SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Yuyang Qiu  
Dept. of Industrial and Systems Engg.  
Rutgers University  
yuyang.qiu@rutgers.edu
Uday V. Shanbhag  
Dept. of Industrial and Manufacturing Engg.  
Pennsylvania State University  
udaybag@psu.edu
Farzad Yousefian  
Dept. of Industrial and Systems Engg.  
Rutgers University  
farzad.yousefian@rutgers.edu
Federated learning (FL) has emerged as an enabling framework for communication-efficient decentralized training. We study three broadly applicable problem classes in FL: (i) Nondifferentiable nonconvex federated optimization; (ii) Federated bilevel optimization; (iii) Federated minimax problems. Notably, in an implicit sense, both (ii) and (iii) are instances of (i). However, the hierarchical problems in (ii) and (iii) are often complicated by the absence of a closed-form expression for the implicit objective function. Unfortunately, research on these problems has been limited and afflicted by reliance on strong assumptions, including the need for differentiability and L-smoothness of the implicit function. We address this shortcoming by making the following contributions. In (i), by leveraging convolution-based smoothing and Clarke's subdifferential calculus, we devise a randomized smoothing-enabled zeroth-order FL method and derive communication and iteration complexity guarantees for computing an approximate Clarke stationary point. To contend with (ii) and (iii), we devise a unified randomized implicit zeroth-order FL framework, equipped with explicit communication and iteration complexities. Importantly, our method utilizes delays during local steps to skip making calls to the inexact lower-level FL oracle. This results in significant reduction in communication overhead when addressing hierarchical problems. We empirically validate the theory on nonsmooth and hierarchical ML problems.
Federated learning (FL) has recently emerged as a promising enabling framework for learning predictive models from a multitude of distributed, privacy-sensitive, and possibly, heterogeneous datasets. This is accomplished through the use of efficiently devised periodic communications between a central server and a collection of clients. The FL algorithmic framework allows for addressing several key obstacles in the development and implementation of standard machine learning methods in a distributed and parallel manner. For instance, the conventional parallel stochastic gradient descent (SGD) method requires the exchange of information among the computing nodes at every single time step, resulting in excessive communication overhead. In contrast, FL methods including FedAvg [34] and Local SGD [46] overcome this onerous communication bottleneck by provably attaining the linear speedup of parallel SGD by using a significantly fewer communication rounds [18, 54, 24, 7]. These guarantees have been further complemented by recent efforts [26,
38] where the presence of both data heterogeneity (i.e., variability of local datasets) and device heterogeneity (i.e., variability of edge devices in computational power, memory, and bandwidth) have been addressed. Despite recent advances, much needs to be understood about designing communication-efficient decentralized methods for resolving three broadly applicable problem classes, each of which is presented next.
(a) Nondifferentiable nonconvex locally constrained FL. Consider the prototypical FL setting:
$$
\min  _ {x} \left\{f (x) \triangleq \frac {1}{m} \sum_ {i = 1} ^ {m} \mathbb {E} _ {\xi_ {i} \in \mathcal {D} _ {i}} [ \tilde {f} _ {i} (x, \xi_ {i}) ] | x \in X \triangleq \bigcap_ {i = 1} ^ {m} X _ {i} \right\}, \quad (\mathbf {F L} _ {n n})
$$
where  $f$  is a nonsmooth nonconvex function and is associated with a group of  $m$  clients indexed by  $i \in [m] \triangleq \{1, \ldots, m\}$ ,  $\mathcal{D}_i$  denotes the local dataset,  $\tilde{f}_i : \mathbb{R}^n \times \mathcal{D}_i \to \mathbb{R}$  is the local loss function, and  $X_i \subseteq \mathbb{R}^n$  is an easy-to-project local constraint set. Notably, local datasets may vary across clients, allowing for data heterogeneity. We also consider client-specific local sets to induce personalization.
(b) Nondifferentiable nonconvex bilevel  $FL$ . Overlaying a bilevel term in  $(\mathbf{FL}_{nn})$  leads to
$$
\min  _ {x \in X \triangleq \bigcap_ {i = 1} ^ {m} X _ {i}} \left\{f (x) \mid y (x) \in \arg \min  _ {y \in \mathbb {R} ^ {\tilde {n}}} \frac {1}{m} \sum_ {i = 1} ^ {m} \mathbb {E} _ {\zeta_ {i} \in \tilde {\mathcal {D}} _ {i}} [ \tilde {h} _ {i} (x, y, \zeta_ {i}) ] \right\}, \quad (\mathbf {F L} _ {b l})
$$
where  $f(\bullet) \triangleq \frac{1}{m} \sum_{i=1}^{m} \mathbb{E}_{\xi_i \in \mathcal{D}_i}[\tilde{f}_i(\bullet, y(\bullet), \xi_i)]$  denotes the implicit objective function and  $y(\bullet): \mathbb{R}^n \to \mathbb{R}^{\tilde{n}}$  is a single-valued map returning the unique solution to the lower-level problem at  $x$ .
(c) Nondifferentiable nonconvex minimax  $FL$ . Finally, we consider the minimax setting, defined as
$$
\min  _ {x \in X ^ {\triangle} \cap_ {i = 1} ^ {m} X _ {i}} \max  _ {y \in \mathbb {R} ^ {\bar {n}}} \frac {1}{m} \sum_ {i = 1} ^ {m} \mathbb {E} _ {\xi_ {i} \in \mathcal {D} _ {i}} [ \tilde {f} _ {i} (x, y, \xi_ {i}) ]. \quad (\mathbf {F L} _ {m m})
$$
where we assume that  $y(x) \in \arg \max_{y \in \mathbb{R}^n} \frac{1}{m} \sum_{i=1}^{m} \mathbb{E}_{\zeta_i \in \tilde{\mathcal{D}}_i}[\tilde{f}_i(x,y,\xi_i)]$  is unique for all  $x$ . Let  $f(\bullet) \triangleq \frac{1}{m} \sum_{i=1}^{m} \mathbb{E}_{\xi_i \in \mathcal{D}_i}[\tilde{f}_i(\bullet,y(\bullet),\xi_i)]$  denote the implicit objective function. Indeed, problem  $(\mathbf{FL}_{bl})$  subsumes this minimax formulation when we choose  $\tilde{h}_i := -\tilde{f}_i$  and  $\tilde{\mathcal{D}}_i := \mathcal{D}_i$ .
Notably, in an implicit sense, both (b) and (c) are instances of problem (a). However, these hierarchical problems are often complicated by the absence of a closed-form expression for the implicit objective, denoted by  $f(\bullet)$ . Indeed,  $f(\bullet)$  is often nonsmooth, nonconvex, and unavailable. As such, the absence of both zeroth and first-order information of  $f(\bullet)$  in problems (b) and (c) makes the design and analysis of FL methods for these problems more challenging than that for (a).
Gaps. To the best of our knowledge, there are no known efficient FL algorithms that can contend with both nonsmoothness and nonconvexity in an unstructured sense. Generalizations that can accommodate either a bilevel or a minimax structure also remain unaddressed in FL.
Goal. To develop a unified FL framework accommodating nondifferentiable nonconvex settings with extensions allowing for bilevel or minimax interactions. We now describe the proposed framework.
1.1 A Smoothed Sampled Zeroth-order Framework
We consider a smoothed framework for contending with constrained, nonsmooth, and nonconvex regimes. Specifically, given that  $f$  is an expectation-valued function and  $X$  is a closed and convex set, both of which are defined in  $(\mathbf{FL}_{nn})$ , a smoothed unconstrained approximation is given as follows.
$$
\left\{ \begin{array}{l} \min  \frac {1}{m} \sum_ {i = 1} ^ {m} \mathbb {E} _ {\xi_ {i}} [ \tilde {f} _ {i} (x, \xi_ {i}) ] \\ \text {s u b j e c t t o} x \in \cap_ {i = 1} ^ {m} X _ {i}. \end{array} \right\} \equiv \left\{\min  \frac {1}{m} \sum_ {i = 1} ^ {m} [ \mathbb {E} _ {\xi_ {i}} [ \tilde {f} _ {i} (x, \xi_ {i}) ] + \underbrace {\mathbb {I} _ {X _ {i}} (x)} _ {\text {I n d i c a t o r f u n c t i o n}} ] \right\}
$$
$$
\underset {\approx} {\text {S m o o t h i n g}} \left\{\min  _ {m} \sum_ {i = 1} ^ {m} \underbrace {\mathbb {E} _ {u _ {i} \in \mathbb {B}} [ \mathbb {E} _ {\xi_ {i}} [ \tilde {f} _ {i} (x + \eta u _ {i} , \xi_ {i}) ] ]} _ {\text {C o n v o l u t i o n s m o o t h i n g}} + \underbrace {\mathbb {I} _ {X _ {i}} ^ {\eta} (x)} _ {\text {M o r e a u s m o o t h i n g}} \right\}. \tag {F L}
$$
If  $f$  is as defined in  $(\mathbf{FL}_{nn})$  and  $d(x) \triangleq \frac{1}{m} \sum_{i=1}^{m} \mathbb{I}_{X_i}(x)$ , then  $\mathbf{f}$  and its smoothing  $\mathbf{f}^\eta$  are defined as
$$
\mathbf {f} (x) \triangleq f (x) + d (x) \text {a n d} \mathbf {f} ^ {\eta} (x) \triangleq f ^ {\eta} (x) + d ^ {\eta} (x), \tag {1}
$$
where  $f^{\eta}(x) \triangleq \frac{1}{m} \sum_{i=1}^{m} [\mathbb{E}_{u_i \in \mathbb{B}}[\mathbb{E}_{\xi_i}[\tilde{f}_i(x + \eta u_i, \xi_i)]]]$  and  $d^{\eta}(x) \triangleq \frac{1}{m} \sum_{i=1}^{m} \mathbb{I}_{X_i}^{\eta}(x)$ .
(i) Clarke-stationarity. Consider the original problem  $(\mathbf{FL}_{nn})$ . Under the assumption that the objective of  $(\mathbf{FL}_{nn})$  is Lipschitz continuous, then Clarke-stationarity of  $x$  w.r.t.  $(\mathbf{FL}_{nn})$  requires that  $x$  satisfies  $0 \in \partial f(x) + \mathcal{N}_{\cap_{i=1}^{m} X_i}(x)$ , where  $\partial f(x)$  represents the Clarke generalized gradient [3] of  $f$  at  $x$ . However, a negative result has been provided regarding the efficient computability of an  $\epsilon$ -stationary solution in nonsmooth nonconvex regimes [56]. Consequently, we focus on the smoothed counterpart  $(\mathbf{FL}_{nn}^{\eta})$ , a smooth nonconvex problem. In fact, under suitable conditions, it can be shown that stationary point of  $(\mathbf{FL}_{nn}^{\eta})$  is a  $2\eta$ -Clarke stationary point of the original problem, i.e.
$$
[ 0 \in \partial \mathbf {f} ^ {\eta} (x) ] \Longrightarrow [ 0 \in \partial_ {2 \eta} \mathbf {f} (x) ], \tag {2}
$$
where  $\partial_{2\eta}\mathbf{f}(x)$  represents the  $2\eta$ -Clarke generalized gradient of  $\mathbf{f}$  at  $x$ .
(ii) Meta-scheme for efficient resolution of  $(\mathbf{FL}_{nn}^{\eta})$ . We develop zeroth-order stochastic gradient schemes for resolving  $(\mathbf{FL}_{nn}^{\eta})$ . This requires a zeroth-order gradient estimator for  $f^{\eta}(x)$ , denoted by  $\frac{1}{m}\sum_{i=1}^{m}g_i^\eta(x,\xi_i,v_i)$  where  $v_i \in \eta\mathbb{S}$  for  $i \in [m]$  and  $\mathbb{S}$  denotes the surface of the unit ball. Note that the Moreau smoothing of the indicator function of  $X_i$ , denoted by  $\mathbb{I}_{X_i}^\eta(x)$ , admits a gradient, defined as  $\nabla_x\mathbb{I}_{X_i}^\eta(x) = \frac{1}{\eta}(x - \mathcal{P}_{X_i}(x))$ , where  $\mathcal{P}_{X_i}(x) \triangleq \operatorname*{arg\min}_{y \in X_i} \|y - x\|^2$ . The resulting meta-scheme is defined next.
$$
x _ {k + 1} = x _ {k} - \gamma \left(\frac {1}{m} \sum_ {i = 1} ^ {m} \left(g _ {i} ^ {\eta} \left(x _ {k}, \xi_ {i, k}, v _ {i, k}\right) + \frac {1}{\eta} \left(x _ {k} - \mathcal {P} _ {X _ {i}} \left(x _ {k}\right)\right)\right)\right), k \geq 0. \quad (\textbf {M e t a - Z O})
$$
(iii) Inexact implicit generalizations for  $(\mathbf{FL}_{bl})$  and  $(\mathbf{FL}_{mm})$ . In addressing the bilevel problem, unlike in  $(\mathbf{FL}_{nn})$ , the clients in  $(\mathbf{FL}_{bl})$  may not have access to the exact evaluation of the implicit local objective  $f_{i}(\bullet, y(\bullet), \xi_{i})$ . This makes a direct extension of FL schemes challenging. This is because the evaluation of the implicit local function necessitates exact resolution of the lower-level problem. We address this challenge by developing inexact implicit variants of the zeroth-order scheme, where clients compute only an  $\varepsilon$ -approximation of  $y(x)$ , denoted by  $y_{\varepsilon}(x)$  in a federated fashion. This inexact framework, described next, is crucial in addressing hierarchy in bilevel FL formulations. Let  $f_{i}^{\eta}(\bullet)$  denote the smoothed implicit local function. We estimate  $\nabla_{x}f_{i}^{\eta}(x)$  by approximating the expectation by sampling in (a.1), as follows, while in (a.2), we replace  $y(x)$  by an inexact form  $y_{\varepsilon}(x)$ . This leads to the introduction of  $g_{i}^{\eta,\varepsilon}(x, \xi, v)$  as captured below.
$$
\nabla_{x}f_{i}^{\eta}(x) = \underbrace{\mathbb{E}_{\xi_{i},v}\left[g_{i}^{\eta}(x,\xi_{i},v)\right]}_{\text{cannot be tractably evaluated}}\stackrel {(a.1)}{\approx}\underbrace{g_{i}^{\eta}(x,\xi_{i,k},v_{T_{r}})}_{\text{intractable since $y(x)$ is unavailable}}\stackrel{(a.2)}{\approx}g_{i}^{\eta ,\varepsilon}(x,\xi_{i,k},v_{T_{r}}),
$$
where  $k$  is the local time index,  $T_{r}$  is the global time index of communication round  $r$  ( $k \geq T_{r}$ ), and  $g_{i}^{\eta, \varepsilon}(x, \xi, v) \triangleq \frac{n}{\eta} (\tilde{f}_{i}(x + v, y_{\varepsilon}(x + v), \xi) - \tilde{f}_{i}(x, y_{\varepsilon}(x), \xi)) \frac{v}{\|v\|}$  denotes an inexact implicit zeroth-order gradient. Note that at each round of communication at the upper level,  $y_{\varepsilon}(x)$  can be computed using calls to a standard FL method, e.g., FedAvg, in the lower level. Notably, such calls to an FL oracle should be made only at the global step to preserve the communication efficiency of the scheme. It follows that  $g_{i}^{\eta, \varepsilon}(x, \xi_{i,k}, v_{T_r}) = \nabla_x f_{i}^{\eta}(x) + \tilde{e}_{i,\varepsilon}$  where the approximation error  $\tilde{e}_{i,\varepsilon}$  is a possibly biased random variable. This bias can be then controlled by carefully by updating the accuracy level  $\varepsilon$  at each communication round, as we will address in this work.
Our goal lies in extending (Meta-ZO) to federated nonsmooth nonconvex optimization and then provide generalizations to bilevel and minimax regimes. In each instance, we intend to provide iteration and communication-complexity bounds for computing an  $\epsilon$ -accurate  $\eta$ -Clarke stationary point of the original problem. Accordingly, we make the following contributions.
(i)  $FL$  for nondifferentiable nonconvex problems. To address  $(\mathbf{FL}_{nn})$  with heterogeneous datasets, we develop a Randomized Zeroth-Order Locally-Projected Federated Averaging method  $(\mathrm{FedRZO_{nn}})$ . We derive iteration complexity of  $\mathcal{O}\left(\frac{1}{m\epsilon^2}\right)$  and communication complexity of  $\mathcal{O}\left(m^{3/4}K^{3/4}\right)$  for computing an approximate Clarke stationary point. Such guarantees appear to be new in the context of resolving nondifferentiable nonconvex FL problems, e.g. in training of ReLU neural networks (see Table 2). This is distinct from existing zeroth-order methods, including FedZO [12], that rely on differentiability and  $L$ -smoothness of the local loss functions.
(ii) Federated bilevel optimization. In addressing  $(\mathbf{FL}_{bl})$ , we develop  $\mathrm{FedRZO_{b1}}$ , an inexact implicit extension of  $\mathrm{FedRZO_{nn}}$ . By skipping local calls to the lower-level FL oracle,  $\mathrm{FedRZO_{b1}}$  is a
Table 1: Communication complexity for nonsmooth nonconvex, bilevel, and minimax FL problems.  $K$  and  $\widetilde{K}$  denote the maximum number of iteration used in upper level and lower level, respectively.
heterogeneous upper level (this work)	lower level (standard FL schemes are employed)	total for bilevel and minimax (this work)
O((mK)3/4)(Prop. 1, Thm. 1)	i.i.d. (Local SGD [26])	O(m)	O(m7/4K3/4)
i.i.d. (FedAC [54])	O(m1/3)	O(m13/12K3/4)
heterogeneous (LFD [20])	O(m13/3K1/3)	O((mK)11/12)
novel communication-efficient FL scheme with single-timescale local steps, resulting in significant reduction in communication overhead. Table 1 summarizes the communication complexity of this scheme. In all cases, we assume heterogeneous data at the upper level. In the lower level, depending on which conventional FL scheme is employed, we obtain the communication complexity accordingly.
(iii) Federated minimax optimization.  $\mathrm{FedRZO_{b1}}$  can be employed for addressing  $(\mathbf{FL}_{mm})$  where  $\tilde{h}_i \coloneqq -\tilde{f}_i$ . As such, the complexity results in (ii) hold for solving (nondifferentiable nonconvex)-(strongly concave) FL minimax problems. Such results are new for this class of FL problems.
Remark 1. There has been recent progress in addressing bilevel and minimax problems in FL, including Local SGDA and FedNest [48, 43]. Our work in (ii) and (iii) has two main distinctions with existing FL methods, described as follows. (1) We do not require the differentiability and  $L$ -smoothness of the implicit objective function. This assumption may fail to hold, e.g., in constrained hierarchical FL problems. (2) The existing FL methods for bilevel and minimax problems assume that the lower-level problem is unconstrained. In fact, even in centralized regimes, addressing hierarchical problems where the lower-level constraints depend on  $x$  have remained challenging. For example, consider the problem  $\min_{x \in [-1,1]} \max_{y \in [-1,1]}$ ,  $x + y \leq 0$ $x^{2} + y$  that admits a unique solution  $(x^{*}, y^{*}) = (0.5, -0.5)$ . Now consider a reversal of min and max in this problem, i.e.,  $\max_{y \in [-1,1]} \min_{x \in [-1,1]}$ ,  $x + y \leq 0$ $x^{2} + y$ , admitting the unique solution  $(x^{*}, y^{*}) = (-1,1)$ . As a consequence, the well-known primal-dual gradient methods, that have been extensively employed for addressing minimax problems with independent constraint sets, may fail to converge to a saddle-point in minimax problems with coupling constrains. Our proposed algorithmic framework allows for accommodating these challenging problems in FL.
Table 2: Comparison of our scheme with other FL schemes for nonconvex settings
Ref.	Nonconvex	Metric	Rate	Comm. rounds	Assumption
[53]	Smooth	||∇xf(x)||2	O(G2/√mK)	O(m3/4K3/4)	Bounded gradients,L-smooth functions
[50]	Smooth	||∇xf(x)||2	O(1/√mK)	O(m3/2K1/2)	L-smooth functions
[18]	Smooth, PL-cond	f(x) - f*	O(1/mK)	O(m1/3K1/3)	L-smooth functions, PL-condition
This work	Nonsmooth	||∇xfη(x)||2	O(1/√mK)	O(m3/4K3/4)	Lipschitz functions
(i) Nondifferentiable nonconvex optimization. Nonsmooth and nonconvex optimization has been studied extensively with convergence guarantees to Clarke-stationary points via gradient sampling [1, 2] and difference-of-convex approaches [5]. Most complexity and rate guarantees necessitate smoothness of the nonconvex term [14, 51, 8, 9, 28] or convexity of the nonsmooth term [13], while only a few results truly consider nonsmooth nonconvex objective function [32, 4, 42]. (ii) Nondifferentiable nonconvex federated learning. The research on FL was initially motivated by decentralized neural networks where local functions are nondifferentiable and nonconvex [34]. Nevertheless, theoretical guarantees that emerged after FedAvg required either nonsmooth convex or smooth nonconvex local costs, under either iid [46, 58, 50, 47] or non-iid datasets [30, 26], while provable guarantees for FL methods under nonconvexity [58, 53, 19] require  $L$ -smoothness of local functions. Unfortunately, these assumptions do not hold either for ReLU neural networks or risk-averse learning and necessitate the use of Clarke calculus [3]. Moreover, existing work on zeroth-order FL methods in convex [31] and nonconvex settings [12] rely on the smoothness properties of the objective function. However, there appear to be no provably convergent FL schemes with complexity guarantees for computing approximate Clarke stationary points of nondifferentiable nonconvex problems. (iii) Federated bilevel optimization. Hyperparameter tuning [21] and its federated counterpart [23] is a crucial, and yet, computationally complex integrant of machine learning (ML) pipeline. Bilevel models
where the lower-level is a parameterized training model while the upper-level requires selecting the best configuration for the unknown hyperparameters [15, 22, 49]. Solving such hierarchical problems is challenging because of nondifferentiable nonconvex terms and absence of an analytical form for the implicit objective. These challenges exacerbate the development of provable guarantees for privacy-aware and communication-efficient schemes. (iv) Federated minimax optimization. Minimax optimization has assumed relevance in adversarial learning [17, 40, 44] and fairness in ML [57], amongst other efforts. Recently, FL was extended to distributed minimax problems [36, 10, 43], but relatively little exists in nonconvex-strongly concave settings [48, 43].
3 A Zeroth-order FL Framework for Nondifferentiable Nonconvex Settings
In this section, we introduce our framework for  $(\mathbf{FL}_{nn})$ , where we impose the following assumption.
Assumption 1. Consider problem  $(\mathbf{FL}_{nn})$ . The following hold.
(i) The function  $f_{i}$  is Lipschitz continuous with parameter  $L_0 > 0$  for all  $i \in [m]$ .
(ii) For any  $i \in [m]$ , client  $i$  has access to a zeroth-order oracle  $\tilde{f}_i(x, \xi_i)$  satisfying the following for every  $x$  in an almost-sure sense:
(ii-a)  $\mathbb{E}[\tilde{f}_i(x,\xi_i)\mid x] = f_i(x)$ ; (ii-b) There exists  $\nu >0$  such that  $\mathbb{E}[|\tilde{f}_i(x,\xi_i) - f_i(x)|^2\mid x]\leq \nu^2$
(iii) The set  $X_{i}$  is nonempty, closed, and convex for all  $i \in [m]$ . In addition, the following bounded set-dissimilarity condition holds for all  $x \in \mathbb{R}^n$  and some scalars  $B_{1}$  and  $B_{2}$ .
$$
\frac {1}{m} \sum_ {i = 1} ^ {m} \operatorname {d i s t} ^ {2} (x, X _ {i}) \leq B _ {1} ^ {2} + B _ {2} ^ {2} \| x - \frac {1}{m} \sum_ {i = 1} ^ {m} \mathcal {P} _ {X _ {i}} (x) \| ^ {2}. \tag {3}
$$
We note that the bounded set-dissimilarity condition is naturally analogous to the so-called bounded gradient-dissimilarity condition that has been employed in the literature, e.g., in [25]. In particular, when the bounded gradient-dissimilarity condition is stated for the Moreau smoothing of the indicator function of  $X_{i}$ , denoted by  $\mathbb{I}_{X_i}^\eta (x)$ , we reach to (3). Notably, condition (3) holds for the generated iterates by the algorithm when, for example, the iterates remain bounded.
Nonsmooth unconstrained reformulation. Consider an unconstrained reformulation of  $(\mathbf{FL}_{nn})$  given by  $\min_{x\in \mathbb{R}^n}\mathbf{f}(x)$  (see (1)), where the nonsmoothness of  $\mathbf{f}$  arises from that of  $f$  and the local indicator functions  $\mathbb{I}_{X_i}$ . The minimization of  $\mathbf{f}$  is challenging, as noted by recent findings on nonsmooth analysis where it is shown [56] that for a suitable class of nonsmooth functions, computing an  $\epsilon$ -stationary point, i.e., a point  $\bar{x}$  for which  $\mathrm{dist}(0_n,\partial \mathbf{f}(\bar{x}))\leq \epsilon$ , is impossible in finite time.
Approximate Clarke stationarity. To circumvent this challenge, as a weakening of  $\epsilon$ -stationarity, a notion of  $(\delta, \epsilon)$ -stationarity is introduced [56] for a vector  $\bar{x}$  when  $\mathrm{dist}(0_n, \partial_\delta \mathbf{f}(\bar{x})) \leq \epsilon$ , where the set
$$
\partial_ {\delta} \mathbf {f} (x) \triangleq \operatorname {c o n v} \left\{\zeta : \zeta \in \partial \mathbf {f} (y), \| x - y \| \leq \delta \right\}
$$
denotes the  $\delta$ -Clarke generalized gradient of  $\mathbf{f}$  at  $x$  [16]; i.e. if  $x$  is  $(\delta, \epsilon)$ -stationary, then there exists a convex combination of gradients in a  $\delta$ -neighborhood of  $x$  that has a norm of at most  $\epsilon$  [41].
This discussion naturally leads to the following key question: Can we devise provably convergent FL methods for computing approximate Clarke stationary points of minimization of  $\mathbf{f}$ ? The aim of this section is to provide an affirmative answer to this question by proposing a zeroth-order FL method that employs smoothing. To contend with the nonsmoothness, we employ the Moreau-smoothed variant of  $\mathbb{I}_X(x)$ , where  $X \triangleq \bigcap_{i=1}^{m} X_i$ , and a randomized smoothed variant of  $f$ , as shown next.