{array}z\; ^\; \{\backslash star\}\; (x,\; y)\; =\; -\; \backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\; ^\; \{-\; 1\}\; \backslash partial\_\; \{y\}\; f\; (x,\; y).\; \backslash tag\; \{25\}$$

Recalling the expression of $z^{\star}(x,y)$ , the following holds:

$$\begin{array}{c}{z}^{\star }(x,y)-z^{\star }(x^{\mathrm{\prime }},y^{\mathrm{\prime }})={\mathrm{\partial }}_{yy}g{(x^{\mathrm{\prime }},y^{\mathrm{\prime }})}^{-1}{\mathrm{\partial }}_{y}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})-{\mathrm{\partial }}_{yy}g(x,y{)}^{-1}{\mathrm{\partial }}_{y}f(x,y)\\ =({\mathrm{\partial }}_{yy}g{(x^{\mathrm{\prime }},y^{\mathrm{\prime }})}^{-1}-{\mathrm{\partial }}_{yy}g(x,y{)}^{-1}){\mathrm{\partial }}_{y}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\\ +{\mathrm{\partial }}_{yy}g(x,y{)}^{-1}({\mathrm{\partial }}_{y}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})-{\mathrm{\partial }}_{y}f(x,y))\\ ={\mathrm{\partial }}_{yy}g{(x^{\mathrm{\prime }},y^{\mathrm{\prime }})}^{-1}({\mathrm{\partial }}_{yy}g(x,y)-{\mathrm{\partial }}_{yy}g(x^{\mathrm{\prime }},y^{\mathrm{\prime }})){\mathrm{\partial }}_{yy}g(x,y{)}^{-1}{\mathrm{\partial }}_{y}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\\ +{\mathrm{\partial }}_{yy}g(x,y{)}^{-1}({\mathrm{\partial }}_{y}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})-{\mathrm{\partial }}_{y}f(x,y))\end{array}\backslash begin\{array\}\{l\}\; z\; ^\; \{\backslash star\}\; (x,\; y)\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; =\; \backslash partial\_\; \{y\; y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; ^\; \{-\; 1\}\; \backslash partial\_\; \{y\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; -\; \backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\; ^\; \{-\; 1\}\; \backslash partial\_\; \{y\}\; f\; (x,\; y)\; \backslash \backslash \; =\; \backslash left(\backslash partial\_\; \{y\; y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; ^\; \{-\; 1\}\; -\; \backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\; ^\; \{-\; 1\}\backslash right)\; \backslash partial\_\; \{y\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash \backslash \; +\; \backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\; ^\; \{-\; 1\}\; \backslash left(\backslash partial\_\; \{y\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; -\; \backslash partial\_\; \{y\}\; f\; (x,\; y)\backslash right)\; \backslash \backslash \; =\; \backslash partial\_\; \{y\; y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; ^\; \{-\; 1\}\; \backslash left(\backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\; -\; \backslash partial\_\; \{y\; y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\backslash right)\; \backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\; ^\; \{-\; 1\}\; \backslash partial\_\; \{y\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash \backslash \; +\; \backslash partial\_\; \{y\; y\}\; g\; (x,\; y)\; ^\; \{-\; 1\}\; \backslash left(\backslash partial\_\; \{y\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; -\; \backslash partial\_\; \{y\}\; f\; (x,\; y)\backslash right)\; \backslash \backslash \; \backslash end\{array\}$$

Hence, by taking the norm of the above quantity a triangular inequality followed by operator inequalities, it follows that:

$$\begin{array}{c}\parallel z^{\star }(x,y)-z^{\star }(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\parallel \le {\parallel H_2^{-1}\parallel }_{op}{\parallel H_1-H_2\parallel }_{op}{\parallel H_1^{-1}\parallel }_{op}\mathrm{\parallel }{\mathrm{\partial }}_{y}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\parallel }\\ +\parallel H_1^{-1}\parallel \parallel {\mathrm{\partial }}_{y}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})-{\mathrm{\partial }}_{y}f(x,y)\parallel \mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash left\backslash |\; z\; ^\; \{\backslash star\}\; (x,\; y)\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash right\backslash |\; \backslash leq\; \backslash left\backslash |\; H\; \_\; \{2\}\; ^\; \{-\; 1\}\; \backslash right\backslash |\; \_\; \{o\; p\}\; \backslash left\backslash |\; H\; \_\; \{1\}\; -\; H\; \_\; \{2\}\; \backslash right\backslash |\; \_\; \{o\; p\}\; \backslash left\backslash |\; H\; \_\; \{1\}\; ^\; \{-\; 1\}\; \backslash right\backslash |\; \_\; \{o\; p\}\; \backslash |\; \backslash partial\_\; \{y\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash |\; \backslash \backslash \; +\; \backslash left\backslash |\; H\; \_\; \{1\}\; ^\; \{-\; 1\}\; \backslash right\backslash |\; \backslash left\backslash |\; \backslash partial\_\; \{y\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; -\; \backslash partial\_\; \{y\}\; f\; (x,\; y)\; \backslash right\backslash |.\; \backslash \backslash \; \backslash end\{array\}$$

where we introduced $H_{1} = \partial_{yy}g(x,y)$ and $H_{2} = \partial_{yy}g(x^{\prime},y^{\prime})$ for conciseness. Using Assumption 1, we can upper-bound $\left| H_1^{-1}\right|{op}$ and $\left| H_2^{-1}\right|{op}$ by $\mu_g^{-1}$ . By Assumption 3, we know that $| H_1 - H_2|_{op}\leq M_g| (x,y) - (x',y')|$ . Finally by Assumption 2, we also have that $| \partial_yf(x',y') - \partial_yf(x,y)| \leq L_f| (x,y) - (x',y')|$ and that $| \partial_yf(x',y')| \leq B$ ensuring that:

$$\parallel z^{\star }(x,y)-z^{\star }(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\parallel \le ({\mu }_g^{-2}{M}_{g}B+{\mu }_g^{-1}{L}_f)\parallel (x,y)-(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\parallel \mathrm{.}\backslash left\backslash |\; z\; ^\; \{\backslash star\}\; (x,\; y)\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash right\backslash |\; \backslash leq\; \backslash left(\backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; M\; \_\; \{g\}\; B\; +\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; L\; \_\; \{f\}\backslash right)\; \backslash left\backslash |\; (x,\; y)\; -\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash right\backslash |.$$

Hence, we conclude that $z^{\star}$ is $L_{z}$ -Lipschitz continuous with $L_{z}$ defined as in (13).

boundedness of $z^{\star}(x,y)$ . Recalling the expression of $z^{\star}$ in (25), it is easy to see that $| z^{\star}(x,y)|$ is upper-bounded by $\mu_g^{-1}B$ since $\partial_{yy}g(x,y)$ is $\mu_g$ -strongly convex in $y$ by Assumption 1 and $\partial_yf(x,y)$ is bounded by $B$ by Assumption 2.

Regularity of $\Psi$

$$\begin{array}{c}\mathrm{\Psi }(x,y,z)-\mathrm{\Psi }(x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }})={\mathrm{\partial }}_{x}f(x,y)-{\mathrm{\partial }}_{x}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})+{\mathrm{\partial }}_{xy}g(x,y)z-{\mathrm{\partial }}_{xy}g(x^{\mathrm{\prime }},y^{\mathrm{\prime }})z^{\mathrm{\prime }}\\ ={\mathrm{\partial }}_{x}f(x,y)-{\mathrm{\partial }}_{x}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})+{\mathrm{\partial }}_{xy}g(x,y)(z-z^{\mathrm{\prime }})\\ +({\mathrm{\partial }}_{xy}g(x,y)-{\mathrm{\partial }}_{xy}g(x^{\mathrm{\prime }},y^{\mathrm{\prime }}))z^{\mathrm{\prime }}\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash Psi\; (x,\; y,\; z)\; -\; \backslash Psi\; (x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\},\; z\; ^\; \{\backslash prime\})\; =\; \backslash partial\_\; \{x\}\; f\; (x,\; y)\; -\; \backslash partial\_\; \{x\}\; f\; (x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\})\; +\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y)\; z\; -\; \backslash partial\_\; \{x\; y\}\; g\; (x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\})\; z\; ^\; \{\backslash prime\}\; \backslash \backslash \; =\; \backslash partial\_\; \{x\}\; f\; (x,\; y)\; -\; \backslash partial\_\; \{x\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; +\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y)\; \backslash left(z\; -\; z\; ^\; \{\backslash prime\}\backslash right)\; \backslash \backslash \; +\; \backslash left(\backslash partial\_\; \{x\; y\}\; g\; (x,\; y)\; -\; \backslash partial\_\; \{x\; y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\backslash right)\; z\; ^\; \{\backslash prime\}.\; \backslash \backslash \; \backslash end\{array\}$$

By taking the norm of the above expression and applying a triangular inequality followed by operator inequalities, it follows that:

$$\begin{array}{cccc}& \begin{array}{c}\parallel \mathrm{\Psi }(x,y,z)-\mathrm{\Psi }(x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }})\parallel \le \parallel {\mathrm{\partial }}_{x}f(x,y)-{\mathrm{\partial }}_{x}f(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\parallel +{\parallel {\mathrm{\partial }}_{xy}g(x,y)\parallel }_{op}\mathrm{\parallel }z-z^{\mathrm{\prime }}\mathrm{\parallel }\\ +{\parallel {\mathrm{\partial }}_{xy}g(x,y)-{\mathrm{\partial }}_{xy}g(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\parallel }_{op}\mathrm{\parallel }{z}^{\mathrm{\prime }}\mathrm{\parallel }\\ \le L_f(\mathrm{\parallel }x-x^{\mathrm{\prime }}\mathrm{\parallel }+\mathrm{\parallel }y-y^{\mathrm{\prime }}\mathrm{\parallel })+L_g^{\mathrm{\prime }}\mathrm{\parallel }z-z^{\mathrm{\prime }}\mathrm{\parallel }\\ +M_g\parallel z^{\mathrm{\prime }}\parallel (\parallel x-x^{\mathrm{\prime }}\parallel +\parallel y-y^{\mathrm{\prime }}\parallel )\mathrm{.}\end{array}& & \text{(26)}\end{array}\backslash begin\{array\}\{l\}\; \backslash left\backslash |\; \backslash Psi\; (x,\; y,\; z)\; -\; \backslash Psi\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\},\; z\; ^\; \{\backslash prime\}\backslash right)\; \backslash right\backslash |\; \backslash leq\; \backslash left\backslash |\; \backslash partial\_\; \{x\}\; f\; (x,\; y)\; -\; \backslash partial\_\; \{x\}\; f\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash right\backslash |\; +\; \backslash left\backslash |\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y)\; \backslash right\backslash |\; \_\; \{o\; p\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash prime\}\; \backslash |\; \backslash tag\; \{26\}\; \backslash \backslash \; +\; \backslash left\backslash |\; \backslash partial\_\; \{x\; y\}\; g\; (x,\; y)\; -\; \backslash partial\_\; \{x\; y\}\; g\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash prime\}\backslash right)\; \backslash right\backslash |\; \_\; \{o\; p\}\; \backslash |\; z\; ^\; \{\backslash prime\}\; \backslash |\; \backslash \backslash \; \backslash leq\; L\; \_\; \{f\}\; \backslash left(\backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash |\; +\; \backslash |\; y\; -\; y\; ^\; \{\backslash prime\}\; \backslash |\backslash right)\; +\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash prime\}\; \backslash |\; \backslash \backslash \; +\; M\; \_\; \{g\}\; \backslash left\backslash |\; z\; ^\; \{\backslash prime\}\; \backslash right\backslash |\; \backslash left(\backslash left\backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash right\backslash |\; +\; \backslash left\backslash |\; y\; -\; y\; ^\; \{\backslash prime\}\; \backslash right\backslash |\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

To get the first term of the last inequality above, we used that $\partial_x f$ is $L_{f}$ -Lipschitz by Assumption 2. To get the second term, we used that $\partial_{xy}g(x,y)$ is bounded since $\partial_y g(x,y)$ is $L_{g}^{\prime}$ -Lipschitz by Assumption 3. Finally, for the last term, we used that $\partial_{xy}g(x,y)$ is $M_g$ -Lipschitz by Assumption 3.

By choosing $x' = x$ , $y' = y^{\star}(x)$ and $z' = z^{\star}(x, y^{\star}(x))$ , it is easy to see from Proposition 1 that $\Psi(x, y^{\star}(x), z^{\star}(x, y^{\star}(x))) = \nabla \mathcal{L}(x)$ . Hence, applying the above inequality yields:

$$\begin{array}{c}\mathrm{\parallel }\mathrm{\Psi }(x,y,z)-\mathrm{\nabla }\mathcal{L}(x)\mathrm{\parallel }\le (L_f+M_g\mathrm{\parallel }{z}^{\star }(x,y^{\star }(x))\mathrm{\parallel })\mathrm{\parallel }y-y^{\star }(x)\mathrm{\parallel }+L_g^{\mathrm{\prime }}\mathrm{\parallel }z-z^{\star }(x,y^{\star }(x))\mathrm{\parallel }\\ \le (L_f+M_g\mathrm{\parallel }{z}^{\star }(x,y^{\star }(x))\mathrm{\parallel })\mathrm{\parallel }y-y^{\star }(x)\mathrm{\parallel }+L_g^{\mathrm{\prime }}\mathrm{\parallel }z-z^{\star }(x,y)\mathrm{\parallel }\\ +L_g^{\mathrm{\prime }}\mathrm{\parallel }{z}^{\star }(x,y)-z^{\star }(x,y^{\star }(x))\mathrm{\parallel }\end{array}\backslash begin\{array\}\{l\}\; \backslash |\; \backslash Psi\; (x,\; y,\; z)\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x)\; \backslash |\; \backslash leq\; \backslash left(L\; \_\; \{f\}\; +\; M\; \_\; \{g\}\; \backslash |\; z\; ^\; \{\backslash star\}\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; \backslash |\backslash right)\; \backslash |\; y\; -\; y\; ^\; \{\backslash star\}\; (x)\; \backslash |\; +\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash star\}\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; \backslash |\; \backslash \backslash \; \backslash leq\; \backslash left(L\; \_\; \{f\}\; +\; M\; \_\; \{g\}\; \backslash |\; z\; ^\; \{\backslash star\}\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; \backslash |\backslash right)\; \backslash |\; y\; -\; y\; ^\; \{\backslash star\}\; (x)\; \backslash |\; +\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash star\}\; (x,\; y)\; \backslash |\; \backslash \backslash \; +\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash |\; z\; ^\; \{\backslash star\}\; (x,\; y)\; -\; z\; ^\; \{\backslash star\}\; (x,\; y\; ^\; \{\backslash star\}\; (x))\; \backslash |\; \backslash \backslash \; \backslash end\{array\}$$

As shown earlier, $| z^{\star}(x,y^{\star}(x))|$ is upper-bounded by $\mu_g^{-1}B$ , while $| z^{\star}(x,y) - z^{\star}(x,y^{\star}(x))|$ is bounded by $L_{z}| y - y^{\star}(x)|$ . This allows to conclude that $| \Psi (x,y,z) - \nabla \mathcal{L}(x)| \leq L_{\psi}$ with $L_{\psi}$ defined in (13).

Lipschitz continuity of $x \mapsto \nabla \mathcal{L}(x)$ . We apply (26) with $(y,z) = (y^{\star}(x),z^{\star}(x,y^{\star}(x)))$ and $(y',z') = (y^{\star}(x'),z^{\star}(x,y^{\star}(x'))))$ which yields:

$$\begin{array}{c}\parallel \mathrm{\nabla }\mathcal{L}(x)-\mathrm{\nabla }\mathcal{L}\left(x^{\mathrm{\prime }}\right)\parallel \le (L_f+M_g\mathrm{\parallel }{z}^{\star }(x^{\mathrm{\prime }},y^{\star }\left(x^{\mathrm{\prime }}\right))\mathrm{\parallel })(\mathrm{\parallel }x-x^{\mathrm{\prime }}\mathrm{\parallel }+\mathrm{\parallel }{y}^{\star }(x)-y^{\star }\left(x^{\mathrm{\prime }}\right)\mathrm{\parallel })\\ +L_g^{\mathrm{\prime }}\parallel z^{\star }(x,y^{\star }(x))-z^{\star }(x^{\mathrm{\prime }},y^{\star }\left(x^{\mathrm{\prime }}\right))\parallel \\ \le (L_f+M_g{\mu }_g^{-1}B+L_g^{\mathrm{\prime }}{L}_z)(1+L_y)\mathrm{\parallel }x-x^{\mathrm{\prime }}\mathrm{\parallel },\end{array}\backslash begin\{array\}\{l\}\; \backslash left\backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x)\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; ^\; \{\backslash prime\}\backslash right)\; \backslash right\backslash |\; \backslash leq\; \backslash left(L\; \_\; \{f\}\; +\; M\; \_\; \{g\}\; \backslash |\; z\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\}\backslash right)\backslash right)\; \backslash |\backslash right)\; \backslash left(\backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash |\; +\; \backslash |\; y\; ^\; \{\backslash star\}\; (x)\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\}\backslash right)\; \backslash |\backslash right)\; \backslash \backslash \; +\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash left\backslash |\; z\; ^\; \{\backslash star\}\; \backslash left(x,\; y\; ^\; \{\backslash star\}\; (x)\backslash right)\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\},\; y\; ^\; \{\backslash star\}\; \backslash left(x\; ^\; \{\backslash prime\}\backslash right)\backslash right)\; \backslash right\backslash |\; \backslash \backslash \; \backslash leq\; \backslash left(L\; \_\; \{f\}\; +\; M\; \_\; \{g\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; B\; +\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; L\; \_\; \{z\}\backslash right)\; \backslash left(1\; +\; L\; \_\; \{y\}\backslash right)\; \backslash |\; x\; -\; x\; ^\; \{\backslash prime\}\; \backslash |,\; \backslash \backslash \; \backslash end\{array\}$$

where we used that $| z^{\star}(x',y^{\star}(x'))|$ is upper-bounded by $\mu_q^{-1}B$ , $z^{\star}$ is $L_{z}$ -Lipschitz and $y^{\star}$ is $L_{y}$ -Lipschitz. Hence, $\nabla \mathcal{L}$ is $L$ -Lipschitz continuous, with $L$ as given by (13).

B.3 CONVERGENCE OF THE ITERATES OF ALGORITHMS $\mathcal{A}_k$ AND $\mathcal{B}_k$

Proof. Controlling the iterates $y^{t}$ of $\mathcal{A}_k$ .

Consider a new batch $\mathcal{D}_g$ of samples $\xi$ . We have by definition of the update equation of $y^t$ that:

$$\begin{array}{c}{\parallel y^t-y^{\star }\left(x_k\right)\parallel }^2={\parallel y^{t-1}-y^{\star }\left(x_k\right)\parallel }^2+{\alpha }_k^2{\parallel {\mathrm{\partial }}_y\widehat{g}(x_k,y^{t-1},{\mathcal{D}}_g)\parallel }^2\\ -2{\alpha }_k{\mathrm{\partial }}_y\widehat{g}{(x_k,y^{t-1},{\mathcal{D}}_g)}^{\mathrm{\top }}(y^{t-1}-y^{\star }\left(x_k\right))\end{array}\backslash begin\{array\}\{l\}\; \backslash left\backslash |\; y\; ^\; \{t\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; =\; \backslash left\backslash |\; y\; ^\; \{t\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; +\; \backslash alpha\_\; \{k\}\; ^\; \{2\}\; \backslash left\backslash |\; \backslash partial\_\; \{y\}\; \backslash hat\; \{g\}\; \backslash left(x\; \_\; \{k\},\; y\; ^\; \{t\; -\; 1\},\; \backslash mathcal\; \{D\}\; \_\; \{g\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash \backslash \; -\; 2\; \backslash alpha\_\; \{k\}\; \backslash partial\_\; \{y\}\; \backslash hat\; \{g\}\; \backslash left(x\; \_\; \{k\},\; y\; ^\; \{t\; -\; 1\},\; \backslash mathcal\; \{D\}\; \_\; \{g\}\backslash right)\; ^\; \{\backslash top\}\; \backslash left(y\; ^\; \{t\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\}\backslash right)\backslash right)\; \backslash \backslash \; \backslash end\{array\}$$

Taking the expectation conditionally on $x_{k}$ and $y^{t - 1}$ , we get:

$$\begin{array}{c}\mathbb{E}[{\mid \mid y^t-y^{\star }\left(x_k\right)\mid \mid }^2\mid x_k,y^{t-1}]=(1-{\alpha }_k{\mu }_g){\mid \mid y^{t-1}-y^{\star }\left(x_k\right)\mid \mid }^2\\ +{\alpha }_k^2\mathbb{E}[{\mid \mid {\mathrm{\partial }}_y\widehat{g}(x_k,y^{t-1},{\mathcal{D}}_g)-{\mathrm{\partial }}_{y}g(x_k,y^{t-1})\mid \mid }^2\mid x_k,y^{t-1}]\\ -2{\alpha }_k{\mathrm{\partial }}_{y}g(x_k,y^{t-1}{)}^{\mathrm{\top }}(y^{t-1}-y^{\star }(x_k)-\frac{{\alpha }_k}{2}{\mathrm{\partial }}_{y}g(x_k,y^{t-1}))\\ \le (1-{\alpha }_k{\mu }_g){\parallel y^{t-1}-y^{\star }(x_k)\parallel }^2+2{\alpha }_k^2{\sigma }_g^2\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left|\; \backslash left|\; y\; ^\; \{t\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; \backslash right|\; \backslash right|\; ^\; \{2\}\; \backslash mid\; x\; \_\; \{k\},\; y\; ^\; \{t\; -\; 1\}\; \backslash right]\; =\; \backslash left(1\; -\; \backslash alpha\_\; \{k\}\; \backslash mu\_\; \{g\}\backslash right)\; \backslash left|\; \backslash left|\; y\; ^\; \{t\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; \backslash right|\; \backslash right|\; ^\; \{2\}\; \backslash \backslash \; +\; \backslash alpha\_\; \{k\}\; ^\; \{2\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash right.\backslash left|\backslash left|\; \backslash partial\_\; \{y\}\; \backslash hat\; \{g\}\; (x\; \_\; \{k\},\; y\; ^\; \{t\; -\; 1\},\; \backslash mathcal\; \{D\}\; \_\; \{g\})\; -\; \backslash partial\_\; \{y\}\; g\; (x\; \_\; \{k\},\; y\; ^\; \{t\; -\; 1\})\; \backslash right|\backslash right|\; ^\; \{2\}\; \backslash left.\; \backslash right|\; x\; \_\; \{k\},\; y\; ^\; \{t\; -\; 1\}\; \backslash left.\; \backslash right]\; \backslash \backslash \; -\; 2\; \backslash alpha\_\; \{k\}\; \backslash partial\_\; \{y\}\; g\; (x\; \_\; \{k\},\; y\; ^\; \{t\; -\; 1\})\; ^\; \{\backslash top\}\; \backslash left(y\; ^\; \{t\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; -\; \backslash frac\; \{\backslash alpha\_\; \{k\}\}\{2\}\; \backslash partial\_\; \{y\}\; g\; (x\; \_\; \{k\},\; y\; ^\; \{t\; -\; 1\})\backslash right)\; \backslash \backslash \; \backslash leq\; \backslash left(1\; -\; \backslash alpha\_\; \{k\}\; \backslash mu\_\; \{g\}\backslash right)\; \backslash left\backslash |\; y\; ^\; \{t\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash right\backslash |\; ^\; \{2\}\; +\; 2\; \backslash alpha\_\; \{k\}\; ^\; \{2\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\}\; \backslash \backslash \; \backslash end\{array\}$$

The first line uses that $\partial_y\hat{g} (x_k,y^{t - 1},\mathcal{D}g)$ is an unbiased estimator of $\partial_yg(x_k,y^{t - 1})$ . For the second line, we use Assumption 4 which allows to upper-bound the variance of $\partial_y\hat{g}$ by $\sigma_g^2$ . Moreover, since $g$ is convex and $L{g}$ -smooth and since $\alpha_{k}\leq L_{g}^{-1}$ , it follows that the last term in the above inequality is non-positive and can thus be upper-bounded by 0. By unrolling the resulting inequality recursively for $1 < t\leq k$ , we obtain the desired result.

Controlling the iterates $z^n$ of $\mathcal{B}_k$ . The proof follows by direct application of Proposition 15 with $\beta = \beta_k$ and the following choices for $A_n, A, \hat{b}, b$ :

$$A_n={\mathrm{\partial }}_{yy}\widehat{g}(x_k,y_k,{\mathcal{D}}_{g_{yy}}),A\; \_\; \{n\}\; =\; \backslash partial\_\; \{y\; y\}\; \backslash hat\; \{g\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{g\; \_\; \{y\; y\}\}\backslash right),$$

$$A={\mathrm{\partial }}_{yy}g(x_k,y_k)A\; =\; \backslash partial\_\; \{y\; y\}\; g\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)$$

$$\widehat{b}={\mathrm{\partial }}_y\widehat{f}(x_k,y_k,{\mathcal{D}}_f)\backslash hat\; \{b\}\; =\; \backslash partial\_\; \{y\}\; \backslash hat\; \{f\}\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})$$

$$b={\mathrm{\partial }}_{y}f(x_k,y_k)\mathrm{.}b\; =\; \backslash partial\_\; \{y\}\; f\; (x\; \_\; \{k\},\; y\; \_\; \{k\}).$$

This directly yields the following inequalities:

$$\begin{array}{c}\mathbb{E}[\mathrm{\parallel }{z}_k-z^{\star }(x_k,y_k){\mathrm{\parallel }}^2]\le {\tilde{\mathrm{\Pi }}}_k\mathbb{E}\left[{\parallel z_k^0-z^{\star }(x_k,y_k)\parallel }^2\right]+{\tilde{R}}_k^z,\\ \mathbb{E}\left[{\parallel {\bar{z}}_k-z^{\star }(x_k,y_k)\parallel }^2\right]\le {\tilde{\mathrm{\Pi }}}_k\mathbb{E}\left[{\parallel z_k^0-z^{\star }(x_k,y_k)\parallel }^2\right]\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; z\; \_\; \{k\}\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash tilde\; \{\backslash Pi\}\; \_\; \{k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; \_\; \{k\}\; ^\; \{0\}\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; +\; \backslash tilde\; \{R\}\; \_\; \{k\}\; ^\; \{z\},\; \backslash \backslash \; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash bar\; \{z\}\; \_\; \{k\}\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash tilde\; \{\backslash Pi\}\; \_\; \{k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; \_\; \{k\}\; ^\; \{0\}\; -\; z\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right].\; \backslash \backslash \; \backslash end\{array\}$$

where $\tilde{\Pi}_k$ and $\tilde{R}_k^z$ are given by:

$${\tilde{\mathrm{\Pi }}}_k:={(1-{\beta }_k{\mu }_g)}^N,{\tilde{R}}_k^z:={\beta }_k^2\left({\sigma }_g^2\mathbb{E}\right[\mathrm{\parallel }{z}^{\star }(x_k,y_k){\mathrm{\parallel }}^2]+3(N\wedge \frac{1}{{\beta }_k{\mu }_g}\left){\sigma }_f^2\right)(N\wedge \frac{1}{{\beta }_k{\mu }_g})\mathrm{.}\backslash tilde\; \{\backslash Pi\}\; \_\; \{k\}\; :=\; \backslash left(1\; -\; \backslash beta\_\; \{k\}\; \backslash mu\_\; \{g\}\backslash right)\; ^\; \{N\},\; \backslash qquad\; \backslash tilde\; \{R\}\; \_\; \{k\}\; ^\; \{z\}\; :=\; \backslash beta\_\; \{k\}\; ^\; \{2\}\; \backslash Big\; (\backslash sigma\_\; \{g\}\; ^\; \{2\}\; \backslash mathbb\; \{E\}\; \backslash big\; [\; \backslash |\; z\; ^\; \{\backslash star\}\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash big\; ]\; +\; 3\; \backslash Big\; (N\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\_\; \{k\}\; \backslash mu\_\; \{g\}\}\; \backslash Big)\; \backslash sigma\_\; \{f\}\; ^\; \{2\}\; \backslash Big)\; \backslash Big\; (N\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\_\; \{k\}\; \backslash mu\_\; \{g\}\}\; \backslash Big).$$

First we have that $\tilde{\Pi}_k\leq \Pi_k$ . Moreover, Proposition 6, we have that $| z^{\star}(x_k,y_k)| \leq B\mu_g^{-1}$ hence, $\tilde{R}_k^z\leq R_k^z$ thus yielding the desired inequalities. Finally (15a) also follows similarly using (45) from Proposition 15.

B.4 CONTROLLING THE BIAS AND VARIANCE $E_k^\psi$ AND $V_k^\psi$

Proof of Proposition 4. Recall that the expressions of $E_k^\psi$ and $V_k^\psi$ in (6) involves the conditional expectation $\mathbb{E}_k[\hat{\psi}k]$ knowing $x_k, y_k$ and $z{k-1}$ . This can be also expressed using $\Psi$ as follows:

$$\begin{array}{c}{\mathbb{E}}_k\left[{\widehat{\psi }}_k\right]=\mathbb{E}[\mathbb{E}[{\widehat{\psi }}_k\mathrm{\mid }{x}_k,y_k,z_k]\mathrm{\mid }{x}_k,y_k,z_{k-1}]\\ =\mathbb{E}[\mathrm{\Psi }(x_k,y_k,z_k)\mid x_k,y_k,z_{k-1}]\\ =\mathbb{E}\left[\mathrm{\Psi }(x_k,y_k,\mathbb{E}[z_k\mid \mid x_k,y_k,z_{k-1}])\right]\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \_\; \{k\}\; \backslash left[\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\}\; \backslash right]\; =\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\}\; |\; x\; \_\; \{k\},\; y\; \_\; \{k\},\; z\; \_\; \{k\}\; \backslash right]\; |\; x\; \_\; \{k\},\; y\; \_\; \{k\},\; z\; \_\; \{k\; -\; 1\}\; \backslash right]\; \backslash \backslash \; =\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash Psi\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\},\; z\; \_\; \{k\}\backslash right)\; \backslash mid\; x\; \_\; \{k\},\; y\; \_\; \{k\},\; z\; \_\; \{k\; -\; 1\}\; \backslash right]\; \backslash \backslash \; =\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash Psi\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathbb\; \{E\}\; \backslash left[\; z\; \_\; \{k\}\; \backslash mid\; \backslash left|\; x\; \_\; \{k\},\; y\; \_\; \{k\},\; z\; \_\; \{k\; -\; 1\}\; \backslash right.\backslash right]\backslash right)\; \backslash right]\; \backslash \backslash \; \backslash end\{array\}$$

where we used the tower property for conditional expectations in the first line, then the fact that the expectation of $\psi_{k}$ conditionally on $x_{k},y_{k}$ and $z_{k}$ is simply $\Psi (x_k,y_k,z_k)$ . Finally, for the last line, we use the independence of the noise and the linearity of $\Psi$ w.r.t. the last variable. In all what follows, we write $\bar{z}k = \mathbb{E}[z_k|x_k,y_k,z{k - 1}]$ which is the same object as defined in Proposition 7. We then treat $E_{k}^{\psi}$ and $V_{k}^{\psi}$ separately.

Bounding $E_k^{\psi}$ . Using Propositions 6 and 7 we directly get the desired inequality:

$$\begin{array}{c}{E}_k^{\psi }\le 2L_{\psi }^2(\mathbb{E}[\mathrm{\parallel }{y}_k-y^{\star }(x_k){\mathrm{\parallel }}^2]+\mathbb{E}[\mathrm{\parallel }{\bar{z}}_k-z^{\star }(x_k,y_k){\mathrm{\parallel }}^2])\\ \le 2L_{\psi }^2({\mathrm{\Lambda }}_k{E}_k^y+{\mathrm{\Pi }}_k{E}_k^z+R_k^y)\end{array}\backslash begin\{array\}\{l\}\; E\; \_\; \{k\}\; ^\; \{\backslash psi\}\; \backslash leq\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash left(\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; +\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; \backslash bar\; \{z\}\; \_\; \{k\}\; -\; z\; ^\; \{\backslash star\}\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash right]\backslash right)\; \backslash \backslash \; \backslash leq\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; (\backslash Lambda\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{y\}\; +\; \backslash Pi\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{z\}\; +\; R\; \_\; \{k\}\; ^\; \{y\})\; \backslash \backslash \; \backslash end\{array\}$$

Bound on $V_{k}^{\psi}$ . We decompose $V_{k}^{\psi}$ into a sum of three terms $W_{k}, W_{k}^{\prime}$ and $W_{k}^{\prime \prime}$ given by:

$$\begin{array}{c}{W}_k:=\mathbb{E}\left[{\parallel {\mathrm{\partial }}_x\widehat{f}(x_k,y_k,{\mathcal{D}}_f)-{\mathrm{\partial }}_{x}f(x_k,y_k)\parallel }^2\right],\\ W_k^{\mathrm{\prime }}:=\mathbb{E}\left[{\parallel {\mathrm{\partial }}_{xy}\widehat{g}(x_k,y_k,{\tilde{\xi }}_{N+1,k})z_k-{\mathrm{\partial }}_{xy}g(x_k,y_k){\bar{z}}_k\parallel }^2\right]\\ W_k^{\mathrm{\prime }\mathrm{\prime }}:=\mathbb{E}\left[\right({\mathrm{\partial }}_x\widehat{f}(x_k,y_k,{\mathcal{D}}_f)-{\mathrm{\partial }}_{x}f(x_k,y_k){)}^{\mathrm{\top }}{\mathrm{\partial }}_{xy}g(x_k,y_k)(z_k-{\bar{z}}_k)]\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; W\; \_\; \{k\}\; :=\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash partial\_\; \{x\}\; \backslash hat\; \{f\}\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; -\; \backslash partial\_\; \{x\}\; f\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right],\; \backslash \backslash \; W\; \_\; \{k\}\; ^\; \{\backslash prime\}\; :=\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash partial\_\; \{x\; y\}\; \backslash hat\; \{g\}\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash tilde\; \{\backslash xi\}\; \_\; \{N\; +\; 1,\; k\}\backslash right)\; z\; \_\; \{k\}\; -\; \backslash partial\_\; \{x\; y\}\; g\; \backslash left(x\; \_\; \{k\},\; y\; \_\; \{k\}\backslash right)\; \backslash bar\; \{z\}\; \_\; \{k\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash \backslash \; W\; \_\; \{k\}\; ^\; \{\backslash prime\; \backslash prime\}\; :=\; \backslash mathbb\; \{E\}\; \backslash bigg\; [\; \backslash Big\; (\backslash partial\_\; \{x\}\; \backslash hat\; \{f\}\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; -\; \backslash partial\_\; \{x\}\; f\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; \backslash Big)\; ^\; \{\backslash top\}\; \backslash partial\_\; \{x\; y\}\; g\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; (z\; \_\; \{k\}\; -\; \backslash bar\; \{z\}\; \_\; \{k\})\; \backslash bigg\; ].\; \backslash \backslash \; \backslash end\{array\}$$

where we used that $\tilde{\xi}{N + 1,k}$ is independent from $z{k}$ and $\mathcal{D}f$ to get the last term. Hence, using Assumption 4 to bound the first term of the above relation, we get $V{k}^{\psi} \leq \sigma_{f}^{2} + W_{k}^{\prime} + 2W_{k}^{\prime \prime}$ . Thus, it remains to control each of $W_{k}^{\prime}$ and $W_{k}^{\prime \prime}$ .

Bound on $W_{k}^{\prime \prime}$ . Using that $\mathcal{D}f$ is independent from $\tilde{\xi}{n,k}$ , we can apply Proposition 14 to write:

$$W_k^{\mathrm{\prime }\mathrm{\prime }}={\beta }_k\mathbb{E}[{\mathrm{\partial }}_x(\widehat{f}-f)(x_k,y_k,{\mathcal{D}}_f{)}^{\mathrm{\top }}{\mathrm{\partial }}_{xy}g(x_k,y_k)(\sum _{t=1}^N(I-{\beta }_{k}A{)}^{N-t}){\mathrm{\partial }}_y(\widehat{f}-f)(x_k,y_k,{\mathcal{D}}_f)]W\; \_\; \{k\}\; ^\; \{\backslash prime\; \backslash prime\}\; =\; \backslash beta\_\; \{k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash partial\_\; \{x\}\; (\backslash hat\; \{f\}\; -\; f)\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; ^\; \{\backslash top\}\; \backslash partial\_\; \{x\; y\}\; g\; (x\; \_\; \{k\},\; y\; \_\; \{k\})\; \backslash left(\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{N\}\; (I\; -\; \backslash beta\_\; \{k\}\; A)\; ^\; \{N\; -\; t\}\backslash right)\; \backslash partial\_\; \{y\}\; (\backslash hat\; \{f\}\; -\; f)\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; \backslash right]$$

where we used the simplifying notion $(\hat{f} - f)(x_k, y_k, \mathcal{D}f) = \hat{f}(x_k, y_k, \mathcal{D}f) - f(x_k, y_k)$ . Using Assumption 3 to bound $| \partial{xy} g(x_k, y_k) |{op}$ by $L_g'$ , Assumption 1 to upper-bound $\left| \left( \sum_{t=1}^{N} (I - \beta_k A)^{N-t} \right) \right|{op}$ by $\left( \sum{t=1}^{N} (1 - \beta_k \mu_g)^{N-t} \right)$ we get

$$\begin{array}{c}\mid W_k^{\mathrm{\prime }\mathrm{\prime }}\mid \le {\beta }_k{L}_g^{\mathrm{\prime }}{\sum }_{t=0}^{N-1}(1-{\beta }_k{\mu }_g{)}^t\mathbb{E}\left[\mid {\mathrm{\partial }}_x(\widehat{f}-f)(x_k,y_k,{\mathcal{D}}_f{)}^{\mathrm{\top }}{\mathrm{\partial }}_y(\widehat{f}-f)(x_k,y_k,{\mathcal{D}}_f)\mid \right]\\ \le L_g^{\mathrm{\prime }}{\mu }_g^{-1}\mathbb{E}[\mid {\mathrm{\partial }}_x(\widehat{f}-f)(x_k,y_k,{\mathcal{D}}_f{)}^{\mathrm{\top }}{\mathrm{\partial }}_y(\widehat{f}-f)(x_k,y_k,{\mathcal{D}}_f)\mid ]\\ \le L_g^{\mathrm{\prime }}{\mu }_g^{-1}\mathbb{E}{\left[\parallel {\mathrm{\partial }}_x(\widehat{f}-f)(x_k,y_k,{\mathcal{D}}_f{)}^2\parallel \right]}^{\frac{1}{2}}\mathbb{E}{\left[{\parallel {\mathrm{\partial }}_y(\widehat{f}-f)(x_k,y_k,{\mathcal{D}}_f)\parallel }^2\right]}^{\frac{1}{2}}\le L_g^{\mathrm{\prime }}{\mu }_g^{-1}{\sigma }_f^2\end{array}\backslash begin\{array\}\{l\}\; \backslash left|\; W\; \_\; \{k\}\; ^\; \{\backslash prime\; \backslash prime\}\; \backslash right|\; \backslash leq\; \backslash beta\_\; \{k\}\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash sum\_\; \{t\; =\; 0\}\; ^\; \{N\; -\; 1\}\; (1\; -\; \backslash beta\_\; \{k\}\; \backslash mu\_\; \{g\})\; ^\; \{t\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left|\; \backslash partial\_\; \{x\}\; (\backslash hat\; \{f\}\; -\; f)\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; ^\; \{\backslash top\}\; \backslash partial\_\; \{y\}\; (\backslash hat\; \{f\}\; -\; f)\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; \backslash right|\; \backslash right]\; \backslash \backslash \; \backslash leq\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash mathbb\; \{E\}\; \backslash Big\; [\; \backslash Big\; |\; \backslash partial\_\; \{x\}\; (\backslash hat\; \{f\}\; -\; f)\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; ^\; \{\backslash top\}\; \backslash partial\_\; \{y\}\; (\backslash hat\; \{f\}\; -\; f)\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; \backslash Big\; |\; \backslash Big\; ]\; \backslash \backslash \; \backslash leq\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash partial\_\; \{x\}\; (\backslash hat\; \{f\}\; -\; f)\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; ^\; \{2\}\; \backslash right\backslash |\; \backslash right]\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash partial\_\; \{y\}\; (\backslash hat\; \{f\}\; -\; f)\; (x\; \_\; \{k\},\; y\; \_\; \{k\},\; \backslash mathcal\; \{D\}\; \_\; \{f\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash leq\; L\; \_\; \{g\}\; ^\; \{\backslash prime\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash sigma\_\; \{f\}\; ^\; \{2\}\; \backslash \backslash \; \backslash end\{array\}$$

where we used that $\sum_{t=0}^{N-1}(1 - \beta_k\mu_g) \leq \frac{1}{\beta_k\mu_g}$ for the second line, Cauchy-Schwarz inequality to get the third line and Assumption 4 to get the last line.

Bound on $W_{k}^{\prime}$ Using that $\tilde{\xi}{N + 1,k}$ is independent from $z{k}$ , we write:

$$Wk′=E[∥∂xy(g^−g)(xk,yk,ξ~N+1,k)zk∥2]+E[∥∂xyg(xk,yk)(zk−zˉk)∥2]σgxy2E[∥zk∥2]+(Lg′)2E[∥zk−zˉk∥2]2σgxy2(E[∥zk−z⋆(xk,yk)∥2]+E[∥z⋆(xk,yk)∥2])+(Lg′)2E[∥zk−zˉk∥2]2σgxy2E[∥zk−z⋆(xk,yk)∥2]+(Lg′)2E[∥zk−zˉk∥2]+2σgxy2B2μg−22σgxy2(ΠkE[∥zk0−z⋆(xk,yk)∥2+Rkz])+(Lg′)2(4σgyy2μg−2ΠkE[∥zk0−z⋆(xk,yk)∥2]+2Rkz)+2σgxy2B2μg−2≤2(σgxy2+2(Lg′)2μg−2σgyy2)ΠkE[∥zk0−z⋆(xk,yk)∥2]+2(σgxy2+(Lg′)2)Rkz+2σgxy2B2μg−2 \begin{array}{l} W _ {k} ^ {\prime} = \mathbb {E} \left[ \left\| \partial_ {x y} (\hat {g} - g) (x _ {k}, y _ {k}, \tilde {\xi} _ {N + 1, k}) z _ {k} \right\| ^ {2} \right] + \mathbb {E} \left[ \| \partial_ {x y} g (x _ {k}, y _ {k}) (z _ {k} - \bar {z} _ {k}) \| ^ {2} \right] \\ \stackrel {(i)} {\leq} \sigma_ {g _ {x y}} ^ {2} \mathbb {E} \left[ \| z _ {k} \| ^ {2} \right] + \left(L _ {g} ^ {\prime}\right) ^ {2} \mathbb {E} \left[ \| z _ {k} - \bar {z} _ {k} \| ^ {2} \right] \\ \stackrel {(i i)} {\leq} 2 \sigma_ {g _ {x y}} ^ {2} \left(\mathbb {E} \left[ \| z _ {k} - z ^ {\star} (x _ {k}, y _ {k}) \| ^ {2} \right] + \mathbb {E} \left[ \| z ^ {\star} (x _ {k}, y _ {k}) \| ^ {2} \right]\right) + (L _ {g} ^ {\prime}) ^ {2} \mathbb {E} \left[ \| z _ {k} - \bar {z} _ {k} \| ^ {2} \right] \\ \stackrel {(i i i)} {\leq} 2 \sigma_ {g _ {x y}} ^ {2} \mathbb {E} \left[ \| z _ {k} - z ^ {\star} (x _ {k}, y _ {k}) \| ^ {2} \right] + \left(L _ {g} ^ {\prime}\right) ^ {2} \mathbb {E} \left[ \| z _ {k} - \bar {z} _ {k} \| ^ {2} \right] + 2 \sigma_ {g _ {x y}} ^ {2} B ^ {2} \mu_ {g} ^ {- 2} \\ \stackrel {(i v)} {\leq} 2 \sigma_ {g _ {x y}} ^ {2} \left(\Pi_ {k} \mathbb {E} \left[ \left\| z _ {k} ^ {0} - z ^ {\star} \left(x _ {k}, y _ {k}\right) \right\| ^ {2} + R _ {k} ^ {z} \right]\right) \\ + \left(L _ {g} ^ {\prime}\right) ^ {2} \left(4 \sigma_ {g _ {y y}} ^ {2} \mu_ {g} ^ {- 2} \Pi_ {k} \mathbb {E} \left[ \left\| z _ {k} ^ {0} - z ^ {\star} \left(x _ {k}, y _ {k}\right) \right\| ^ {2} \right] + 2 R _ {k} ^ {z}\right) + 2 \sigma_ {g _ {x y}} ^ {2} B ^ {2} \mu_ {g} ^ {- 2} \\ \leq 2 \left(\sigma_ {g _ {x y}} ^ {2} + 2 \left(L _ {g} ^ {\prime}\right) ^ {2} \mu_ {g} ^ {- 2} \sigma_ {g _ {y y}} ^ {2}\right) \Pi_ {k} \mathbb {E} \left[ \left\| z _ {k} ^ {0} - z ^ {\star} \left(x _ {k}, y _ {k}\right) \right\| ^ {2} \right] \\ + 2 \left(\sigma_ {g _ {x y}} ^ {2} + \left(L _ {g} ^ {\prime}\right) ^ {2}\right) R _ {k} ^ {z} + 2 \sigma_ {g _ {x y}} ^ {2} B ^ {2} \mu_ {g} ^ {- 2} \\ \end{array} $$

(i) follows from Assumptions 3 and 5, (ii) uses that $| z_k|^2 \leq 2\left(| z_k - z|^2 + | z|^2\right)$ , (iii) uses that $| z^{\star}(x_k,y_k)| \leq B\mu_g^{-1}$ by Proposition 6. Finally (iv) follows by application of Proposition 7. We further have by definition of $R_k^z$ that:

$$\begin{array}{cccc}& R_k^z\le B^2{L}_g^{-1}{\mu }_g^{-3}{\sigma }_{g_{yy}}^2+3{\mu }_g^{-2}{\sigma }_f^2& & \text{(29)}\end{array}R\; \_\; \{k\}\; ^\; \{z\}\; \backslash leq\; B\; ^\; \{2\}\; L\; \_\; \{g\}\; ^\; \{-\; 1\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 3\}\; \backslash sigma\_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; 3\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{f\}\; ^\; \{2\}\; \backslash tag\; \{29\}$$

Combining the inequalities on $W_{k}^{\prime}, W_{k}^{\prime \prime}$ and (29), we get that $V_{k}^{\psi} \leq w_{x}^{2} + \sigma_{x}^{2}\Pi_{k}E_{k}^{z}$ , with $w_{x}^{2}$ and $\sigma_x^2$ given by (16).

C GENERAL ANALYSIS OF AMIGO

C.1 ANALYSIS OF THE OUTER-LOOP

Proof of Proposition 2. We treat both cases $\mu \geq 0$ and $\mu < 0$ separately. For simplicity we denote by $\mathbb{E}k$ the conditional expectation knowing the iterates $x{k},y_{k}$ and $z_{k - 1}$ and write $\psi_{k} = \mathbb{E}{k}\Big[\hat{\psi}{k}\Big]$ .

Case $\mu \geq 0$ . Recall that $E_k^x$ is given by:

$$E_k^x=\frac{{\eta }_k}{2}\mathbb{E}\left[{\parallel x_k-x^{\star }\parallel }^2\right]+(1-u)\mathbb{E}[\mathcal{L}\left(x_k\right)-{\mathcal{L}}^{\star }]\mathrm{.}E\; \_\; \{k\}\; ^\; \{x\}\; =\; \backslash frac\; \{\backslash eta\_\; \{k\}\}\{2\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; x\; \_\; \{k\}\; -\; x\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; +\; (1\; -\; u)\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\; \backslash right].$$

For simplicity define $\epsilon_{k} = u\delta_{k} + (1 - u)$ , $e_k = (1 - u)(\mathcal{L}(x_k) - \mathcal{L}^\star) + \frac{\eta_k}{2}| x_k - x^\star |^2$ and $e_k' = u\delta_k(\mathcal{L}(x_k) - \mathcal{L}^\star)$ . It is then easy to see that $\mathbb{E}[e_k]$ is equal to the l.h.s of (10), i.e. $\mathbb{E}[e_k] = E_k^x$ . We

will start by bounding the difference between two successive iterates of $e_k$ :

$$\begin{array}{c}{e}_k^{\mathrm{\prime }}+e_k-e_{k-1}\le u{\delta }_k(\mathcal{L}(x_k)-{\mathcal{L}}^{\star })+(1-u)(\mathcal{L}(x_k)-\mathcal{L}(x_{k-1}))\\ +\frac{{\eta }_k}{2}{\parallel x_k-x^{\star }\parallel }^2-\frac{{\eta }_{k-1}}{2}{\parallel x_{k-1}-x^{\star }\parallel }^2\end{array}\backslash begin\{array\}\{l\}\; e\; \_\; \{k\}\; ^\; \{\backslash prime\}\; +\; e\; \_\; \{k\}\; -\; e\; \_\; \{k\; -\; 1\}\; \backslash leq\; u\; \backslash delta\_\; \{k\}\; (\backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\})\; +\; (1\; -\; u)\; (\backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\}))\; \backslash \backslash \; +\; \backslash frac\; \{\backslash eta\_\; \{k\}\}\{2\}\; \backslash left\backslash |\; x\; \_\; \{k\}\; -\; x\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}\; -\; \backslash frac\; \{\backslash eta\_\; \{k\; -\; 1\}\}\{2\}\; \backslash left\backslash |\; x\; \_\; \{k\; -\; 1\}\; -\; x\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash \backslash \; \backslash end\{array\}$$

$$\begin{array}{c}u{\delta }_k(\mathcal{L}(x_k)-{\mathcal{L}}^{\star })+(1-u)(\mathcal{L}(x_k)-\mathcal{L}(x_{k-1}))\\ -\frac{{\delta }_k{\eta }_{k-1}}{2}\mathrm{\parallel }{x}_{k-1}-x^{\star }{\mathrm{\parallel }}^2+{\delta }_k(\frac{\mu }{2}\mathrm{\parallel }{x}_{k-1}-x^{\star }{\mathrm{\parallel }}^2-\mathrm{\nabla }\mathcal{L}{\left(x_{k-1}\right)}^{\mathrm{\top }}(x_{k-1}-x^{\star }))\\ +\frac{{\eta }_k}{2}({\gamma }_k^2{\parallel {\widehat{\psi }}_{k-1}\parallel }^2-2{\gamma }_k{({\widehat{\psi }}_{k-1}-\mathrm{\nabla }\mathcal{L}\left(x_{k-1}\right))}^{\mathrm{\top }}(x_{k-1}-x^{\star }))\end{array}\backslash begin\{array\}\{l\}\; \backslash stackrel\; \{(i)\}\; \{\backslash leq\}\; u\; \backslash delta\_\; \{k\}\; (\backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\})\; +\; (1\; -\; u)\; (\backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\}))\; \backslash \backslash \; -\; \backslash frac\; \{\backslash delta\_\; \{k\}\; \backslash eta\_\; \{k\; -\; 1\}\}\{2\}\; \backslash |\; x\; \_\; \{k\; -\; 1\}\; -\; x\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; \backslash delta\_\; \{k\}\; \backslash left(\backslash frac\; \{\backslash mu\}\{2\}\; \backslash |\; x\; \_\; \{k\; -\; 1\}\; -\; x\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\; ^\; \{\backslash top\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\; -\; x\; ^\; \{\backslash star\}\backslash right)\backslash right)\; \backslash \backslash \; +\; \backslash frac\; \{\backslash eta\_\; \{k\}\}\{2\}\; \backslash left(\backslash gamma\_\; \{k\}\; ^\; \{2\}\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; -\; 2\; \backslash gamma\_\; \{k\}\; \backslash left(\backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\backslash right)\; ^\; \{\backslash top\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\; -\; x\; ^\; \{\backslash star\}\backslash right)\backslash right)\; \backslash \backslash \; \backslash end\{array\}$$

$$uδk(L(xk)−L⋆)+(1−u)(L(xk)−L(xk−1))−δkηk−12∥xk−1−x⋆∥2−δk(L(xk−1)−L⋆)+ηk2(γk2∥ψ^k−1∥2−2γk(ψ^k−1−∇L(xk−1))⊤(xk−1−x⋆))≤(uδk+(1−u))(L(xk)−L(xk−1))−δkηk−12∥xk−1−x⋆∥2−δk(1−u)(L(xk−1)−L⋆)+ηk2(γk2∥ψ^k−1∥2−2γk(ψ^k−1−∇L(xk−1))⊤(xk−1−x⋆))≤ϵk(L(xk)−L(xk−1))−δkek−1+ηk2(γk2∥ψ^k−1∥2−2γk(ψ^k−1−∇L(xk−1))⊤(xk−1−x⋆)) \begin{array}{l} \stackrel {(i i)} {\leq} u \delta_ {k} (\mathcal {L} (x _ {k}) - \mathcal {L} ^ {\star}) + (1 - u) (\mathcal {L} (x _ {k}) - \mathcal {L} (x _ {k - 1})) \\ - \frac {\delta_ {k} \eta_ {k - 1}}{2} \| x _ {k - 1} - x ^ {\star} \| ^ {2} - \delta_ {k} \left(\mathcal {L} \left(x _ {k - 1}\right) - \mathcal {L} ^ {\star}\right) \\ + \frac {\eta_ {k}}{2} \left(\gamma_ {k} ^ {2} \left\| \hat {\psi} _ {k - 1} \right\| ^ {2} - 2 \gamma_ {k} \left(\hat {\psi} _ {k - 1} - \nabla \mathcal {L} \left(x _ {k - 1}\right)\right) ^ {\top} \left(x _ {k - 1} - x ^ {\star}\right)\right) \\ \leq \left(u \delta_ {k} + (1 - u)\right) \left(\mathcal {L} \left(x _ {k}\right) - \mathcal {L} \left(x _ {k - 1}\right)\right) \\ - \frac {\delta_ {k} \eta_ {k - 1}}{2} \| x _ {k - 1} - x ^ {\star} \| ^ {2} - \delta_ {k} (1 - u) \left(\mathcal {L} \left(x _ {k - 1}\right) - \mathcal {L} ^ {\star}\right) \\ + \frac {\eta_ {k}}{2} \left(\gamma_ {k} ^ {2} \left\| \hat {\psi} _ {k - 1} \right\| ^ {2} - 2 \gamma_ {k} \left(\hat {\psi} _ {k - 1} - \nabla \mathcal {L} \left(x _ {k - 1}\right)\right) ^ {\top} \left(x _ {k - 1} - x ^ {\star}\right)\right) \\ \leq \epsilon_ {k} (\mathcal {L} (x _ {k}) - \mathcal {L} (x _ {k - 1})) - \delta_ {k} e _ {k - 1} \\ + \frac {\eta_ {k}}{2} \left(\gamma_ {k} ^ {2} \left\| \hat {\psi} _ {k - 1} \right\| ^ {2} - 2 \gamma_ {k} \left(\hat {\psi} _ {k - 1} - \nabla \mathcal {L} \left(x _ {k - 1}\right)\right) ^ {\top} \left(x _ {k - 1} - x ^ {\star}\right)\right) \\ \end{array} $$

$$\begin{array}{c}-{\delta }_k{e}_{k-1}+{ϵ}_k\mathrm{\nabla }\mathcal{L}(x_{k-1}{)}^{\mathrm{\top }}(x_k-x_{k-1})+\frac{{ϵ}_{k}L}{2}\mathrm{\parallel }{x}_k-x_{k-1}{\mathrm{\parallel }}^2\\ +\frac{{\eta }_k}{2}({\gamma }_k^2{\parallel {\widehat{\psi }}_{k-1}\parallel }^2-2{\gamma }_k{({\widehat{\psi }}_{k-1}-\mathrm{\nabla }\mathcal{L}\left(x_{k-1}\right))}^{\mathrm{\top }}(x_{k-1}-x^{\star }))\\ =-{\delta }_k{e}_{k-1}-{\gamma }_k{ϵ}_k\mathrm{\nabla }\mathcal{L}(x_{k-1}{)}^{\mathrm{\top }}{\widehat{\psi }}_{k-1}+\frac{{ϵ}_k{\gamma }_k^{2}L}{2}{\parallel {\widehat{\psi }}_{k-1}\parallel }^2\\ +\frac{{\eta }_k}{2}({\gamma }_k^2{\parallel {\widehat{\psi }}_{k-1}\parallel }^2-2{\gamma }_k{({\widehat{\psi }}_{k-1}-\mathrm{\nabla }\mathcal{L}\left(x_{k-1}\right))}^{\mathrm{\top }}(x_{k-1}-x^{\star }))\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash stackrel\; \{(i\; i\; i)\}\; \{\backslash leq\}\; -\; \backslash delta\_\; \{k\}\; e\; \_\; \{k\; -\; 1\}\; +\; \backslash epsilon\_\; \{k\}\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; ^\; \{\backslash top\}\; (x\; \_\; \{k\}\; -\; x\; \_\; \{k\; -\; 1\})\; +\; \backslash frac\; \{\backslash epsilon\_\; \{k\}\; L\}\{2\}\; \backslash |\; x\; \_\; \{k\}\; -\; x\; \_\; \{k\; -\; 1\}\; \backslash |\; ^\; \{2\}\; \backslash \backslash \; +\; \backslash frac\; \{\backslash eta\_\; \{k\}\}\{2\}\; \backslash left(\backslash gamma\_\; \{k\}\; ^\; \{2\}\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; -\; 2\; \backslash gamma\_\; \{k\}\; \backslash left(\backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\backslash right)\; ^\; \{\backslash top\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\; -\; x\; ^\; \{\backslash star\}\backslash right)\backslash right)\; \backslash \backslash \; =\; -\; \backslash delta\_\; \{k\}\; e\; \_\; \{k\; -\; 1\}\; -\; \backslash gamma\_\; \{k\}\; \backslash epsilon\_\; \{k\}\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; ^\; \{\backslash top\}\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; +\; \backslash frac\; \{\backslash epsilon\_\; \{k\}\; \backslash gamma\_\; \{k\}\; ^\; \{2\}\; L\}\{2\}\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash \backslash \; +\; \backslash frac\; \{\backslash eta\_\; \{k\}\}\{2\}\; \backslash left(\backslash gamma\_\; \{k\}\; ^\; \{2\}\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; -\; 2\; \backslash gamma\_\; \{k\}\; \backslash left(\backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\backslash right)\; ^\; \{\backslash top\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\; -\; x\; ^\; \{\backslash star\}\backslash right)\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

(i) follows from the update expression $x_{k} = x_{k - 1} - \gamma_{k}\hat{\psi}{k - 1}$ , (ii) follows from the convexity of $\mathcal{L}$ and (iii) follows by $L$ -smoothness of $\mathcal{L}$ . Taking the expectation conditionally on the randomness at iteration $k - 1$ and using that $\mathbb{E}{k - 1}\left[\hat{\psi}{k - 1}\right] = \psi{k - 1}$ , we therefore get

$$Ek−1[ek′+ek−ek−1]≤−δkek−1−γkϵk∇L(xk−1)⊤ψk−1+γk2(δk+ϵk)Ek−1[∥ψ^k−1∥2]−δk(ψk−1−∇L(xk−1))⊤(xk−1−x⋆)=−δkek−1+γksk(Ek−1[∥ψ^k−1−ψk−1∥]2+∥ψk−1−∇L(xk−1)∥2)−δk(ψk−1−∇L(xk−1))⊤(xk−1−γk∇L(xk−1)−x⋆)−γk2(ϵk−δk)∥∇L(xk−1)∥2 \begin{array}{l} \mathbb {E} _ {k - 1} \left[ e _ {k} ^ {\prime} + e _ {k} - e _ {k - 1} \right] \leq - \delta_ {k} e _ {k - 1} - \gamma_ {k} \epsilon_ {k} \nabla \mathcal {L} \left(x _ {k - 1}\right) ^ {\top} \psi_ {k - 1} + \frac {\gamma_ {k}}{2} \left(\delta_ {k} + \epsilon_ {k}\right) \mathbb {E} _ {k - 1} \left[ \left\| \hat {\psi} _ {k - 1} \right\| ^ {2} \right] \\ - \delta_ {k} \left(\psi_ {k - 1} - \nabla \mathcal {L} \left(x _ {k - 1}\right)\right) ^ {\top} \left(x _ {k - 1} - x ^ {\star}\right) \\ = - \delta_ {k} e _ {k - 1} + \gamma_ {k} s _ {k} \left(\mathbb {E} _ {k - 1} \left[ \left\| \hat {\psi} _ {k - 1} - \psi_ {k - 1} \right\| \right] ^ {2} + \| \psi_ {k - 1} - \nabla \mathcal {L} (x _ {k - 1}) \| ^ {2}\right) \\ - \delta_ {k} \left(\psi_ {k - 1} - \nabla \mathcal {L} \left(x _ {k - 1}\right)\right) ^ {\top} \left(x _ {k - 1} - \gamma_ {k} \nabla \mathcal {L} \left(x _ {k - 1}\right) - x ^ {\star}\right) \\ - \frac {\gamma_ {k}}{2} \left(\epsilon_ {k} - \delta_ {k}\right) \| \nabla \mathcal {L} \left(x _ {k - 1}\right) \| ^ {2} \\ \end{array} $$

$$\begin{array}{c}-{\delta }_k{e}_{k-1}+{\gamma }_k{s}_k({\mathbb{E}}_{k-1}{\left[\parallel {\widehat{\psi }}_{k-1}-{\psi }_{k-1}\parallel \right]}^2+\mathrm{\parallel }{\psi }_{k-1}-\mathrm{\nabla }\mathcal{L}(x_{k-1}){\mathrm{\parallel }}^2)\\ -{\delta }_k{({\psi }_{k-1}-\mathrm{\nabla }\mathcal{L}\left(x_{k-1}\right))}^{\mathrm{\top }}(x_{k-1}-{\gamma }_k\mathrm{\nabla }\mathcal{L}\left(x_{k-1}\right)-x^{\star })\end{array}\backslash begin\{array\}\{l\}\; \backslash stackrel\; \{(i)\}\; \{\backslash leq\}\; -\; \backslash delta\_\; \{k\}\; e\; \_\; \{k\; -\; 1\}\; +\; \backslash gamma\_\; \{k\}\; s\; \_\; \{k\}\; \backslash left(\backslash mathbb\; \{E\}\; \_\; \{k\; -\; 1\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; -\; \backslash psi\_\; \{k\; -\; 1\}\; \backslash right\backslash |\; \backslash right]\; ^\; \{2\}\; +\; \backslash |\; \backslash psi\_\; \{k\; -\; 1\}\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash |\; ^\; \{2\}\backslash right)\; \backslash \backslash \; -\; \backslash delta\_\; \{k\}\; \backslash left(\backslash psi\_\; \{k\; -\; 1\}\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\backslash right)\; ^\; \{\backslash top\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\; -\; \backslash gamma\_\; \{k\}\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\; -\; x\; ^\; \{\backslash star\}\backslash right)\; \backslash \backslash \; \backslash end\{array\}$$

where (i) follows from $\delta_k \leq \epsilon_k$ since by construction $\delta_k \leq 1$ . Taking the expectation w.r.t. all the randomness and applying Cauchy-Schwarz inequality to the last term yields the following inequality:

$$\begin{array}{cccc}& \begin{array}{c}u{\delta }_k(\mathcal{L}\left(x_k\right)-{\mathcal{L}}^{\star })+E_k^x\le (1-{\delta }_k)E_{k-1}^x+{\gamma }_k{s}_k(V_{k-1}^{\psi }+E_{k-1}^{\psi })\\ +{\delta }_k{\left(E_{k-1}^{\psi }\right)}^{\frac{1}{2}}\mathbb{E}{[\mathrm{\mid }\mathrm{\mid }{x}_{k-1}-{\gamma }_k\mathrm{\nabla }\mathcal{L}(x_{k-1})-x^{\star }\mathrm{\mid }{\mathrm{\mid }}^2]}^{\frac{1}{2}}\mathrm{.}\end{array}& & \text{(30)}\end{array}\backslash begin\{array\}\{l\}\; u\; \backslash delta\_\; \{k\}\; \backslash left(\backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\}\backslash right)\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\backslash right)\; +\; E\; \_\; \{k\}\; ^\; \{x\}\; \backslash leq\; \backslash left(1\; -\; \backslash delta\_\; \{k\}\backslash right)\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; +\; \backslash gamma\_\; \{k\}\; s\; \_\; \{k\}\; \backslash left(V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\backslash right)\; \backslash tag\; \{30\}\; \backslash \backslash \; +\; \backslash delta\_\; \{k\}\; \backslash left(E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash mathbb\; \{E\}\; \backslash left[\; |\; |\; x\; \_\; \{k\; -\; 1\}\; -\; \backslash gamma\_\; \{k\}\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; -\; x\; ^\; \{\backslash star\}\; |\; |\; ^\; \{2\}\; \backslash right]\; ^\; \{\backslash frac\; \{1\}\{2\}\}.\; \backslash \backslash \; \backslash end\{array\}$$

Since $\mathcal{L}$ is convex, we have the inequality: $| x_{k - 1} - \gamma_k\nabla \mathcal{L}(x_{k - 1}) - x^\star | ^2\leq | x_{k - 1} - x^\star | ^2$ . Hence, we can deduce that:

$${\delta }_k{\parallel x_{k-1}-{\gamma }_k\mathrm{\nabla }\mathcal{L}\left(x_{k-1}\right)-x^{\star }\parallel }^2\le {\delta }_k{\parallel x_{k-1}-x^{\star }\parallel }^2\le 2{\gamma }_k{\eta }_k{\eta }_{k-1}^{-1}{E}_{k-1}^x\le 2{\gamma }_k{E}_{k-1}^x,\backslash delta\_\; \{k\}\; \backslash left\backslash |\; x\; \_\; \{k\; -\; 1\}\; -\; \backslash gamma\_\; \{k\}\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\; -\; x\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash leq\; \backslash delta\_\; \{k\}\; \backslash left\backslash |\; x\; \_\; \{k\; -\; 1\}\; -\; x\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash leq\; 2\; \backslash gamma\_\; \{k\}\; \backslash eta\_\; \{k\}\; \backslash eta\_\; \{k\; -\; 1\}\; ^\; \{-\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; \backslash leq\; 2\; \backslash gamma\_\; \{k\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\},$$

where we used that $\eta_{k}$ is non-increasing by construction. Combining the above inequality with (30) yields:

$$F_k+E_k^x\le (1-{\delta }_k)E_{k-1}^x+{\gamma }_k{s}_k(V_{k-1}^{\psi }+E_{k-1}^{\psi })+\sqrt{2}{\gamma }_k^{\frac{1}{2}}{\delta }_k^{\frac{1}{2}}\left(E_{k-1}^{\psi }{)}^{\frac{1}{2}}\right(E_{k-1}^x{)}^{\frac{1}{2}}\mathrm{.}F\; \_\; \{k\}\; +\; E\; \_\; \{k\}\; ^\; \{x\}\; \backslash leq\; (1\; -\; \backslash delta\_\; \{k\})\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; +\; \backslash gamma\_\; \{k\}\; s\; \_\; \{k\}\; \backslash Big\; (V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; \backslash Big)\; +\; \backslash sqrt\; \{2\}\; \backslash gamma\_\; \{k\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash delta\_\; \{k\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash Big\; (E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; \backslash Big)\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash big\; (E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; \backslash big)\; ^\; \{\backslash frac\; \{1\}\{2\}\}.$$

Case $\mu < 0$ . Recall that for $\mu < 0$ , we set $E_k^x = \frac{1}{L} \mathbb{E}\left[|\nabla \mathcal{L}(x_k)|^2\right]$ . Using that $\mathcal{L}$ is $L$ -smooth, we have that:

$$\begin{array}{c}\mathcal{L}(x_k)-\mathcal{L}(x_{k-1})\le \mathrm{\nabla }\mathcal{L}(x_{k-1}{)}^{\mathrm{\top }}(x_k-x_{k-1})+\frac{L}{2}\mathrm{\parallel }{x}_k-x_{k-1}{\mathrm{\parallel }}^2\\ \le -{\gamma }_k\mathrm{\nabla }\mathcal{L}(x_{k-1}{)}^{\mathrm{\top }}{\widehat{\psi }}_{k-1}+\frac{L{\gamma }_k^2}{2}{\parallel {\widehat{\psi }}_{k-1}\parallel }^2\\ \le -{\gamma }_k\mathrm{\parallel }\mathrm{\nabla }\mathcal{L}(x_{k-1}){\mathrm{\parallel }}^2-{\gamma }_k\mathrm{\nabla }\mathcal{L}(x_{k-1}{)}^{\mathrm{\top }}({\widehat{\psi }}_{k-1}-\mathrm{\nabla }\mathcal{L}(x_{k-1}))\\ +\frac{L{\gamma }_k^2}{2}({\parallel {\widehat{\psi }}_{k-1}-{\psi }_{k-1}\parallel }^2+2{({\widehat{\psi }}_{k-1}-{\psi }_{k-1})}^{\mathrm{\top }}{\psi }_{k-1}+\mathrm{\parallel }{\psi }_{k-1}{\mathrm{\parallel }}^2)\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\})\; -\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash leq\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; ^\; \{\backslash top\}\; (x\; \_\; \{k\}\; -\; x\; \_\; \{k\; -\; 1\})\; +\; \backslash frac\; \{L\}\{2\}\; \backslash |\; x\; \_\; \{k\}\; -\; x\; \_\; \{k\; -\; 1\}\; \backslash |\; ^\; \{2\}\; \backslash \backslash \; \backslash leq\; -\; \backslash gamma\_\; \{k\}\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; ^\; \{\backslash top\}\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; +\; \backslash frac\; \{L\; \backslash gamma\_\; \{k\}\; ^\; \{2\}\}\{2\}\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash \backslash \; \backslash leq\; -\; \backslash gamma\_\; \{k\}\; \backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash |\; ^\; \{2\}\; -\; \backslash gamma\_\; \{k\}\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; ^\; \{\backslash top\}\; \backslash left(\backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\backslash right)\; \backslash \backslash \; +\; \backslash frac\; \{L\; \backslash gamma\_\; \{k\}\; ^\; \{2\}\}\{2\}\; \backslash left(\backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; -\; \backslash psi\_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; +\; 2\; \backslash left(\backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; -\; \backslash psi\_\; \{k\; -\; 1\}\backslash right)\; ^\; \{\backslash top\}\; \backslash psi\_\; \{k\; -\; 1\}\; +\; \backslash |\; \backslash psi\_\; \{k\; -\; 1\}\; \backslash |\; ^\; \{2\}\backslash right).\; \backslash \backslash \; \backslash end\{array\}$$

Taking the expectation w.r.t. all randomness in the algorithm in the above inequality, we get:

$$E[L(xk)−L(xk−1)]≤−γkE[∥∇L(xk−1)∥2]−γkE[∇L(xk−1)⊤(ψk−1−∇L(xk−1))]+Lγk22(E[(∥ψ^k−1−ψk−1∥2)]+E[∥ψk−1∥2])=−γk(1−Lγk2)E[∥∇L(xk−1)∥2]+Lγk22(Vk−1ψ+Ek−1ψ)−γk(1−Lγk)E[∇L(xk−1)⊤(ψk−1−∇L(xk−1))]≤−γk2E[∥∇L(xk−1)∥2]+Lγk22(Vk−1ψ+Ek−1ψ)+γkE[∥∇L(xk−1)∥2]12(Ek−1ψ)12.=−δkEk−1x+δkγk2(Vk−1ψ+Ek−1ψ)+2δk12γk12(Ek−1x)12(Ek−1ψ)12. \begin{array}{l} \mathbb {E} [ \mathcal {L} (x _ {k}) - \mathcal {L} (x _ {k - 1}) ] \leq - \gamma_ {k} \mathbb {E} \left[ \| \nabla \mathcal {L} (x _ {k - 1}) \| ^ {2} \right] - \gamma_ {k} \mathbb {E} \left[ \nabla \mathcal {L} (x _ {k - 1}) ^ {\top} (\psi_ {k - 1} - \nabla \mathcal {L} (x _ {k - 1})) \right] \\ + \frac {L \gamma_ {k} ^ {2}}{2} \left(\mathbb {E} \left[ \left(\left\| \hat {\psi} _ {k - 1} - \psi_ {k - 1} \right\| ^ {2}\right) \right] + \mathbb {E} \left[ \| \psi_ {k - 1} \| ^ {2} \right]\right) \\ = - \gamma_ {k} \left(1 - \frac {L \gamma_ {k}}{2}\right) \mathbb {E} \left[ \| \nabla \mathcal {L} (x _ {k - 1}) \| ^ {2} \right] + \frac {L \gamma_ {k} ^ {2}}{2} \left(V _ {k - 1} ^ {\psi} + E _ {k - 1} ^ {\psi}\right) \\ \left. - \gamma_ {k} (1 - L \gamma_ {k}) \mathbb {E} \left[ \nabla \mathcal {L} \left(x _ {k - 1}\right) ^ {\top} \left(\psi_ {k - 1} - \nabla \mathcal {L} \left(x _ {k - 1}\right)\right) \right] \right. \\ \leq - \frac {\gamma_ {k}}{2} \mathbb {E} \left[ \| \nabla \mathcal {L} \left(x _ {k - 1}\right) \| ^ {2} \right] + \frac {L \gamma_ {k} ^ {2}}{2} \left(V _ {k - 1} ^ {\psi} + E _ {k - 1} ^ {\psi}\right) \\ + \gamma_ {k} \mathbb {E} \left[ \| \nabla \mathcal {L} (x _ {k - 1}) \| ^ {2} \right] ^ {\frac {1}{2}} \left(E _ {k - 1} ^ {\psi}\right) ^ {\frac {1}{2}}. \\ = - \delta_ {k} E _ {k - 1} ^ {x} + \frac {\delta_ {k} \gamma_ {k}}{2} \left(V _ {k - 1} ^ {\psi} + E _ {k - 1} ^ {\psi}\right) + \sqrt {2} \delta_ {k} ^ {\frac {1}{2}} \gamma_ {k} ^ {\frac {1}{2}} \left(E _ {k - 1} ^ {x}\right) ^ {\frac {1}{2}} \left(E _ {k - 1} ^ {\psi}\right) ^ {\frac {1}{2}}. \\ \end{array} $$

where we used that $1 - \frac{L\gamma_k}{2} \geq \frac{1}{2}$ and $0 \leq 1 - L\gamma_k \leq 1$ to get the last inequality. Using the definition of $F_{k}$ yields an inequality of the form:

$$F_k+E_k^x\le (1-{\delta }_k)E_{k-1}^x+{\gamma }_k{s}_k(V_{k-1}^{\psi }+E_{k-1}^{\psi })+\sqrt{2}{\gamma }_k^{\frac{1}{2}}{\delta }_k^{\frac{1}{2}}\left(E_{k-1}^{\psi }{)}^{\frac{1}{2}}\right(E_{k-1}^x{)}^{\frac{1}{2}}\mathrm{.}F\; \_\; \{k\}\; +\; E\; \_\; \{k\}\; ^\; \{x\}\; \backslash leq\; (1\; -\; \backslash delta\_\; \{k\})\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; +\; \backslash gamma\_\; \{k\}\; s\; \_\; \{k\}\; \backslash Big\; (V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; \backslash Big)\; +\; \backslash sqrt\; \{2\}\; \backslash gamma\_\; \{k\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash delta\_\; \{k\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash Big\; (E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; \backslash Big)\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash Big\; (E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; \backslash Big)\; ^\; \{\backslash frac\; \{1\}\{2\}\}.$$

Hence, in both cases $\mu \geq 0$ and $\mu < 0$ we get an inequality of the same form, but with different expressions for $F_{k}$ and $s_k$ . We get the desired result using Young's inequality, to upper-bound the last term in the r.h.s. of the above inequality. More precisely, we use that for any $0 < \rho_{k} < 1$ :

$$\sqrt{2}{\gamma }_k^{\frac{1}{2}}{\delta }_k^{\frac{1}{2}}{\left(E_{k-1}^{\psi }\right)}^{\frac{1}{2}}{\left(E_{k-1}^x\right)}^{\frac{1}{2}}\le \frac{1}{2}{\rho }_k{\delta }_k{E}_{k-1}^x+{\rho }_k^{-1}{\gamma }_k{E}_{k-1}^{\psi }\mathrm{.}\backslash sqrt\; \{2\}\; \backslash gamma\_\; \{k\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash delta\_\; \{k\}\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash left(E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash left(E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash leq\; \backslash frac\; \{1\}\{2\}\; \backslash rho\_\; \{k\}\; \backslash delta\_\; \{k\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash gamma\_\; \{k\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}.$$

C.2 INNER-LEVEL ERROR BOUND

In this section we prove Proposition 5 which controls the evolutions of the warm-start errors $E_{k}^{y}$ and $E_{k}^{z}$ . As a first step, in Proposition 12, we provide a result controlling the mean squared error between two successive iterates $x_{k-1}, x_{k}$ and $y_{k-1}, y_{k}$ which will be used in the proof of Proposition 5.

Proposition 12 (Control of the increments of $x_{k}$ and $y_{k}$ ). Consider $\zeta_{k}, \phi_{k}$ and $\tilde{R}_{k}^{y}$ as defined in Proposition 8 for some fixed $0 \leq v \leq 1$ . Then, the following holds:

$$\begin{array}{c}{\gamma }_k^2\mathbb{E}\left[{\parallel {\widehat{\psi }}_{k-1}\parallel }^2\right]=\mathbb{E}[\mathrm{\parallel }{x}_k-x_{k-1}{\mathrm{\parallel }}^2]\le {\gamma }_k^2(V_{k-1}^{\psi }+2E_{k-1}^{\psi }+2{\zeta }_k{E}_{k-1}^x)\\ \mathbb{E}\left[{\parallel y_k-y_{k-1}\parallel }^2\right]\le 2{\varphi }_k{E}_k^y+2{\tilde{R}}_k^y\end{array}\backslash begin\{array\}\{l\}\; \backslash gamma\_\; \{k\}\; ^\; \{2\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; =\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; x\; \_\; \{k\}\; -\; x\; \_\; \{k\; -\; 1\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash gamma\_\; \{k\}\; ^\; \{2\}\; \backslash Big\; (V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; 2\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; 2\; \backslash zeta\_\; \{k\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; \backslash Big)\; \backslash \backslash \; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; y\; \_\; \{k\}\; -\; y\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; 2\; \backslash phi\_\; \{k\}\; E\; \_\; \{k\}\; ^\; \{y\}\; +\; 2\; \backslash tilde\; \{R\}\; \_\; \{k\}\; ^\; \{y\}\; \backslash \backslash \; \backslash end\{array\}$$

Proof. Proof of Proposition 12 We prove each inequality separately.

Increments of $x_{k}$ . By the update equation, we have that $x_{k} = x_{k - 1} - \gamma_{k}\hat{\psi}{k - 1}$ , hence we only need to control $\mathbb{E}\left[\left| \hat{\psi}{k - 1}\right| ^2\right]$ . We have the following:

$$\begin{array}{c}\mathbb{E}\left[{\parallel {\widehat{\psi }}_{k-1}\parallel }^2\right]\le \mathbb{E}\left[{\parallel {\widehat{\psi }}_{k-1}-{\psi }_{k-1}\parallel }^2\right]+2\mathbb{E}[\mathrm{\parallel }{\psi }_{k-1}-\mathrm{\nabla }\mathcal{L}(x_{k-1}){\mathrm{\parallel }}^2]+2\mathbb{E}[\mathrm{\parallel }\mathrm{\nabla }\mathcal{L}(x_{k-1}){\mathrm{\parallel }}^2]\\ =V_{k-1}^{\psi }+2E_{k-1}^{\psi }+2\mathbb{E}\left[{\parallel \mathrm{\nabla }\mathcal{L}(x_{k-1})\parallel }^2\right]\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; -\; \backslash psi\_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; +\; 2\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; \backslash psi\_\; \{k\; -\; 1\}\; -\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; +\; 2\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash \backslash \; =\; V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; 2\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; 2\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right].\; \backslash \backslash \; \backslash end\{array\}$$

In the case $(\mu < 0)$ , we have $E_{k - 1}^{x} = \frac{1}{2L}\mathbb{E}\Big[| \nabla \mathcal{L}(x_{k - 1})|^{2}\Big]$ , hence by setting $\zeta_k\coloneqq 2L$ , we get the desired inequality. In the convex case $(\mu \geq 0)$ , since $\mathcal{L}$ is $L$ -smooth, we have that:

$${\parallel \mathrm{\nabla }\mathcal{L}\left(x_{k-1}\right)\parallel }^2\le 2L(\mathcal{L}\left(x_{k-1}\right)-{\mathcal{L}}^{\star })\le 2L(1-u{)}^{-1}{E}_{k-1}^x,\backslash left\backslash |\; \backslash nabla\; \backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash leq\; 2\; L\; \backslash left(\backslash mathcal\; \{L\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\; -\; \backslash mathcal\; \{L\}\; ^\; \{\backslash star\}\backslash right)\; \backslash leq\; 2\; L\; (1\; -\; u)\; ^\; \{-\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\},$$

provided that $u < 1$ . We also have that $(\mathcal{L}(x_{k - 1}) - \mathcal{L}^{\star}) \leq \frac{L}{2}| x_{k - 1} - x^{\star}|^{2} \leq L\eta_{k - 1}^{-1}E_{k - 1}^{x}$ which yields $|\nabla \mathcal{L}(x_{k - 1})|^2 \leq 2L^2\eta_{k - 1}^{-1}E_{k - 1}^{x}$ . Hence, we can set $\zeta_k = 2L\min \left((1 - u)^{-1}, L\eta_{k - 1}^{-1}\right)$ .

Increments of $y_{k}$ . Denoting by $\mathcal{D}_g^t$ a batch of samples at time iteration $t$ of algorithm $\mathcal{A}_k$ and using the update equation of $y^{t}$ we get the following inequality by application of the triangular inequality:

$$\begin{array}{c}\mathbb{E}{\left[{\parallel y_k-y_{k-1}\parallel }^2\right]}^{\frac{1}{2}}\le {\alpha }_k{\sum }_{t=0}^{T-1}\mathbb{E}{\left[{\parallel {\mathrm{\partial }}_y\widehat{g}(x_k,y^t,{\mathcal{D}}_g^t)\parallel }^2\right]}^{\frac{1}{2}}\le {\alpha }_k{\sum }_{t=0}^{T-1}{({\sigma }_g^2+L_g^2\mathbb{E}\left[{\parallel y^t-y^{\star }(x_k)\parallel }^2\right])}^{\frac{1}{2}}\\ \le {\alpha }_k{\sum }_{t=0}^{T-1}{({\sigma }_g^2+L_g^2{R}_k+L_g^2{\mathrm{\Lambda }}_{t,k}{E}_k^y)}^{\frac{1}{2}}\le {\alpha }_{k}T{({\sigma }_g^2(1+2L_g^2{\alpha }_k{\mu }_g^{-1})+L_g^2{E}_k^y)}^{\frac{1}{2}}\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; y\; \_\; \{k\}\; -\; y\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash leq\; \backslash alpha\_\; \{k\}\; \backslash sum\_\; \{t\; =\; 0\}\; ^\; \{T\; -\; 1\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash partial\_\; \{y\}\; \backslash hat\; \{g\}\; (x\; \_\; \{k\},\; y\; ^\; \{t\},\; \backslash mathcal\; \{D\}\; \_\; \{g\}\; ^\; \{t\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash leq\; \backslash alpha\_\; \{k\}\; \backslash sum\_\; \{t\; =\; 0\}\; ^\; \{T\; -\; 1\}\; \backslash left(\backslash sigma\_\; \{g\}\; ^\; \{2\}\; +\; L\; \_\; \{g\}\; ^\; \{2\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; y\; ^\; \{t\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\backslash right)\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash \backslash \; \backslash leq\; \backslash alpha\_\; \{k\}\; \backslash sum\_\; \{t\; =\; 0\}\; ^\; \{T\; -\; 1\}\; \backslash left(\backslash sigma\_\; \{g\}\; ^\; \{2\}\; +\; L\; \_\; \{g\}\; ^\; \{2\}\; R\; \_\; \{k\}\; +\; L\; \_\; \{g\}\; ^\; \{2\}\; \backslash Lambda\_\; \{t,\; k\}\; E\; \_\; \{k\}\; ^\; \{y\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash leq\; \backslash alpha\_\; \{k\}\; T\; \backslash left(\backslash sigma\_\; \{g\}\; ^\; \{2\}\; \backslash left(1\; +\; 2\; L\; \_\; \{g\}\; ^\; \{2\}\; \backslash alpha\_\; \{k\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\backslash right)\; +\; L\; \_\; \{g\}\; ^\; \{2\}\; E\; \_\; \{k\}\; ^\; \{y\}\backslash right)\; ^\; \{\backslash frac\; \{1\}\{2\}\}\; \backslash \backslash \; \backslash end\{array\}$$

where we applied Proposition 7 for every $0 < t \leq T - 1$ to get the second line with $\Lambda_{t,k} := (1 - \alpha_k\mu_g)^t$ . This directly implies the following bound:

$$\begin{array}{cccc}& \mathbb{E}[\mathrm{\parallel }{\tilde{y}}_k-y_{k-1}{\mathrm{\parallel }}^2]\le 2{\alpha }_k^2{T}^2({\sigma }_g^2(1+2L_g^2{\mu }_g^{-1}{\alpha }_k)+L_g^2{E}_k^y)\mathrm{.}& & \text{(31)}\end{array}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; \backslash tilde\; \{y\}\; \_\; \{k\}\; -\; y\; \_\; \{k\; -\; 1\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; 2\; \backslash alpha\_\; \{k\}\; ^\; \{2\}\; T\; ^\; \{2\}\; \backslash left(\backslash sigma\_\; \{g\}\; ^\; \{2\}\; \backslash left(1\; +\; 2\; L\; \_\; \{g\}\; ^\; \{2\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash alpha\_\; \{k\}\backslash right)\; +\; L\; \_\; \{g\}\; ^\; \{2\}\; E\; \_\; \{k\}\; ^\; \{y\}\backslash right).\; \backslash tag\; \{31\}$$

On the other hand, using a triangular inequality and applying Proposition 7, we also have that:

$$\begin{array}{cccc}& \mathbb{E}[\mathrm{\parallel }{y}_k-y_{k-1}{\mathrm{\parallel }}^2]\le 2\mathbb{E}[\mathrm{\parallel }{y}_k-y^{\star }(x_k){\mathrm{\parallel }}^2]+2\mathbb{E}[\mathrm{\parallel }{y}_{k-1}-y^{\star }(x_k){\mathrm{\parallel }}^2]\le (4E_k^y+2R_k^y)\mathrm{.}& & \text{(32)}\end{array}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; \_\; \{k\}\; -\; y\; \_\; \{k\; -\; 1\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; 2\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; +\; 2\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; \_\; \{k\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; \backslash left(4\; E\; \_\; \{k\}\; ^\; \{y\}\; +\; 2\; R\; \_\; \{k\}\; ^\; \{y\}\backslash right).\; \backslash tag\; \{32\}$$

The result follows by combining (31) and (32) using coefficients $1 - v$ and $v$ .

C.3 PROOF OF PROPOSITION 5

Proof of Proposition 5. We will control each of $E_k^y$ and $E_k^z$ separately.

Upper-bound on $E_k^y$ . Let $r_k$ be a non-increasing sequence between 0 and 1. The following holds:

$$\begin{array}{c}{E}_k^y=\mathbb{E}\left[{\parallel y_{k-1}-y^{\star }\left(x_{k-1}\right)\parallel }^2\right]+\mathbb{E}\left[\parallel y^{\star }{(x_k-y^{\star }\left(x_{k-1}\right)\parallel }^2\right]\\ +2\mathbb{E}[{(y_{k-1}-y^{\star }\left(x_{k-1}\right))}^{\mathrm{\top }}\left(y^{\star }(x_k-y^{\star }\left(x_{k-1}\right))\right]\end{array}\backslash begin\{array\}\{l\}\; E\; \_\; \{k\}\; ^\; \{y\}\; =\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; y\; \_\; \{k\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; +\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right.\; \backslash right]\; \backslash \backslash \; +\; 2\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left(y\; \_\; \{k\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\backslash right)\; ^\; \{\backslash top\}\; \backslash left(y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; \backslash left(x\; \_\; \{k\; -\; 1\}\backslash right)\backslash right)\; \backslash right]\; \backslash right.\; \backslash \backslash \; \backslash end\{array\}$$

$$\begin{array}{cccc}& \begin{array}{c}(1+r_k)\mathbb{E}[\mathrm{\parallel }{y}_{k-1}-y^{\star }(x_{k-1}){\mathrm{\parallel }}^2]+(1+r_k^{-1})\mathbb{E}[\mathrm{\parallel }{y}^{\star }(x_k-y^{\star }(x_{k-1}){\mathrm{\parallel }}^2]\\ (1+r_k)({\mathrm{\Lambda }}_{k-1}{E}_{k-1}^y+R_{k-1}^y)+2r_k^{-1}\mathbb{E}[\mathrm{\parallel }{y}^{\star }(x_k)-y^{\star }(x_{k-1}){\mathrm{\parallel }}^2]\\ (1+r_k)({\mathrm{\Lambda }}_{k-1}{E}_{k-1}^y+R_{k-1}^y)+2L_y^2{r}_k^{-1}\mathbb{E}[\mathrm{\parallel }{x}_k-x_{k-1}{\mathrm{\parallel }}^2]\\ (1+r_k)({\mathrm{\Lambda }}_{k-1}{E}_{k-1}^y+R_{k-1}^y)+2L_y^2{r}_k^{-1}{\gamma }_k^2\mathbb{E}\left[{\parallel {\widehat{\psi }}_{k-1}\parallel }^2\right]\end{array}& & \text{(33a)}\end{array}\backslash begin\{array\}\{l\}\; \backslash stackrel\; \{(i)\}\; \{\backslash leq\}\; (1\; +\; r\; \_\; \{k\})\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; \_\; \{k\; -\; 1\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; +\; (1\; +\; r\; \_\; \{k\}\; ^\; \{-\; 1\})\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\}\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash \backslash \; \backslash stackrel\; \{(i\; i)\}\; \{\backslash leq\}\; \backslash left(1\; +\; r\; \_\; \{k\}\backslash right)\; \backslash left(\backslash Lambda\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\backslash right)\; +\; 2\; r\; \_\; \{k\}\; ^\; \{-\; 1\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\})\; -\; y\; ^\; \{\backslash star\}\; (x\; \_\; \{k\; -\; 1\})\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash \backslash \; \backslash stackrel\; \{(i\; i\; i)\}\; \{\backslash leq\}\; (1\; +\; r\; \_\; \{k\})\; \backslash big\; (\backslash Lambda\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; \backslash big)\; +\; 2\; L\; \_\; \{y\}\; ^\; \{2\}\; r\; \_\; \{k\}\; ^\; \{-\; 1\}\; \backslash mathbb\; \{E\}\; \backslash Big\; [\; \backslash |\; x\; \_\; \{k\}\; -\; x\; \_\; \{k\; -\; 1\}\; \backslash |\; ^\; \{2\}\; \backslash Big\; ]\; \backslash \backslash \; \backslash stackrel\; \{(i\; v)\}\; \{\backslash leq\}\; \backslash left(1\; +\; r\; \_\; \{k\}\backslash right)\; \backslash left(\backslash Lambda\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\backslash right)\; +\; 2\; L\; \_\; \{y\}\; ^\; \{2\}\; r\; \_\; \{k\}\; ^\; \{-\; 1\}\; \backslash gamma\_\; \{k\}\; ^\; \{2\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash tag\; \{33a\}\; \backslash \backslash \; \backslash end\{array\}$$

(i) follows by Young's inequality, (ii) uses Proposition 7 to bound the first term and that $(1 + r_k^{-1})\leq$ $2r_{k}^{-1}$ for the second term, (iii) uses that $y^{\star}$ is $L_{y}$ -Lipschitz by Proposition 6 and (iv) uses the update equation $x_{k} = x_{k - 1} - \gamma_{k}\hat{\psi}_{k - 1}$

Upper-bound on $E_k^z$ . Similarly, for a non-increasing sequence $0 < \theta_k \leq 1$ , we have that:

$$Ekz=E[∥zk−1−z⋆(xk−1,yk−1)∥2]+E[∥z⋆(xk,yk)−z⋆(xk−1,yk−1)2∥]+2E[(zk−1−z⋆(xk−1,yk−1))⊤(z⋆(xk,yk)−z⋆(xk−1,yk−1))](1+θk)E[∥zk−1−z⋆(xk−1,yk−1)∥2]+(1+θk−1)E[∥z⋆(xk,yk)−z⋆(xk−1,yk−1)∥2](1+θk)(Πk−1Ek−1z+Rk−1z)+2θk−1E[∥z⋆(xk,yk)−z⋆(xk−1,yk−1)∥2](1+θk)(Πk−1Ek−1z+Rk−1z)+4Lz2θk−1(E[∥xk−xk−1∥2+∥yk−yk−1∥2])(1+θk)(Πk−1Ek−1z+Rk−1z)+4Lz2θk−1(γk2E[∥ψ^k−1∥2]+2ϕkEky+2R~ky)(34a) \begin{array}{l} E _ {k} ^ {z} = \mathbb {E} \left[ \left\| z _ {k - 1} - z ^ {\star} \left(x _ {k - 1}, y _ {k - 1}\right) \right\| ^ {2} \right] + \mathbb {E} \left[ \left\| z ^ {\star} \left(x _ {k}, y _ {k}\right) - z ^ {\star} \left(x _ {k - 1}, y _ {k - 1}\right) ^ {2} \right\| \right] \\ + 2 \mathbb {E} \left[ \left(z _ {k - 1} - z ^ {\star} \left(x _ {k - 1}, y _ {k - 1}\right)\right) ^ {\top} \left(z ^ {\star} \left(x _ {k}, y _ {k}\right) - z ^ {\star} \left(x _ {k - 1}, y _ {k - 1}\right)\right) \right] \\ \stackrel {(i)} {\leq} (1 + \theta_ {k}) \mathbb {E} \left[ \| z _ {k - 1} - z ^ {\star} \left(x _ {k - 1}, y _ {k - 1}\right) \| ^ {2} \right] + (1 + \theta_ {k} ^ {- 1}) \mathbb {E} \left[ \| z ^ {\star} \left(x _ {k}, y _ {k}\right) - z ^ {\star} \left(x _ {k - 1}, y _ {k - 1}\right) \| ^ {2} \right] \\ \stackrel {(i i)} {\leq} \left(1 + \theta_ {k}\right) \left(\Pi_ {k - 1} E _ {k - 1} ^ {z} + R _ {k - 1} ^ {z}\right) + 2 \theta_ {k} ^ {- 1} \mathbb {E} \left[ \| z ^ {\star} (x _ {k}, y _ {k}) - z ^ {\star} (x _ {k - 1}, y _ {k - 1}) \| ^ {2} \right] \\ \stackrel {(i i i)} {\leq} \left(1 + \theta_ {k}\right) \left(\Pi_ {k - 1} E _ {k - 1} ^ {z} + R _ {k - 1} ^ {z}\right) + 4 L _ {z} ^ {2} \theta_ {k} ^ {- 1} \left(\mathbb {E} \left[ \| x _ {k} - x _ {k - 1} \| ^ {2} + \| y _ {k} - y _ {k - 1} \| ^ {2} \right]\right) \\ \stackrel {(i v)} {\leq} \left(1 + \theta_ {k}\right) \left(\Pi_ {k - 1} E _ {k - 1} ^ {z} + R _ {k - 1} ^ {z}\right) + 4 L _ {z} ^ {2} \theta_ {k} ^ {- 1} \left(\gamma_ {k} ^ {2} \mathbb {E} \left[ \left\| \hat {\psi} _ {k - 1} \right\| ^ {2} \right] + 2 \phi_ {k} E _ {k} ^ {y} + 2 \tilde {R} _ {k} ^ {y}\right) \tag {34a} \\ \end{array} $$

(i) follows by Young's inequality, (ii) uses Proposition 7 to bound the first term and that $(1 + \theta_k^{-1})\leq 2\theta_k^{-1}$ for the second term, (iii) uses that $z^{\star}(x,y)$ is $L_{z}$ -Lipschitz in $x$ and $y$ by Proposition 6 and, finally, (iv) uses the update equation $x_{k} = x_{k - 1} - \gamma_{k}\hat{\psi}{k - 1}$ for the term $\mathbb{E}\Big[| x_k - x{k - 1}| ^2\Big]$ and Proposition 12 to control the increments $\mathbb{E}\big[| y_k - y_{k - 1}| ^2\big]$ .

In order to express the upper-bound on $E_k^z$ in terms of $E_{k-1}^y$ instead of $E_k^y$ , we substitute $E_k^y$ in (34a) by its upper-bound in (33a) and use that $(1 + r_k) \leq 2$ to write:

$$\begin{array}{cccc}& \begin{array}{c}{E}_k^z\le (1+{\theta }_k)({\mathrm{\Pi }}_{k-1}{E}_{k-1}^z+R_{k-1}^z)+4L_z^2{\theta }_k^{-1}{\gamma }_k^2(1+4L_y^2{\varphi }_k{r}_k^{-1})\mathbb{E}\left[{\parallel {\widehat{\psi }}_{k-1}\parallel }^2\right]\\ +8L_z^2{\theta }_k^{-1}(2{\varphi }_k({\mathrm{\Lambda }}_{k-1}{E}_{k-1}^y+R_{k-1}^y)+{\tilde{R}}_k^y)\end{array}& & \text{(35)}\end{array}\backslash begin\{array\}\{l\}\; E\; \_\; \{k\}\; ^\; \{z\}\; \backslash leq\; \backslash left(1\; +\; \backslash theta\_\; \{k\}\backslash right)\; \backslash left(\backslash Pi\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\backslash right)\; +\; 4\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash gamma\_\; \{k\}\; ^\; \{2\}\; \backslash left(1\; +\; 4\; L\; \_\; \{y\}\; ^\; \{2\}\; \backslash phi\_\; \{k\}\; r\; \_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash \backslash \; +\; 8\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left(2\; \backslash phi\_\; \{k\}\; \backslash left(\backslash Lambda\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\backslash right)\; +\; \backslash tilde\; \{R\}\; \_\; \{k\}\; ^\; \{y\}\backslash right)\; \backslash tag\; \{35\}\; \backslash \backslash \; \backslash end\{array\}$$

We can then express (33a) and (35) jointly in matrix form as follows:

$$\left(\begin{array}{c}{E}_k^y\\ E_k^z\end{array}\right)\le {\mathit{P}}_{\mathit{k}}\left(\begin{array}{c}{\mathrm{\Lambda }}_{k-1}{E}_{k-1}^y+R_{k-1}^y\\ {\mathrm{\Pi }}_{k-1}{E}_{k-1}^z+R_{k-1}^z\end{array}\right)+{\gamma }_k\mathbb{E}\left[{\parallel {\widehat{\psi }}_{k-1}\parallel }^2\right]{\mathit{U}}_{\mathit{k}}+{\mathit{V}}_{\mathit{k}}\backslash left(\; \backslash begin\{array\}\{c\}\; E\; \_\; \{k\}\; ^\; \{y\}\; \backslash \backslash \; E\; \_\; \{k\}\; ^\; \{z\}\; \backslash end\{array\}\; \backslash right)\; \backslash leq\; \backslash boldsymbol\; \{P\}\; \_\; \{\backslash boldsymbol\; \{k\}\}\; \backslash left(\; \backslash begin\{array\}\{c\}\; \backslash Lambda\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; \backslash \backslash \; \backslash Pi\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; \backslash end\{array\}\; \backslash right)\; +\; \backslash gamma\_\; \{k\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{\backslash psi\}\; \_\; \{k\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash boldsymbol\; \{U\}\; \_\; \{\backslash boldsymbol\; \{k\}\}\; +\; \backslash boldsymbol\; \{V\}\; \_\; \{\backslash boldsymbol\; \{k\}\}$$

where the $P_{k}$ is a $2\times 2$ matrix and $\pmb{U}{k}$ and $\pmb{V}{k}$ are 2-dimensional vectors given by (17). The desired result follows directly by substituting $\mathbb{E}\left[\left| \hat{\psi}_{k - 1}\right| ^2\right]$ by its upper-bound from Proposition 12 in the above inequality.

C.4 GENERAL ERROR BOUND

Proof of Proposition 9. First note that, by assumption, we have that $\delta_k r_k^{-1} \leq \delta_{k-1} r_{k-1}^{-1}$ and $\delta_k \theta_k^{-1} \leq \delta_{k-1} \theta_{k-1}^{-1}$ . Moreover, since $\alpha_k$ and $\beta_k$ are non-increasing, we also have that $\Lambda_{k-1} \leq \Lambda_k$ and $\Pi_{k-1} \leq \Pi_k$ . This implies the following inequalities which will be used in the rest of the proof:

$$\begin{array}{cccc}& a_{k-1}^{-1}{\mathrm{\Lambda }}_{k-1}\le a_k^{-1}{\mathrm{\Lambda }}_k,b_{k-1}^{-1}{\mathrm{\Pi }}_{k-1}\le b_k^{-1}{\mathrm{\Pi }}_k\mathrm{.}& & \text{(36)}\end{array}a\; \_\; \{k\; -\; 1\}\; ^\; \{-\; 1\}\; \backslash Lambda\_\; \{k\; -\; 1\}\; \backslash leq\; a\; \_\; \{k\}\; ^\; \{-\; 1\}\; \backslash Lambda\_\; \{k\},\; \backslash quad\; b\; \_\; \{k\; -\; 1\}\; ^\; \{-\; 1\}\; \backslash Pi\_\; \{k\; -\; 1\}\; \backslash leq\; b\; \_\; \{k\}\; ^\; \{-\; 1\}\; \backslash Pi\_\; \{k\}.\; \backslash tag\; \{36\}$$

Now, let $\rho_{k}$ be a non-increasing sequence with $0 < \rho_{k} < 1$ . By Proposition 2, it follows that $E_{k}^{x}$ satisfies the inequality:

$$\begin{array}{cccc}& F_k+E_k^x\le (1-(1-\frac{1}{2}{\rho }_k){\delta }_k)E_{k-1}^x+{\gamma }_k{s}_k{V}_{k-1}^{\psi }+{\gamma }_k(s_k+{\rho }_k^{-1})E_{k-1}^{\psi }& & \text{(37)}\end{array}F\; \_\; \{k\}\; +\; E\; \_\; \{k\}\; ^\; \{x\}\; \backslash leq\; \backslash left(1\; -\; \backslash left(1\; -\; \backslash frac\; \{1\}\{2\}\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\backslash right)\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; +\; \backslash gamma\_\; \{k\}\; s\; \_\; \{k\}\; V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; \backslash gamma\_\; \{k\}\; \backslash left(s\; \_\; \{k\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; \backslash tag\; \{37\}$$

On the other hand, by Proposition 5 we know that $E_k^y$ and $E_k^z$ satisfy:

$$\begin{array}{cccc}& \left(\genfrac{}{}{0px}{}{E_k^y}{E_k^z}\right)\le {\mathit{P}}_{\mathit{k}}\left(\genfrac{}{}{0px}{}{{\mathrm{\Lambda }}_{k-1}{E}_{k-1}^y+R_{k-1}^y}{{\mathrm{\Pi }}_{k-1}{E}_{k-1}^z+R_{k-1}^z}\right)+{\gamma }_k(V_{k-1}^{\psi }+2E_{k-1}^{\psi }+2{\zeta }_k{E}_{k-1}^x){\mathit{U}}_{\mathit{k}}+{\mathit{V}}_{\mathit{k}}& & \text{(38)}\end{array}\backslash binom\; \{E\; \_\; \{k\}\; ^\; \{y\}\}\; \{E\; \_\; \{k\}\; ^\; \{z\}\}\; \backslash leq\; \backslash boldsymbol\; \{P\}\; \_\; \{\backslash boldsymbol\; \{k\}\}\; \backslash binom\; \{\backslash Lambda\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\}\; \{\backslash Pi\_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; +\; R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\}\; +\; \backslash gamma\_\; \{k\}\; \backslash left(V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; 2\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; 2\; \backslash zeta\_\; \{k\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\backslash right)\; \backslash boldsymbol\; \{U\}\; \_\; \{\backslash boldsymbol\; \{k\}\}\; +\; \backslash boldsymbol\; \{V\}\; \_\; \{\backslash boldsymbol\; \{k\}\}\; \backslash tag\; \{38\}$$

where the $P_{k}, U_{k}$ and $\pmb{V}_{k}$ are defined in (17). For conciseness, define $\pmb{S}_k$ and $E_k^I$ to be:

By (36), we directly have that:

where the inequality in (39) holds component-wise. Therefore, multiplying (38) by $S_{k}$ and using (39) yields:

$$\begin{array}{cccc}& {\mathit{E}}_k^I\le {\tilde{\mathit{P}}}_k{\mathit{E}}_{k-1}^I+{\mathit{S}}_k({\mathit{P}}_k\left(\genfrac{}{}{0px}{}{R_{k-1}^y}{R_{k-1}^z}\right)+{\gamma }_k(V_{k-1}^{\psi }+2E_{k-1}^{\psi }+2{\zeta }_k{E}_{k-1}^x){\mathit{U}}_k+{\mathit{V}}_k)& & \text{(40)}\end{array}\backslash boldsymbol\; \{E\}\; \_\; \{k\}\; ^\; \{I\}\; \backslash leq\; \backslash tilde\; \{\backslash boldsymbol\; \{P\}\}\; \_\; \{k\}\; \backslash boldsymbol\; \{E\}\; \_\; \{k\; -\; 1\}\; ^\; \{I\}\; +\; \backslash boldsymbol\; \{S\}\; \_\; \{k\}\; \backslash left(\backslash boldsymbol\; \{P\}\; \_\; \{k\}\; \backslash binom\; \{R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\}\; \{R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\}\; +\; \backslash gamma\_\; \{k\}\; \backslash left(V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; 2\; E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; +\; 2\; \backslash zeta\_\; \{k\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\backslash right)\; \backslash boldsymbol\; \{U\}\; \_\; \{k\}\; +\; \backslash boldsymbol\; \{V\}\; \_\; \{k\}\backslash right)\; \backslash tag\; \{40\}$$

Furthermore, by Proposition 4 we can bound $E_{k - 1}^{\psi}$ and $V_{k - 1}^{\psi}$ as follows:

$$E_{k-1}^{\psi }\le 2L_{\psi }^2({\mathrm{\Lambda }}_k(a_k{)}^{-1}{a}_{k-1}{E}_{k-1}^y+{\mathrm{\Pi }}_k(b_k{)}^{-1}{b}_{k-1}{E}_{k-1}^z)+2L_{\psi }^2{R}_{k-1}^{y}E\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; \backslash leq\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash Big\; (\backslash Lambda\_\; \{k\}\; (a\; \_\; \{k\})\; ^\; \{-\; 1\}\; a\; \_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; \backslash Pi\_\; \{k\}\; (b\; \_\; \{k\})\; ^\; \{-\; 1\}\; b\; \_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; \backslash Big)\; +\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}$$

$$V_{k-1}^{\psi }\le w_x^2+{\sigma }_x^2{\mathrm{\Pi }}_k(b_k{)}^{-1}{b}_{k-1}{E}_{k-1}^z,V\; \_\; \{k\; -\; 1\}\; ^\; \{\backslash psi\}\; \backslash leq\; w\; \_\; \{x\}\; ^\; \{2\}\; +\; \backslash sigma\_\; \{x\}\; ^\; \{2\}\; \backslash Pi\_\; \{k\}\; (b\; \_\; \{k\})\; ^\; \{-\; 1\}\; b\; \_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{z\},$$

where we used (36) a second time to replace $\Lambda_{k - 1}(a_{k - 1})^{-1}$ and $\Pi_{k - 1}(b_{k - 1})^{-1}$ by $\Lambda_k(a_k)^{-1}$ and $\Pi_k(b_k)^{-1}$ . By summing both inequalities (37) and (40) and substituting all terms $E_{k - 1}^{\psi}$ and $V_{k - 1}^{\psi}$ by their upper-bounds we obtain an inequality of the form:

$$F_k+E_k^{tot}\le A_k^x{E}_{k-1}^x+A_k^y{a}_{k-1}{E}_{k-1}^y+A_k^z{b}_{k-1}{E}_{k-1}^z+V_k^{tot}F\; \_\; \{k\}\; +\; E\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; \backslash leq\; A\; \_\; \{k\}\; ^\; \{x\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{x\}\; +\; A\; \_\; \{k\}\; ^\; \{y\}\; a\; \_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; +\; A\; \_\; \{k\}\; ^\; \{z\}\; b\; \_\; \{k\; -\; 1\}\; E\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; +\; V\; \_\; \{k\}\; ^\; \{t\; o\; t\}$$

where $A_{k}^{x}, A_{k}^{y}, A_{k}^{z}$ are the components of the vector $A_{k}$ defined in (18) and $V_{k}^{tot}$ is the variance term also defined in (18). The desired inequality follows by upper-bounding $A_{k}^{x}, A_{k}^{y}, A_{k}^{z}$ by their maximum value $| A_{k}|_{\infty}$ .

D CONTROLLING THE PRECISION OF THE INNER-LEVEL ALGORITHMS.

In this section, we prove Proposition 10. To achieve this, we first provide general conditions on $\Lambda_{k}$ and $\Pi_{k}$ for controlling the rate $| A_k|_{\infty}$ and which hold regardless of the choice of step-sizes. This is achieved in Proposition 13 of Appendix D.1. Then we prove Proposition 10 in Appendix D.2.

D.1 CONTROLLING $\Pi_k$ AND $\Lambda_k$ .

We introduce the following quantities:

$$\begin{array}{cccc}& D_k^{(1)}:=\frac{1}{1-s}[\log \left[\frac{1-(1-{\rho }_k){\delta }_k}{1+2r_k[1+2L_{\psi }^2{\gamma }_k{\delta }_k^{-1}[s_k+1+{\left[2{\rho }_k\right]}^{-1}]]}\right]]& & \text{(41a)}\end{array}D\; \_\; \{k\}\; ^\; \{(1)\}\; :=\; \backslash frac\; \{1\}\{1\; -\; s\}\; \backslash left[\; \backslash log\; \backslash left[\; \backslash frac\; \{1\; -\; \backslash left(1\; -\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\}\{1\; +\; 2\; r\; \_\; \{k\}\; \backslash left[\; 1\; +\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash gamma\_\; \{k\}\; \backslash delta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left[\; s\; \_\; \{k\}\; +\; 1\; +\; \backslash left[\; 2\; \backslash rho\_\; \{k\}\; \backslash right]\; ^\; \{-\; 1\}\; \backslash right]\; \backslash right]\}\; \backslash right]\; \backslash right]\; \backslash tag\; \{41a\}$$

$$\begin{array}{cccc}& D_k^{(2)}:=\frac{1}{s}\log ((16L_y^2{)}^{-1}{\rho }_k{\zeta }_k^{-1}{\gamma }_k^{-2}{r}_k^2)& & \text{(41b)}\end{array}D\; \_\; \{k\}\; ^\; \{(2)\}\; :=\; \backslash frac\; \{1\}\{s\}\; \backslash log\; \backslash left((1\; 6\; L\; \_\; \{y\}\; ^\; \{2\})\; ^\; \{-\; 1\}\; \backslash rho\_\; \{k\}\; \backslash zeta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash gamma\_\; \{k\}\; ^\; \{-\; 2\}\; r\; \_\; \{k\}\; ^\; \{2\}\backslash right)\; \backslash tag\; \{41b\}$$

$$\begin{array}{cccc}& D_k^{(3)}:=-\frac{1}{s}\log \left(4L_y^2{\delta }_k{\gamma }_k{r}_k^{-2}\right)& & \text{(41c)}\end{array}D\; \_\; \{k\}\; ^\; \{(3)\}\; :=\; -\; \backslash frac\; \{1\}\{s\}\; \backslash log\; \backslash left(4\; L\; \_\; \{y\}\; ^\; \{2\}\; \backslash delta\_\; \{k\}\; \backslash gamma\_\; \{k\}\; r\; \_\; \{k\}\; ^\; \{-\; 2\}\backslash right)\; \backslash tag\; \{41c\}$$

$$\begin{array}{cccc}& D_k^{(4)}:=\frac{1}{1-s}\log \left[\frac{1-(1-{\rho }_k){\delta }_k}{1+{\theta }_k[1+2{\gamma }_k{\delta }_k^{-1}(2L_{\psi }^2+{\sigma }_x^2)[s_k+1+{\left(2{\rho }_k\right)}^{-1}]]}\right]& & \text{(41d)}\end{array}D\; \_\; \{k\}\; ^\; \{(4)\}\; :=\; \backslash frac\; \{1\}\{1\; -\; s\}\; \backslash log\; \backslash left[\; \backslash frac\; \{1\; -\; \backslash left(1\; -\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\}\{1\; +\; \backslash theta\_\; \{k\}\; \backslash left[\; 1\; +\; 2\; \backslash gamma\_\; \{k\}\; \backslash delta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left(2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; +\; \backslash sigma\_\; \{x\}\; ^\; \{2\}\backslash right)\; \backslash left[\; s\; \_\; \{k\}\; +\; 1\; +\; \backslash left(2\; \backslash rho\_\; \{k\}\backslash right)\; ^\; \{-\; 1\}\; \backslash right]\; \backslash right]\}\; \backslash right]\; \backslash tag\; \{41d\}$$

$$\begin{array}{cccc}& D_k^{(5)}:=-\frac{1}{s}\log \left(16L_z^2{\theta }_k^{-2}{\varphi }_k\right)& & \text{(41e)}\end{array}D\; \_\; \{k\}\; ^\; \{(5)\}\; :=\; -\; \backslash frac\; \{1\}\{s\}\; \backslash log\; \backslash left(1\; 6\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; \backslash phi\_\; \{k\}\backslash right)\; \backslash tag\; \{41e\}$$

$$\begin{array}{cccc}& D_k^{(6)}:=-\frac{1}{s}\log \left(2L_z^2{L}_y^{-2}{\theta }_k^{-2}{r}_k^2(1+4L_y^2{r}_k^{-1}{\varphi }_k)\right)& & \text{(41f)}\end{array}D\; \_\; \{k\}\; ^\; \{(6)\}\; :=\; -\; \backslash frac\; \{1\}\{s\}\; \backslash log\; \backslash left(2\; L\; \_\; \{z\}\; ^\; \{2\}\; L\; \_\; \{y\}\; ^\; \{-\; 2\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; r\; \_\; \{k\}\; ^\; \{2\}\; \backslash left(1\; +\; 4\; L\; \_\; \{y\}\; ^\; \{2\}\; r\; \_\; \{k\}\; ^\; \{-\; 1\}\; \backslash phi\_\; \{k\}\backslash right)\backslash right)\; \backslash tag\; \{41f\}$$

Proposition 13. Let $\rho_{k}$ be a non-increasing sequence of positive numbers smaller than 1. Consider $\Lambda_{k}$ and $\Pi_{k}$ so that:

$$\begin{array}{cccc}& \log {\mathrm{\Lambda }}_k\le \min (D_k^{(1)},D_k^{(2)},D_k^{(3)}),& & \text{(42a)}\end{array}\backslash log\; \backslash Lambda\_\; \{k\}\; \backslash leq\; \backslash min\; \backslash left(D\; \_\; \{k\}\; ^\; \{(1)\},\; D\; \_\; \{k\}\; ^\; \{(2)\},\; D\; \_\; \{k\}\; ^\; \{(3)\}\backslash right),\; \backslash tag\; \{42a\}$$

$$\begin{array}{cccc}& \log {\mathrm{\Pi }}_k\le \min (D_k^{(1)},D_k^{(2)},\log {\mathrm{\Lambda }}_k+D_k^{(3)})\mathrm{.}& & \text{(42b)}\end{array}\backslash log\; \backslash Pi\_\; \{k\}\; \backslash leq\; \backslash min\; \backslash left(D\; \_\; \{k\}\; ^\; \{(1)\},\; D\; \_\; \{k\}\; ^\; \{(2)\},\; \backslash log\; \backslash Lambda\_\; \{k\}\; +\; D\; \_\; \{k\}\; ^\; \{(3)\}\backslash right).\; \backslash tag\; \{42b\}$$

Then, the following inequalities hold:

$${\parallel A_k\parallel }_{\mathrm{\infty }}\le (1-(1-{\rho }_k){\delta }_k),V_k^{tot}\le {\tilde{V}}_k^{tot}\mathrm{.}\backslash left\backslash |\; A\; \_\; \{k\}\; \backslash right\backslash |\; \_\; \{\backslash infty\}\; \backslash leq\; \backslash left(1\; -\; \backslash left(1\; -\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\backslash right),\; \backslash qquad\; V\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; \backslash leq\; \backslash tilde\; \{V\}\; \_\; \{k\}\; ^\; \{t\; o\; t\}.$$

where $A_{k}$ and $V_{k}^{tot}$ are defined in (18) of Proposition 9 and $\tilde{V}_k^{tot}$ is defined as:

$$\begin{array}{c}{\tilde{V}}_k^{tot}:={\delta }_k(3r_k^{-1}+2L_{\psi }^2{\gamma }_k{\delta }_k^{-1}(2+(s_k+{\rho }_k^{-1})))R_{k-1}^y\\ +{\delta }_k{\mathrm{\Pi }}_k^s(2{\theta }_k^{-1}{R}_{k-1}^z+8L_z^2{\theta }_k^{-2}{\tilde{R}}_k^y)+{\gamma }_k(s_k+4L_y^2{\mathrm{\Lambda }}_k^s{\delta }_k{\gamma }_k{r}_k^{-2})w_x^2\end{array}\backslash begin\{array\}\{l\}\; \backslash tilde\; \{V\}\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; :=\; \backslash delta\_\; \{k\}\; \backslash left(3\; r\; \_\; \{k\}\; ^\; \{-\; 1\}\; +\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash gamma\_\; \{k\}\; \backslash delta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left(2\; +\; \backslash left(s\; \_\; \{k\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\backslash right)\backslash right)\; R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; \backslash \backslash \; +\; \backslash delta\_\; \{k\}\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; \backslash left(2\; \backslash theta\_\; \{k\}\; ^\; \{-\; 1\}\; R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; +\; 8\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; \backslash tilde\; \{R\}\; \_\; \{k\}\; ^\; \{y\}\backslash right)\; +\; \backslash gamma\_\; \{k\}\; \backslash left(s\; \_\; \{k\}\; +\; 4\; L\; \_\; \{y\}\; ^\; \{2\}\; \backslash Lambda\_\; \{k\}\; ^\; \{s\}\; \backslash delta\_\; \{k\}\; \backslash gamma\_\; \{k\}\; r\; \_\; \{k\}\; ^\; \{-\; 2\}\backslash right)\; w\; \_\; \{x\}\; ^\; \{2\}\; \backslash \backslash \; \backslash end\{array\}$$

Proof. We first prove that $u_{k}^{I} \leq u_{k}^{+} \leq 1$ and $p_{k}^{+} \leq 1$ under (42a) and (42b) with $u_{k}^{+}, p_{k}^{+}$ given by:

$$u_k^{+}:=4L_y^2{\delta }_k{\gamma }_k{r}_k^{-2}{\mathrm{\Lambda }}_k^s,p_k^{+}=16L_z^2{\mathrm{\Pi }}_k^s{\theta }_k^{-2}{\varphi }_k\mathrm{.}u\; \_\; \{k\}\; ^\; \{+\}\; :=\; 4\; L\; \_\; \{y\}\; ^\; \{2\}\; \backslash delta\_\; \{k\}\; \backslash gamma\_\; \{k\}\; r\; \_\; \{k\}\; ^\; \{-\; 2\}\; \backslash Lambda\_\; \{k\}\; ^\; \{s\},\; \backslash qquad\; p\; \_\; \{k\}\; ^\; \{+\}\; =\; 1\; 6\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; \backslash phi\_\; \{k\}.$$

A direct calculation shows $u_{k}^{+} \leq 1$ whenever (42a) holds. Moreover, recall that $u_{k}^{I} = a_{k}U_{k}^{(1)} + b_{k}U_{k}^{(2)}$ with $U_{k}^{(1)}$ and $U_{k}^{(2)}$ being the components of the vector $U_{k}$ defined in (17). Thus by direct substitution, we get the following expression for $u_{k}^{I}$ :

$$u_k^I=2L_y^2{\delta }_k{\gamma }_k{r}_k^{-2}{\mathrm{\Lambda }}_k^s(1+2L_z^2{L}_y^{-2}{\theta }_k^{-2}{r}_k^2(1+4L_y^2{r}_k^{-1}{\varphi }_k\left)\frac{{\mathrm{\Pi }}_k^s}{{\mathrm{\Lambda }}_k^s}\right)\mathrm{.}u\; \_\; \{k\}\; ^\; \{I\}\; =\; 2\; L\; \_\; \{y\}\; ^\; \{2\}\; \backslash delta\_\; \{k\}\; \backslash gamma\_\; \{k\}\; r\; \_\; \{k\}\; ^\; \{-\; 2\}\; \backslash Lambda\_\; \{k\}\; ^\; \{s\}\; \backslash bigg\; (1\; +\; 2\; L\; \_\; \{z\}\; ^\; \{2\}\; L\; \_\; \{y\}\; ^\; \{-\; 2\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; r\; \_\; \{k\}\; ^\; \{2\}\; \backslash big\; (1\; +\; 4\; L\; \_\; \{y\}\; ^\; \{2\}\; r\; \_\; \{k\}\; ^\; \{-\; 1\}\; \backslash phi\_\; \{k\}\; \backslash big)\; \backslash frac\; \{\backslash Pi\_\; \{k\}\; ^\; \{s\}\}\{\backslash Lambda\_\; \{k\}\; ^\; \{s\}\}\; \backslash bigg).$$

Therefore, (42b) suffices to ensure that $u_{k}^{I} \leq u_{k}^{+}$ . Finally, (42b) implies directly that $p_{k}^{+} \leq 1$ .

We will control each component $A_{k}^{x}, A_{k}^{y}$ and $A_{k}^{z}$ of the vector $A_{k}$ separately.

Controlling $A_{k}^{x}$ . Recalling the expression of $A_{k}^{x}$ , the first component of $A_{k}$ in (18), it holds that:

$$\begin{array}{c}{A}_k^x=1-(1-\frac{1}{2}{\rho }_k){\delta }_k+2{\zeta }_k{\lambda }_k{u}_k^{I}1-(1-\frac{1}{2}{\rho }_k){\delta }_k+2{\zeta }_k{\gamma }_k{u}_k^{+}\\ 1-(1-\frac{1}{2}{\rho }_k){\delta }_k+\frac{1}{2}{\rho }_k{\delta }_k=1-(1-{\rho }_k){\delta }_k\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; A\; \_\; \{k\}\; ^\; \{x\}\; =\; 1\; -\; \backslash left(1\; -\; \backslash frac\; \{1\}\{2\}\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\; +\; 2\; \backslash zeta\_\; \{k\}\; \backslash lambda\_\; \{k\}\; u\; \_\; \{k\}\; ^\; \{I\}\; \backslash stackrel\; \{(i)\}\; \{\backslash leq\}\; 1\; -\; \backslash left(1\; -\; \backslash frac\; \{1\}\{2\}\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\; +\; 2\; \backslash zeta\_\; \{k\}\; \backslash gamma\_\; \{k\}\; u\; \_\; \{k\}\; ^\; \{+\}\; \backslash \backslash \; \backslash stackrel\; \{(i\; i)\}\; \{\backslash leq\}\; 1\; -\; \backslash left(1\; -\; \backslash frac\; \{1\}\{2\}\; \backslash rho\_\; \{k\}\backslash right)\; \backslash delta\_\; \{k\}\; +\; \backslash frac\; \{1\}\{2\}\; \backslash rho\_\; \{k\}\; \backslash delta\_\; \{k\}\; =\; 1\; -\; (1\; -\; \backslash rho\_\; \{k\})\; \backslash delta\_\; \{k\}.\; \backslash \backslash \; \backslash end\{array\}$$

(i) holds since $u_{k}^{I}\leq u_{k}^{+}$ while (ii) follows from (42a) which ensures that $2\zeta_k\gamma_ku_k^+\leq \frac{1}{2}\rho_k\delta_k$ .

Controlling $A_{k}^{y}$ . Recall the expression of second component of $A_{k}^{y}$ in (18), we have:

$$\begin{array}{c}{A}_k^y={\mathrm{\Lambda }}_k^{1-s}(1+r_k(1+16L_z^2{\mathrm{\Pi }}_k^s{\theta }_k^{-2}{\varphi }_k+2L_{\psi }^2{\gamma }_k{\delta }_k^{-1}(2u_k^I+(s_k+{\rho }_k^{-1}))\left)\right)\\ (1+2r_k(1+2L_{\psi }^2{\gamma }_k{\delta }_k^{-1}(s_k+1+(2{\rho }_k{)}^{-1}))){\mathrm{\Lambda }}_k^{1-s}1-(1-{\rho }_k){\delta }_k\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; A\; \_\; \{k\}\; ^\; \{y\}\; =\; \backslash Lambda\_\; \{k\}\; ^\; \{1\; -\; s\}\; \backslash big\; (1\; +\; r\; \_\; \{k\}\; \backslash big\; (1\; +\; 1\; 6\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; \backslash theta\_\; \{k\}\; ^\; \{-\; 2\}\; \backslash phi\_\; \{k\}\; +\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash gamma\_\; \{k\}\; \backslash delta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash big\; (2\; u\; \_\; \{k\}\; ^\; \{I\}\; +\; (s\; \_\; \{k\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\})\; \backslash big)\; \backslash big)\; \backslash big)\; \backslash \backslash \; \backslash stackrel\; \{(i)\}\; \{\backslash leq\}\; \backslash Big\; (1\; +\; 2\; r\; \_\; \{k\}\; \backslash Big\; (1\; +\; 2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash gamma\_\; \{k\}\; \backslash delta\_\; \{k\}\; ^\; \{-\; 1\}\; (s\; \_\; \{k\}\; +\; 1\; +\; (2\; \backslash rho\_\; \{k\})\; ^\; \{-\; 1\})\; \backslash Big)\; \backslash Big)\; \backslash Lambda\_\; \{k\}\; ^\; \{1\; -\; s\}\; \backslash stackrel\; \{(i\; i)\}\; \{\backslash leq\}\; 1\; -\; (1\; -\; \backslash rho\_\; \{k\})\; \backslash delta\_\; \{k\}.\; \backslash \backslash \; \backslash end\{array\}$$

(i) holds since $p_k^+ \coloneqq 16L_z^2\Pi_k^s\theta_k^{-2}\phi_k \leq 1$ and $u_{k}^{I} \leq 1$ while (ii) is a consequence of (42a).

Controlling $A_{k}^{z}$ . Similarly, recalling the expression of the third component of $A_{k}$ we get that:

$$\begin{array}{c}{A}_k^z={\mathrm{\Pi }}_k^{1-s}(1+{\theta }_k(1+{\gamma }_k{\delta }_k^{-1}(2L_{\psi }^2+{\sigma }_x^2)(2u_k^I+(s_k+{\rho }_k^{-1}))))\\ {\mathrm{\Pi }}_k^{1-s}(1+{\theta }_k(1+2{\gamma }_k{\delta }_k^{-1}(2L_{\psi }^2+{\sigma }_x^2)(1+(s_k+(2{\rho }_k{)}^{-1})\left)\right))1-(1-{\rho }_k){\delta }_k,\end{array}\backslash begin\{array\}\{l\}\; A\; \_\; \{k\}\; ^\; \{z\}\; =\; \backslash Pi\_\; \{k\}\; ^\; \{1\; -\; s\}\; \backslash left(1\; +\; \backslash theta\_\; \{k\}\; \backslash left(1\; +\; \backslash gamma\_\; \{k\}\; \backslash delta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash left(2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; +\; \backslash sigma\_\; \{x\}\; ^\; \{2\}\backslash right)\; \backslash left(2\; u\; \_\; \{k\}\; ^\; \{I\}\; +\; \backslash left(s\; \_\; \{k\}\; +\; \backslash rho\_\; \{k\}\; ^\; \{-\; 1\}\backslash right)\backslash right)\backslash right)\backslash right)\; \backslash \backslash \; \backslash stackrel\; \{(i)\}\; \{\backslash leq\}\; \backslash Pi\_\; \{k\}\; ^\; \{1\; -\; s\}\; \backslash big\; (1\; +\; \backslash theta\_\; \{k\}\; \backslash big\; (1\; +\; 2\; \backslash gamma\_\; \{k\}\; \backslash delta\_\; \{k\}\; ^\; \{-\; 1\}\; \backslash big\; (2\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; +\; \backslash sigma\_\; \{x\}\; ^\; \{2\}\; \backslash big)\; \backslash big\; (1\; +\; (s\; \_\; \{k\}\; +\; (2\; \backslash rho\_\; \{k\})\; ^\; \{-\; 1\})\; \backslash big)\; \backslash big)\; \backslash big)\; \backslash stackrel\; \{(i\; i)\}\; \{\backslash leq\}\; 1\; -\; (1\; -\; \backslash rho\_\; \{k\})\; \backslash delta\_\; \{k\},\; \backslash \backslash \; \backslash end\{array\}$$

where (i) uses that $u_{k}^{I}\leq 1$ and (ii) follows from (42b).

Controlling $V_{k}^{tot}$ . Recalling the expression of $V_{k}^{tot}$ from (18), we have that:

$$Vktot:=δk(Λks(1+rk−1)+16Lz2ϕkθk−2Πks+2Lψ2γkδk−1(2ukI+(sk+ρk−1)))Rk−1y+δk((1+θk−1)ΠksRk−1z+8Lz2θk−2ΠksR~ky)+γk(sk+ukI)wx2≤δk(3rk−1+2Lψ2γkδk−1(2+(sk+ρk−1)))Rk−1y+δk(2θk−1ΠksRk−1z+8Lz2θk−2ΠksR~ky)+γk(sk+uk+)wx2=V~ktot. \begin{array}{l} V _ {k} ^ {t o t} := \delta_ {k} \left(\Lambda_ {k} ^ {s} \left(1 + r _ {k} ^ {- 1}\right) + 1 6 L _ {z} ^ {2} \phi_ {k} \theta_ {k} ^ {- 2} \Pi_ {k} ^ {s} + 2 L _ {\psi} ^ {2} \gamma_ {k} \delta_ {k} ^ {- 1} \left(2 u _ {k} ^ {I} + \left(s _ {k} + \rho_ {k} ^ {- 1}\right)\right)\right) R _ {k - 1} ^ {y} \\ + \delta_ {k} \left((1 + \theta_ {k} ^ {- 1}) \Pi_ {k} ^ {s} R _ {k - 1} ^ {z} + 8 L _ {z} ^ {2} \theta_ {k} ^ {- 2} \Pi_ {k} ^ {s} \tilde {R} _ {k} ^ {y}\right) + \gamma_ {k} \left(s _ {k} + u _ {k} ^ {I}\right) w _ {x} ^ {2} \\ \leq \delta_ {k} \left(3 r _ {k} ^ {- 1} + 2 L _ {\psi} ^ {2} \gamma_ {k} \delta_ {k} ^ {- 1} \left(2 + \left(s _ {k} + \rho_ {k} ^ {- 1}\right)\right)\right) R _ {k - 1} ^ {y} \\ + \delta_ {k} \left(2 \theta_ {k} ^ {- 1} \Pi_ {k} ^ {s} R _ {k - 1} ^ {z} + 8 L _ {z} ^ {2} \theta_ {k} ^ {- 2} \Pi_ {k} ^ {s} \tilde {R} _ {k} ^ {y}\right) + \gamma_ {k} \left(s _ {k} + u _ {k} ^ {+}\right) w _ {x} ^ {2} = \tilde {V} _ {k} ^ {t o t}. \\ \end{array} $$

where we use $16L_{z}^{2}\phi_{k}\theta_{k}^{-2}\Pi_{k}^{s}\leq 1$ and $u_{k}^{I}\leq 1$ for the first line and $u_{k}^{I}\leq u_{k}^{+}$ for the last line.

D.2 CONTROLLING THE NUMBER OF INNER-LEVEL ITERATIONS

We provide now a proof of Proposition 10 which is a consequence of Proposition 13.

Proof of Proposition 10. We first provide conditions on the number of iterations $T$ and $N$ of algorithms $\mathcal{A}_k$ and $\mathcal{B}k$ to control the rate $| A_k|{\infty}$ and then provide an upper-bound on $V_k^{tot}$ .

Conditions on $T$ and $N$ . We consider the setting with constant step-size $\gamma_{k} = \gamma$ , $\alpha_{k} = \alpha$ and $\beta_{k} = \beta$ and choose $r_k = \theta_k = 1$ and $\delta_k = \delta_0$ for some $0 < \delta_0 < 1$ . We also take $v = 1$ so that $\phi_k = 2$ and $\tilde{R}_k^y = R_k^y$ . By direct substitution of the parameters $r_k$ , $\theta_k$ , $\phi_k$ , $\gamma_k$ , $\delta_k$ , $\rho_k$ and $\zeta_k$ , in the expressions of $D_k^{(1)}$ , $D_k^{(2)}$ , $D_k^{(3)}$ , $D_k^{(4)}$ , $D_k^{(5)}$ and $D_k^{(6)}$ defined in (41a) to (41f), we verify that:

$$-C_1\le D_k^{(1)},-C_1^{\mathrm{\prime }}\le D_k^{(4)},-C_2\le \min (D_k^{(2)},D_k^{(3)}),-C_2^{\mathrm{\prime }}\le \min (D_k^{(5)},D_k^{(6)})\mathrm{.}-\; C\; \_\; \{1\}\; \backslash leq\; D\; \_\; \{k\}\; ^\; \{(1)\},\; \backslash qquad\; -\; C\; \_\; \{1\}\; ^\; \{\backslash prime\}\; \backslash leq\; D\; \_\; \{k\}\; ^\; \{(4)\},\; \backslash qquad\; -\; C\; \_\; \{2\}\; \backslash leq\; \backslash min\; \backslash left(D\; \_\; \{k\}\; ^\; \{(2)\},\; D\; \_\; \{k\}\; ^\; \{(3)\}\backslash right),\; \backslash qquad\; -\; C\; \_\; \{2\}\; ^\; \{\backslash prime\}\; \backslash leq\; \backslash min\; \backslash left(D\; \_\; \{k\}\; ^\; \{(5)\},\; D\; \_\; \{k\}\; ^\; \{(6)\}\backslash right).$$

Hence, we can ensure the conditions of Proposition 13 hold by choosing $T$ and $N$ so that:

$$\log {\mathrm{\Lambda }}_k\le -\max (1,C_1,C_2),\log {\mathrm{\Pi }}_k\le -\log {\mathrm{\Lambda }}_k-\max (1,C_1^{\mathrm{\prime }},C_2^{\mathrm{\prime }})\mathrm{.}\backslash log\; \backslash Lambda\_\; \{k\}\; \backslash leq\; -\; \backslash max\; \backslash left(1,\; C\; \_\; \{1\},\; C\; \_\; \{2\}\backslash right),\; \backslash quad\; \backslash log\; \backslash Pi\_\; \{k\}\; \backslash leq\; -\; \backslash log\; \backslash Lambda\_\; \{k\}\; -\; \backslash max\; \backslash left(1,\; C\; \_\; \{1\}\; ^\; \{\backslash prime\},\; C\; \_\; \{2\}\; ^\; \{\backslash prime\}\backslash right).$$

This is achieved by for the following choice:

$$\begin{array}{cccc}& T=\lfloor {\alpha }^{-1}{\mu }_g^{-1}\max (C_1,C_2,C_3)\rfloor +1,& & \text{(43)}\end{array}T\; =\; \backslash left\backslash lfloor\; \backslash alpha^\; \{-\; 1\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash max\; \backslash left(C\; \_\; \{1\},\; C\; \_\; \{2\},\; C\; \_\; \{3\}\backslash right)\; \backslash right\backslash rfloor\; +\; 1,\; \backslash tag\; \{43\}$$

$$N=\lfloor 2{\beta }^{-1}{\mu }_g^{-1}(\max (C_1,C_2,C_3)+\max (C_1^{\mathrm{\prime }},C_2^{\mathrm{\prime }},C_3^{\mathrm{\prime }}))\rfloor +1N\; =\; \backslash left\backslash lfloor\; 2\; \backslash beta^\; \{-\; 1\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash left(\backslash max\; \backslash left(C\; \_\; \{1\},\; C\; \_\; \{2\},\; C\; \_\; \{3\}\backslash right)\; +\; \backslash max\; \backslash left(C\; \_\; \{1\}\; ^\; \{\backslash prime\},\; C\; \_\; \{2\}\; ^\; \{\backslash prime\},\; C\; \_\; \{3\}\; ^\; \{\backslash prime\}\backslash right)\backslash right)\; \backslash right\backslash rfloor\; +\; 1$$

Hence, for such choice, we are guaranteed by Proposition 13 that $| A_k|_{\infty}\leq 1 - (1 - \rho_k)\delta_k$

Bound on the variance $V_{k}^{tot}$ . By choosing $T$ and $N$ as in (43), we know that $\Lambda_{k}$ and $\Pi_{k}$ satisfy (42), so that the variance term $V_{k}^{tot}$ is upper-bounded by $\tilde{V}{k}^{tot}$ . Moreover, by direct substitution of the sequences appearing in the expression of $\tilde{V}{k}^{tot}$ by their values, we get:

$$Vktot≤V~ktot=δk(Λks(1+rk−1)+16Lz2ϕkθk−2Πks+2Lψ2ηk−1(2ukI+(sk+ρk−1)))Rk−1y+δk((1+θk−1)ΠksRk−1z+8Lz2θk−2ΠksR~ky)+γk(sk+ukI)wx2≤δ0(η0−11−u2+(1+u2γ+4Ly2Λksγ2))wx2+δ0(2Λks+32Lz2Πks+10Lψ2η0−1)Rk−1y+δ0(2ΠksRk−1z+8Lz2ΠksR~ky) \begin{array}{l} V _ {k} ^ {t o t} \leq \tilde {V} _ {k} ^ {t o t} = \delta_ {k} \left(\Lambda_ {k} ^ {s} \left(1 + r _ {k} ^ {- 1}\right) + 1 6 L _ {z} ^ {2} \phi_ {k} \theta_ {k} ^ {- 2} \Pi_ {k} ^ {s} + 2 L _ {\psi} ^ {2} \eta_ {k} ^ {- 1} \left(2 u _ {k} ^ {I} + \left(s _ {k} + \rho_ {k} ^ {- 1}\right)\right)\right) R _ {k - 1} ^ {y} \\ + \delta_ {k} \left((1 + \theta_ {k} ^ {- 1}) \Pi_ {k} ^ {s} R _ {k - 1} ^ {z} + 8 L _ {z} ^ {2} \theta_ {k} ^ {- 2} \Pi_ {k} ^ {s} \tilde {R} _ {k} ^ {y}\right) + \gamma_ {k} \left(s _ {k} + u _ {k} ^ {I}\right) w _ {x} ^ {2} \\ \leq \delta_ {0} \left(\eta_ {0} ^ {- 1} \frac {1 - u}{2} + \left(\frac {1 + u}{2} \gamma + 4 L _ {y} ^ {2} \Lambda_ {k} ^ {s} \gamma^ {2}\right)\right) w _ {x} ^ {2} \\ + \delta_ {0} \left(2 \Lambda_ {k} ^ {s} + 3 2 L _ {z} ^ {2} \Pi_ {k} ^ {s} + 1 0 L _ {\psi} ^ {2} \eta_ {0} ^ {- 1}\right) R _ {k - 1} ^ {y} + \delta_ {0} \left(2 \Pi_ {k} ^ {s} R _ {k - 1} ^ {z} + 8 L _ {z} ^ {2} \Pi_ {k} ^ {s} \tilde {R} _ {k} ^ {y}\right) \\ \end{array} $$

Furthermore, by definition of $R_{k - 1}^{y}$ and $R_{k - 1}^{z}$ , we have that:

$$R_{k-1}^y\le 2{\mu }_g^{-1}{L}_g^{-1}{\sigma }_g^2,R_{k-1}^z\le {\mu }_g^{-3}(B^2{L}_g^{-1}{\sigma }_{g_{yy}}^2+3{\mu }_g{\sigma }_f^2)\mathrm{.}R\; \_\; \{k\; -\; 1\}\; ^\; \{y\}\; \backslash leq\; 2\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; L\; \_\; \{g\}\; ^\; \{-\; 1\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\},\; R\; \_\; \{k\; -\; 1\}\; ^\; \{z\}\; \backslash leq\; \backslash mu\_\; \{g\}\; ^\; \{-\; 3\}\; \backslash Big\; (B\; ^\; \{2\}\; L\; \_\; \{g\}\; ^\; \{-\; 1\}\; \backslash sigma\_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; 3\; \backslash mu\_\; \{g\}\; \backslash sigma\_\; \{f\}\; ^\; \{2\}\; \backslash Big).$$

Moreover, recall that $\tilde{R}_k^y = R_k^y$ since we chose $v = 1$ . Thus $\tilde{R}_k^y \leq 2\mu_g^{-1}\alpha \sigma_g^2$ . This implies that:

$$\begin{array}{cccc}& \begin{array}{c}{V}_k^{\text{t o t}}{\delta }_0^{-1}{\gamma }^{-1}\le ({\delta }_0^{-1}\frac{1-u}{2}+(1+4L_y^2{\mathrm{\Lambda }}_k^s\gamma ))w_x^2+2{\mathrm{\Pi }}_k^s{\gamma }^{-1}{\mu }_g^{-3}(B^2\beta {\sigma }_{g_{yy}}^2+3{\mu }_g{\sigma }_f^2)\\ +(4{\mathrm{\Lambda }}_k^s+80L_z^2{\mathrm{\Pi }}_k^s+20L_{\psi }^2{\eta }_0^{-1}){\mu }_g^{-1}\alpha {\gamma }^{-1}{\sigma }_g^2\end{array}& & \text{(44)}\end{array}\backslash begin\{array\}\{l\}\; V\; \_\; \{k\}\; ^\; \{\backslash text\; \{t\; o\; t\}\}\; \backslash delta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash gamma^\; \{-\; 1\}\; \backslash leq\; \backslash left(\backslash delta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash frac\; \{1\; -\; u\}\{2\}\; +\; \backslash left(1\; +\; 4\; L\; \_\; \{y\}\; ^\; \{2\}\; \backslash Lambda\_\; \{k\}\; ^\; \{s\}\; \backslash gamma\backslash right)\backslash right)\; w\; \_\; \{x\}\; ^\; \{2\}\; +\; 2\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; \backslash gamma^\; \{-\; 1\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 3\}\; \backslash left(B\; ^\; \{2\}\; \backslash beta\; \backslash sigma\_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; 3\; \backslash mu\_\; \{g\}\; \backslash sigma\_\; \{f\}\; ^\; \{2\}\backslash right)\; \backslash tag\; \{44\}\; \backslash \backslash \; +\; \backslash left(4\; \backslash Lambda\_\; \{k\}\; ^\; \{s\}\; +\; 8\; 0\; L\; \_\; \{z\}\; ^\; \{2\}\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; +\; 2\; 0\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash eta\_\; \{0\}\; ^\; \{-\; 1\}\backslash right)\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; \backslash alpha\; \backslash gamma^\; \{-\; 1\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\}\; \backslash \backslash \; \backslash end\{array\}$$

By choosing $T$ and $N$ as in (43), the following conditions hold:

$${\mathrm{\Lambda }}_k^s\le \frac{L}{4L_y^2},{\mathrm{\Lambda }}_k^s\le 5L_{\psi }^2{\eta }_0^{-1},{\mathrm{\Pi }}_k^s\le \gamma ({\sigma }_{g_{xy}}^2+(L_g^{\mathrm{\prime }}{)}^2),{\mathrm{\Pi }}_k^s\le \frac{1}{4L_z^2}{L}_{\psi }^2{\eta }_0^{-1}\mathrm{.}\backslash Lambda\_\; \{k\}\; ^\; \{s\}\; \backslash leq\; \backslash frac\; \{L\}\{4\; L\; \_\; \{y\}\; ^\; \{2\}\},\; \backslash qquad\; \backslash Lambda\_\; \{k\}\; ^\; \{s\}\; \backslash leq\; 5\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash eta\_\; \{0\}\; ^\; \{-\; 1\},\; \backslash qquad\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; \backslash leq\; \backslash gamma\; \backslash Big\; (\backslash sigma\_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; (L\; \_\; \{g\}\; ^\; \{\backslash prime\})\; ^\; \{2\}\; \backslash Big),\; \backslash qquad\; \backslash Pi\_\; \{k\}\; ^\; \{s\}\; \backslash leq\; \backslash frac\; \{1\}\{4\; L\; \_\; \{z\}\; ^\; \{2\}\}\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash eta\_\; \{0\}\; ^\; \{-\; 1\}.$$

By applying these inequalities in (44), we get:

$$\begin{array}{c}{V}_k^{tot}{\delta }_0^{-1}{\gamma }^{-1}\le ({\delta }_0^{-1}\frac{1-u}{2}+2)w_x^2+2{\mu }_g^{-3}({\sigma }_{g_{xy}}^2+{\left(L_g^{\mathrm{\prime }}\right)}^2)(B^2\beta {\sigma }_{g_{yy}}^2+3{\mu }_g{\sigma }_f^2)\\ +60L_{\psi }^2{\eta }_0^{-1}{\mu }_g^{-1}{L}_g^{-1}{\gamma }^{-1}{\sigma }_g^2\\ \le ({\delta }_0^{-1}\frac{1-u}{2}+3)w_x^2+\frac{60L_{\psi }^2}{{\delta }_0{\mu }_g{L}_g}{\sigma }_g^2={\mathcal{W}}^2\end{array}\backslash begin\{array\}\{l\}\; V\; \_\; \{k\}\; ^\; \{t\; o\; t\}\; \backslash delta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash gamma^\; \{-\; 1\}\; \backslash leq\; \backslash left(\backslash delta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash frac\; \{1\; -\; u\}\{2\}\; +\; 2\backslash right)\; w\; \_\; \{x\}\; ^\; \{2\}\; +\; 2\; \backslash mu\_\; \{g\}\; ^\; \{-\; 3\}\; \backslash left(\backslash sigma\_\; \{g\; \_\; \{x\; y\}\}\; ^\; \{2\}\; +\; \backslash left(L\; \_\; \{g\}\; ^\; \{\backslash prime\}\backslash right)\; ^\; \{2\}\backslash right)\; \backslash left(B\; ^\; \{2\}\; \backslash beta\; \backslash sigma\_\; \{g\; \_\; \{y\; y\}\}\; ^\; \{2\}\; +\; 3\; \backslash mu\_\; \{g\}\; \backslash sigma\_\; \{f\}\; ^\; \{2\}\backslash right)\; \backslash \backslash \; +\; 6\; 0\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\; \backslash eta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash mu\_\; \{g\}\; ^\; \{-\; 1\}\; L\; \_\; \{g\}\; ^\; \{-\; 1\}\; \backslash gamma^\; \{-\; 1\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\}\; \backslash \backslash \; \backslash leq\; \backslash left(\backslash delta\_\; \{0\}\; ^\; \{-\; 1\}\; \backslash frac\; \{1\; -\; u\}\{2\}\; +\; 3\backslash right)\; w\; \_\; \{x\}\; ^\; \{2\}\; +\; \backslash frac\; \{6\; 0\; L\; \_\; \{\backslash psi\}\; ^\; \{2\}\}\{\backslash delta\_\; \{0\}\; \backslash mu\_\; \{g\}\; L\; \_\; \{g\}\}\; \backslash sigma\_\; \{g\}\; ^\; \{2\}\; =\; \backslash mathcal\; \{W\}\; ^\; \{2\}\; \backslash \backslash \; \backslash end\{array\}$$

where we used that $2\mu_{g}^{-3}\Bigl (\sigma_{g_{xy}}^{2} + (L_{g}^{\prime})^{2}\Bigr)\Bigl (B^{2}\beta \sigma_{g_{yy}}^{2} + 3\mu_{g}\sigma_{f}^{2}\Bigr)\leq w_{x}^{2}$ by definition of $w_{x}^{2}$ in (16a). Therefore, we have shown that $V_{k}^{tot}\leq \gamma \delta_{0}\mathcal{W}^{2}$ , with $\mathcal{W}^2$ given by (20).

E STOCHASTIC LINEAR DYNAMICAL SYSTEM WITH CORRELATED NOISE

Let $A$ be a positive definite matrix in $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0 < \mu_{g}\geq |\sigma_{i}(A)|\leq L_{g}$ and $b$ a vector in $\mathbb{R}^d$ . We denote by $z^{\star} = -A^{-1}b$ . Consider $A_{m}$ be a sequence of i.i.d. positive symmetric matrices in $\mathbb{R}^d\times \mathbb{R}^d$ such that $\mathbb{E}[A_m] = A$ , and $\hat{b}$ a random vector in $\mathbb{R}^d$ such that $\mathbb{E}\big[\hat{b}\big] = b$ , with $A_{m}$ and $\hat{b}$ being mutually independent. Define $\Sigma_A = \mathbb{E}\Big[(A_n - A)^\top (A_n - A)\Big]$ and denote by $\sigma_{A}$ and $L_{A}$ the largest singular values of $\Sigma_{A}$ and $A^{-1}\Sigma_{A}A^{-1}$ . Let $\beta$ be such that $\beta \leq \frac{1}{L_g}$ . Finally let $\sigma_c^2$ be an upper-bound on $\mathbb{E}\left[\left| \hat{b} -b\right| ^2\right]$ . Let $z$ and $z^{\prime}$ be two vectors in $\mathbb{R}^d$ and define the iterates $z^n$ and $\bar{z}^n$ such that $z^0 = z$ and $\bar{z}^0 = z'$ and using the recursion:

$$z^n=(I-\beta A_n)z^{n-1}-\beta \widehat{b},{\bar{z}}^n=(I-\beta A){\bar{z}}^{n-1}-\beta b\mathrm{.}z\; ^\; \{n\}\; =\; (I\; -\; \backslash beta\; A\; \_\; \{n\})\; z\; ^\; \{n\; -\; 1\}\; -\; \backslash beta\; \backslash hat\; \{b\},\; \backslash bar\; \{z\}\; ^\; \{n\}\; =\; (I\; -\; \backslash beta\; A)\; \backslash bar\; \{z\}\; ^\; \{n\; -\; 1\}\; -\; \backslash beta\; b.$$

Hence, from the definition of $z^n$ and $\bar{z}^n$ we directly have that:

$$z^n=\prod _{t=1}^n(I-\beta A_n)z-\sum _{t=1}^n\prod _{j=t+1}^n\beta (I-\beta A_j)\widehat{b},{\bar{z}}^n=\prod _{t=1}^n(I-\beta A)z^{\mathrm{\prime }}-\sum _{t=1}^n\prod _{j=t+1}^n\beta (I-\beta A)b\mathrm{.}z\; ^\; \{n\}\; =\; \backslash prod\_\; \{t\; =\; 1\}\; ^\; \{n\}\; (I\; -\; \backslash beta\; A\; \_\; \{n\})\; z\; -\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{n\}\; \backslash prod\_\; \{j\; =\; t\; +\; 1\}\; ^\; \{n\}\; \backslash beta\; (I\; -\; \backslash beta\; A\; \_\; \{j\})\; \backslash hat\; \{b\},\; \backslash quad\; \backslash bar\; \{z\}\; ^\; \{n\}\; =\; \backslash prod\_\; \{t\; =\; 1\}\; ^\; \{n\}\; (I\; -\; \backslash beta\; A)\; z\; ^\; \{\backslash prime\}\; -\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{n\}\; \backslash prod\_\; \{j\; =\; t\; +\; 1\}\; ^\; \{n\}\; \backslash beta\; (I\; -\; \backslash beta\; A)\; b.$$

The next proposition computes the bias $\mathbb{E}\left[z^n -\bar{z}^n\big|\hat{b}\right]$

Proposition 14. The following identities hold:

$$\begin{array}{c}\mathbb{E}[z^n-{\bar{z}}^n\mid \widehat{b}]=(I-\beta A{)}^n(z-z^{\mathrm{\prime }})-\beta ({\sum }_{t=1}^n(I-\beta A{)}^{n-t})(\widehat{b}-b)\\ \mathbb{E}[z^n-{\bar{z}}^n]=(I-\beta A{)}^n(z-z^{\mathrm{\prime }})\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash left[\; z\; ^\; \{n\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\}\; \backslash mid\; \backslash hat\; \{b\}\; \backslash right]\; =\; (I\; -\; \backslash beta\; A)\; ^\; \{n\}\; (z\; -\; z\; ^\; \{\backslash prime\})\; -\; \backslash beta\; \backslash left(\backslash sum\_\; \{t\; =\; 1\}\; ^\; \{n\}\; (I\; -\; \backslash beta\; A)\; ^\; \{n\; -\; t\}\backslash right)\; (\backslash hat\; \{b\}\; -\; b)\; \backslash \backslash \; \backslash mathbb\; \{E\}\; \backslash left[\; z\; ^\; \{n\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\}\; \backslash right]\; =\; (I\; -\; \backslash beta\; A)\; ^\; \{n\}\; \backslash left(z\; -\; z\; ^\; \{\backslash prime\}\backslash right)\; \backslash \backslash \; \backslash end\{array\}$$

Proof. The proof is a consequence of $A_{n}$ and $\hat{b}$ being i.i.d. and unbiased estimates of $A$ and $b$ .

The next proposition controls the mean squared errors $\mathbb{E}\left[| z^n -z^\star | ^2\right]$ and $\mathbb{E}\left[| z^n -\bar{z}^n| ^2\right]$ .

Proposition 15. Define $n^{\star}(n,\beta) = \min (n,\frac{1}{\beta\mu_g})$ . Let $\beta$ such that:

$$\beta \le \frac{1}{2L_g}\min (1,\frac{2L_g}{{\mu }_g(1+{\mu }_g^{-2}{\sigma }_A^2)})\backslash beta\; \backslash leq\; \backslash frac\; \{1\}\{2\; L\; \_\; \{g\}\}\; \backslash min\; \backslash left(1,\; \backslash frac\; \{2\; L\; \_\; \{g\}\}\{\backslash mu\_\; \{g\}\; \backslash left(1\; +\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{A\}\; ^\; \{2\}\backslash right)\}\backslash right)$$

Then, the following inequalities hold:

$$\begin{array}{c}\mathbb{E}[\mathrm{\parallel }{z}^n-z^{\star }{\mathrm{\parallel }}^2]\le (1-\beta {\mu }_g{)}^n\mathrm{\parallel }z-z^{\star }{\mathrm{\parallel }}^2+{\beta }^2({\sigma }_A^2\mathrm{\parallel }{z}^{\star }{\mathrm{\parallel }}^2+3(n\wedge \frac{1}{\beta {\mu }_g}){\sigma }_c^2)(n\wedge \frac{1}{\beta {\mu }_g}),\\ {\parallel {\bar{z}}^n-z^{\star }\parallel }^2\le (1-\beta {\mu }_g{)}^n{\parallel z^{\mathrm{\prime }}-z^{\star }\parallel }^2\mathrm{.}\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; z\; ^\; \{n\}\; -\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; (1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; ^\; \{n\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; \backslash beta^\; \{2\}\; \backslash left(\backslash sigma\_\; \{A\}\; ^\; \{2\}\; \backslash |\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; 3\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right)\; \backslash sigma\_\; \{c\}\; ^\; \{2\}\backslash right)\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right),\; \backslash \backslash \; \backslash left\backslash |\; \backslash bar\; \{z\}\; ^\; \{n\}\; -\; z\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash leq\; (1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; ^\; \{n\}\; \backslash left\backslash |\; z\; ^\; \{\backslash prime\}\; -\; z\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}.\; \backslash \backslash \; \backslash end\{array\}$$

Moreover, if $z' = z$ , then we have:

$$\begin{array}{cccc}& \begin{array}{c}\mathbb{E}[\mathrm{\parallel }{z}^n-{\bar{z}}^n{\mathrm{\parallel }}^2]\le 4{\mu }_g^{-2}{\sigma }_A^2{(1-\frac{\beta {\mu }_g}{2})}^n\mathrm{\parallel }z-z^{\star }{\mathrm{\parallel }}^2\\ +2{\beta }^2({\sigma }_A^2\mathrm{\parallel }{z}^{\star }{\mathrm{\parallel }}^2+3(n\wedge \frac{1}{\beta {\mu }_g}\left){\sigma }_c^2\right)(n\wedge \frac{1}{\beta {\mu }_g})\mathrm{.}\end{array}& & \text{(45)}\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; z\; ^\; \{n\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; 4\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{A\}\; ^\; \{2\}\; \backslash left(1\; -\; \backslash frac\; \{\backslash beta\; \backslash mu\_\; \{g\}\}\{2\}\backslash right)\; ^\; \{n\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; \backslash tag\; \{45\}\; \backslash \backslash \; +\; 2\; \backslash beta^\; \{2\}\; \backslash Big\; (\backslash sigma\_\; \{A\}\; ^\; \{2\}\; \backslash |\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; 3\; \backslash Big\; (n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\; \backslash Big)\; \backslash sigma\_\; \{c\}\; ^\; \{2\}\; \backslash Big)\; \backslash Big\; (n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\; \backslash Big).\; \backslash \backslash \; \backslash end\{array\}$$

Proof. It is straightforward to see that:

$${\parallel {\bar{z}}^n-z^{\star }\parallel }^2\le (1-\beta {\mu }_g){\parallel z^{\mathrm{\prime }}-z^{\star }\parallel }^2\mathrm{.}\backslash left\backslash |\; \backslash bar\; \{z\}\; ^\; \{n\}\; -\; z\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash leq\; \backslash left(1\; -\; \backslash beta\; \backslash mu\_\; \{g\}\backslash right)\; \backslash left\backslash |\; z\; ^\; \{\backslash prime\}\; -\; z\; ^\; \{\backslash star\}\; \backslash right\backslash |\; ^\; \{2\}.$$

Now, let's control $\mathbb{E}\left[| z^n -\bar{z}^n|^2\right]$ . The following identity holds by definition of $z^n$ and $\bar{z}^n$ :

$$E[∥zn−zˉn∥2]=E[(zn−1−zˉn−1)⊤((I−βA)2+β2ΣA)(zn−1−zˉn−1)]−2βE[(zn−1−zˉn−1)⊤(I−βA)(b^−b)]+β2(E[∥b^−b∥2]+E[(zˉn−1)⊤ΣAzˉn−1])=E[(zn−1−zˉn−1)⊤((I−βA)2+β2ΣA)(zn−1−zˉn−1)]+β2((zˉn−1)⊤ΣAzˉn−1+E[(b^−b)⊤(I+2(I−βA)Dn)(b^−b)]) \begin{array}{l} \mathbb {E} \left[ \| z ^ {n} - \bar {z} ^ {n} \| ^ {2} \right] = \mathbb {E} \left[ \left(z ^ {n - 1} - \bar {z} ^ {n - 1}\right) ^ {\top} \left((I - \beta A) ^ {2} + \beta^ {2} \Sigma_ {A}\right) \left(z ^ {n - 1} - \bar {z} ^ {n - 1}\right) \right] \\ - 2 \beta \mathbb {E} \Big [ (z ^ {n - 1} - \bar {z} ^ {n - 1}) ^ {\top} (I - \beta A) (\hat {b} - b) \Big ] \\ + \beta^ {2} \left(\mathbb {E} \left[ \left\| \hat {b} - b \right\| ^ {2} \right] + \mathbb {E} \left[ \left(\bar {z} ^ {n - 1}\right) ^ {\top} \Sigma_ {A} \bar {z} ^ {n - 1} \right]\right) \\ = \mathbb {E} \left[ \left(z ^ {n - 1} - \bar {z} ^ {n - 1}\right) ^ {\top} \left((I - \beta A) ^ {2} + \beta^ {2} \Sigma_ {A}\right) \left(z ^ {n - 1} - \bar {z} ^ {n - 1}\right) \right] \\ + \beta^ {2} \Bigg (\left(\bar {z} ^ {n - 1}\right) ^ {\top} \Sigma_ {A} \bar {z} ^ {n - 1} + \mathbb {E} \bigg [ \left(\hat {b} - b\right) ^ {\top} (I + 2 (I - \beta A) D _ {n}) \left(\hat {b} - b\right) \bigg ] \Bigg) \\ \end{array} $$

where $\Sigma_A = \mathbb{E}\left[(A_n - A)^\top (A_n - A)\right]$ and $D_{n} = \sum_{t = 0}^{n - 1}(I - \beta A)^{t}$ . By simple calculation we can upper-bound the last term by:

$${\beta }^2\mathbb{E}[(\widehat{b}-b{)}^{\mathrm{\top }}(I+2(I-\beta A)D_n)(\widehat{b}-b)]\le 3(n\wedge \frac{1}{\beta {\mu }_g}){\beta }^2\mathbb{E}\left[{\parallel \widehat{b}-b\parallel }^2\right]\mathrm{.}\backslash beta^\; \{2\}\; \backslash mathbb\; \{E\}\; \backslash left[\; (\backslash hat\; \{b\}\; -\; b)\; ^\; \{\backslash top\}\; (I\; +\; 2\; (I\; -\; \backslash beta\; A)\; D\; \_\; \{n\})\; (\backslash hat\; \{b\}\; -\; b)\; \backslash right]\; \backslash leq\; 3\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right)\; \backslash beta^\; \{2\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{b\}\; -\; b\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right].$$

Moreover, provided that $\beta \leq \frac{1}{L_g(1 + L_A)}$ , where $L_{A}$ is the highest eigenvalue of $A^{-1}\Sigma_{A}A^{-1}$ , then we have the following:

$$\begin{array}{c}\mathbb{E}\left[{\parallel z^n-{\bar{z}}^n\parallel }^2\right]\le (1-\beta {\mu }_g)\mathbb{E}\left[{\parallel z^{n-1}-{\bar{z}}^{n-1}\parallel }^2\right]\\ +{\beta }^2({\left({\bar{z}}^{n-1}\right)}^{\mathrm{\top }}{\mathrm{\Sigma }}_A{\bar{z}}^{n-1}+3(n\wedge \frac{1}{\beta {\mu }_g})\mathbb{E}\left[{\parallel \widehat{b}-b\parallel }^2\right])\\ \le (1-\beta {\mu }_g)\mathbb{E}\left[{\parallel z^{n-1}-{\bar{z}}^{n-1}\parallel }^2\right]+{\beta }^2({\sigma }_A^2{\parallel {\bar{z}}^{n-1}\parallel }^2+3(n\wedge \frac{1}{\beta {\mu }_g}){\sigma }_c^2),\end{array}\backslash begin\{array\}\{l\}\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; ^\; \{n\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; (1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; ^\; \{n\; -\; 1\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; \backslash \backslash \; +\; \backslash beta^\; \{2\}\; \backslash left(\backslash left(\backslash bar\; \{z\}\; ^\; \{n\; -\; 1\}\backslash right)\; ^\; \{\backslash top\}\; \backslash Sigma\_\; \{A\}\; \backslash bar\; \{z\}\; ^\; \{n\; -\; 1\}\; +\; 3\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right)\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; \backslash hat\; \{b\}\; -\; b\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\backslash right)\; \backslash \backslash \; \backslash leq\; (1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; \backslash mathbb\; \{E\}\; \backslash left[\; \backslash left\backslash |\; z\; ^\; \{n\; -\; 1\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; \backslash right]\; +\; \backslash beta^\; \{2\}\; \backslash left(\backslash sigma\_\; \{A\}\; ^\; \{2\}\; \backslash left\backslash |\; \backslash bar\; \{z\}\; ^\; \{n\; -\; 1\}\; \backslash right\backslash |\; ^\; \{2\}\; +\; 3\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right)\; \backslash sigma\_\; \{c\}\; ^\; \{2\}\backslash right),\; \backslash \backslash \; \backslash end\{array\}$$

Unrolling the recursion, it follows that:

$$\begin{array}{cccc}& \mathbb{E}[\mathrm{\parallel }{z}^n-{\bar{z}}^n{\mathrm{\parallel }}^2]\le (1-\beta {\mu }_g{)}^n\mathrm{\parallel }z-z^{\mathrm{\prime }}{\mathrm{\parallel }}^2+{\beta }^2\sum _{t=1}^n(1-\beta {\mu }_g{)}^{n-t}({\sigma }_A^2\mathrm{\parallel }{\bar{z}}^{n-1}{\mathrm{\parallel }}^2+3(n\wedge \frac{1}{\beta {\mu }_g}){\sigma }_c^2)& & \text{(46)}\end{array}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; z\; ^\; \{n\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; (1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; ^\; \{n\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash prime\}\; \backslash |\; ^\; \{2\}\; +\; \backslash beta^\; \{2\}\; \backslash sum\_\; \{t\; =\; 1\}\; ^\; \{n\}\; (1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; ^\; \{n\; -\; t\}\; \backslash left(\backslash sigma\_\; \{A\}\; ^\; \{2\}\; \backslash |\; \backslash bar\; \{z\}\; ^\; \{n\; -\; 1\}\; \backslash |\; ^\; \{2\}\; +\; 3\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right)\; \backslash sigma\_\; \{c\}\; ^\; \{2\}\backslash right)\; \backslash tag\; \{46\}$$

In particular, if $z' = z^{\star}$ , then $\bar{z}^{n} = z^{\star}$ and we get:

$$\mathbb{E}[\mathrm{\parallel }{z}^n-z^{\star }{\mathrm{\parallel }}^2]\le (1-\beta {\mu }_g{)}^n\mathrm{\parallel }z-z^{\star }{\mathrm{\parallel }}^2+{\beta }^2({\sigma }_A^2\mathrm{\parallel }{z}^{\star }{\mathrm{\parallel }}^2+3(n\wedge \frac{1}{\beta {\mu }_g}){\sigma }_c^2)(n\wedge \frac{1}{\beta {\mu }_g})\mathrm{.}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; z\; ^\; \{n\}\; -\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; (1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; ^\; \{n\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; \backslash beta^\; \{2\}\; \backslash left(\backslash sigma\_\; \{A\}\; ^\; \{2\}\; \backslash |\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; 3\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right)\; \backslash sigma\_\; \{c\}\; ^\; \{2\}\backslash right)\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right).$$

To get the last inequality, we simply choose $z' = z$ and recall that:

$$\parallel {\bar{z}}^{t-1}\parallel \le 2((1-\beta {\mu }_g{)}^{t-1}\mathrm{\parallel }{z}^{\mathrm{\prime }}-z^{\star }\mathrm{\parallel }+\mathrm{\parallel }{z}^{\star }\mathrm{\parallel })\mathrm{.}\backslash left\backslash |\; \backslash bar\; \{z\}\; ^\; \{t\; -\; 1\}\; \backslash right\backslash |\; \backslash leq\; 2\; \backslash bigl\; ((1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; ^\; \{t\; -\; 1\}\; \backslash |\; z\; ^\; \{\backslash prime\}\; -\; z\; ^\; \{\backslash star\}\; \backslash |\; +\; \backslash |\; z\; ^\; \{\backslash star\}\; \backslash |\; \backslash bigr).$$

Using the above in in (46) yields:

$$\mathbb{E}[\mathrm{\parallel }{z}^n-{\bar{z}}^n{\mathrm{\parallel }}^2]\le 2n{\beta }^2{\sigma }_A^2(1-\beta {\mu }_g{)}^{n-1}\mathrm{\parallel }z-z^{\star }{\mathrm{\parallel }}^2+2{\beta }^2({\sigma }_A^2\mathrm{\parallel }{z}^{\star }{\mathrm{\parallel }}^2+3(n\wedge \frac{1}{\beta {\mu }_g}){\sigma }_c^2\left)\right(n\wedge \frac{1}{\beta {\mu }_g})\mathrm{.}\backslash mathbb\; \{E\}\; \backslash Big\; [\; \backslash |\; z\; ^\; \{n\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\}\; \backslash |\; ^\; \{2\}\; \backslash Big\; ]\; \backslash leq\; 2\; n\; \backslash beta^\; \{2\}\; \backslash sigma\_\; \{A\}\; ^\; \{2\}\; (1\; -\; \backslash beta\; \backslash mu\_\; \{g\})\; ^\; \{n\; -\; 1\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; 2\; \backslash beta^\; \{2\}\; \backslash Big\; (\backslash sigma\_\; \{A\}\; ^\; \{2\}\; \backslash |\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; 3\; \backslash Big\; (n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\; \backslash Big)\; \backslash sigma\_\; \{c\}\; ^\; \{2\}\; \backslash Big)\; \backslash Big\; (n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\; \backslash Big).$$

Moreover, by Lemma 3 we know that $n\beta^2\mu_g^2 (1 - \beta \mu_g)^{n - 1}\leq (1 - \frac{\beta\mu_g}{2})^{n - 1}$ and since $\beta \mu_{g}\leq 1$ , we have that $(1 - \frac{\beta\mu_g}{2})^{-1}\leq 2$ so that $(1 - \frac{\beta\mu_g}{2})^{n - 1}\leq 2(1 - \frac{\beta\mu_g}{2})^n$ . Hence, we can write:

$$\mathbb{E}[\mathrm{\parallel }{z}^n-{\bar{z}}^n{\mathrm{\parallel }}^2]\le 4{\mu }_g^{-2}{\sigma }_A^2(1-\frac{\beta {\mu }_g}{2}{)}^n\mathrm{\parallel }z-z^{\star }{\mathrm{\parallel }}^2+2{\beta }^2({\sigma }_A^2\mathrm{\parallel }{z}^{\star }{\mathrm{\parallel }}^2+3(n\wedge \frac{1}{\beta {\mu }_g}){\sigma }_c^2)(n\wedge \frac{1}{\beta {\mu }_g})\mathrm{.}\backslash mathbb\; \{E\}\; \backslash left[\; \backslash |\; z\; ^\; \{n\}\; -\; \backslash bar\; \{z\}\; ^\; \{n\}\; \backslash |\; ^\; \{2\}\; \backslash right]\; \backslash leq\; 4\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{A\}\; ^\; \{2\}\; (1\; -\; \backslash frac\; \{\backslash beta\; \backslash mu\_\; \{g\}\}\{2\})\; ^\; \{n\}\; \backslash |\; z\; -\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; 2\; \backslash beta^\; \{2\}\; \backslash left(\backslash sigma\_\; \{A\}\; ^\; \{2\}\; \backslash |\; z\; ^\; \{\backslash star\}\; \backslash |\; ^\; \{2\}\; +\; 3\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right)\; \backslash sigma\_\; \{c\}\; ^\; \{2\}\backslash right)\; \backslash left(n\; \backslash wedge\; \backslash frac\; \{1\}\{\backslash beta\; \backslash mu\_\; \{g\}\}\backslash right).$$

Lemma 2. Let $A$ and $\Sigma_A$ be symmetric positive matrix in $\mathbb{R}^d\times \mathbb{R}^d$ with $\sigma_A^2$ its largest singular value of $\Sigma_{A}$ and $0 < \mu_{g}\leq \sigma_{i}(A)\leq L_{g}$ . Let $\beta$ be a positive number such that:

$$\beta \le \frac{1}{2L_g}\min (1,\frac{2L_g}{{\mu }_g(1+{\mu }_g^{-2}{\sigma }_A^2)})\backslash beta\; \backslash leq\; \backslash frac\; \{1\}\{2\; L\; \_\; \{g\}\}\; \backslash min\; \backslash left(1,\; \backslash frac\; \{2\; L\; \_\; \{g\}\}\{\backslash mu\_\; \{g\}\; \backslash left(1\; +\; \backslash mu\_\; \{g\}\; ^\; \{-\; 2\}\; \backslash sigma\_\; \{A\}\; ^\; \{2\}\backslash right)\}\backslash right)$$

Then the following holds:

$${\parallel (I-\beta A{)}^2+{\beta }^2{\mathrm{\Sigma }}_A\parallel }_{op}\le 1-\beta {\mu }_g\mathrm{.}\backslash left.\; \backslash left\backslash |\; (I\; -\; \backslash beta\; A)\; ^\; \{2\}\; +\; \backslash beta^\; \{2\}\; \backslash Sigma\_\; \{A\}\; \backslash right\backslash |\; \_\; \{o\; p\}\; \backslash leq\; 1\; -\; \backslash beta\; \backslash mu\_\; \{g\}.\; \backslash right.$$

Proof. First note that $\beta \leq \frac{1}{L_g}$ , so that $I - \beta A$ is positive. Now, we observe that $\left| (I - \beta A)^2 + \beta^2 \Sigma_A \right|_{op} \leq (1 - \beta \mu_g)^2 + \beta^2 \sigma_A^2$ which holds since $I - \beta A$ is positive. And since $\beta \leq \frac{\mu_g}{\mu_g^2 + \sigma_A^2}$ , we further have $(1 - \beta \mu_g)^2 + \beta^2 \sigma_A^2 \leq 1 - \beta \mu_g$ , which yields the desired result.

Lemma 3. Let $0 \leq b < 1$ and $n \geq 1$ , then the following inequality holds:

$$n{b}^2(1-b{)}^{n-1}\le {(1-\frac{b}{2})}^{n-1}\mathrm{.}n\; b\; ^\; \{2\}\; (1\; -\; b)\; ^\; \{n\; -\; 1\}\; \backslash leq\; \backslash left(1\; -\; \backslash frac\; \{b\}\{2\}\backslash right)\; ^\; \{n\; -\; 1\}.$$

Proof. We consider the function $h(n, b)$ defined by:

$$h(n,b):=(n-1)\log \left(\frac{1-\frac{b}{2}}{1-b}\right)-\log (n{b}^2)\mathrm{.}h\; (n,\; b)\; :=\; (n\; -\; 1)\; \backslash log\; \backslash left(\backslash frac\; \{1\; -\; \backslash frac\; \{b\}\{2\}\}\{1\; -\; b\}\backslash right)\; -\; \backslash log\; (n\; b\; ^\; \{2\}).$$

We need to show that $h(n, b)$ is non-negative for any $n \geq 1$ and $0 \leq b < 1$ . For this purpose, we fix $b$ and consider the variations of $h(n, b)$ in $n$ :

$${\mathrm{\partial }}_{n}h(n,b)=\log \left(\frac{1-\frac{b}{2}}{1-b}\right)-\frac{1}{n}\mathrm{.}\backslash partial\_\; \{n\}\; h\; (n,\; b)\; =\; \backslash log\; \backslash left(\backslash frac\; \{1\; -\; \backslash frac\; \{b\}\{2\}\}\{1\; -\; b\}\backslash right)\; -\; \backslash frac\; \{1\}\{n\}.$$

$\partial_{n}h(n,b)$ is non-negative for $n \geq n^{\star} = \log \left(\frac{1 - \frac{b}{2}}{1 - b}\right)^{-1}$ and non-positive for all $n \leq n^{\star}$ . Hence, $h(n,b)$ achieves its minimum value in $n^{\star}$ over the $(0, +\infty)$ . We distinguish two cases depending on whether $n^{\star}$ is greater of smaller than 1.

Case $n^{\star} \leq 1$ . In this case $n \mapsto h(n, b)$ is increasing on the interval $[1, +\infty)$ since $\partial_n h(n, b) \geq 0$ for $n \geq n^{\star}$ . Hence, $h(n, b) \geq h(1, b)$ for all $n \geq 1$ . Moreover, since $h(1, b) = -\log (b^{2}) \geq 0$ the result follows directly.

Case $n^{\star} > 1$ . In this case we still have $h(n,b) \geq h(n^{\star},b)$ for all $n \geq 1$ , since $n^{\star}$ achieves the minimum value of $h$ . Thus we only need to show that $h(n^{\star},b) \geq 0$ . Using the expression of $n^{\star}$ , we have:

$$\begin{array}{c}h(n^{\star },b)=1-\log \left(\frac{1-\frac{b}{2}}{1-b}\right)-\log \left(n^{\star }{b}^2\right)\\ =1-\frac{1}{n^{\star }}-\log \left(n^{\star }{b}^2\right)\end{array}\backslash begin\{array\}\{l\}\; h\; (n\; ^\; \{\backslash star\},\; b)\; =\; 1\; -\; \backslash log\; \backslash left(\backslash frac\; \{1\; -\; \backslash frac\; \{b\}\{2\}\}\{1\; -\; b\}\backslash right)\; -\; \backslash log\; \backslash left(n\; ^\; \{\backslash star\}\; b\; ^\; \{2\}\backslash right)\; \backslash \backslash \; =\; 1\; -\; \backslash frac\; \{1\}\{n\; ^\; \{\backslash star\}\}\; -\; \backslash log\; \backslash left(n\; ^\; \{\backslash star\}\; b\; ^\; \{2\}\backslash right)\; \backslash \backslash \; \backslash end\{array\}$$

Since $n^{\star} > 1$ , the first term $1 - \frac{1}{n^{\star}}$ is non-negative, thus we only need to show that $n^{\star}b^{2} \leq 1$ so that the last term is also non-negative. It is easy to see that $n^{\star}b^{2} \leq 1$ is equivalent to having $\tilde{h}(b) \geq 0$ , where we define the function $\tilde{h}(b)$ as:

$$\tilde{h}(b)=\log \frac{1-\frac{b}{2}}{1-b}-b^2\mathrm{.}\backslash tilde\; \{h\}\; (b)\; =\; \backslash log\; \backslash frac\; \{1\; -\; \backslash frac\; \{b\}\{2\}\}\{1\; -\; b\}\; -\; b\; ^\; \{2\}.$$

We can analyze the variations of $\tilde{b}$ be computing its derivative which is given by:

$${\mathrm{\partial }}_b\tilde{h}(b)=\frac{1}{(1-b)(2-b)}-2b\mathrm{.}\backslash partial\_\; \{b\}\; \backslash tilde\; \{h\}\; (b)\; =\; \backslash frac\; \{1\}\{(1\; -\; b)\; (2\; -\; b)\}\; -\; 2\; b.$$

Hence, we have the following equivalence:

$${\mathrm{\partial }}_b\tilde{h}(b)\ge 0⟺2b(1-b)(2-b)\le 1\backslash partial\_\; \{b\}\; \backslash tilde\; \{h\}\; (b)\; \backslash geq\; 0\; \backslash iff\; 2\; b\; (1\; -\; b)\; (2\; -\; b)\; \backslash leq\; 1$$

This is always true for $0 \leq b < 1$ since $b(1 - b) \leq \frac{1}{4}$ so that $2b(1 - b)(2 - b) \leq \frac{2 - b}{2} \leq 1$ . Thus we have shown that $\tilde{h}$ is increasing over $[0,1)$ so that $\tilde{h}(b) \geq \tilde{h}(0) = 0$ . As discussed above, this is equivalent to having $n^{\star}b^{2} \leq 1$ , so that $h(n^{\star}, b) \geq 0$ which concludes the proof.

F EXPERIMENTS

F.1 DETAILS OF THE SYNTHETIC EXAMPLE

We choose the functions $f$ and $g$ to be of the form: $f(x,y) := \frac{1}{2} x^{\top}A_{f}x + y^{\top}C_{f}$ and $g(x,y) := \frac{1}{2} y^{\top}A_{g}y + y^{\top}B_{g}x$ where $A_{f}$ and $A_{g}$ are symmetric definite positive matrices of size $d_x \times d_x$ and $d_y \times d_y$ , $B_{g}$ is a $d_y \times d_x$ matrix and $C_{f}$ is a $d_y$ vector with $d_x = 2000$ and $d_y = 1000$ .

We generate the parameters of the problem so that the smoothness constants $L$ and $L_{g}$ are fixed to $1, \kappa_{\mathcal{L}} = 10$ and $\kappa_{g}$ taking values in ${10^i, i \in {0,..,7}}$ . We then solve each problem using different methods and perform a grid-search on the number of iterations $T$ and $M$ of algorithms $\mathcal{A}_k$ and $\mathcal{B}_k$ .

We fix the step-sizes to $\gamma_{k} = 1 / L$ and $\alpha_{k} = \beta_{k} = 1 / L_{g}$ and perform a grid-search on the number of iterations $T$ and $M$ of algorithms $\mathcal{A}_k$ and $\mathcal{B}_k$ from ${10^i, i \in 0, 1, 2, 3}$ . For AID methods without warm-start in $\mathcal{B}_k$ , we consider an additional setting where $M$ increases logarithmically with $k$ , as suggested in Ji et al. (2021), with $M = \lfloor 10^{3} \log(k) \rfloor$ . Similarly, for (ITD) and (Reverse), we additionally use an increasing $T$ of the same form.

F.2 EXPERIMENTAL DETAILS FOR LOGISTIC REGRESSION

The inner-level and outer-level cost functions for such task take the following form:

$$f(x,y)=\frac{1}{\mathrm{\mid }{\mathcal{D}}_{val}\mathrm{\mid }}\sum _{\xi \in {\mathcal{D}}_{val}}L(y,\xi ),g(x,y)=\frac{1}{\mathrm{\mid }{\mathcal{D}}_{tr}\mathrm{\mid }}\sum _{\xi \in {\mathcal{D}}_{tr}}L(y,\xi )+\frac{1}{pd}\sum _{i=1}\exp (x_i)\mathrm{\parallel }{y}_{\mathrm{.},i}{\mathrm{\parallel }}^{2}f\; (x,\; y)\; =\; \backslash frac\; \{1\}\{|\; \backslash mathcal\; \{D\}\; \_\; \{v\; a\; l\}\; |\}\; \backslash sum\_\; \{\backslash xi\; \backslash in\; \backslash mathcal\; \{D\}\; \_\; \{v\; a\; l\}\}\; L\; (y,\; \backslash xi),\; \backslash qquad\; g\; (x,\; y)\; =\; \backslash frac\; \{1\}\{|\; \backslash mathcal\; \{D\}\; \_\; \{t\; r\}\; |\}\; \backslash sum\_\; \{\backslash xi\; \backslash in\; \backslash mathcal\; \{D\}\; \_\; \{t\; r\}\}\; L\; (y,\; \backslash xi)\; +\; \backslash frac\; \{1\}\{p\; d\}\; \backslash sum\_\; \{i\; =\; 1\}\; \backslash exp\; (x\; \_\; \{i\})\; \backslash |\; y\; \_\; \{.,\; i\}\; \backslash |\; ^\; \{2\}$$

For the default setting, we use the well-chosen parameters reported in Grazzi et al. (2020); Ji et al. (2021) where $\alpha_{k} = \gamma_{k} = 100$ , $\beta_{k} = 0.5$ , and $T = N = 10$ . For the grid-search setting, we select the best performing parameters $T$ , $M$ and $\beta_{k}$ from a grid ${10,20} \times {5,10} \times {0.5,10}$ , while the batch-size (chosen to be the same for all steps of the algorithms) varies from $10 * {0.1,1,2,4}$ . We also compared with VRBO (Yang et al., 2021) using the implementation available online and noticed instabilities for large values of $T$ and $N$ , as reported by the authors, but also a drop in performance compared to stocBiO for smaller $T$ and $N$ due to inexact estimates of the gradient.

Figure 2: Evolution of the relative error vs. time in seconds for different AID based methods on the synthetic example. Each column corresponds to a method (AID-CG, AmIGO-CG, AmIGO-GD, AID-N, AID-FP) and each row corresponds to a choice of the conditioning number $\kappa_{g}$ . For each method we consider $T$ and $N$ from a grid ${1, 10, 10^{2}, 10^{3}} \times {1, 10, 10^{2}, 10^{3}}$ . Lightest colors correspond to smaller values of $N$ while nuances within each color correspond to increasing values of $T$ .

Figure 3: Evolution of the relative error vs. time in seconds for different ITD based methods on the synthetic example. From the left to the right, the first two columns correspond to Reverse and ITD method small conditioning numbers $\kappa_{g} \in {1, 10, 10^{3}}$ , last two columns are for higher conditioning numbers $\kappa_{g} \in {10^{4}, 10^{5}, 10^{7}}$ . For each method we consider $T \in {1, 10, 10^{2}, 10^{3}}$ . Lightest colors correspond to smaller values of $T$ .

F.3 DATASET DISTILLATION

Dataset distillation (Wang et al., 2018; Lorraine et al., 2020) consists in learning a small synthetic dataset such that a model trained on this dataset achieves a small error on the training set. Specifically, we consider a classification problem of $C$ classes using a linear model and a training dataset $\mathcal{D}{tr}$ where each training point $\xi \in \mathcal{D}{tr}$ is a $d$ -dimensional vector with a class $c_{\xi} \in {1,\dots,C}$ . The linear model is represented by a matrix $y \in \mathbb{R}^{c\times d}$ multiplying a data point $y\xi$ and providing the logits of each class. The dataset distillation can be cas as a bilevel problem of the form:

$$\underset{x\in {\mathbb{R}}^{c\times d},\lambda \in {\mathbb{R}}^d}{\min }\frac{1}{\mathrm{\mid }{\mathcal{D}}_{tr}\mathrm{\mid }}\sum _{\xi \in {\mathcal{D}}_{tr}}CE(y^{\star }(x,\lambda )\xi ,c_{\xi }),\backslash min\; \_\; \{x\; \backslash in\; \backslash mathbb\; \{R\}\; ^\; \{c\; \backslash times\; d\},\; \backslash lambda\; \backslash in\; \backslash mathbb\; \{R\}\; ^\; \{d\}\}\; \backslash frac\; \{1\}\{|\; \backslash mathcal\; \{D\}\; \_\; \{t\; r\}\; |\}\; \backslash sum\_\; \{\backslash xi\; \backslash in\; \backslash mathcal\; \{D\}\; \_\; \{t\; r\}\}\; C\; E\; (y\; ^\; \{\backslash star\}\; (x,\; \backslash lambda)\; \backslash xi\; ,\; c\; \_\; \{\backslash xi\}),$$

$$y^{\star }(x,\lambda )\in \arg \underset{y\in {\mathbb{R}}^{c\times d}}{\min }\frac{1}{C}\sum _{c=1}^{C}CE(y{x}_c,c)+\frac{1}{Cd}\sum _{i=1}^d\exp ({\lambda }_i)\mathrm{\parallel }{y}_{\mathrm{.},i}{\mathrm{\parallel }}^2\mathrm{.}y\; ^\; \{\backslash star\}\; (x,\; \backslash lambda)\; \backslash in\; \backslash arg\; \backslash min\; \_\; \{y\; \backslash in\; \backslash mathbb\; \{R\}\; ^\; \{c\; \backslash times\; d\}\}\; \backslash frac\; \{1\}\{C\}\; \backslash sum\_\; \{c\; =\; 1\}\; ^\; \{C\}\; C\; E\; (y\; x\; \_\; \{c\},\; c)\; +\; \backslash frac\; \{1\}\{C\; d\}\; \backslash sum\_\; \{i\; =\; 1\}\; ^\; \{d\}\; \backslash exp\; (\backslash lambda\_\; \{i\})\; \backslash |\; y\; \_\; \{.,\; i\}\; \backslash |\; ^\; \{2\}.$$

where $\lambda \in \mathbb{R}^d$ is a vector of hyper-parameter for regularizing the inner-level problem which we found beneficial to add.

Experimental setup. We perform the distillation task on MNIST dataset. We set the step-sizes $\alpha_{k} = \beta_{k} = 0.1$ and $T = N = 10$ . We perform a grid-search on the outer-level step-size $\gamma_{k}\in {0.01,0.001,0.0001}$ and run the algorithms for $k = 10000$ iterations.

Figure 4: Evolution of the validation loss (left column), validation accuracy (middle column) and test accuracy (right column) in time (s) for different methods on the logistic regression task. Each row corresponds to different choices for the size of the batch $|\mathcal{D}| \in {100, 1000, 2000, 4000}$ chosen to be the same for all gradient, Hessian and Jacobian-vector products evaluations. Time is reported in seconds.

Figure 5: Performance of various bi-level algorithms on the dataset distillation task on MNIST dataset.