SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Deeparnab Chakrabarty *
Hanover, USA
deeparnab@dartmouth.edu
Andrei Graur†
Stanford, USA
agraur@stanford.edu
Haotian Jiang
jhtdavid96@gmail.com
Aaron Sidford
sidford@stanford.edu
We consider the parallel complexity of submodular function minimization (SFM). We provide a pair of methods which obtain two new query versus depth trade-offs a submodular function defined on subsets of  $n$  elements that has integer values between  $-M$  and  $M$ . The first method has depth 2 and query complexity  $n^{O(M)}$  and the second method has depth  $\widetilde{O}(n^{1/3}M^{2/3})$  and query complexity  $O(\mathrm{poly}(n, M))$ . Despite a line of work on improved parallel lower bounds for SFM, prior to our work the only known algorithms for parallel SFM either followed from more general methods for sequential SFM or highly-parallel minimization of convex  $\ell_2$ -Lipschitz functions. Interestingly, to obtain our second result we provide the first highly-parallel algorithm for minimizing  $\ell_{\infty}$ -Lipschitz function over the hypercube which obtains near-optimal depth for obtaining constant accuracy.
A function  $f: 2^{[n]} \to \mathbb{Z}$  is submodular if it has the diminishing marginal return property that  $f(S \cup \{i\}) - f(S) \geq f(T \cup \{i\}) - f(T)$  for all elements  $i \in [n]$  and subsets  $S \subseteq T \subseteq [n] \setminus \{i\}$ , i.e., the function value increase of adding  $i$  to a set  $S$  is at least that of adding  $i$  to a superset  $T$  that does not contain  $i$ . Submodular functions are used to model a range of problems arising machine learning [KG05, KNTB09, KC10], operations research [FG88, QS95], and economics [Top98]. The submodular function minimization (SFM) problem consists of finding the subset  $S$  with the smallest  $f(S)$ , given evaluation oracle access to the submodular function  $f$ , using as few queries as possible. SFM has numerous applications. For example, in natural language processing SFM has played a key role in speech analysis where [LB11, JB11] modeled the task of optimally selecting terse, yet diverse, training data out of a large speech dataset as a SFM problem; in computer vision, the task of energy minimization was reduced to SFM [BVZ01, KKT08, KT10, JBS13, KBJ+15].
Seminal work of [GLS81] established that SFM can be solved with a polynomial number of queries and in polynomial time. In the decades since, there has been extensive research characterizing the query and computational complexity of SFM. When  $f$  is integer valued with values between
*Supported in part by NSF grant #2041920.
†Supported in party by NSF CAREER Award CCF-1844855, NSF Grant CCF-1955039, and Stanford Management Science & Engineering Department Nakagawa Fellowship
$^{\ddagger}$ Supported in part by a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF-1844855, NSF Grant CCF-1955039, a PayPal research award, and a Sloan Research Fellowship.
$-M$  and  $M$ , the state-of-the-art query complexities are  $\widetilde{O}(nM^2)^4$  due to [CLSW17] and [ALS20],  $\widetilde{O}(n^2\log M)$  and  $\widetilde{O}(n^3)$ , both due to [LSW15] (the latter query complexity being improved by a factor of  $\log^2 n$  in [Jia21]), and  $O(n^2\log n)$  (with an exponential runtime) due to [Jia22]. The largest query complexity lower bound is  $\Omega(n\log n)$  for deterministic algorithms due to [CGJS22].
The algorithms underlying the above results are highly sequential. State-of-the-art SFM algorithms use at least a linear number of rounds of queries to the evaluation oracle  $(\widetilde{O}(nM^2)$  rounds in [ALS20] and  $O(n\log n)$  rounds in [Jia22]). Although there is a trivial 1-round algorithm that queries all  $2^n$  input of  $f$  in parallel, all known polynomial time SFM algorithms use  $\Omega(n)$  rounds.
With the prevalence of parallel computation in practice and applications of SFM to problems involving massive data sets, recently there has been an increasing number of works which study the parallel complexity of SFM, i.e., the smallest number of rounds a SFM algorithm must take. Starting with [BS20], significant progress has been made in proving lower bounds on the parallel complexity of SFM. It is now known that any SFM algorithm making polynomially many queries needs  $\widetilde{\Omega}(n^{1/3})$  rounds when  $M = \Theta(n)$  [CCK21] and  $\widetilde{\Omega}(n)$  rounds when  $M = \Theta(n^n)$  [CGJS22].
In contrast to the significant progress on lower bounds, less progress has been made on designing SFM algorithms with small parallel complexity. This is due in part to the established lower bounds; as noted above, if we desire range-independent parallel complexity bounds, i.e., independent of the range  $M$ , then it is impossible to obtain an  $O(n^{1 - \varepsilon})$ -round SFM algorithm, for any constant  $\varepsilon > 0$ . The central question motivating this paper is what non-trivial parallel speedups for SFM are possible if we allow methods with a range-dependent parallel complexity.
Is it possible to obtain query-efficient  $o(n)$ -round  $M$ -dependent algorithms for SFM?
In this paper we provide positive answers to the above question. Our first result is an algorithm that runs in  $\widetilde{O}(n^{1/3}M^{2/3})$  rounds with query complexity  $\widetilde{O}(n^2M^2)$ . To achieve this result, we first provide a generic reduction from parallel SFM to parallel convex optimization (a well studied problem discussed below). While naively this approach yields a sublinear bound of  $\widetilde{O}(n^{2/3}M^{2/3})$  rounds, we show how to further improve these convex optimization methods in our setting.
Our second result is a simple 2-round SFM algorithm (Algorithm 1) with query complexity  $n^{O(M)}$ . For constant  $M$ , the parallel complexity of 2 is optimal among the class of query-efficient algorithms that query the minimizer [CLSW17]. It is instructive to contrast our second result to the lower bound in [CCK21], where it is proved that when  $M = n$ , any algorithm with query complexity  $n^{M^{1 - \delta}}$  for any constant  $\delta > 0$  must proceed in  $n^{\Omega(\delta)}$  rounds.
Highly Parallel Convex Optimization. Motivated by applications to distributed and large scale optimization [BPC $^{+}$ 11, GR18], the question of highly parallel convex optimization has received significant attention in the last decade. Formally, the task is to find an (approximate) minimizer of a convex function this is Lipschitz in some norm, using as few rounds of  $O(\mathrm{poly}(n))$  parallel queries to a subgradient oracle as possible. Over the past few years, there has been progress on both upper [DBW12, SBB $^{+}$ 18, BJL $^{+}$ 19] and lower bounds [Nem94, BS18, DG19] for this problem.
This line of work is particularly relevant as SFM reduces to minimizing the Lovász extension (see Fact 2.4) [Lov83] over  $[0,1]^n$ , which is convex and  $O(M)$ -Lipschitz in  $\ell_{\infty}$  (see Definition 2.1). In Section 2.1 we provide a straightforward reduction from this problem to unconstrained minimization of a  $O(L)$ -Lipschitz function in  $\ell_{\infty}$  where the minimizer has  $\ell_{\infty}$  norm  $O(1)$ . Consequently, improved parallel  $\ell_{\infty}$ -Lipschitz convex optimization algorithms can imply improved parallel SFM algorithms.
Many prior algorithms for highly parallel convex optimization focused on convex functions that are  $\ell_2$ -Lipschitz and some were written only for unconstrained optimization problems. Naively using these algorithms for  $\ell_{\infty}$  yields parallel complexities of  $\widetilde{O}(n^{3/4}/\varepsilon)$  [DBW12] and  $\widetilde{O}(n^{2/3}/\varepsilon^{2/3})$  [BJL+19, CJJ+23] $^5$  for convex functions that are  $\ell_{\infty}$ -Lipschitz.
Our first SFM result follows from an improved algorithm that we develop that finds  $\varepsilon$ -approximate minimizers for convex  $\ell_{\infty}$ -Lipschitz functions in  $\widetilde{O}(n^{1/3}/\varepsilon^{2/3})$  rounds. Interestingly, for constant  $\varepsilon > 0$ , the dependence on  $n$  in this improved parallel complexity is optimal up to logarithmic
4 Throughout, we use  $\widetilde{O}(f(n))$  to hide  $\mathrm{poly}(\log f(n), \log n)$  factors.
$^{5}$  [BJL  $^{+}19$ , CJJ  $^{+}23$ ] study the  $\ell_{2}$  setting. To obtain the corresponding  $\widetilde{O}(n^{2/3}/\varepsilon^{2/3})$  result for  $\ell_{\infty}$ , we combine their result with Lemma 2.3 given in this paper.
factors; [Nem94] proved a lower bound of  $\widetilde{\Omega}(n^{1/3}\ln(1/\varepsilon))$  on the round complexity of any algorithm obtaining an  $\varepsilon$ -approximate minimizer over  $[0,1]^n$  for  $\ell_{\infty}$ -Lipschitz convex functions. For constant  $\varepsilon > 0$ , this lower bound also applies to unconstrained  $\ell_{\infty}$ -optimization [DG19].
Paper	Year	Parallel Rounds
[ALS20]	2020	O(nM2)
[JLSW20]	2020	O(n log M)
[Jia22]	2021	O(n)
This paper	2023	O(n1/3M2/3)
Paper	Year	Parallel Rounds
Subgradient Descent	1960s	O(n/ε2)
Cutting Plane	1965	O(n log(1/ε))
[BJL+19, CJJ+23]*	2019	O(n2/3/ε2/3)
This paper	2023	O(n1/3/ε2/3)
Table 1: State-of-the-art parallel complexity for SFM and  $\ell_{\infty}$ -optimization. See Section 1.2 for references on cutting plane methods. *The  $\widetilde{O}(n^{2/3}/\varepsilon^{2/3})$  result uses Lemma 2.3 from this paper.
Notation. We let  $[n] := \{1, \dots, n\}$  for any  $n \in \mathbb{Z}_{>0}$ . We let  $\mathbf{I}_n$  denote the identity matrix in  $\mathbb{R}^{n \times n}$ .
1.1 Problems, Results, and Approach
Here we formally define the problems we consider, present our results, and discuss some of the key insights of our approach. This section is organized as follows. We begin by presenting the parallel computation model. Then, we introduce the parallel  $\ell_p$ -Lipschitz convex optimization problem, which is closely tied to parallel SFM. We present our improvement for the  $\ell_{\infty}$ -Lipschitz setting and offer a brief overview of our techniques in obtaining this result. Finally, we formally introduce the SFM setup, along with the new results we obtain (Theorem 1.3 and Theorem 1.4).
Parallel Complexity Model. We consider the standard black-box query model for optimization, where we are given a convex function  $f: \mathcal{D} \to \mathbb{R}$ , with domain  $\mathcal{D} \subset \mathbb{R}^n$ , accessed through an oracle. Parallel algorithms for minimizing  $f$  proceed in rounds, where in each round the algorithm can submit a set of queries to the oracle in parallel. The parallel complexity of an algorithm is the total number of rounds it uses and it captures how "sequential" the algorithm is. Additionally, we consider the query complexity of an algorithm which is the total number of queries it makes.
Parallel  $\ell_p$ -Lipschitz Convex Optimization. In Section 2 we consider the problem of minimizing a convex function  $f: \mathbb{R}^n \to \mathbb{R}$  given access to a subgradient oracle  $g: \mathbb{R}^n \to \mathbb{R}^n$ , which when queried with point  $x \in \mathbb{R}^n$  returns a vector  $g(x)$  that is a subgradient, denote  $g(x) \in \partial(f(x))$  where  $\partial(f(x))$  is the set of all subgradients of  $f$  at  $x$ , i.e.,  $v \in \partial f(x)$  if and only if  $f(y) \geq f(x) + v^\top(y - x)$  for all  $y \in \mathbb{R}^n$ . Furthermore, we assume that  $f$  is  $L$ -Lipschitz with respect to a norm  $\|\cdot\|$ .
Definition 1.1 (L-Lipschitzness with respect to a given norm). We say that  $f: \mathbb{R}^n \to \mathbb{R}$  is  $L$ -Lipschitz with respect to  $\| \cdot \|$  for norm  $\| \cdot \|$  if  $|f(x) - f(y)| \leq L \| x - y \|$  for all  $x, y \in \mathbb{R}^n$ . If  $f$  is  $O(1)$ -Lipschitz with respect to a norm  $\| \cdot \|$ , we say that  $f$  is  $\| \cdot \|$ -Lipschitz. When the norm is  $\| \cdot \|_p$  we alternatively say that  $f$  is  $L$ -Lipschitz in  $\ell_p$  and that  $f$  is  $\ell_p$ -Lipschitz respectively.
There is a broad line of work on studying the parallel complexity of  $\ell_p$ -Lipschitz convex optimization in which the goal is to efficiently compute an  $\varepsilon$ -approximate minimizer (i.e., a point  $y$  with  $f(y) \leq \inf_x f(x)$ ) of a  $\ell_p$ -Lipschitz convex function where the minimizer either has  $\ell_p$ -norm  $O(1)$  or the problem is constrained to the  $\ell_p$ -norm ball of radius  $O(1)$ . The case when  $p = 2$  is perhaps the most well studied and our new result regarding  $\ell_\infty$ -Lipschitz convex optimization builds upon a result in [CJJ+23], Theorem 1, which considers this setting. The statement of our result is below:
Theorem 1.2 (Parallel Convex Optimization in  $\ell_{\infty}$ ). There is an algorithm that when given a subgradient oracle for convex  $f:\mathbb{R}^n\to \mathbb{R}$  that is 1-Lipschitz in  $\ell_{\infty}$  and has a minimizer  $x_{*}$  with  $\| x_{*}\|_{\infty}\leq 1$  computes an  $\varepsilon$ -approximate minimizer of  $f$  in  $\widetilde{O}(n^{1/3}\varepsilon^{-2/3})$  rounds and  $\widetilde{O}(n\varepsilon^{-2})$  queries.
In fact, we obtain a more general result that solves stochastic variants of parallel  $\ell_{\infty}$ -optimization (see Theorem 2.2) and Theorem 1.2 is an important corollary of this more general result.
Our parallel convex optimization algorithms build on machinery developed for highly-parallel algorithms for minimizing convex functions that are  $\ell_2$ -Lipschitz. These methods consider a convolution of  $f$  with a centered Gaussian with covariance  $\rho^2\mathbf{I}_n$  (also referred to as Gaussian smoothing), and then apply optimization methods [CJJ+23, ACJ+21, BJL+19] to this smooth function. By leveraging
the properties of this smoothing and the convergence rate of the associated optimization methods, they obtain their parallel complexities. Since functions which are  $\ell_{\infty}$ -Lipschitz are also  $\ell_{2}$ -Lipschitz, the above algorithms also apply to  $\ell_{\infty}$ -Lipschitz functions but they give us a suboptimal dependence of  $n^{2/3}$  on the dimension. We improve the dependence on dimension by utilizing the  $\ell_{\infty}$ -Lipschitzness of our function to add more Gaussian smoothing. This allows us to obtain a  $n^{1/3}$  dependence on the dimension, which is optimal up to polylogarithmic factors for constant  $\varepsilon$  [Nem94, DG19]. The key observation is that convolving an  $\ell_{\infty}$ -Lipschitz function with a Gaussian of covariance  $\rho^2\mathbf{I}_n$  changes the function value by no more than  $O(\rho \sqrt{\log n})$  (see Lemma 2.6), whereas for  $\ell_{2}$ -Lipschitz functions it could change the function value by  $O(\rho \sqrt{n})$ .
Submodular Function Minimization. In SFM, we assume the submodular function  $f: 2^{[n]} \to \mathbb{Z}$  is given by an evaluation oracle, which when queried with  $S \subseteq [n]$ , returns the value of  $f(S)$ . Throughout the paper, we assume that  $f(S) \in [-M, M]$  for some  $M \in \mathbb{Z}_{>0}$ , and  $f(\emptyset) = 0$ . The assumption that  $f(\emptyset) = 0$  can be made without loss of generality by instead minimizing  $\tilde{f}(S) := f(S) - f(\emptyset)$ ; this transformation can be implemented with one query and moves the range of  $f$  by at most  $\pm M$ , turning any dependence on  $M$  in an algorithms' complexity to  $2M$ .
Note that this submodular function minimization setup is different from the setup of parallel convex optimization, as  $f$  only defined on the vertices of the unit hypercube. Nonetheless, it is known that there is a convex function  $f_{\mathrm{Lov}}$  defined on  $[0,1]^n$ , known as the Lovász Extension, such that optimizing  $f_{\mathrm{Lov}}$  suffices for optimizing  $f$ . Additionally, it is known how to compute a subgradient of  $f_{\mathrm{Lov}}$  at any point  $x$  in 1 round using at most  $n$  evaluation queries to  $f$  (as highlighted in Fact 2.4).
Now, we are ready to present our first result on parallel SFM. Later, in Section 2.1, we provide a more general version of this theorem, Theorem 2.5, which gives improved parallel complexities for approximately minimizing bounded, real-valued submodular functions.
Theorem 1.3 (Sublinear Parallel SFM). There is an algorithm that, when given an evaluation oracle for submodular  $f:2^{[n]} \to \mathbb{Z}$  with  $f(\emptyset) = 0$  and  $|f(S)| \leq M$  for all  $S \subseteq [n]$ , finds a minimizer of  $f$  in  $\widetilde{O}(n^{1/3}M^{2/3})$  rounds and  $\widetilde{O}(n^2M^2)$  queries.
As discussed, Theorem 1.3, is obtained by using and enhancing tools for optimizing Lipschitz convex functions with a subgradient oracle. We in fact prove a more general result, namely that  $\widetilde{O}(n^{1/3}/\varepsilon^{2/3})$  rounds and  $\widetilde{O}(n^2/\varepsilon^2)$  queries are sufficient to find an  $\varepsilon M$ -approximate minimizer (Theorem 2.5). Since the function is integer valued, approximating the scaled Lovász extension to  $\varepsilon \approx \Theta(1/M)$  gives the exact minimizer to the submodular function. Our proof of Theorem 2.5 follows from our new result on  $\ell_{\infty}$ -convex optimization (Theorem 1.2). By applying it to a scaled version of the Lovász extension; it is known that if  $f$  is a submodular function with range  $\mathbb{Z} \cap [-M, +M]$ , then the Lovász extension scaled by  $O(1/M)$  is a convex function which is  $\ell_{\infty}$ -Lipschitz. However, it is important to note that SFM is only equivalent to constrained minimization of the Lovász extension in  $[0,1]^n$ , while Theorem 1.2 below is unconstrained (e.g. applies for minimizing over  $\mathbb{R}^n$ ). To apply Theorem 1.2 in the context of SFM, we give a general reduction from constrained to unconstrained optimization by adding a regularizer that restricts the minimizer of the regularized function to the constrained set (see Lemma 2.3). This reduction is a generic technique and might be of independent utility.
Parallel SFM in Two Rounds. Our second SFM result is a simple combinatorial 2-round algorithm which is efficient for functions of constant range.
Theorem 1.4 (Two-round Parallel SFM). There is an algorithm (Algorithm 1) that when given an evaluation oracle for submodular  $f:2^{[n]} \to \mathbb{Z}$  with  $f(\emptyset) = 0$  and  $|f(S)| \leq M$  for all  $S \subseteq [n]$  finds a minimizer of  $f$  in 2 rounds and  $O(n^{M + 1})$  queries.
The algorithm proving Theorem 1.4 relies on two key observations. First, if  $S_*$  is the minimizer of maximum size, for every subset  $T \subseteq S_*$  and  $i \in [n]$  with  $f(T \cup \{i\}) \leq f(T)$ , we have  $i \in S_*$ . In other words, every element with a non-positive marginal at a subset  $T \subseteq S^*$  is also contained in  $S_*$ . This leads to the idea of augmenting a set  $T$  by the set  $T' = T \cup \{i : f(T \cup \{i\}) \leq f(T)\}$ . Secondly, every subgradient  $g \in \partial f(x)$  has at most  $M$  entries that are strictly positive. This ensures that there exists an  $M$ -sparse subset  $T \subseteq S_*$  with the property that  $f(T \cup \{i\}) \leq f(T)$  for all  $i \in S_* \setminus T$ . Consequently, our algorithm proceeds by augmenting all  $M$ -sparse sets, as it is guaranteed that one of these augmented sets is the maximal minimizer (see Section 3 for more details).
Parallel Convex Optimization. As mentioned in Section 1.1, there are a number of works studying the parallel complexity of  $\ell_p$ -Lipschitz convex optimization. Perhaps, the most well-studied case is that of  $p = 2$ . In this case, the classic subgradient descent algorithm achieves a parallel complexity of  $O(\varepsilon^{-2})$ -rounds and the standard cutting-plane methods achieve a parallel complexity of  $O(n\log(1/\varepsilon))$ -rounds. [DBW12, BJL+19, CJJ+23] improved upon this rate, achieving parallel complexities of  $\widetilde{O}(n^{1/4}\varepsilon^{-1})$  [DBW12] and  $\widetilde{O}(n^{1/3}\epsilon^{-2/3})$  [BJL+19, CJJ+23] respectively. The implications of these results for the  $p = \infty$  case, which is the object of study in our paper, were discussed earlier and we are unaware of works on alternative upper bounds for  $p = \infty$ .
In terms of lower bounds, the  $p = \infty$  case was studied in the prescient paper of [Nem94] which obtains a  $\widetilde{\Omega}(n^{1/3}\ln(1/\varepsilon))$  lower bound for minimizing  $\ell_{\infty}$ -Lipschitz functions over the  $\ell_{\infty}$ -ball (see also [DG19]). When  $\varepsilon$  is a constant, our upper bound matches this lower bound, though our dependence on  $\varepsilon$  is polynomial instead of logarithmic. For the  $p = 2$  case, [Nem94, BS18] proved a tight lower bound of  $\Omega(1/\varepsilon^2)$  on the parallel complexity when  $1/\varepsilon^2 \leq \widetilde{O}(n^{1/3})$ , i.e., the parallel complexity of subgradient descent is optimal up to  $\widetilde{O}(n^{1/3})$  rounds of queries. This was later improved by [BJL+19], which showed that subgradient descent is optimal up to  $\widetilde{O}(n^{1/2})$  rounds. [DG19] considered the general  $p$  case (and other non-Euclidean settings) and proved a lower bound of  $\Omega(\varepsilon^{-p})$  lower bound on the parallel complexity for  $2 \leq p < \infty$ ,  $\Omega(\varepsilon^{-2})$  for  $1 < p \leq 2$ , and  $\Omega(\varepsilon^{-2/3})$  for  $p = 1$ . This paper also has lower bounds for smooth convex functions.
Submodular Function Minimization. We now expand on the history of SFM upper and lower bounds for parallel and sequential algorithms touched upon earlier. Since the seminal work of Edmonds in 1970 [Edm70], there has been extensive work [GLS81, GLS88, Sch00, IFF01, Iwa03, FI00, Vyg03, Orl09, IO09, LSW15, CLSW17, ALS20, DVZ21, Jia21, Jia22] on developing query-efficient algorithms for SFM. [GLS81, GLS88] gave the first polynomial time algorithms using the ellipsoid method. The state-of-the-art SFM algorithms include a  $\widetilde{O}(n^2)$ -query exponential time algorithm due to [Jia22],  $\widetilde{O}(n^3)$ -query polynomial time algorithms due to [Jia21, JLSZ23]; a  $\widetilde{O}(n^2\log M)$ -query polynomial time algorithm due to [LSW15], and a  $\widetilde{O}(nM^2)$ -query polynomial time algorithm due to [ALS20]. Despite these algorithmic improvements, limited progress has been made on lower bounding the query complexity of SFM and the best known lower bound has been  $\Omega(n)$  for decades [Har08, CLSW17, GPRW20]. Very recently, [CGJS22] proved an  $\Omega(n\log n)$ -lower bound for deterministic SFM algorithms.
All the algorithms above are highly sequential and proceed in at least  $n$  rounds. The question of parallel complexity for SFM was first studied in [BS20] where an  $\Omega (\log n / \log \log n)$ -lower bound on the number of rounds required by any query-efficient SFM algorithm was given. The range  $M$  in their construction is  $M = n^{\Theta (n)}$ . Subsequently, [CCK21] proved a  $\widetilde{\Omega} (n^{1 / 3})$  lower bound on the round-complexity and the range is  $M = n$  for their functions. Recently, [CGJS22] described a  $\widetilde{\Omega} (n)$  lower bound with functions of range  $M = n^{\Theta (n)}$ .
Cutting plane methods. Cutting plane methods are a class of optimization methods that minimize a convex function by iteratively refining a convex set containing the minimizer. Since the center of gravity method was developed independently in [Lev65, New65], there have been many developments of faster cutting plane methods over the last six decades [Sho77, YN76, Kha80, KTE88, NN89, Vai89, BV04, LSW15], with the state-of-the-art due to [JLSW20].
2 Minimizing  $\ell_{\infty}$ -Lipschitz Functions and Submodular Functions
In this section we provide a new, improved parallel algorithm for convex optimization in  $\ell_{\infty}$  and show how to use these algorithms to obtain an improved parallel algorithm for SFM. In much of this section, we consider the following optimization problem which we term stochastic convex optimization in  $\ell_{\infty}$ . As we discussed, this problem generalizes parallel convex optimization in  $\ell_{\infty}$ . The problem is more general in terms of the norms it considers and how it allows for stochastic gradients; we consider it as it could be useful more broadly and as it perhaps more tightly captures the performance of our optimization algorithm.