Original: Near-Optimal Multi-Agent Learning for Safe Coverage Control
Revised: Near-Optimal Multi-Agent Learning for Safe Coverage Control

Original: Abstract
Revised: Abstract

Original: In multi-agent coverage control problems, agents navigate their environment to reach locations that maximize the coverage of some density.
Revised: In multi-agent coverage control problems, agents navigate their environment to reach locations that maximize the coverage of some density.

Original: In practice, the density is rarely known a priori , further complicating the original NP-hard problem.
Revised: In practice, the density is rarely known a priori , further complicating the original NP-hard problem.

Original: Moreover, in many applications, agents cannot visit arbitrary locations due to a priori unknown safety constraints.
Revised: Moreover, in many applications, agents cannot visit arbitrary locations due to a priori unknown safety constraints.

Original: In this paper, we aim to efficiently learn the density to approximately solve the coverage problem while preserving the agents’ safety.
Revised: In this paper, we aim to efficiently learn the density to approximately solve the coverage problem while preserving the agents’ safety.

Original: We first propose a conditionally linear submodular coverage function that facilitates theoretical analysis.
Revised: We first propose a conditionally linear submodular coverage function that facilitates theoretical analysis.

Original: Utilizing this structure, we develop M A CO PT , a novel algorithm that efficiently trades off the exploration-exploitation dilemma due to partial observability, and show that it achieves sublinear regret.
Revised: Utilizing this structure, we develop M A CO PT , a novel algorithm that efficiently trades off the exploration-exploitation dilemma due to partial observability, and show that it achieves sublinear regret.

Original: Next, we extend results on single-agent safe exploration to our multi-agent setting and propose S AFE M A C for safe coverage and exploration.
Revised: Next, we extend results on single-agent safe exploration to our multi-agent setting and propose S AFE M A C for safe coverage and exploration.

Original: We analyze S AFE M A C and give first of its kind results: near optimal coverage in finite time while provably guaranteeing safety.
Revised: We analyze S AFE M A C and give first of its kind results: near optimal coverage in finite time while provably guaranteeing safety.

Original: We extensively evaluate our algorithms on synthetic and real problems, including a biodiversity monitoring task under safety constraints, where
Revised: We extensively evaluate our algorithms on synthetic and real problems, including a biodiversity monitoring task under safety constraints, where

Original: S AFE M A C outperforms competing methods.
Revised: S AFE M A C outperforms competing methods.

Original:  Introduction
Revised:  Introduction

Original: In multi-agent coverage control ( M A C ) problems, multiple agents coordinate to maximize coverage over some spatially distributed events.
Revised: In multi-agent coverage control ( M A C ) problems, multiple agents coordinate to maximize coverage over some spatially distributed events.

Original: Their applications abound, from collaborative mapping [1], environmental monitoring [2], inspection robotics [3] to sensor networks [4].
Revised: Their applications abound, from collaborative mapping [1], environmental monitoring [2], inspection robotics [3] to sensor networks [4].

Original: In addition, the coverage formulation can address core challenges in cooperative multi-agent RL [5, 6], e.g., exploration [7], by providing high-level goals.
Revised: In addition, the coverage formulation can address core challenges in cooperative multi-agent RL [5, 6], e.g., exploration [7], by providing high-level goals.

Original: In these applications, agents often encounter safety constraints that may lead to critical accidents when ignored, e g., obstacles [8] or extreme weather conditions [9, 10].
Revised: In these applications, agents often encounter safety constraints that may lead to critical accidents when ignored, e g., obstacles [8] or extreme weather conditions [9, 10].

Original: Deploying coverage control solutions in the real world presents many challenges: ( i ) for a given density of relevant events, this is an NP hard problem [11]; ( ii ) such density is rarely known in practice [2] and must be learned from data, which presents a complex active learning problem as the quantity we measure (the density) differs from the one we want to optimize (its coverage); ( iii ) agents often operate under safety-critical conditions, [8–10], that may be unknown a priori .
Revised: Deploying coverage control solutions in the real world presents many challenges: ( i ) for a given density of relevant events, this is an NP hard problem [11]; ( ii ) such density is rarely known in practice [2] and must be learned from data, which presents a complex active learning problem as the quantity we measure (the density) differs from the one we want to optimize (its coverage); ( iii ) agents often operate under safety-critical conditions, [8–10], that may be unknown a priori .

Original: This requires cautious exploration of the environment to prevent catastrophic outcomes.
Revised: This requires cautious exploration of the environment to prevent catastrophic outcomes.

Original: While prior work addresses subsets of these challenges (see Section 7), we are not aware of methods that address them jointly.
Revised: While prior work addresses subsets of these challenges (see Section 7), we are not aware of methods that address them jointly.

Original: [Figure]
Revised: [Figure]

Original: This work makes the following contributions toward efficiently solving safe coverage control with a-priori unknown objectives and constraints.
Revised: This work makes the following contributions toward efficiently solving safe coverage control with a-priori unknown objectives and constraints.

Original: Firstly , we model this multi-agent learning task as a conditionally linear coverage function.
Revised: Firstly , we model this multi-agent learning task as a conditionally linear coverage function.

Original: We use the monotonocity and the submodularity of this function to propose M A CO PT , a new algorithm for the unconstrained setting that enjoys sublinear cumulative regret and efficiently recommends a near-optimal solution.
Revised: We use the monotonocity and the submodularity of this function to propose M A CO PT , a new algorithm for the unconstrained setting that enjoys sublinear cumulative regret and efficiently recommends a near-optimal solution.

Original: Secondly , we extend G O OSE [12], an algorithm for single agent safe exploration, to the multi-agent case.
Revised: Secondly , we extend G O OSE [12], an algorithm for single agent safe exploration, to the multi-agent case.

Original: Combining our extension of G O OSE with M A CO PT , we propose S AFE M A C , a novel algorithm for safe multi-agent coverage control.
Revised: Combining our extension of G O OSE with M A CO PT , we propose S AFE M A C , a novel algorithm for safe multi-agent coverage control.

Original: We analyze it and show it attains a near-optimal solution in a finite time.
Revised: We analyze it and show it attains a near-optimal solution in a finite time.

Original: Finally , we demonstrate our algorithms on a synthetic and two real world applications: safe biodiversity monitoring and obstacle avoidance.
Revised: Finally , we demonstrate our algorithms on a synthetic and two real world applications: safe biodiversity monitoring and obstacle avoidance.

Original: We show S AFE M A C finds better solutions than algorithms that do not actively explore the feasible region and is more sample efficient than competing near-optimal safe algorithms.
Revised: We show S AFE M A C finds better solutions than algorithms that do not actively explore the feasible region and is more sample efficient than competing near-optimal safe algorithms.

Original:  Problem Statement
Revised:  Problem Statement

Original: We present the safety-constrained multi-agent coverage control problem (Fig. 1) that we aim to solve.
Revised: We present the safety-constrained multi-agent coverage control problem (Fig. 1) that we aim to solve.

Original: Coverage control .
Revised: Coverage control .

Original: Coverage control models situations where we want deploy a swarm of dynamic agents to maximize the coverage of a quantity of interest, see Fig.
Revised: Coverage control models situations where we want deploy a swarm of dynamic agents to maximize the coverage of a quantity of interest, see Fig.

Original: Formally, given a finite 1 set of possible locations V , the goal of coverage control is to maximize a function F : 2 V → R that assigns to each subset, X ⊆ V , the corresponding coverage value.
Revised: Formally, given a finite 1 set of possible locations V , the goal of coverage control is to maximize a function F : 2 V → R that assigns to each subset, X ⊆ V , the corresponding coverage value.

Original: For N agents, the resulting problem is arg max X : | X |≤ N F ( X ) .
Revised: For N agents, the resulting problem is arg max X : | X |≤ N F ( X ) .

Original: The discrete domain V can be represented by a graph, where nodes represent locations in the domain, and an edge connects node v to v ′ if the agent can go from v to v ′ .
Revised: The discrete domain V can be represented by a graph, where nodes represent locations in the domain, and an edge connects node v to v ′ if the agent can go from v to v ′ .

Original: This corresponds to a deterministic MDP where locations are states and edges represent transitions.
Revised: This corresponds to a deterministic MDP where locations are states and edges represent transitions.

Original: Sensing region .
Revised: Sensing region .

Original: Depending on the application, we may use different definitions of F .
Revised: Depending on the application, we may use different definitions of F .

Original: Here, we model cases where agent i at location x
Revised: Here, we model cases where agent i at location x

Original: i covers a limited sensing region around it, D i .
Revised: i covers a limited sensing region around it, D i .

Original: While D i can be any connected subset of V , in practice it is often a ball centered at x i .
Revised: While D i can be any connected subset of V , in practice it is often a ball centered at x i .

Original: Given a function ρ : V → R denoting the density of a quantity of interest at each v ∈ V , our coverage objective is
Revised: Given a function ρ : V → R denoting the density of a quantity of interest at each v ∈ V , our coverage objective is

Original: [Equation]
Revised: [Equation]

Original: where D i − :
Revised: where D i − :

Original: = D i \ D 1: i − 1 indicates the elements in V covered by agent
Revised: = D i \ D 1: i − 1 indicates the elements in V covered by agent

Original: i but not agents 1: i − 1 ,
Revised: i but not agents 1: i − 1 ,

Original: D 1: i − 1 = ∪ i − 1 j =1 D j and | V | denotes cardinality of the domain V .
Revised: D 1: i − 1 = ∪ i − 1 j =1 D j and | V | denotes cardinality of the domain V .

Original: Safety .
Revised: Safety .

Original: In many real-world problems, agents cannot go to arbitrary locations due to safety concerns.
Revised: In many real-world problems, agents cannot go to arbitrary locations due to safety concerns.

Original: To model this, we introduce a constraint function q : V → R and we consider safe all locations v satisfying q ( v ) ≥ 0 .
Revised: To model this, we introduce a constraint function q : V → R and we consider safe all locations v satisfying q ( v ) ≥ 0 .

Original: Such constraint restricts the space of possible solutions of our problem in two ways.
Revised: Such constraint restricts the space of possible solutions of our problem in two ways.

Original: First, it prevents agents from monitoring from unsafe locations.
Revised: First, it prevents agents from monitoring from unsafe locations.

Original: Second, depending on its dynamics, agent i may be unable to safely reach a disconnected safe area starting from x i 0 , see
Revised: Second, depending on its dynamics, agent i may be unable to safely reach a disconnected safe area starting from x i 0 , see

Original: Appendix A.3.
Revised: Appendix A.3.

Original: We denote with ¯ R ϵ q ( { x i 0 } ) the largest safely reachable region starting from x i 0 and with B a collection of batches of agents such that all agents in the same batch B share the same safely reachable set, ∀ i, j ∈ B : ¯ R ϵ q ( { x i 0 } )
Revised: We denote with ¯ R ϵ q ( { x i 0 } ) the largest safely reachable region starting from x i 0 and with B a collection of batches of agents such that all agents in the same batch B share the same safely reachable set, ∀ i, j ∈ B : ¯ R ϵ q ( { x i 0 } )

Original: ∩ ¯
Revised: ∩ ¯

Original: R ϵ q
Revised: R ϵ q

Original: ( { x j 0 } ) ̸
Revised: ( { x j 0 } ) ̸

Original: = ∅ , see Appendix A for formal definitions.
Revised: = ∅ , see Appendix A for formal definitions.

Original: Based on this, we define the safely reachable control problem
Revised: Based on this, we define the safely reachable control problem

Original: [Equation]
Revised: [Equation]

Original: where X B 

Revised: where X B

Original: = { x i 0 } i ∈ B are the starting locations of all agents in B and ¯ R ϵ q ( X B 0 ) =
Revised: = { x i 0 } i ∈ B are the starting locations of all agents in B and ¯ R ϵ q ( X B 0 ) =

Original: (cid:83)
Revised: (cid:83)

Original: i ∈
Revised: i ∈

Original: B ¯ R ϵ q ( { x i 0 } ) indicates the largest safely reachable region from any point x i 0 for all i in B (since the agents have the same dynamics, ¯ R ϵ q ( X B 0 )
Revised: B ¯ R ϵ q ( { x i 0 } ) indicates the largest safely reachable region from any point x i 0 for all i in B (since the agents have the same dynamics, ¯ R ϵ q ( X B 0 )

Original: =
Revised: =

Original: ¯ R ϵ q ( { x i 0 } ) , ∀ i ∈ B ).
Revised: ¯ R ϵ q ( { x i 0 } ) , ∀ i ∈ B ).

Original: In safety-critical monitoring, there may be unreachable safe regions.
Revised: In safety-critical monitoring, there may be unreachable safe regions.

Original: However, since agents should be able to collect measurements if required, we focus only on covering the safely reachable region.
Revised: However, since agents should be able to collect measurements if required, we focus only on covering the safely reachable region.

Original: Unknown density and constraint .
Revised: Unknown density and constraint .

Original: In practice, the density ρ and the constraint q are often unknown a priori .
Revised: In practice, the density ρ and the constraint q are often unknown a priori .

Original: However, the agents can iteratively obtain noisy measurements of their values at target locations.
Revised: However, the agents can iteratively obtain noisy measurements of their values at target locations.

Original: We consider synchronous measurements, i.e., we wait until all agents have collected the desired measurement for the current iteration before moving to the next one.
Revised: We consider synchronous measurements, i.e., we wait until all agents have collected the desired measurement for the current iteration before moving to the next one.

Original: Here, we focus on the high-level problem of choosing informative locations, rather than the design of low-level motion planning 2 .
Revised: Here, we focus on the high-level problem of choosing informative locations, rather than the design of low-level motion planning 2 .

Original: Therefore, our goal is to find an approximate solution to the problem in Eq.
Revised: Therefore, our goal is to find an approximate solution to the problem in Eq.

Original: (2) preserving safety throughout exploration, i.e., at every location visited by the agents, while taking as few measurements as possible in case the dynamics of the agents are deterministic and known as in [12].
Revised: (2) preserving safety throughout exploration, i.e., at every location visited by the agents, while taking as few measurements as possible in case the dynamics of the agents are deterministic and known as in [12].

Original:  Background
Revised:  Background

Original: This section presents foundational ideas that our method builds on.
Revised: This section presents foundational ideas that our method builds on.

Original: In particular, it discusses ( i ) monotone submodular functions and ( ii ) previous work on single-agent safe exploration.
Revised: In particular, it discusses ( i ) monotone submodular functions and ( ii ) previous work on single-agent safe exploration.

Original: Submodularity .
Revised: Submodularity .

Original: Optimizing a function defined over the power set of a finite domain, V , scales combinatorially with the size of V in general.
Revised: Optimizing a function defined over the power set of a finite domain, V , scales combinatorially with the size of V in general.

Original: In special cases, we can exploit the structure of the objective to find approximate solutions efficiently.
Revised: In special cases, we can exploit the structure of the objective to find approximate solutions efficiently.

Original: Monotone submodular functions are one example of this.
Revised: Monotone submodular functions are one example of this.

Original: A set function F : 2 V → R is monotone if for all A ⊆ B ⊂
Revised: A set function F : 2 V → R is monotone if for all A ⊆ B ⊂

Original: V
Revised: V

Original: we have F ( A ) ≤ F ( B ) .
Revised: we have F ( A ) ≤ F ( B ) .

Original: It is submodular if ∀ A ⊆ B ⊆ V, v ∈ V \ B , we have, F ( A ∪ { v } ) − F ( A ) ≥ F
Revised: It is submodular if ∀ A ⊆ B ⊆ V, v ∈ V \ B , we have, F ( A ∪ { v } ) − F ( A ) ≥ F

Original: ( B ∪ { v } ) − F ( B ) .
Revised: ( B ∪ { v } ) − F ( B ) .

Original: In coverage control, this means adding v to A yields at least as much increase in coverage than adding v to B , if A ⊆ B .
Revised: In coverage control, this means adding v to A yields at least as much increase in coverage than adding v to B , if A ⊆ B .

Original: Crucially, [13] guarantees that the greedy algorithm produces a solution within a factor of (1 − 1 /e ) of the optimal solution for problems of the type arg max X : | X |≤ N F ( X ; ρ, V ) , when F is monotone and submodular.
Revised: Crucially, [13] guarantees that the greedy algorithm produces a solution within a factor of (1 − 1 /e ) of the optimal solution for problems of the type arg max X : | X |≤ N F ( X ; ρ, V ) , when F is monotone and submodular.

Original: In practice, the greedy algorithm often outperforms this worst-case guarantee [14] and guaranteeing a solution better than (1 − 1 /e ) factor is NP hard [15].
Revised: In practice, the greedy algorithm often outperforms this worst-case guarantee [14] and guaranteeing a solution better than (1 − 1 /e ) factor is NP hard [15].

Original: The coverage function in Eq.
Revised: The coverage function in Eq.

Original: (1) is a conditionally linear, monotone and submodular function (proof in Appendix B), which lets us use the results above to design our algorithm for safe coverage control.
Revised: (1) is a conditionally linear, monotone and submodular function (proof in Appendix B), which lets us use the results above to design our algorithm for safe coverage control.

Original: Goal-oriented safe exploration .
Revised: Goal-oriented safe exploration .

Original: G O OSE [12] is a single-agent safe exploration algorithm that extends unconstrained methods to safety-critical cases.
Revised: G O OSE [12] is a single-agent safe exploration algorithm that extends unconstrained methods to safety-critical cases.

Original: Concretely, it maintains under- and overapproximations of the feasible set, called pessimistic and optimistic safe sets.
Revised: Concretely, it maintains under- and overapproximations of the feasible set, called pessimistic and optimistic safe sets.

Original: It preserves safety by restricting the agent to the pessimistic safe set.
Revised: It preserves safety by restricting the agent to the pessimistic safe set.

Original: It efficiently explores the objective by letting the original unconstrained algorithm recommend locations within the optimistic safe set.
Revised: It efficiently explores the objective by letting the original unconstrained algorithm recommend locations within the optimistic safe set.

Original: If such recommendations are provably safe, the agent evaluates the objective there.
Revised: If such recommendations are provably safe, the agent evaluates the objective there.

Original: Otherwise, it evaluates the constraint at a sequence of safe locations to prove that such recommendation is either safe, which allows it to evaluate the objective, or unsafe, which triggers the unconstrained algorithm to provide a new recommendation.
Revised: Otherwise, it evaluates the constraint at a sequence of safe locations to prove that such recommendation is either safe, which allows it to evaluate the objective, or unsafe, which triggers the unconstrained algorithm to provide a new recommendation.

Original: Assumptions .
Revised: Assumptions .

Original: To guarantee safety, G O OSE makes two main assumptions.
Revised: To guarantee safety, G O OSE makes two main assumptions.

Original: First, it assumes there is an initial set of safe locations, X 0 , from where the agent can start exploring.
Revised: First, it assumes there is an initial set of safe locations, X 0 , from where the agent can start exploring.

Original: Second, it assumes the constraint is sufficiently well-behaved, so that we can use data to infer the safety of unvisited locations.
Revised: Second, it assumes the constraint is sufficiently well-behaved, so that we can use data to infer the safety of unvisited locations.

Original: Formally, it assumes the domain V is endowed with a positive definite kernel k q ( · , · ) , and that the constraint’s norm in the associated Reproducing Kernel Hilbert Space [16] is bounded, ∥ q ∥ k q ≤ B q .
Revised: Formally, it assumes the domain V is endowed with a positive definite kernel k q ( · , · ) , and that the constraint’s norm in the associated Reproducing Kernel Hilbert Space [16] is bounded, ∥ q ∥ k q ≤ B q .

Original: This lets us use Gaussian Processes (GPs) [17]to construct high-probability confidence intervals for q .
Revised: This lets us use Gaussian Processes (GPs) [17]to construct high-probability confidence intervals for q .

Original: We specify the GP prior over q through a mean function, which we assume to be zero everywhere w.l.o.g., µ ( v ) = 0 , ∀ v ∈ V , and a kernel function, k , that captures the covariance between different locations.
Revised: We specify the GP prior over q through a mean function, which we assume to be zero everywhere w.l.o.g., µ ( v ) = 0 , ∀ v ∈ V , and a kernel function, k , that captures the covariance between different locations.

Original: If we have access to T measurements, at V T = { v t } Tt =1 perturbed by i.i.d.
Revised: If we have access to T measurements, at V T = { v t } Tt =1 perturbed by i.i.d.

Original: Gaussian noise, y T =
Revised: Gaussian noise, y T =

Original: { q ( v t )
Revised: { q ( v t )

Original: + η t } Tt =

Revised: + η t } Tt =

Original: with η t ∼ N (0 , σ 2 ) , we can compute the posterior mean and covariance over the constraint at unseen locations v , v ′ as µ T ( v ) =
Revised: with η t ∼ N (0 , σ 2 ) , we can compute the posterior mean and covariance over the constraint at unseen locations v , v ′ as µ T ( v ) =

Original: k ⊤ T ( v )( K T + σ 2 I ) − 1 y T and k t ( v, v ′ )
Revised: k ⊤ T ( v )( K T + σ 2 I ) − 1 y T and k t ( v, v ′ )

Original: =
Revised: =

Original: k ( v, v ′ )
Revised: k ( v, v ′ )

Original: − k ⊤ T ( v )( K T + σ 2 I ) − 1 k T ( v ′ ) , where k T ( v ) =
Revised: − k ⊤ T ( v )( K T + σ 2 I ) − 1 k T ( v ′ ) , where k T ( v ) =

Original: [ k ( v 1 , v ) , ..., k ( v T , v )]
Revised: [ k ( v 1 , v ) , ..., k ( v T , v )]

Original: ⊤ , K T is the positive definite kernel matrix [ k ( v, v ′ )] v,v ′ ∈ V T and I ∈ R T × T denotes the identity matrix.
Revised: ⊤ , K T is the positive definite kernel matrix [ k ( v, v ′ )] v,v ′ ∈ V T and I ∈ R T × T denotes the identity matrix.

Original: In this work, we make the same assumptions about the safe seed and the regularity of q and ρ .
Revised: In this work, we make the same assumptions about the safe seed and the regularity of q and ρ .

Original: Approximations of the feasible set .
Revised: Approximations of the feasible set .

Original: Based on the GP posterior above, G O OSE builds monotonic confidence intervals for the constraint at each iteration t as l qt ( v ) :
Revised: Based on the GP posterior above, G O OSE builds monotonic confidence intervals for the constraint at each iteration t as l qt ( v ) :

Original: = max { l qt − 1 ( v ) , µ qt − 1 ( v ) − β qt σ qt − 1 ( v ) } and u qt ( v ) :
Revised: = max { l qt − 1 ( v ) , µ qt − 1 ( v ) − β qt σ qt − 1 ( v ) } and u qt ( v ) :

Original: = min { u qt − 1 ( v ) , µ qt − 1 ( v ) + β qt σ qt − 1 ( v ) } , which contain the true constraint function for every v ∈ V and t ≥ 1 , with high probability if β qt is selected as in [18] or Section 5.
Revised: = min { u qt − 1 ( v ) , µ qt − 1 ( v ) + β qt σ qt − 1 ( v ) } , which contain the true constraint function for every v ∈ V and t ≥ 1 , with high probability if β qt is selected as in [18] or Section 5.

Original: G O OSE uses these confidence intervals within a set S ⊆ V together with the L q -Lipschitz continuity of q , to define operators that determine which locations are safe in plausible worst- and best-case scenarios, p
Revised: G O OSE uses these confidence intervals within a set S ⊆ V together with the L q -Lipschitz continuity of q , to define operators that determine which locations are safe in plausible worst- and best-case scenarios, p

Original: [Equation]
Revised: [Equation]

Original: −
Revised: −

Original: [Equation]
Revised: [Equation]

Original: Notice that the pessimistic operator relies on the lower bound, l q , while the optimistic one on the upper bound, u q .
Revised: Notice that the pessimistic operator relies on the lower bound, l q , while the optimistic one on the upper bound, u q .

Original: Moreover, the optimistic one uses a margin ϵ q to exclude "barely" safe locations as the agent might get stuck learning about them.
Revised: Moreover, the optimistic one uses a margin ϵ q to exclude "barely" safe locations as the agent might get stuck learning about them.

Original: Finally, to disregard locations the agent could not safely reach or from where it could not safely return, G O OSE introduces the R ergodic ( · , · ) operator.
Revised: Finally, to disregard locations the agent could not safely reach or from where it could not safely return, G O OSE introduces the R ergodic ( · , · ) operator.

Original: R ergodic ( p t ( S ) , S ) indicates locations in S or locations in p t ( S ) reachable from S and from where the agent can return to S along a path contained in p t ( S ) .
Revised: R ergodic ( p t ( S ) , S ) indicates locations in S or locations in p t ( S ) reachable from S and from where the agent can return to S along a path contained in p t ( S ) .

Original: Combining p t ( S ) and R ergodic ( · , · ) , G O OSE defines the pessimistic and ergodic operator ˜ P t ( · ) , which it uses to update the pessimistic safe set.
Revised: Combining p t ( S ) and R ergodic ( · , · ) , G O OSE defines the pessimistic and ergodic operator ˜ P t ( · ) , which it uses to update the pessimistic safe set.

Original: Similarly, it defines ˜ O t ( · ) using o ϵ q t ( · ) to compute the optimistic safe set.
Revised: Similarly, it defines ˜ O t ( · ) using o ϵ q t ( · ) to compute the optimistic safe set.

Original:  M A CO PT and S AFE M A C
Revised:  M A CO PT and S AFE M A C

Original: This section presents M A CO PT and S AFE M A C , our algorithms for unconstrained and safety-constrained multi-agent coverage control, which we then formally analyze in Section 5.
Revised: This section presents M A CO PT and S AFE M A C , our algorithms for unconstrained and safety-constrained multi-agent coverage control, which we then formally analyze in Section 5.

Original: 4. M A CO PT : unconstrained multi-agent coverage control
Revised: 4. M A CO PT : unconstrained multi-agent coverage control

Original: Greedy sensing regions .
Revised: Greedy sensing regions .

Original: In sequential optimization, it is crucial to balance exploration and exploitation.
Revised: In sequential optimization, it is crucial to balance exploration and exploitation.

Original: GP-UCB [19] is a theoretically sound strategy to strike such a trade-off that works well in practice.
Revised: GP-UCB [19] is a theoretically sound strategy to strike such a trade-off that works well in practice.

Original: Agents evaluate the objective at locations that maximize an upper confidence bound over the objective given by the GP model such that locations with either a high posterior mean (exploitation) or standard deviation (exploration) are visited.
Revised: Agents evaluate the objective at locations that maximize an upper confidence bound over the objective given by the GP model such that locations with either a high posterior mean (exploitation) or standard deviation (exploration) are visited.

Original: We construct a valid upper confidence bound for the coverage F ( X ) starting from our confidence intervals on ρ , by replacing the true density ρ with its upper bound u ρ t in Eq.
Revised: We construct a valid upper confidence bound for the coverage F ( X ) starting from our confidence intervals on ρ , by replacing the true density ρ with its upper bound u ρ t in Eq.

Original: Next, we apply the greedy algorithm to this upper bound (Line 3 of Algorithm 1) to select N candidate locations for evaluating the density.
Revised: Next, we apply the greedy algorithm to this upper bound (Line 3 of Algorithm 1) to select N candidate locations for evaluating the density.

Original: However, this simple exploration strategy may perform poorly, due to the fact that in order to reduce the uncertainty over the coverage F at X , we must learn the density ρ at all locations inside the sensing region, (cid:83) Ni
Revised: However, this simple exploration strategy may perform poorly, due to the fact that in order to reduce the uncertainty over the coverage F at X , we must learn the density ρ at all locations inside the sensing region, (cid:83) Ni

Original: =
Revised: =

Original: 1 D i , rather than simply at X .
Revised: 1 D i , rather than simply at X .

Original: It is a form of partial monitoring [20], where the objective F differs from the quantity we measure, i.e., the density ρ .
Revised: It is a form of partial monitoring [20], where the objective F differs from the quantity we measure, i.e., the density ρ .

Original: Next, we explain how to choose locations where to observe the density for a given X .
Revised: Next, we explain how to choose locations where to observe the density for a given X .

Original: Uncertainty sampling .
Revised: Uncertainty sampling .

Original: Given location assignments X for the agents, we measure the density to efficiently learn the function F ( X ) .
Revised: Given location assignments X for the agents, we measure the density to efficiently learn the function F ( X ) .

Original: Intuitively, agent i observes the density where it’s most uncertain within the area it covers that is not covered by agents { 1 , . . .
Revised: Intuitively, agent i observes the density where it’s most uncertain within the area it covers that is not covered by agents { 1 , . . .

Original: i − 1 } , i.e., D i − t
Revised: i − 1 } , i.e., D i − t

Original: (Line 4 of Alg. 2, Fig. 2a).
Revised: (Line 4 of Alg. 2, Fig. 2a).

Original: Stopping criterion .
Revised: Stopping criterion .

Original: The algorithm terminates when a near-optimal solution is achieved.
Revised: The algorithm terminates when a near-optimal solution is achieved.

Original: Intuitively, this occurs when the uncertainty about the coverage value of the greedy recommendation is low.
Revised: Intuitively, this occurs when the uncertainty about the coverage value of the greedy recommendation is low.

Original: Formally, we require the sum of the uncertainties over the sampling targets to be below a threshold, i.e. , w t =
Revised: Formally, we require the sum of the uncertainties over the sampling targets to be below a threshold, i.e. , w t =

Original: [Equation]
Revised: [Equation]

Original: ( x g,i
Revised: ( x g,i

Original: [Equation]
Revised: [Equation]

Original: ) −
Revised: ) −

Original: [Equation]
Revised: [Equation]

Original: ( x g,i
Revised: ( x g,i

Original: [Equation]
Revised: [Equation]

Original: ) ≤ ϵ ρ (Line 3 of Algorithm 2).
Revised: ) ≤ ϵ ρ (Line 3 of Algorithm 2).

Original: Importantly, this stopping criterion requires the confidence intervals to shrink only at regions that potentially maximize the coverage.
Revised: Importantly, this stopping criterion requires the confidence intervals to shrink only at regions that potentially maximize the coverage.

Original: M A CO PT .
Revised: M A CO PT .

Original: Now, we introduce M A CO PT in Algorithm 2.
Revised: Now, we introduce M A CO PT in Algorithm 2.

Original: At round t , we select the sensing locations for the agents, X t , by greedily optimizing the upper confidence bound of the coverage.
Revised: At round t , we select the sensing locations for the agents, X t , by greedily optimizing the upper confidence bound of the coverage.

Original: Then, each flaticon.com x g, 1 t x g, 2 t x g, 3 t flaticon.com (a) Uncertainty sampling in M A CO PT
Revised: Then, each flaticon.com x g, 1 t x g, 2 t x g, 3 t [Figure] a [Figure]

Revised: in

Revised: [Figure]

Original: S o,ϵt
Revised: S o,ϵt

Original: S p,B 1 t
Revised: S p,B 1 t

Original: S p,B 2 t x 1 t x 2 t x 3 t x g, 1 t x g, 2 t x g, 3 t (b) Illustration of multi-agent G O OSE agent i collects noisy density measurements at the points of highest uncertainty within D i − t .
Revised: S p,B 2 t x 1 t x 2 t x 3 t x g, 1 t x g, 2 t x g, 3 t ( [Figure] ) Illustration of multi-agent G O OSE agent i collects noisy density measurements at the points of highest uncertainty within D i − t .

Original: Finally, we update our GP over the density and, if the sum of maximum uncertainties within each sensing region is small, we stop the algorithm.
Revised: Finally, we update our GP over the density and, if the sum of maximum uncertainties within each sensing region is small, we stop the algorithm.

Original: 4. S AFE M A C : safety-constrained multi-agent coverage control
Revised: 4. S AFE M A C : safety-constrained multi-agent coverage control

Original: Intuition .
Revised: Intuition .

Original: We adopt a perspective similar to G O OSE as we separate the exploration of the safe set from the maximization of the coverage.
Revised: We adopt a perspective similar to G O OSE as we separate the exploration of the safe set from the maximization of the coverage.

Original: Given an over and under approximation of the safe set (whose computation is discussed later) we want to explore optimistically optimal goals for each agent, similar to M A CO PT .
Revised: Given an over and under approximation of the safe set (whose computation is discussed later) we want to explore optimistically optimal goals for each agent, similar to M A CO PT .

Original: To this end, we find the maximizers of the density upper bound in the optimistic safe set with the G REEDY algorithm.
Revised: To this end we find the maximizers of the density upper bound in the optimistic safe set with the G REEDY algorithm.

Original: Then, we define sampling goals to learn the coverage at those locations.
Revised: Then we define sampling goals to learn the coverage at those locations.

Original: Phases of S AFE M A C .
Revised: Phases of S AFE M A C .

Original: Coverage values depend both on the density and the feasible region (Eq.
Revised: Coverage values depend both on the density and the feasible region (Eq.

Original: Thus, there are two sensible sampling goals given a disk assignment: i) optimistic coverage : if we are uncertain about the density within the disks, we target locations with the highest density uncertainty (Line 6 of Algorithm 4); ii) optimistic exploration : if we know the density within the disk but there are locations under it that we cannot classify as either safe (in S p ) or unsafe (in V \ S o,ϵ q ), we target those with the highest constraint uncertainty among them (Line 8).
Revised: Thus, there are two sensible sampling goals given a disk assignment: i) optimistic coverage : if we are uncertain about the density within the disks, we target locations with the highest density uncertainty (Line 6 of Algorithm 4); ii) optimistic exploration : if we know the density within the disk but there are locations under it that we cannot classify as either safe (in S p ) or unsafe (in V \ S o,ϵ q ), we target those with the highest constraint uncertainty among them (Line 8).

Original: If all the goal locations are safe with high probability, which can only happen during optimistic coverage , we safely evaluate the density there (Line 19).
Revised: If all the goal locations are safe with high probability, which can only happen during optimistic coverage , we safely evaluate the density there (Line 19).

Original: Otherwise, we explore the constraint with a goal directed strategy that aims at classifying them as either safe or unsafe similar to G O OSE (Line 9-12).
Revised: Otherwise, we explore the constraint with a goal directed strategy that aims at classifying them as either safe or unsafe similar to G O OSE (Line 9-12).

Original: In case this changes the topological connection of the optimistic feasible set, we recompute the disks as this may change G REEDY ’s output (Line 15-17).
Revised: In case this changes the topological connection of the optimistic feasible set, we recompute the disks as this may change G REEDY ’s output (Line 15-17).

Original: We repeat this loop until we know the feasibility of all the points under the disks recommended by G REEDY and their density uncertainty is low (Line 4).
Revised: We repeat this loop until we know the feasibility of all the points under the disks recommended by G REEDY and their density uncertainty is low (Line 4).

Original: Next, we explain how the multiple agents coordinate their individual safe regions to evaluate a goal ( M A CO PT in batches ), how the agents progress toward their goals ( safe expansion ) and finally we describe S AFE M A C convergence .
Revised: Next, we explain how the multiple agents coordinate their individual safe regions to evaluate a goal ( M A CO PT in batches ), how the agents progress toward their goals ( safe expansion ) and finally we describe S AFE M A C convergence .

Original: M A CO PT in batches .
Revised: M A CO PT in batches .

Original: In the multi-agent setting of G O OSE (see Fig. 2b), each agent i maintains S p,it a pessimistic (or S o,ϵ q ,i t an optimistic) belief of the safe locations, obtained by iteratively applying ˜
Revised: In the multi-agent setting of G O OSE (see Fig. 2b), each agent i maintains S p,it a pessimistic (or S o,ϵ q ,i t an optimistic) belief of the safe locations, obtained by iteratively applying ˜

Original: P t ( · ) the pessimistic ( or ˜ O t ( · ) the optimistic) ergodic operators (see Section 3) to the previous pessimistic belief S p,it − 1 (Line 11 of Algorithm 4).
Revised: P t ( · ) the pessimistic ( or ˜ O t ( · ) the optimistic) ergodic operators (see Section 3) to the previous pessimistic belief S p,it − 1 (Line 11 of Algorithm 4).

Original: Since the agents cannot navigate to an arbitrary location in the constrained case, S AFE M A C computes coverage maximizers on a restricted region, obtained by ignoring the known unsafe locations.
Revised: Since the agents cannot navigate to an arbitrary location in the constrained case, S AFE M A C computes coverage maximizers on a restricted region, obtained by ignoring the known unsafe locations.

Original: To denote such a restricted region, we define a union set S u,it :=
Revised: To denote such a restricted region, we define a union set S u,it :=

Original: S o,ϵ q ,i t ∪ S p,i t , which is the largest set known to be optimistically or pessimistically safe up to time t .
Revised: S o,ϵ q ,i t ∪ S p,i t , which is the largest set known to be optimistically or pessimistically safe up to time t .

Original: Moreover, if the agents are topologically disconnected, they cannot travel from one safe region to another and the best strategy for any batch of agents is to maximize coverage locally.
Revised: Moreover, if the agents are topologically disconnected, they cannot travel from one safe region to another and the best strategy for any batch of agents is to maximize coverage locally.

Original: For this, we form a collection of batches B t , such that any batch B ∈ B t contains agents that lie in topologically connected regions determined by the union set (Line 13-14 ).
Revised: For this, we form a collection of batches B t , such that any batch B ∈ B t contains agents that lie in topologically connected regions determined by the union set (Line 13-14 ).

Original: S AFE M A C computes a G REEDY solution for each
Revised: S AFE M A C computes a G REEDY solution for each

Original: B ∈ B t in their corresponding S u,Bt := ∪ i ∈ B S u,it .
Revised: B ∈ B t in their corresponding S u,Bt := ∪ i ∈ B S u,it .

Original: This is the largest set where the agents can find an
Revised: This is the largest set where the agents can find an

Original: 1: Inputs u ρt − 1 , l ρt − 1 , B, S ut 2: for i = 1 , 2 , ..., | B | do 3: x it ← arg max x i (cid:80) v ∈ D i \ D 1: i − 1 t ∩ S ut u ρt − 1 ( v ) 4: x g,it ← arg max v ∈ D i \ D 1: i − 1 t ∩ S ut u ρt − 1 ( v ) − l ρt − 1 ( v ) 5: w t ← (cid:80) | B | i =1 u ρt − 1 ( x g,it ) − l ρt − 1 ( x g,it ) 6: Return X Bt , w t
Revised: 1: Inputs u ρt − 1 , l ρt − 1 , B, S ut 2: for i = 1 , 2 , ..., | B | do 3: x it ← arg max x i (cid:80) v ∈ D i \ D 1: i − 1 t ∩ S ut u ρt − 1 ( v ) 4: x g,it ← arg max v ∈ D i \ D 1: i − 1 t ∩ S ut u ρt − 1 ( v ) − l ρt − 1 ( v ) 5: w t ← (cid:80) | B | i =1 u ρt − 1 ( x g,it ) − l ρt − 1 ( x g,it ) 6: Return X Bt , w t

Original: 1: Inputs X 0 , ϵ ρ , V , GP ρ , t ←2: X 1 , w 1 ← G REEDY ( u ρ 0 , l ρ 0 , [ N ] , V ) 3: while w t > ϵ ρ do 4: ∀
Revised: 1: Inputs X 0 , ϵ ρ , V , GP ρ , t ←2: X 1 , w 1 ← G REEDY ( u ρ 0 , l ρ 0 , [ N ] , V ) 3: while w t > ϵ ρ do 4: ∀

Original: i, x g,it ← arg max v ∈ D i − t u ρt − 1 ( v ) − l ρt − 1 ( v ) 5: ∀ i , y iρ t
Revised: i, x g,it ← arg max v ∈ D i − t u ρt − 1 ( v ) − l ρt − 1 ( v ) 5: ∀ i , y iρ t

Original: = ρ ( x g,it ) + η ρ , Update GP 6: t ← t +7:
Revised: = ρ ( x g,it ) + η ρ , Update GP 6: t ← t +7:

Original: X t ,w t ← G REEDY ( u ρt − 1 , l ρt − 1 , [ N ] ,V ) 8:
Revised: X t ,w t ← G REEDY ( u ρt − 1 , l ρt − 1 , [ N ] ,V ) 8:

Original: Recommend X t
Revised: Recommend X t

Original: 1: Inputs S o,ϵ q t , S pt , x gt 2: A t ( p ) ←{ v ∈ S o,ϵ q t \ p t ( S pt ) | h ( v ) = p } 3: W ϵ q t ← { v ∈ S pt | u qt ( v ) − l qt ( v ) >
Revised: 1: Inputs S o,ϵ q t , S pt , x gt 2: A t ( p ) ←{ v ∈ S o,ϵ q t \ p t ( S pt ) | h ( v ) = p } 3: W ϵ q t ← { v ∈ S pt | u qt ( v ) − l qt ( v ) >

Original: ϵ q } 4: α ⋆ ← max α s.t.
Revised: ϵ q } 4: α ⋆ ← max α s.t.

Original: | G ϵ q t ( α ) | >5: if Optimization problem feasible then 6: v t ← arg max v ∈ G ϵqt ( α ⋆ ) u qt ( v ) − l qt ( v ) 7:
Revised: | G ϵ q t ( α ) | >5: if Optimization problem feasible then 6: v t ← arg max v ∈ G ϵqt ( α ⋆ ) u qt ( v ) − l qt ( v ) 7:

Original: Update GP with y t
Revised: Update GP with y t

Original: = q ( v t )
Revised: = q ( v t )

Original: + η q
Revised: + η q

Original: 1: Inputs X 0 , L q , ϵ ρ , V , GP ρ , GP q 2: ∀ i , S p,i 

Revised: 1: Inputs X 0 , L q , ϵ ρ , V , GP ρ , GP q 2: ∀ i , S p,i

Original: ←− X 0 , S o,ϵ q ,i←− V , t ←3:
Revised: ←− X 0 , S o,ϵ q ,i←− V , t ←3:

Original: X 1 , w 1 ← G REEDY ( u ρ 0 , l ρ 0 , [ N ] , V ) 4: while ∀ i, ( S o,ϵ q ,i t − 1 \ S p,it − 1 )
Revised: X 1 , w 1 ← G REEDY ( u ρ 0 , l ρ 0 , [ N ] , V ) 4: while ∀ i, ( S o,ϵ q ,i t − 1 \ S p,it − 1 )

Original: ∩
Revised: ∩

Original: D it ̸
Revised: D it ̸

Original: = ∅ or w t > ϵ ρ do 5: if w t > ϵ ρ then 6: ∀ i, x g,it ← arg max v ∈ D i − t u ρt − 1 ( v ) − l ρt − 1 ( v ) 7: else 8: ∀ i, x g,it ← arg max v ∈ ( S o,ϵq,it − 1 \ S p,it − 1 )
Revised: = ∅ or w t > ϵ ρ do 5: if w t > ϵ ρ then 6: ∀ i, x g,it ← arg max v ∈ D i − t u ρt − 1 ( v ) − l ρt − 1 ( v ) 7: else 8: ∀ i, x g,it ← arg max v ∈ ( S o,ϵq,it − 1 \ S p,it − 1 )

Original: ∩ D it u qt − 1 ( v ) − l qt − 1 ( v ) 9: if ∃ i ∈ [ N ] , x g,it ̸∈ S p,it then 10:
Revised: ∩ D it u qt − 1 ( v ) − l qt − 1 ( v ) 9: if ∃ i ∈ [ N ] , x g,it ̸∈ S p,it then 10:

Original: SE ( S o,ϵ q ,i t − 1 , S p,it − 1 , x g,it ) , ∀ i : x g,it ̸∈ S p,it 11:
Revised: SE ( S o,ϵ q ,i t − 1 , S p,it − 1 , x g,it ) , ∀ i : x g,it ̸∈ S p,it 11:

Original: S p,it ← ˜ P t ( S p,it − 1 ) , S o,ϵ q ,i t ← ˜ O
Revised: S p,it ← ˜ P t ( S p,it − 1 ) , S o,ϵ q ,i t ← ˜ O

Original: ϵ q t ( S p,it − 1 ) , ∀ i 12: t ← t +13: ∀ i, B ′ t ( i ) =
Revised: ϵ q t ( S p,it − 1 ) , ∀ i 12: t ← t +13: ∀ i, B ′ t ( i ) =

Original: { j ∈ [ N ] | S u,it ∩ S u,jt ̸ = ∅} 14:
Revised: { j ∈ [ N ] | S u,it ∩ S u,jt ̸ = ∅} 14:

Original: B t = (cid:83) i ∈
Revised: B t = (cid:83) i ∈

Original: [ N ] B ′ t ( i ) 15: if for any B ∈ B t , S u,Bt ̸ = S u,Bt − 1 then 16:
Revised: [ N ] B ′ t ( i ) 15: if for any B ∈ B t , S u,Bt ̸ = S u,Bt − 1 then 16:

Original: X t , w t ← G REEDY ( u ρt − 1 , l ρt − 1 , B, S u,Bt ) 17: ∀ i, x g,it ← arg max v ∈ D
Revised: X t , w t ← G REEDY ( u ρt − 1 , l ρt − 1 , B, S u,Bt ) 17: ∀ i, x g,it ← arg max v ∈ D

Original: i − t u ρt − 1 ( v ) − l ρt − 1 ( v ) 18: if ∀ i, x g,it ∈ S p,it and w t > ϵ ρ then 19: ∀ i, y iρ t
Revised: i − t u ρt − 1 ( v ) − l ρt − 1 ( v ) 18: if ∀ i, x g,it ∈ S p,it and w t > ϵ ρ then 19: ∀ i, y iρ t

Original: = ρ ( x g,it ) + η ρ 20:
Revised: = ρ ( x g,it ) + η ρ 20:

Original: Update GP i.e, compute u ρt , l ρt 21: t ← t +22:
Revised: Update GP i.e, compute u ρt , l ρt 21: t ← t +22:

Original: X t ,w t ← G REEDY ( u ρt − 1 , l ρt − 1 ,B, S u,Bt − 1 ) 23:
Revised: X t ,w t ← G REEDY ( u ρt − 1 , l ρt − 1 ,B, S u,Bt − 1 ) 23:

Original: Recommend X t optimistically safe path to travel.
Revised: Recommend X t optimistically safe path to travel.

Original: Analogous to B t , we define B pt as collection of batches where any
Revised: Analogous to B t , we define B pt as collection of batches where any

Original: B ∈ B pt contains agents which are topologically connected in pessimistic set and S p,Bt
Revised: B ∈ B pt contains agents which are topologically connected in pessimistic set and S p,Bt

Original: [Equation]
Revised: [Equation]

Original: Safe expansion .
Revised: Safe expansion .

Original: Safe expansion is the sub-routine inspired by G O OSE for goal-oriented exploration of the safe set that we use to learn about the feasibility of sampling targets.
Revised: Safe expansion is the sub-routine inspired by G O OSE for goal-oriented exploration of the safe set that we use to learn about the feasibility of sampling targets.

Original: It uses a heuristic h to assign priority scores p to points that are optimistically but not pessimistically safe.
Revised: It uses a heuristic h to assign priority scores p to points that are optimistically but not pessimistically safe.

Original: Those determine locations whose feasibility is relevant to learn that of the sampling targets ( Line 2 of Algorithm 3).
Revised: Those determine locations whose feasibility is relevant to learn that of the sampling targets ( Line 2 of Algorithm 3).

Original: A simple and effective choice for the heuristic is the inverse of the distance to the targets.
Revised: A simple and effective choice for the heuristic is the inverse of the distance to the targets.

Original: Then, it identifies safe locations where the constraint is not yet known ϵ q
Revised: Then, it identifies safe locations where the constraint is not yet known ϵ q

Original: -accurately (Line 3).
Revised: -accurately (Line 3).

Original: Among them, it determines the α -immediate expanders, i.e., those that could potentially add locations with priority α to the pessimistic set, G ϵ q t ( α ) = { v ∈ W ϵ q t |∃ z ∈
Revised: Among them, it determines the α -immediate expanders, i.e., those that could potentially add locations with priority α to the pessimistic set, G ϵ q t ( α ) = { v ∈ W ϵ q t |∃ z ∈

Original: A t ( α ) : u qt  − L q d ( v, z ) ≥ 0 } .
Revised: A t ( α ) : u qt ( v ) − L q d ( v, z ) ≥ 0 } .

Original: In Line 4, it selects the non-empty α -expander set with the highest priority.
Revised: In Line 4, it selects the non-empty α -expander set with the highest priority.

Original: In Line 6 - 7, the agent evaluates the constraint at the location with the highest uncertainty in this set (see [12] for details).
Revised: In Line 6 - 7, the agent evaluates the constraint at the location with the highest uncertainty in this set (see [12] for details).

Original: S AFE M A C convergence .
Revised: S [Equation]

Original: The optimistic coverage phase switches to optimistic exploration phase, when density uncertaintyis under the disks is low ( w t ≤ ϵ ρ ).
Revised: The optimistic coverage phase switches to optimistic exploration phase, when density uncertainty under the disks is low ( w t ≤ ϵ ρ ).

Original: In the exploration, either the topological connection of the optimistic feasible set changes or will classify the uncertain region as pessimistically safe.
Revised: In the exploration, either the topological connection of the optimistic feasible set changes or will classify the uncertain region as pessimistically safe.

Original: In the former case, S AFE M A C will recompute a new coverage location and switch to the coverage phase.
Revised: In the former case, S AFE M A C will recompute a new coverage location and switch to the coverage phase.

Original: Alternatively, if the uncertain region is pessimistically safe, S AFE M A C has converged since the density uncertainty in the exploration phase is already low.
Revised: Alternatively, if the uncertain region is pessimistically safe, S AFE M A C has converged since the density uncertainty in the exploration phase is already low.

Original: The phases show an interesting dynamics;
Revised: The phases show an interesting dynamics;

Original: S AFE M A C continuously iterates between the optimistic exploration and the optimistic coverage phase until we know about the feasibility of the disk and their uncertainty is low.
Revised: S AFE M A C continuously iterates between the optimistic exploration and the optimistic coverage phase until we know about the feasibility of the disk and their uncertainty is low.

Original: In the worst case,
Revised: In the worst case,

Original: S AFE M A C might explore the entire environment.
Revised: S AFE M A C might explore the entire environment.

Original: In this case the sample complexity will be similar to a two-stage algorithm, where we explore the whole domain and then optimize coverage in the resulting known environment.
Revised: In this case the sample complexity will be similar to a two-stage algorithm, where we explore the whole domain and then optimize coverage in the resulting known environment.

Original: However, in practice, S AFE M A C is much better than this worst case.
Revised: However, in practice, S AFE M A C is much better than this worst case.

Original:  Analysis
Revised:  Analysis

Original: We now analyze M A CO PT ’s convergence and S AFE M A C ’s optimality and safety properties.
Revised: We now analyze M A CO PT ’s convergence and S AFE M A C ’s optimality and safety properties.

Original: M A CO PT .
Revised: M A CO PT .

Original: To measure the progress of M A CO PT , we study its regret, i.e., the difference between its solution and the one we could find if we knew the true density.
Revised: To measure the progress of M A CO PT , we study its regret, i.e., the difference between its solution and the one we could find if we knew the true density.

Original: Since control coverage consists in maximizing a monotone submodular function, we cannot efficiently compute the true optimum even for known densities.
Revised: Since control coverage consists in maximizing a monotone submodular function, we cannot efficiently compute the true optimum even for known densities.

Original: However, we can efficiently find a solution that is at least (1 − 1 /e ) within the optimum.
Revised: However, we can efficiently find a solution that is at least (1 − 1 /e ) within the optimum.

Original: Thus, we quantify performance using the following notion of cumulative regret,
Revised: Thus, we quantify performance using the following notion of cumulative regret,

Original: [Equation]
Revised: [Equation]

Original: where F ( X ⋆ ; ρ, V ) is the optimal coverage.
Revised: where F ( X ⋆ ; ρ, V ) is the optimal coverage.

Original: We now state one of our main results, which guarantees that the cumulative regret of M A CO PT grows sublinearly in time (proof in Appendix D).
Revised: We now state one of our main results, which guarantees that the cumulative regret of M A CO PT grows sublinearly in time (proof in Appendix D).

Original: Theorem 1.
Revised: Theorem 1.

Original: Let δ ∈ (0 , 1) ,
Revised: Let δ ∈ (0 , 1) ,

Original: β ρt 1 / 2 =
Revised: β ρt 1 / 2 =

Original: B ρ + 4 σ ρ (cid:112) γ ρNt + ln(1 /δ ) and C D = max x i ∈ V | D i | / | V | ≤ 1 .
Revised: B ρ + 4 σ ρ (cid:112) γ ρNt + ln(1 /δ ) and C D = max x i ∈ V | D i | / | V | ≤ 1 .

Original: With probability at least 1 − δ , M A CO PT ’s regret defined in Eq.
Revised: With probability at least 1 − δ , M A CO PT ’s regret defined in Eq.

Original: (5) is bounded by O ( (cid:112)
Revised: (5) is bounded by O ( (cid:112)

Original: Tβ ρT γ ρNT ) ,
Revised: Tβ ρT γ ρNT ) ,

Original: [Equation]
Revised: [Equation]

Original: The proof of 1 builds on two key ideas.
Revised: The proof of 1 builds on two key ideas.

Original: First, we exploit the conditional linearity of the submodular objective to bound the cumulative regret defined in Eq.
Revised: First, we exploit the conditional linearity of the submodular objective to bound the cumulative regret defined in Eq.

Original: (5) with a sum of per agent regrets.
Revised: (5) with a sum of per agent regrets.

Original: Secondly, we bound the per agent regret with the information capacity γ ρNT , a quantity that measures the largest reduction in uncertainty about the density that can be obtained from NT noisy evaluations of it.
Revised: Secondly, we bound the per agent regret with the information capacity γ ρNT , a quantity that measures the largest reduction in uncertainty about the density that can be obtained from NT noisy evaluations of it.

Original: Since γ ρNT [21] grows sublinearly with T for commonly used kernels, so does M A CO PT ’s regret in Eq.
Revised: Since γ ρNT [21] grows sublinearly with T for commonly used kernels, so does M A CO PT ’s regret in Eq.

Original: The immediate corollary of the above theorem, when the M A CO PT stopping criteria is reached (Line 3 of Algorithm 2) guarantees a near optimal solution up to ϵ ρ precision.
Revised: The immediate corollary of the above theorem, when the M A CO PT stopping criteria is reached (Line 3 of Algorithm 2) guarantees a near optimal solution up to ϵ ρ precision.

Original: Corollary 1.
Revised: Corollary 1.

Original: Let t
Revised: Let t

Original: ⋆ρ be the smallest integer, such that
Revised: ⋆ρ be the smallest integer, such that

Original: [Equation]
Revised: [Equation]

Original: ρ , then there exists a t < t
Revised: ρ , then there exists a t < t

Original: ⋆ρ such that w.h.p, M A CO PT terminates and achieves, F ( X t ; ρ, V ) ≥ (1 − 1 e ) F ( X ⋆ ; ρ, V ) − ϵ ρ .
Revised: ⋆ρ such that w.h.p, M A CO PT terminates and achieves, F ( X t ; ρ, V ) ≥ (1 − 1 e ) F ( X ⋆ ; ρ, V ) − ϵ ρ .

Original: S AFE M A C .
Revised: S AFE M A C .

Original: This section presents our main result for safety-constrained multi-agent coverage control.
Revised: This section presents our main result for safety-constrained multi-agent coverage control.

Original: In particular, Theorem 2 (proof in Appendix E) guarantees that S AFE M A C safely achieves near-optimal safe coverage in finite time.
Revised: In particular, Theorem 2 (proof in Appendix E) guarantees that S AFE M A C safely achieves near-optimal safe coverage in finite time.

Original: Theorem 2.
Revised: Theorem 2.

Original: Let δ ∈ (0 , 1) , ϵ ρ ≥ 0 , ∥ ρ ∥ k ρ ≤ B ρ , β ρt 1 / 2 =
Revised: Let δ ∈ (0 , 1) , ϵ ρ ≥ 0 , ∥ ρ ∥ k ρ ≤ B ρ , β ρt 1 / 2 =

Original: B ρ + 4 σ ρ (cid:112) γ ρNt + 1 + ln(1 /δ ) , γ ρNt denote the information capacity associated with the kernel k ρ .
Revised: B ρ + 4 σ ρ (cid:112) γ ρNt + 1 + ln(1 /δ ) , γ ρNt denote the information capacity associated with the kernel k ρ .

Original: Let q ( · ) be L q -Lipschitz continuous and ϵ q , β qt , γ qNt be defined analogously.
Revised: Let q ( · ) be L q -Lipschitz continuous and ϵ q , β qt , γ qNt be defined analogously.

Original: Given X 0 ̸ = ∅ , q ( x i 0 ) ≥ 0 for all i ∈ [ N ] .
Revised: Given X 0 ̸ = ∅ , q ( x i 0 ) ≥ 0 for all i ∈ [ N ] .

Original: Then, for any heuristic h t : V → R , with probability at least 1 − δ , we have q ( x ) ≥ 0 , for any x along the state trajectory pursued by any agent in S AFE M A C .
Revised: Then, for any heuristic h t : V → R , with probability at least 1 − δ , we have q ( x ) ≥ 0 , for any x along the state trajectory pursued by any agent in S AFE M A C .

Original: Moreover, let t ⋆ρ be the smallest integer such that
Revised: Moreover, let t ⋆ρ be the smallest integer such that

Original: [Equation]
Revised: [Equation]

Original: ρ , with C D = max x i ∈ V | D i | | V | ≤ 1 and let t ⋆q be the smallest integer such that
Revised: ρ , with C D = max x i ∈ V | D i | | V | ≤ 1 and let t ⋆q be the smallest integer such that

Original: [Equation]
Revised: [Equation]

Original: q , with C = 8 / log(1 + σ − 2 q ) then, there exists t ≤ t ⋆q + t ⋆ρ , such that with probability at least 1 − δ ,
Revised: q , with C = 8 / log(1 + σ − 2 q ) then, there exists t ≤ t ⋆q + t ⋆ρ , such that with probability at least 1 − δ ,

Original: [Equation]
Revised: [Equation]

Original: The theoretical analysis has two components: ( i ) we show S AFE M A C ’s coverage is near-optimal at convergence (Lemma 10), and ( ii ) we prove it converges in finite time.
Revised: The theoretical analysis has two components: ( i ) we show S AFE M A C ’s coverage is near-optimal at convergence (Lemma 10), and ( ii ) we prove it converges in finite time.

Original: Since S AFE M A C learns the constraint and the density, we must bound the sample complexity for both to prove ( ii ).
Revised: Since S AFE M A C learns the constraint and the density, we must bound the sample complexity for both to prove ( ii ).

Original: For the constraint, we extend the results for single-agent G O OSE to our multi-agent setting (Appendix F).
Revised: For the constraint, we extend the results for single-agent G O OSE to our multi-agent setting (Appendix F).

Original: For the density, we use results from Theorem 1 to show that, within a coverage phase, the cumulative regret is sublinear.
Revised: For the density, we use results from Theorem 1 to show that, within a coverage phase, the cumulative regret is sublinear.

Original: Next, we use additivity of the information gain (Lemma 13) between any pair of coverage phases to bound the sample complexity of density for the subsequent coverage phases.
Revised: Next, we use additivity of the information gain (Lemma 13) between any pair of coverage phases to bound the sample complexity of density for the subsequent coverage phases.

Original: Combining these results, we obtain Theorem 2.
Revised: Combining these results, we obtain Theorem 2.

Original: Intermediate recommendation .
Revised: Intermediate recommendation .

Original: Theorem 2 guarantees that S AFE M A C converges to a safe and nearoptimal solution.
Revised: Theorem 2 guarantees that S AFE M A C converges to a safe and nearoptimal solution.

Original: Can it also make sensible recommendations before the stopping criteria are met?
Revised: Can it also make sensible recommendations before the stopping criteria are met?

Original: [Figure]
Revised: [Figure]

Original: M A CO PT does a more principled exploration of the coverage and does not stick to minimum.
Revised: M A CO PT does a more principled exploration of the coverage and does not stick to minimum.

Original: Ideally, such recommendations should ( i ) be safely reachable and ( ii ) ensure a minimum coverage.
Revised: Ideally, such recommendations should ( i ) be safely reachable and ( ii ) ensure a minimum coverage.

Original: To satisfy ( i ), they should be in the pessimistic safe set, S pt .
Revised: To satisfy ( i ), they should be in the pessimistic safe set, S pt .

Original: To satisfy ( ii ), their coverage should be computed according to F ( · ; l ρ t − 1 , S pt ) , i.e., assuming a worst-case density, l ρ t − 1 , and a worst-case feasible set, S pt .
Revised: To satisfy ( ii ), their coverage should be computed according to F ( · ; l ρ t − 1 , S pt ) , i.e., assuming a worst-case density, l ρ t − 1 , and a worst-case feasible set, S pt .

Original: If the greedy recommendation X t is in S pt , we can recommend it at intermediate steps.
Revised: If the greedy recommendation X t is in S pt , we can recommend it at intermediate steps.

Original: However, this is not always the case and we need an alternative.
Revised: However, this is not always the case and we need an alternative.

Original: To this end, we compute X l,Bt , i.e., the greedy solution w.r.t.
Revised: To this end, we compute X l,Bt , i.e., the greedy solution w.r.t.

Original: the worst-case objective, F ( · ; l ρt − 1 , S p,Bt ) ∀ B ∈ B pt .
Revised: the worst-case objective, F ( · ; l ρt − 1 , S p,Bt ) ∀ B ∈ B pt .

Original: At any time T ,
Revised: At any time T ,

Original: S AFE M A C recommends the best of either strategy up to time T according to the worst-case objective.
Revised: S AFE M A C recommends the best of either strategy up to time T according to the worst-case objective.

Original: In Appendix E.1, we show that such recommendation is also near optimal at convergence.
Revised: In Appendix E.1, we show that such recommendation is also near optimal at convergence.

Original:  Experiments
Revised:  Experiments

Original: This section compares M A CO PT and S AFE M A C to existing methods (or their extensions) on synthetic and real-world problems.
Revised: This section compares M A CO PT and S AFE M A C to existing methods (or their extensions) on synthetic and real-world problems.

Original: We validate our theoretical claims and observe their superiority.
Revised: We validate our theoretical claims and observe their superiority.

Original: We set β q = 3 and β ρ = 3 for all t ≥ 1 , it ensures safety as well as efficient exploration in practice [12].
Revised: We set β q = 3 and β ρ = 3 for all t ≥ 1 , it ensures safety as well as efficient exploration in practice [12].

Original: Experiment details and extended empirical analysis are in Appendix G.
Revised: Experiment details and extended empirical analysis are in Appendix G.

Original: Environments .
Revised: Environments .

Original: We perform our experiments with N = 3 agents in a 30 × 30 grid world where states are evenly spaced over [0 , 3] 2 .
Revised: We perform our experiments with N = 3 agents in a 30 × 30 grid world where states are evenly spaced over [0 , 3] 2 .

Original: Each agent’s disk is defined as the region an agent can reach in r = 5 steps in the defined grid.
Revised: Each agent’s disk is defined as the region an agent can reach in r = 5 steps in the defined grid.

Original: We normalize coverage with a maximum value (cid:80) v ∈ ¯ R 

Revised: We normalize coverage with a maximum value (cid:80) v ∈ ¯ R

Original: ( X 0 ) ρ ( v ) / | V | .
Revised: ( X 0 ) ρ ( v ) / | V | .

Original: Below, we present the 3 environments we consider.
Revised: Below, we present the 3 environments we consider.

Original: i)
Revised: i)

Original: In synthetic data , both the density ρ and the constrain q are sampled from a GP with zero mean and
Revised: In synthetic data , both the density ρ and the constrain q are sampled from a GP with zero mean and

Original: Matérn Kernel with ν = 2 . 5 , scale σ k = 1 , and lengthscale l = 2 .
Revised: Matérn Kernel with ν = 2 . 5 , scale σ k = 1 , and lengthscale l = 2 .

Original: The observations are perturbed by i.i.d noise N (0 , 10 − 3 ) .
Revised: The observations are perturbed by i.i.d noise N (0 , 10 − 3 ) .

Original: ii)
Revised: ii)

Original: In obstacles , we sample maps with several block-shaped obstacles (Fig. 3a) and we aim to maximize coverage while avoiding dangerous collisions.
Revised: In obstacles , we sample maps with several block-shaped obstacles (Fig. 3a) and we aim to maximize coverage while avoiding dangerous collisions.

Original: At v , each agent senses the distance to the nearest obstacle d m ( v ) , which could be given by sensors such as 1D-Lidars.
Revised: At v , each agent senses the distance to the nearest obstacle d m ( v ) , which could be given by sensors such as 1D-Lidars.

Original: We use q ′ ( v )
Revised: We use q ′ ( v )

Original: =
Revised: =

Original: 1 / (1 + exp( − 1 .
Revised: 1 / (1 + exp( − 1 .

Original: 5 d m ( v ))) , to map the distance between [0 , 3] and saturate the constraint value for large distances, and we set q ( v ) =
Revised: 5 d m ( v ))) , to map the distance between [0 , 3] and saturate the constraint value for large distances, and we set q ( v ) =

Original: q ′ ( v ) − 0 .
Revised: q ′ ( v ) − 0 .

Original: 5 to avoid collisions.
Revised: 5 to avoid collisions.

Original: The density is sampled from the same GP as the synthetic case.
Revised: The density is sampled from the same GP as the synthetic case.

Original: iii)
Revised: iii)

Original: In gorilla nest , we simulate a bio-diversity monitoring task, where we aim to cover areas with high density of gorilla nests with a quadrotor in the Kagwene
Revised: In gorilla nest , we simulate a bio-diversity monitoring task, where we aim to cover areas with high density of gorilla nests with a quadrotor in the Kagwene

Original: Gorilla Sanctuary (Fig. 3b) .
Revised: Gorilla Sanctuary (Fig. 3b) .

Original: Regions affected by adverse weather (e.g. rain and storms) are unsafe for the drone due to higher chances of crashes and should be avoided.
Revised: Regions affected by adverse weather (e.g. rain and storms) are unsafe for the drone due to higher chances of crashes and should be avoided.

Original: As a proxy for bad weather, we use the cloud coverage data over the KGS from OpenWeather [22].
Revised: As a proxy for bad weather, we use the cloud coverage data over the KGS from OpenWeather [22].

Original: The nest density is obtained by fitting a smooth rate function [23] over Gorilla nest counts [24].
Revised: The nest density is obtained by fitting a smooth rate function [23] over Gorilla nest counts [24].

Original: M A CO PT .
Revised: M A CO PT .

Original: We compare M A CO PT to UCB , a baseline that skips the uncertainty sampling step from
Revised: We compare M A CO PT to UCB , a baseline that skips the uncertainty sampling step from

Original: Section 4.1 and obtains measurements at the centers of the G REEDY sensing regions.
Revised: Section 4.1 and obtains measurements at the centers of the G REEDY sensing regions.

Original: We further develop two sample-efficient extensions of M A CO PT : i)
Revised: We further develop two sample-efficient extensions of M A CO PT : i)

Original: Correlated upper bound (CUB), a variant of M A CO PT that constructs tighter upper confidence bound of the coverage function utilizing the covariance of density, instead of using the sum of density UCB . ii)
Revised: Correlated upper bound (CUB), a variant of M A CO PT that constructs tighter upper confidence bound of the coverage function utilizing the covariance of density, instead of using the sum of density UCB . ii)

Original: Hallucinated uncertainty sampling (H), a variant of M A CO PT that samples at the most informative location for each agent i , after hallucinating sampling locations of { 1 , . . .
Revised: Hallucinated uncertainty sampling (H), a variant of M A CO PT that samples at the most informative location for each agent i , after hallucinating sampling locations of { 1 , . . .

Original: i − 1 } agents.
Revised: i − 1 } agents.

Original: Please see Appendix D.1 for theoretical analysis.
Revised: Please see Appendix D.1 for theoretical analysis.

Original: Fig.
Revised: Fig.

Original: 3c shows a comparison in the gorilla environment on a day of good weather, i.e. when all locations are safe.
Revised: 3c shows a comparison in the gorilla environment on a day of good weather, i.e. when all locations are safe.

Original: Here, UCB gets stuck in a local optimum as it does not reduce the uncertainty
Revised: Here, UCB gets stuck in a local optimum as it does not reduce the uncertainty

Original: [Figure]
Revised: [Figure]

Original: C ov
Revised: C ov

Original: e r a g e
Revised: e r a g e

Original: [Figure]
Revised: [Figure]

Original: S a m p
Revised: S a m p

Original: [Figure]
Revised: [Figure]

Original: s
Revised: s

Original: [Figure]
Revised: [Figure]

Original: Samples 0 .0 .0 .
Revised: Samples 0 .0 .0 .

Original: F ( X T ; ρ , ¯ R 

Revised: F ( X T ; ρ , ¯ R

Original: ( X 0 )
Revised: ( X 0 )

Original: SafeMac
Revised: SafeMac

Original: Two-Stage
Revised: Two-Stage

Original: PassiveMac
Revised: PassiveMac

Original: [Figure]
Revised: [Figure]

Original: of the density
Revised: of the density

Original: , whereas M A CO PT explores more and achieves a higher coverage value up to 25%.
Revised: , whereas M A CO PT explores more and achieves a higher coverage value up to 25%.

Original: Moreover, variants of M A CO PT account for correlation and condition on other agents’ measurement locations, which results in achieving the same coverage but more efficiently.
Revised: Moreover, variants of M A CO PT account for correlation and condition on other agents’ measurement locations, which results in achieving the same coverage but more efficiently.

Original: S AFE M A C .
Revised: S AFE M A C .

Original: We compare S AFE M A C with two baselines: i) a two-stage algorithm [25], that first fully explores the feasible region, and then uses M A CO PT to maximize the coverage; ii) P ASSIVE M A C , a baseline inspired by [26] that runs M A CO PT in the pessimistic set and passively measures the constraint in the process.
Revised: We compare S AFE M A C with two baselines: i) a two-stage algorithm [25], that first fully explores the feasible region, and then uses M A CO PT to maximize the coverage; ii) P ASSIVE M A C , a baseline inspired by [26] that runs M A CO PT in the pessimistic set and passively measures the constraint in the process.

Original: Figs. 4a and 4b show the coverage at convergence and the number of samples to converge for S AFE M A C and the two baselines across all the environments.
Revised: Figs. 4a and 4b show the coverage at convergence and the number of samples to converge for S AFE M A C and the two baselines across all the environments.

Original: The results are averaged over 50 instances produced using different seeds and samples for every environment.
Revised: The results are averaged over 50 instances produced using different seeds and samples for every environment.

Original: In Fig.
Revised: In Fig.

Original: 4b, the y-axis is normalized with the maximum number of samples in the instance and then averaged over all instances.
Revised: 4b, the y-axis is normalized with the maximum number of samples in the instance and then averaged over all instances.

Original: P ASSIVE M A C converges quickly but gets stuck in a local optimum as it does not actively explore the constraint.
Revised: P ASSIVE M A C converges quickly but gets stuck in a local optimum as it does not actively explore the constraint.

Original: S AFE M A C and Two-Stage converge to much higher coverage values.
Revised: S AFE M A C and Two-Stage converge to much higher coverage values.

Original: However, S AFE M A C is up to 50% more sample efficient thanks to its goal-oriented exploration.
Revised: However, S AFE M A C is up to 50% more sample efficient thanks to its goal-oriented exploration.

Original: Fig.
Revised: Fig.

Original: 4c shows the coverage value of the intermediate safe recommendations (Section 5) in the gorilla environment as a function of the number of samples.
Revised: 4c shows the coverage value of the intermediate safe recommendations (Section 5) in the gorilla environment as a function of the number of samples.

Original: It confirms the previous results: S AFE M A C finds solutions comparable to Two-Stage more efficiently and P
Revised: It confirms the previous results: S AFE M A C finds solutions comparable to Two-Stage more efficiently and P

Original: ASSIVE M A C gets stuck in a local optimum.
Revised: ASSIVE M A C gets stuck in a local optimum.

Original: Scalability .
Revised: Scalability .

Original: S AFE M A C utilizes the G REEDY algorithm, which is linear in the number of nodes (domain size).
Revised: S AFE M A C utilizes the G REEDY algorithm, which is linear in the number of nodes (domain size).

Original: In each iteration, S AFE M A C computes a greedy solution N times (one for each agent), which makes it linear in the number of agents.
Revised: In each iteration, S AFE M A C computes a greedy solution N times (one for each agent), which makes it linear in the number of agents.

Original: We model density and constraint using GP , which scales cubically with the number of samples.
Revised: We model density and constraint using GP , which scales cubically with the number of samples.

Original: To demonstrate scalability in practice, we conducted experiments with N = 3 , 6 , 10 , 15 agents each with domain length of 30 , 40 , 50 and 60 in Appendix G.
Revised: To demonstrate scalability in practice, we conducted experiments with N = 3 , 6 , 10 , 15 agents each with domain length of 30 , 40 , 50 and 60 in Appendix G.

Original:  Related work
Revised:  Related work

Original: Our work relates to multiple fields.
Revised: Our work relates to multiple fields.

Original: We highlight the most relevant connections, referencing surveys where possible; an exhaustive overview is beyond the scope of this paper.
Revised: We highlight the most relevant connections, referencing surveys where possible; an exhaustive overview is beyond the scope of this paper.

Original: Bayesian optimization .
Revised: Bayesian optimization .

Original: In BO, an agent sequentially evaluates a noisy objective, seeking to maximize it [27].
Revised: In BO, an agent sequentially evaluates a noisy objective, seeking to maximize it [27].

Original: In contrast, the quantity we measure differs from our objective.
Revised: In contrast, the quantity we measure differs from our objective.

Original: Partial monitoring [28] addresses such issues in an abstract setting [20, 29].
Revised: Partial monitoring [28] addresses such issues in an abstract setting [20, 29].

Original: We exploit special structure in our problem.
Revised: We exploit special structure in our problem.

Original: In coverage control with unknown density, this challenge is often addressed by learning the density uniformly over the domain [30, 31].
Revised: In coverage control with unknown density, this challenge is often addressed by learning the density uniformly over the domain [30, 31].

Original: In contrast, M A CO PT learns the density only at promising locations.
Revised: In contrast, M A CO PT learns the density only at promising locations.

Original: Coverage control .
Revised: Coverage control .

Original: M A C with known densities is a well-studied NP hard [32] problem.
Revised: M A C with known densities is a well-studied NP hard [32] problem.

Original: Many algorithms use efficient heuristics to converge quickly to a local optimum.
Revised: Many algorithms use efficient heuristics to converge quickly to a local optimum.

Original: One popular strategy is Lloyd’s algorithm
Revised: One popular strategy is Lloyd’s algorithm

Original: [33], which has been studied in different settings, e.g., with known densities [34, 35], a-priori unknown densities [31, 36–38], using graph neural networks [39], taking into account agent’s dynamics and constraints [40], or in case of non-identical robots [41].
Revised: [33], which has been studied in different settings, e.g., with known densities [34, 35], a-priori unknown densities [31, 36–38], using graph neural networks [39], taking into account agent’s dynamics and constraints [40], or in case of non-identical robots [41].

Original: These methods apply to continuous state and action spaces and show convergence to local optima, but lack optimality guarantees [30, 31, 40] and sample complexity bounds.
Revised: These methods apply to continuous state and action spaces and show convergence to local optima, but lack optimality guarantees [30, 31, 40] and sample complexity bounds.

Original: Moreover, their extension to non-convex, disconnected domains is not trivial [42].
Revised: Moreover, their extension to non-convex, disconnected domains is not trivial [42].

Original: Coverage control is also studied in the episodic setting to learn the unknown policy or the environment using deep RL methods [43, 44].
Revised: Coverage control is also studied in the episodic setting to learn the unknown policy or the environment using deep RL methods [43, 44].

Original: Submodular optimization .
Revised: Submodular optimization .

Original: Submodular functions are ubiquitous in machine learning [45] as they can be efficiently approximately maximized under different kinds of constraints [46].
Revised: Submodular functions are ubiquitous in machine learning [45] as they can be efficiently approximately maximized under different kinds of constraints [46].

Original: For example, the G REEDY algorithm can be used in case of cardinality constraints [13] to maximize quantities like mutual information [47] or weighted coverage functions [15].
Revised: For example, the G REEDY algorithm can be used in case of cardinality constraints [13] to maximize quantities like mutual information [47] or weighted coverage functions [15].

Original: Online submodular maximization aims at optimizing unknown submodular functions from noisy measurements [48].
Revised: Online submodular maximization aims at optimizing unknown submodular functions from noisy measurements [48].

Original: It has multiple applications, including optimization of numerical solvers [49], information gathering [50] and crowd-sourced image collection summarization [51].
Revised: It has multiple applications, including optimization of numerical solvers [49], information gathering [50] and crowd-sourced image collection summarization [51].

Original: Particularly related to ours is the work in [52], which proposes an algorithm for contextual news recommendation for linear user preferences with strong regret guarantees.
Revised: Particularly related to ours is the work in [52], which proposes an algorithm for contextual news recommendation for linear user preferences with strong regret guarantees.

Original: In contrast to that setting, we consider dynamic agents, safety constraints and partial feedback.
Revised: In contrast to that setting, we consider dynamic agents, safety constraints and partial feedback.

Original: Safety .
Revised: Safety .

Original: Depending on the safety formulation and the assumptions, many algorithms have been proposed for safe learning in dynamical systems, e.g., based on model predictive control [53], curriculum learning [54], Lyapunov functions [55, 56], reachability [57], CMDPs [58], behavioral system theory [59], and more [60–62].
Revised: Depending on the safety formulation and the assumptions, many algorithms have been proposed for safe learning in dynamical systems, e.g., based on model predictive control [53], curriculum learning [54], Lyapunov functions [55, 56], reachability [57], CMDPs [58], behavioral system theory [59], and more [60–62].

Original: Here, we focus on the setting that is most closely related to ours, i.e., one with unknown but sufficiently regular instantaneous constraints that must be satisfied at all times.
Revised: Here, we focus on the setting that is most closely related to ours, i.e., one with unknown but sufficiently regular instantaneous constraints that must be satisfied at all times.

Original: For stateless problems, e.g. BO , [26, 63] propose algorithms with safety and optimality guarantees with different exploration strategies.
Revised: For stateless problems, e.g. BO , [26, 63] propose algorithms with safety and optimality guarantees with different exploration strategies.

Original: For stateful problems, [64] studies the pure exploration case, while [25] extends the two-stage approach from [63].
Revised: For stateful problems, [64] studies the pure exploration case, while [25] extends the two-stage approach from [63].

Original: These approaches may be sample inefficient as they may explore the constraint in regions irrelevant for the objective.
Revised: These approaches may be sample inefficient as they may explore the constraint in regions irrelevant for the objective.

Original: G O OSE [12] addresses this problem for both the stateful and stateless setting.
Revised: G O OSE [12] addresses this problem for both the stateful and stateless setting.

Original: The only work in this context that addresses multi-agent problems is [65].
Revised: The only work in this context that addresses multi-agent problems is [65].

Original: However, their objective differs from ours, and they do not establish safety guarantees.
Revised: However, their objective differs from ours, and they do not establish safety guarantees.

Original:  Conclusion
Revised:  Conclusion

Original: We present two novel algorithms for multi-agent coverage control in unconstrained ( M A CO PT ) and safety critical environments ( S AFE M A C ).
Revised: We present two novel algorithms for multi-agent coverage control in unconstrained ( M A CO PT ) and safety critical environments ( S AFE M A C ).

Original: We show M A CO PT achieves sublinear cumulative regret, despite the challenge of partial observability.
Revised: We show M A CO PT achieves sublinear cumulative regret, despite the challenge of partial observability.

Original: Moreover, we prove S AFE M A C achieves near optimal coverage in finite time while navigating safely.
Revised: Moreover, we prove S AFE M A C achieves near optimal coverage in finite time while navigating safely.

Original: We demonstrate the superiority of our algorithms in terms of sample efficiency and coverage in real-world applications such as safe biodiversity monitoring.
Revised: We demonstrate the superiority of our algorithms in terms of sample efficiency and coverage in real-world applications such as safe biodiversity monitoring.

Original: Currently, our algorithms choose informative targets but do not plan informative trajectories, which is crucial in robotics.
Revised: Currently, our algorithms choose informative targets but do not plan informative trajectories, which is crucial in robotics.

Original: We aim to address this in future work.
Revised: We aim to address this in future work.

Original: Finally, while in many real-world applications the density and the constraints are as regular as assumed here, in some they are not.
Revised: Finally, while in many real-world applications the density and the constraints are as regular as assumed here, in some they are not.

Original: In these cases, our optimality and safety guarantees would not apply.
Revised: In these cases, our optimality and safety guarantees would not apply.

Original: Acknowledgements
Revised: Acknowledgements

Original: Manish Prajapat is supported by an ETH AI Center doctoral fellowship.
Revised: Manish Prajapat is supported by an ETH AI Center doctoral fellowship.

Original: Matteo Turchetta is supported by the Swiss National Science Foundation under NCCR Automation, grant agreement 51NF 180545.
Revised: Matteo Turchetta is supported by the Swiss National Science Foundation under NCCR Automation, grant agreement 51NF 180545.

Original: We would like to thank Pawel Czyz for insightful discussions.
Revised: We would like to thank Pawel Czyz for insightful discussions.