Title: Noisy Dual Mirror Descent: A Near Optimal Algorithm for Jointly-DP Convex Resource Allocation
Venue: NeurIPS 2024
Accepted: yes
Keywords: joint differential privacy

resource allocation

near optimal algorithm

mirror descent
Abstract: We study convex resource allocation problems with $m$ hard constraints under $(\varepsilon,\delta)$-joint differential privacy (Joint-DP or JDP) in an offline setting. To approximately solve the problem, we propose a generic algorithm called Noisy Dual Mirror Descent. The algorithm applies noisy Mirror Descent to a dual problem from relaxing the hard constraints for private shadow prices, and then uses the shadow prices to coordinate allocations in the primal problem. Leveraging weak duality theory, we show that the optimality gap is upper bounded by $\mathcal{O}(\frac{\sqrt{m\ln(1/\delta)}}{\varepsilon})$, and constraint violation is no more than $\mathcal{O}(\frac{\sqrt{m\ln(1/\delta)}}{\varepsilon})$ per constraint. When strong duality holds, both preceding results can be improved to $\widetilde{\mathcal{O}}(\frac{\sqrt{\ln(1/\delta)}}{\varepsilon})$ by better utilizing the geometric structure of the dual space, which is neglected by existing works. To complement our results under strong duality, we derive a minimax lower bound $\Omega(\frac{m}{\varepsilon})$ for any JDP algorithm outputting feasible allocations. The lower bound matches our upper bounds up to some logarithmic factors for $\varepsilon\geq \max(1, 1/(n\gamma))$, where $n\gamma$ is the available resource level. Numerical studies further confirm the effectiveness of our algorithm.

Noisy Dual Mirror Descent: A Near Optimal Algorithm for Jointly-DP Convex Resource Allocation

Abstract

We study convex resource allocation problems with m hard constraints under ( ε, δ ) -joint differential privacy (Joint-DP or JDP) in an offline setting. To approximately solve the problem, we propose a generic algorithm called Noisy Dual Mirror Descent. The algorithm applies noisy Mirror Descent to a dual problem from relaxing the hard constraints for private shadow prices, and then uses the shadow prices to coordinate allocations in the primal problem. Leveraging weak duality theory, we show that the optimality gap is upper bounded by O ( √ m ln (1 /δ ) ε ) , and constraint violation is no more than O ( √ m ln (1 /δ ) ε ) per constraint. When strong duality holds, both preceding results can be improved to ˜ O ( √ ln (1 /δ ) ε ) by better utilizing the geometric structure of the dual space, which is neglected by existing works. To complement our results under strong duality, we derive a minimax lower bound Ω ( m ε ) for any JDP algorithm outputting feasible allocations. The lower bound matches our upper bounds up to some logarithmic factors for ε ≥ max { 1 , 1 / ( nγ ) } , where nγ is the available resource level. Numerical studies further confirm the effectiveness of our algorithm.

1 Introduction

The resource allocation problem is a classic optimization problem and has many applications in machine learning, such as Internet advertising and personalized recommendation, among many others. The problem is typically modeled as a utility maximization problem, where the central decision maker has to properly allocate m types of limited resources to n agents in order to maximize the total utility of all agents. Each agent's data z i := ( u i ( · ) , a i ( · ) , X i ) consists of three elements: (i) a utility function u i : R s + → R + that maps an allocation x i to a utility scalar; (ii) a consumption function a i : R s + → R m + that maps an allocation x i to a consumption vector; and (iii) a feasible set X i ⊆ R s + that may capture the agent's special requirements. With a dataset D := { z i } n i =1 of n agents, the problem is modeled as

where nγ b > 0 is the available level of m types of resources. Evidently, the goal in (1) is to find the best allocation x i for each agent subject to resource-coupling constraints (2) and to a personal

requirement constraint (3). When u i ( · ) is concave, and a i ( · ) , X i are both convex, the problem reduces to a convex constrained problem that has been widely studied in optimization literature.

However, an emerging issue in allocation problems is data privacy. Because constraint (2) couples agents, the allocation decision to one agent would affect the allocation to another agent. When a group of agents collude, they can infer other agents' data by analyzing their received allocations. For example, when allocating limited budgets among education agencies to support students in poverty, a simple rule is to allocate budgets proportional to the number of students in poverty [SLWA22]. Agencies who notice a change in received funding may infer the financial status of students in other agencies. Realizing this potential leakage of its students' financial status, an agency may misreport the number of its students in poverty, compromising the original intention of the financial support.

To overcome privacy concerns in such cases, joint differential privacy (Joint-DP or JDP, for short), which is a relaxation of differential privacy (DP), has been adopted. It is found that JDP is more suitable than DP for the considered allocation problems [HHR + 14]. Essentially, JDP ensures agent i 's data cannot be accurately inferred by a group of collusive agents without i . It guarantees that the allocations received by the collusive agents are insensitive to agent i 's data. Considering that the allocation decision x i is usually private to agent i herself, the privacy guarantee by JDP is hence meaningful and sufficiently strong. The most handy way to achieve JDP is by the well-known Billboard Lemma [HHR + 14]: publish a DP public signal on a billboard accessible to all agents, and compute the allocation for agent i based on the DP public signal and agent i 's own data only; then the final allocations ( x 1 , . . . , x n ) satisfy JDP. The Billboard Lemma has wide applications, ranging from convex optimization [HHR + 14, HHRW16, HZ18], to multi-armed bandits [SS18, HLL + 22], to reinforcement learning [VBKW20]. We also use it for privacy analysis. For clarification purposes, we highlight that, in problem (1), dataset { z i } n i =1 := { ( u i ( · ) , a i ( · ) , X i ) } n i =1 is the private information to protect; resource level nγ b is treated as public information.

While important and fundamental, resource allocation problems under JDP are far from being well understood, in terms of both algorithm design and theoretical analysis. For algorithm design, most existing works adopt a 'dual decomposition' idea [HHRW16, HZ18], which dualizes the coupling constraint (2) and iteratively seeks private minimizers of a dual problem to coordinate the allocation process in the primal problem. However, the primal-dual relationship and the geometric structure of the dual space are not well exploited, leaving substantial room for improvement. In terms of theoretical analysis, both privacy accounting and optimality analysis are short of modern standard. For privacy accounting, existing works [HHRW16, HZ18, HZ19] all use Advanced Composition, which suggests an unnecessarily large noise to be injected, significantly hindering algorithm practicability. For optimality analysis, [HHRW16] fails to take full advantage of strong assumptions made, and the analysis by [HZ18] is specific to linear packing problems. More importantly, there is no formal understanding of the fundamental trade-off between privacy and optimality in resource allocation problems under JDP, i.e. minimax lower bounds. Motivated by these gaps in the literature, we focus our study on the following three questions.

Can we design a generic algorithm for resource allocation problem (1), and provide better privacy and optimality analysis?

Can we better utilize the primal-dual relationship to improve performance?

Is our algorithm (near) optimal? What is the minimax lower bound?

Our contributions First, we propose a generic algorithm Noisy Dual Mirror Descent, which follows a similar dual decomposition idea. We tighten the privacy analysis through Rényi DP, and analyze performance (both optimality and constraint violations) by better utilizing the primal-dual relationship. Second, for many cases where strong duality holds, we further improve performance by leveraging on the ℓ 1 geometry of the dual space. Last, we complement previous analysis with a matching minimax lower bound for ε ≥ max { 1 , 1 / ( nγ ) } , suggesting near optimality of our algorithm. We further conduct numerical experiments to show the effectiveness of our proposed algorithm. Table 1 summarizes our contributions and compares them with results in the literature.

Our interpretation of algorithm performance Because the algorithms considered may output infeasible solutions, when assessing their performance, we should take into account both suboptimality in utility and constraint violations, i.e., the last two columns in Table 1. We therefore treat the sum of them as the ultimate performance. This idea admits a social welfare interpretation: when the central

Notes. All results in the table are stated for ε ≤ ln (1 /δ ) for conciseness; m,n,T are the number of constraints, agents, and iterations, respectively. Tilde symbol ˜ O hides poly-log factors in m ; for results in [HHRW16, HZ18], ˜ O additionally hides poly-log terms in n and T . LB=lower bound, UB=upper bound, w.h.p=with high probability. ‡ The algorithm in [HHRW16] can be practically applied to general convex problems. But one supporting Lemma (Theorem 2.1) they used to derive analytical results is only valid for linear problems with solutions in probability simplex. § An lower bound ˜ Ω( √ m ε ) can be derived from [HZ18, Theorem 1.2]. But their original statement is for the minimal supply, not for suboptimality as we considered. † The lower bound is for algorithms outputting feasible allocations, while upper bounds are achieved by algorithms that may output infeasible allocations. So, the lower bound could be higher. The lower bound holds only for ε ≥ max { 1 , 1 / ( nγ ) } where γ ∈ (0 , 1) .

decision maker (e.g., a government) desires to implement an infeasible allocation, she may purchase additional resources from an emergency supplier to make the allocation feasible. Then, the total loss in social welfare is the sum of (i) the loss in utility of agents and (ii) the decision maker's expenditure on extra resources. If we only look at suboptimality, we may mistakenly conclude that the gap could be arbitrarily small. Following our interpretation of performance, our proposed algorithm is near optimal.

Related work JDP was initially proposed by [KPRU14] as a relaxation of differential privacy [DR14], which better fits the nature of privacy issues in zero-sum games, such as allocation problems [HHR + 14, CKRW15]. Following this stream, the most relevant works to ours are [HHRW16, HZ18], both of which designed their algorithms with a dual decomposition idea. [HHRW16] proposed the first generic method, dual gradient descent, for solving allocation problems. For the special case of linear packing problem, [HZ18] designed a dual multiplicative weight update algorithm, and extended it to online setting. [GU22] further provides an economic interpretation of payoff sharing. Our work is also closely related to Mirror Descent (MD) [NY83], a generalization of projected gradient descent to non-Euclidean settings. Its private version, Noisy MD, has recently found many applications in non-Euclidean DP stochastic convex optimization [AFKT21, BGN21] and saddle-point problems [GGP24]. We further apply Noisy MD to JDP resource allocation problems.

2 Preliminaries

Definition 2.1 (Differential privacy, [DR14]) . A mechanism M : Z n →P is ( ε, δ ) -differentially private if, for any pair of neighboring datasets D ∼ D ′ that differ in one data point, and for any subset of output S ⊆ P , we have P [ M ( D ) ∈ S ] ≤ e ε · P [ M ( D ′ ) ∈ S ] + δ .

Throughout the work, we use the subscript -i to indicates variables without agent i . For example, X -i := X 1 × · · · X i -1 × X i +1 · · · × X n is a feasible space without i , and M ( D ) -i := ( M ( D ) 1 , . . . , M ( D ) i -1 , M ( D ) i +1 , . . . , M ( D ) n ) is the view from other agents without i .

Definition 2.2 (Joint differential privacy, [KPRU14]) . A mechanism M : Z n →X n is ( ε, δ ) -jointly differentially private if for any pair of neighboring datasets D ∼ D ′ that differ in datapoint i ∈ [ n ] , and for any subset of outputs S ⊆ X -i , we have P [ M ( D ) -i ∈ S ] ≤ e ε · P [ M ( D ′ ) -i ∈ S ] + δ .

Lemma 2.3 (Billboard Lemma, [HHR + 14]) . Suppose M : Z n → P is ( ε, δ ) -DP. For any given function f : Z × P → X , denote the output of f on individual i 's data as f i := f ( z i , M ( D )) . Then, ( f 1 , . . . , f n ) is ( ε, δ ) -JDP.

Definition 2.4 ( α -strong convexity) . Let α > 0 . Function ℓ : R m → R is said to be α -strongly convex w.r.t. ∥·∥ p over set W , if ℓ ( x ) ≥ ℓ ( y ) + ⟨∇ ℓ ( y ) , x -y ⟩ + α 2 ∥ x -y ∥ 2 p , ∀ x , y ∈ W .

Primal problem in a condensed form Let x := ( x 1 , . . . , x n ) be the collection of allocations. For a given dataset D , let F ( x ) := ∑ n i =1 u i ( x i ) be the total utility when allocation is x . Let a ( x ) := ∑ n i =1 a i ( x i ) be the total consumed resource, and let X := X 1 ×··· × X n be the feasible region. Then, the resource allocation problem (1), referred to as 'primal problem' later, can be written in a condensed form:

The optimal allocation is denoted as x ∗ := arg max x ∈X { F ( x ) : a ( x ) ≤ nγ b } .

Dual problem By dualizing the coupling constraint with shadow prices p ≥ 0 , we get a Lagrangian function min p ≥ 0 max x ∈X { F ( x ) + ⟨ p , nγ b -a ( x ) ⟩} . Thus, the Lagrangian dual problem is

where D ( p ) := max x ∈X { F ( x ) + ⟨ p , nγ b -a ( x ) ⟩} . It is easy to see that, given p ≥ 0 , the dual problem is decomposable across agents, i.e., D ( p ) = ∑ n i =1 max x i ∈X i { u i ( x ) + ⟨ p , γ b -a i ( x i ) ⟩} . Let us denote the dual problem's minimizer by p ∗ := arg min p ≥ 0 D ( p ) .

When there are multiple x ∗ and p ∗ , we can break ties arbitrarily. Let ∥·∥ p denote the p -norm of a vector, and let [ n ] := { 1 , . . . , n } be a running set. We impose some assumptions on dataset D to make both primal and dual problems interesting.

Assumption 2.5 (Interesting instances) . For a dataset D := { ( u i ( · ) , a i ( · ) , X i ) } n i =1 , we assume

Convexity : for each i ∈ [ n ] , utility function u i is concave on X i , consumption function a i is convex on X i , and X i is a convex set;

Boundedness : ∃ u > 0 and bounded b > 0 with ∥ b ∥ 2 ≤ B such that u i ( x i ) ∈ [0 , u ] and a i ( x i ) ∈ [ 0 , b ] , ∀ x i ∈ X i , i ∈ [ n ] . Specially, if 0 ∈ X i , then u i ( 0 ) = 0 and a i ( 0 ) = 0 ;

̸

Limited resource : the constant γ in (4) is assumed to be in the interval (0 , 1) . The primal problem (4) is feasible, and optimal x ∗ is attainable. Under x ∗ , at least one of m constraints in (4) is binding. Moreover, the optimal shadow prices p ∗ to dual problem (5) are not all zeros, i.e., ∥ p ∗ ∥ 1 = 0 ;

Compulsory request allowed : request i can be compulsory in the sense that X i ̸∋ 0 , only if max x i ∈X i { u i ( x i ) + ⟨ p ∗ , -a i ( x i ) ⟩} ≥ 0 .

These assumptions are very mild. The first assumption restricts our attention to convex problems. The second one assumes both utility and consumption functions are non-negative and bounded from above, and are both zeros if no resource is allocated. The third one assumes the non-private problem is indeed resource-constrained and not ill-posed. The constant γ controls available resource levels, and nγ means the number of agents whose requests can be fully fulfilled. In practice, usually γ ≥ 1 /n . The fourth one allows agents to propose compulsory requests with minimal requirements strictly greater than 0 , as long as allocating resource to the agent is beneficial. Our assumptions are weaker than those in the literature [HZ18, BLM20, BLM23] by allowing compulsory requests.

3 The algorithm and analysis

Our algorithm is a private version of Mirror Descent applied to the dual problem (5). Mirror Descent (MD) [NY83] is a generalization of Projected Gradient Descent (Proj-GD) that can better cater to the geometry of the problem at hand by the proper choice of a strongly convex potential function Φ : W → R + . The potential function chosen should meet some conditions below.

Condition 3.1 (Potential Functions) . The potential function Φ : W → R + chosen should be (i) differentiable on int ( W ) , i.e., the interior of its domain W ⊆ R m ; and (ii) α -strongly convex with respect to ∥·∥ p on W . For a given Φ , let B Φ ( x , y ) := Φ( x ) -Φ( y ) - ⟨∇ Φ( y ) , x -y ⟩ be the Bregman divergence between x ∈ W , y ∈ int ( W ) .

With the primal problem (4) and dual problem (5) in mind, and a well-chosen potential function, we are ready to present the algorithm, Noisy Dual Mirror Descent. The algorithm is an iterative

Algorithm 1 Noisy Dual Mirror Descent for Resource Allocation Problems, A

Parameters: privacy parameters ( ε, δ ) , variance σ 2 ; stepsizes { η ( t ) } T t =1 ; potential function Φ( · ) : W → R + that is α -strongly convex with respect to ℓ p -norm; conjugate index q s.t. 1 /q +1 /p = 1 , number of iterations T , feasible region of shadow prices P := R m + .

1: Set initial point p (1) ∈ P ∩ int ( W )

2: for t = 1 to T do

3: Get intermediate allocation decision x ( t ) ← arg max x ∈X F ( x ) + 〈 p ( t ) , γn b -a ( x ) 〉

4: Draw a noise vector n ( t ) ∼ N ( 0 , σ 2 I m × m )

5: Update private shadow prices according to Noisy Mirror Descent:

where

g

(

t

)

:=

nγ

b

-

a

(

x

(

t

)

)

6: Final allocation decision x A := ( x A 1 , . . . , x A n ) ← 1 T ∑ T t =1 x ( t )

7: Output x A i to agent i , for all i ∈ [ n ]

algorithm, and in each iteration, we first calculate allocation decisions based on current shadow prices p ( t ) , i.e. Step 3. Then, we update the shadow prices by Noisy Mirror Descent update rule (6). The gradient g ( t ) used is exactly the gradient of the dual problem D ( p ( t ) ) according to Danskin's theorem. After T iterations, we output the averaging allocations across all iterations. In other words, we basically apply Noisy Mirror Descent to solve the dual problem, and use the sequence of shadow prices to coordinate allocations in the primal problem. Intuitively, the shadow prices are implicitly pricing each resource to ensure that limited resources are allocated to agents most in need.

Recall the Billboard Lemma, if we can privatize the entire sequence of shadow prices in a DP manner, then the final allocation decisions will be JDP. Compared to existing works invoking Advanced Composition, we provide a tighter privacy accounting through Rényi DP.

Theorem 3.2 (JDP Guarantee) . Given ε > 0 , δ ∈ (0 , 1) and T ≥ 1 , if noise variance σ 2 = T · c ε,δ with c ε,δ := ∥ b ∥ 2 2 · ( 2 ln (1 /δ ) ε 2 + 1 ε ) , then Algorithm 1 A is ( ε, δ ) -JDP.

To better align with expressions in DP literature, we can assume ε ≤ ln (1 /δ ) . Then, the magnitude of c ε,δ becomes ln (1 /δ ) ε 2 , a magnitude of Gaussian Mechanism that frequently appears in DP literature. Moreover, the privacy analysis via Rényi DP significantly lowers the variance level. For example, when ε = 1 , δ = 10 -3 , and ∥ b ∥ 2 2 = 1 , the variance indicated by Theorem 3.2 is 14 . 8 T , compared to approximately 121 . 6 T by [HHRW16, Theorem 3.2], [HZ18, Lemma 3.1].

3.1 Performance upper bounds

We now move on to analyze the utility optimality gap F ( x ∗ ) -E A [ F ( x A ) ] . The following weak duality lemma bridges the primal and dual problems.

Lemma 3.3 (Weak duality) . For any p ≥ 0 , the objective value of dual problem (5) is always greater than or equal to that of primal problem (4) , i.e., D ( p ) ≥ max x ∈X { F ( x ) : a ( x ) ≤ nγ b } , ∀ p ≥ 0 .

An immediate result of weak duality is D ( p ( t ) ) ≥ F ( x ∗ ) , ∀ t . Hence, applying weak duality and Jensen's inequality, the optimality gap F ( x ∗ ) -F ( x A ) can be upper bounded as follows, provided

the randomness dice of A is fixed,

where the first equality is by definition of the dual problem, and the second equality is by definition of gradient g ( t ) . Taking expectation with respect to A removes the subtrahend, since n ( t ) is independent of p ( t ) and zero-mean, which gives F ( x ∗ ) -E A [ F ( x A ) ] ≤ E A [ 1 T ∑ T t =1 〈 p ( t ) , g ( t ) + n ( t ) 〉 ] . The inner term is closely related to stationarity gap [GL13, ABG + 23] in nonconvex optimization. It is well-known that Proj-GD guarantees a small stationarity gap in both non-private [JNG + 21] and private [ABG + 23] cases. Not surprisingly, MD can also achieve a small stationarity gap.

Lemma 3.4 (Cumulative stationarity gap of MD) . Suppose stepsizes η ( t ) = η in MD update rule (6) are the same for all t ∈ [ T ] , and let { ˜ g ( t ) } T t =1 be gradients used for update. Then the cumulative stationarity gap ∑ T t =1 〈 p ( t ) -p , ˜ g ( t ) 〉 for any anchor point p ∈ P ∩ W is upper bounded as

With Lemma 3.4 and proper stepsizes, we can control the optimality gap. Denote γ := max { γ, 1 -γ } , G := γ 2 n 2 ∥ b ∥ 2 q , and let z ∼ N ( 0 , I m × m ) be a standard Gaussian random vector. For a given potential function Φ and an initial point p (1) , denote C Φ ( p (1) ) := √ B Φ ( 0 , p (1) ) /α .

Theorem 3.5 (Utility guarantee) . Set stepsizes η ( t ) = √ αB Φ ( 0 , p (1) ) T · ( G + σ 2 E [ ∥ z ∥ 2 q ]) , ∀ t ∈ [ T ] . Suppose potential function's domain includes 0 , i.e., 0 ∈ W . Then running algorithm A with iterations T ≥ G c ε,δ · E [ ∥ z ∥ 2 q ] and σ 2 chosen in Theorem 3.2 yields

An instantiation of the utility guarantee is by Φ( · ) = 1 2 ∥·∥ 2 2 , which is 1 -strongly convex w.r.t. ℓ 2 -norm on R m . Then, the upper bound in (7) becomes 2 √ 2 m ∥ ∥ p (1) ∥ ∥ 2 · √ c ε,δ , which depends on the choice of initial point p (1) . It seems the bound can be arbitrarily small if p (1) is close to 0 . But in fact, the theorem itself does not reflect the whole picture, since x A could be infeasible. Therefore, we have to further examine A 's feasibility guarantee.

Theorem 3.6 (Feasibility guarantee) . Given allocation decision x A , denote the violation levels as v A := ( a ( x A ) -nγ b ) + , where positive part operator ( · ) + applies element-wisely. In addition to Condition 3.1, additionally assume the domain W of potential function Φ contains 2 ∥ p ∗ ∥ 1 e j , ∀ j ∈ [ m ] . Then, running algorithm A with the same setting as in Theorem 3.5 yields

where C Φ , 1 ( p (1) ) = B Φ ( 0 , p (1) )+max e ∈ E 1 B Φ (2 ∥ p ∗ ∥ 1 e , p (1) ) √ B Φ ( 0 , p (1) ) , set E 1 := { e ∈ { 0 , 1 } m : ⟨ 1 , e ⟩ ≤ 1 } .

The proof idea is to sandwich the cumulative stationarity gap ∑ T t =1 〈 p ( t ) -p , ˜ g ( t ) 〉 at a properly chosen anchor point p := 2 ∥ p ∗ ∥ 1 e j A ∈ P ∩ W (the constant 2 is chosen arbitrarily, it can be any value strictly greater than 1), and then comparing the upper bound and lower bound in the sandwich inequality gives the desired result. The base vector e j A with a single 1 on j A -th position

indicates which constraint j A ∈ [ m ] is most severely violated. One may notice that the anchor point p := 2 ∥ p ∗ ∥ 1 e j A needs to be in P ∩ W , which implicitly imposes one additional condition on the domain W of potential function Φ . While the capability of computationally calculating ∥ p ∗ ∥ 1 is not needed for deriving (8), on a practical note, we may need to adjust W accordingly so that W contains all possible anchor points 2 ∥ p ∗ ∥ 1 e j A , ∀ j A almost surely. When doing so, we should be very careful because p ∗ depends on dataset D and thus, such an adjustment may leak privacy.

Again, one may think Φ( · ) = 1 2 ∥·∥ 2 2 is the ideal potential function, because its domain W = R m contains all anchor points of interest, does not depend on p ∗ , and therefore no privacy leakage risk. As shown in the first row of Table 2, the guarantees under squared ℓ 2 are only comparable to, not better than, those in the literature summarized in Table 1. Nevertheless, being comparable is already a significant improvement because our results are derived from weak duality only, whereas existing works all assume strong duality. Moreover, since anchor points p = 2 ∥ p ∗ ∥ 1 e j A are related to ℓ 1 norm of p ∗ , if we can find a data-independent upper bound for the value of ∥ p ∗ ∥ 1 , we might be able to adjust W accordingly without privacy concerns and may further improve performance. We do so in the next subsection.

Notes. Results for squared ℓ 2 (negative entropy) are from Section 3.1 (Section 3.2). Strong duality is necessary for theoretical guarantees by (parameterized) negative entropy. Abbreviations in column titles: cvx.=convexity, init.pt.=initial point, opt.gap = optimality gap, contrs.viol.=constraint violations. The (scaled) simplex set ∆ K ( b ) with radius K is defined as ∆ K ( b ) := { p ≥ 0 : ⟨ b , p ⟩ ≤ K } ; the standard simplex is ∆ K := ∆ K ( 1 ) . † Squared ℓ 2 function is strongly convex on whole space R m , while negative entropy functions only on their respective domains. Results in the first and second rows are highlighted as the main contributions in Table 1.

3.2 Improvements by strong duality

Indeed, there exists space for improvement, if we assume strong duality holds.

Assumption 3.7 (Strong duality holds) . Strong duality between primal problem (4) and dual problem (5) holds, i.e., F ( x ∗ ) = D ( p ∗ ) , where x ∗ and p ∗ are optimal solutions to (4) and (5) , respectively.

While weak duality always holds, strong duality does not universally hold. However, there are many applications where strong duality naturally holds. For example, when the dualized constraint a ( x ) ≤ nγ b is linear in x , strong duality holds. If the constraint is not linear, one can check strong duality by Slater's condition, which essentially says that if the primal problem (4) has strictly feasible solutions, then strong duality holds. The strong duality assumption here is also very mild, and all examples previously considered in the literature satisfy strong duality, for example [HHRW16, Section 4] and [HZ18]. However, these works fail to take full advantage of strong duality. We fill the gap by noticing that strong duality actually restricts p ∗ to an ℓ 1 space.

Lemma 3.8 (Strong duality implies bounded p ∗ ) . Let p ∗ := arg min p ≥ 0 D ( p ) . Then, under assumptions 2.5 and 3.7, we have ⟨ b , p ∗ ⟩ ≤ u γ and ⟨ 1 , p ∗ ⟩ ≤ u γb with b := min j { b j } .

The above lemma indicates that p ∗ lies in the interior of a scaled simplex, suggesting the best choice of potential function could be the negative entropy function, which better fits the ℓ 1 geometry. However, negative entropy Φ( w ) = ∑ m j =1 w j ln w j is widely known to be 1-strongly convex only on the probability simplex ∆ 1 := { w > 0 : ⟨ 1 , w ⟩ = 1 } . To tailor it for our studied problem, we parameterize the negative entropy Φ( w ; θ ) := ∑ m j =1 ( θ j w j ) ln ( θ j w j ) with θ > 0 , and show that it is also strongly convex in a scaled simplex.

Lemma 3.9 (Strong convexity of parameterized negative entropy) . Let the parameterized negative entropy Φ( w ; θ ) := ∑ m j =1 ( θ j w j ) ln ( θ j w j ) be defined on R m + , where we define 0 0 = 0 and 0 ln 0 = 0

by continuity. Then Φ( w ; θ ) is (min j { θ j } ) 2 /K -strongly convex in w w.r.t. ∥·∥ 1 in a scaled simplex ∆ int K ( θ ) := { w > 0 : ⟨ θ , w ⟩ ≤ K } .

It is immediate to see that if we set θ = 1 and K = 1 , then Lemma 3.9 recovers the well-known result that negative entropy is 1 -strongly convex. For our studied problem, we can let the negative entropy be parameterized by b , and use Φ( w ; b ) = ∑ m j =1 ( b j w j ) ln ( b j w j ) for the Mirror Descent update step (6). This does not compromise privacy, since b is a universal upper bound and is not associated with any specific agent. Now, we are ready to improve the algorithm's performance.

Theorem 3.10 (Improved utility & feasibility guarantees) . Let b := max j { b j } , b := min j { b j } . Under assumptions 2.5 and 3.7, running algorithm 1 A with parameterized negative entropy Φ( w ; b ) with radius K := 2 u/γ , p (1) j = K/ ( mb j ) , ∀ j ∈ [ m ] , and same T , σ 2 as in Theorem 3.5 yields

Compared to the results in Theorems 3.5 and 3.6 where both dependencies on number of constraints are √ m , the results here have better dependencies of ln m . Moreover, following our interpretation of algorithm performance as discussed in the Introduction, Theorem 3.10 implies that the algorithm's ultimate performance is ˜ O ( √ c ε,δ ) + m · ˜ O ( √ c ε,δ ) = ˜ O ( m √ c ε,δ ) . The additional m factor comes from m constraints, since the feasibility guarantee in Theorem 3.10 is for any single constraint.

4 The lower bound

Some post-processing lemmas for JDP are necessary to prove the lower bound.

Lemma 4.1 (Self post-processing for JDP) . Let M : Z n → X n be an ( ε, δ ) -JDP mechanism and denote its output by M ( D ) := ( M ( D ) 1 , . . . , M ( D ) n ) . Let f : X × Z → Y be an arbitrary (randomized) function that can be applied element-wisely to M 's output. Then, ( f ( M ( D ) 1 , z 1 ) , . . . , f ( M ( D ) n , z n )) is ( ε, δ ) -JDP.

Lemma 4.2 (Post-processing for JDP) . Let M : Z n →X n be an ( ε, δ ) -JDP mechanism and denote its output by M ( D ) := ( M ( D ) 1 , . . . , M ( D ) n ) . Let f : X n -1 →Y be an arbitrary (randomized) function that can be applied to any collection of n -1 elements of M 's output. Then, for any k ∈ [ n ] , f ( M ( D ) -k ) and f ( M ( D ′ ) -k ) are ( ε, δ ) -indistinguishable. 2

Lemma 4.1 confirms that processing each JDP output with the agent's own data preserves JDP. Lemma 4.2 says applying any operation to n -1 elements of two n -length JDP outputs obtained from a pair of neighboring datasets will be indistinguishable.

̸

To see how these two lemmas help, we consider the allocations x A := ( x A 1 , . . . , x A n ) outputted by a JDP algorithm A : Z n →X n . By Lemma 4.1, their consumptions ( a 1 ( x A 1 ) , . . . , a n ( x A n )) are also JDP. Furthermore, for any collusive group without agent k , their total consumption ∑ n i = k a i ( x A i ) should be insensitive to agent k 's allocation by Lemma 4.2. To ensure insensitivity, intuitively but informally, any algorithm A outputting feasible allocations should reserve some resource exclusively for agent k . But the reserved resource is wasted in some scenarios, which leads to the lower bound.

Theorem 4.3 (Minimax lower bound) . For ε ≥ max { 1 , 1 / ( nγ ) } , 0 < δ ≤ 1 / 2 , there exists a dataset D satisfying assumptions 2.5 and 3.7 such that any ( ε, δ ) -JDP algorithm A outputting feasible allocations will lead to a utility loss at least m/ (4 ε ) . Therefore, the minimax lower bound is

The lower bound here nearly matches upper bounds given by Theorem 3.10 up to some logarithmic factors in m and 1 /δ . Therefore, Noisy Dual Mirror Descent is near optimal, under our interpretation

of algorithm performance. However, the result is limited to ε ≥ max { 1 , 1 / ( nγ ) } , and it is interesting to derive lower bounds for other ε . We provide a promising direction in the Appendix, which involves a lower bounding optimization problem with hockey-stick divergence constraints. We conjecture that couplings and divergences employed by [BBG18] are promising tools for this purpose.

5 Numerical experiments

5.1 Workforce scheduling

Workforce scheduling is about establishing a shift schedule for a given period to maximize workers' total preference while meeting worker availability and shift coverage requirements. Workers' preference and availability are private data to protect. We consider a simple case of scheduling n = 7 workers for m = 14 days with data publicly available [Opt24]. In this case, (i) each worker has preferences c i ∈ Z m + , and utility function is u i ( x i ) = ⟨ c i , x i ⟩ ; (ii) shift requirement r j ∈ Z + is imposed on each day j ∈ [ m ] , i.e., the number of workers needed for day j , which couples all workers; and (iii) workers' availability and shift limits described by a polyhedral set X i := { x i ∈ [0 , 1] m : l i ≤ ⟨ 1 , x i ⟩ ≤ u i } . The problem is modeled as max { ∑ n i =1 ⟨ c i , x i ⟩ : ∑ i x i = r ; x i ∈ X i , ∀ i } . While this is a small case, it is suitable for visualizing decisions and understanding the impact of JDP. Moreover, since the optimal non-private p ∗ = [0 , 3 , 1 , 0 , 2 , 0 , 0 , 4 , 3 , 2 , 3 , 0 , 0 , 0] is a sparse vector, one can expect MD equipped with negative entropy to perform better.

Figure 1 shows the final decisions under various ε , where the rightmost is the optimal non-private decision where a value of 1 means arranging the person to that day. Private decisions are fractional















































































































































































































































ε

=1



















































































































































































































ε

=5



















































































































































































































ε

=10



















































































































































































































ε

=20



















































































































































































































ε

=

∞













and can be interpreted as probabilities. It is not hard to see that private decisions are in a similar pattern to non-private decisions, and many private decisions are exactly the same as their optimal non-private counterpart. Moreover, private decisions different from their non-private counterpart converge gradually as ε →∞ , see for example, the decision of (Day1, Marisa), (Day2, Marisa), and (Day4, Vincent). Optimality gaps and constraint violations are shown in Figure 2 (left panel). For all













ε





























ε



























curves, the lower the better. It is clear that our algorithms have significantly smaller optimality gaps at slightly higher constraint violations.

One may notice that the constant of K is 1.1 rather than 2 suggested by Theorem 3.10. This is the consequence of parameter tuning, and we show its impact in Figure 2 (right panel). Comparing the two plots in Figure 2, it is obvious that larger K_constant leads to conservative decisions: fewer constraint violations but higher optimality gaps. Moreover, when K_constant is 2, our algorithms achieve lower optimality gaps at the same level of constraint violations.

5.2 Assignment problem

We next consider three large-scale assignment problems with ( m,n ) = (8, 800), (15, 1500), (30, 3000), aiming to maximize aggregated utility by assigning at most one unit to each agent. The problem is modeled as max { ∑ n i =1 ⟨ c i , x i ⟩ : ∑ i I m × n x i ≤ nγ 1 ; x i ∈ X i , ∀ i } with X i := { x i ∈ [0 , 1] n : ⟨ 1 , x i ⟩ ≤ 1 } being a unit simplex and I m × n := [ I m , . . . , I m ] being a consumption matrix concatenated from n/m identity matrices. Raw data of c i are available at [Bea04]. In this assignment problem, p ∗ > 0 is strictly non-zero. We run experiments and report results in Table 3.

Notes. Parameters for our algorithms: K = 1 . 1 u/ ( γb ) , δ = . 01 , T = 10 4 . Other settings follow Theorem 3.10. MD_l2 and MD_ne mean that potential function is squared ℓ 2 and negative entropy, respectively. For three cases n = 800 , 1500 , 3000 , we set resource level γ = 0 . 1 , 0 . 05 , 0 . 02 , respectively. For all values in the table, the lower the better. Bold =better. More results are deferred to Appendix C.

From the table, we observe MD_l2 has lower gaps and reasonable constraint violations for small ε ; and MD_ne almost dominates others for large ε . Both our algorithms outperform existing methods.

6 Discussion and conclusion

Limitation of our work. One significant limitation is the restriction of ε ≥ max { 1 , 1 / ( nγ ) } to make the lower bound hold. However, we identify a promising direction to overcome this limitation in Appendix B.4. Once the limitation is overcame, the analysis may further uncover lower bounds on online cases [SS18], a long-standing open question in private online optimization [CDE + 24].

In this work, we considered convex resource allocation problems under joint differential privacy. To solve the problem approximately, we proposed an algorithm Noisy Dual Mirror Descent, which privatizes dual variables of hard constraints, and then uses private dual variables to coordinate allocations. A significant merit of the algorithm is its ability to better leverage the geometric structure of the dual space; thus, it is provably near optimal for a large range of privacy parameters. There are many interesting directions for future study. For example, getting rid of the limitation discussed previously is a fruitful endeavour. Identifying a proper DP manner to tune the constant of K is of more practical interest.

Acknowledgments and Disclosure of Funding

We sincerely thank the anonymous Area Chair and reviewers for their valuable feedback that significantly improved this work. This research is supported by the Ministry of Education, Singapore, under its MOE AcRF Tier 1 (RG117/23).

References

[ABG + 23] Raman Arora, Raef Bassily, Tomás González, Cristóbal A Guzmán, Michael Menart, and Enayat Ullah. Faster rates of convergence to stationary points in differentially private optimization. In International Conference on Machine Learning , pages 1060-1092. PMLR, 2023. 6

[AFKT21] Hilal Asi, Vitaly Feldman, Tomer Koren, and Kunal Talwar. Private stochastic convex optimization: Optimal rates in l1 geometry. In International Conference on Machine Learning , pages 393-403. PMLR, 2021. 3

[BBG18] Borja Balle, Gilles Barthe, and Marco Gaboardi. Privacy amplification by subsampling: Tight analyses via couplings and divergences. Advances in neural information processing systems , 31, 2018. 9, 23

[Bea04] J E Beasley. Or-library, assignment problem. http://people.brunel.ac.uk/~mastjjb/jeb/ orlib/assigninfo.html , 2004. Accessed: 2024-May-3. 10

[Bec17] Amir Beck. First-order methods in optimization . SIAM, 2017. 14

[BGN21] Raef Bassily, Cristóbal Guzmán, and Anupama Nandi. Non-euclidean differentially private stochastic convex optimization. In Conference on Learning Theory , pages 474-499. PMLR, 2021. 3

[BLM20] Santiago Balseiro, Haihao Lu, and Vahab Mirrokni. Dual mirror descent for online allocation problems. In International Conference on Machine Learning , pages 613-628. PMLR, 2020. 4

[BLM23] Santiago R Balseiro, Haihao Lu, and Vahab Mirrokni. The best of many worlds: Dual mirror descent for online allocation problems. Operations Research , 71(1):101-119, 2023. 4, 18

[CDE + 24] Rachel Cummings, Damien Desfontaines, David Evans, Roxana Geambasu, Yangsibo Huang, Matthew Jagielski, Peter Kairouz, Gautam Kamath, Sewoong Oh, Olga Ohrimenko, et al. Advancing differential privacy: Where we are now and future directions for real-world deployment. 2024. 10

[CKRW15] Rachel Cummings, Michael Kearns, Aaron Roth, and Zhiwei Steven Wu. Privacy and truthful equilibrium selection for aggregative games. In Web and Internet Economics: 11th International Conference, WINE 2015, Amsterdam, The Netherlands, December 9-12, 2015, Proceedings 11 , pages 286-299. Springer, 2015. 3

[DR14] Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Foundations and Trends® in Theoretical Computer Science , 9(3-4):211-407, 2014. 3

[GGP24] Tomás González, Cristóbal Guzmán, and Courtney Paquette. Mirror descent algorithms with nearly dimension-independent rates for differentially-private stochastic saddle-point problems. arXiv preprint arXiv:2403.02912 , 2024. 3

[GL13] Saeed Ghadimi and Guanghui Lan. Stochastic first-and zeroth-order methods for nonconvex stochastic programming. SIAM journal on optimization , 23(4):2341-2368, 2013. 6

[GU22] George Gilliam and Nelson A Uhan. Computing payoff allocations in the approximate core of linear programming games in a privacy-preserving manner. Operations Research Letters , 50(1):64-71, 2022. 3

[HHR + 14] Justin Hsu, Zhiyi Huang, Aaron Roth, Tim Roughgarden, and Zhiwei Steven Wu. Private matchings and allocations. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing , pages 21-30, 2014. 2, 3, 4

[HHRW16] Justin Hsu, Zhiyi Huang, Aaron Roth, and Zhiwei Steven Wu. Jointly private convex programming. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms , pages 580-599. SIAM, 2016. 2, 3, 5, 7, 10, 24, 25, 26

[HLL + 22] Yuxuan Han, Zhicong Liang, Zhipeng Liang, Yang Wang, Yuan Yao, and Jiheng Zhang. Private streaming sco in lp geometry with applications in high dimensional online decision making. In International Conference on Machine Learning , pages 8249-8279. PMLR, 2022. 2

[HZ18] Zhiyi Huang and Xue Zhu. Near optimal jointly private packing algorithms via dual multiplicative weight update. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 343-357. SIAM, 2018. 2, 3, 4, 5, 7

[HZ19] Zhiyi Huang and Xue Zhu. Scalable and jointly differentially private packing. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) . Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2019. 2

[JNG + 21] Chi Jin, Praneeth Netrapalli, Rong Ge, Sham M Kakade, and Michael I Jordan. On nonconvex optimization for machine learning: Gradients, stochasticity, and saddle points. Journal of the ACM (JACM) , 68(2):1-29, 2021. 6

[KPRU14] Michael Kearns, Mallesh Pai, Aaron Roth, and Jonathan Ullman. Mechanism design in large games: Incentives and privacy. In Proceedings of the 5th conference on Innovations in theoretical computer science , pages 403-410, 2014. 3

[Mir17] Ilya Mironov. Rényi differential privacy. In 2017 IEEE 30th computer security foundations symposium (CSF) , pages 263-275. IEEE, 2017. 13

[NY83] Arkadij Semenoviˇ c Nemirovskij and David Borisovich Yudin. Problem complexity and method efficiency in optimization. 1983. 3, 4

[Opt24] Gurobi Optimization. Workforce scheduling. https:// gurobi-optimization-gurobi-optimods.readthedocs-hosted.com/en/latest/ mods/workforce.html , 2024. Accessed: 2024-April-30. 9

[SLWA22] Ryan Steed, Terrance Liu, Zhiwei Steven Wu, and Alessandro Acquisti. Policy impacts of statistical uncertainty and privacy. Science , 377(6609):928-931, 2022. 2

[SS18] Roshan Shariff and Or Sheffet. Differentially private contextual linear bandits. Advances in Neural Information Processing Systems , 31, 2018. 2, 10

[VBKW20] Giuseppe Vietri, Borja Balle, Akshay Krishnamurthy, and Steven Wu. Private reinforcement learning with pac and regret guarantees. In International Conference on Machine Learning , pages 9754-9764. PMLR, 2020. 2

A Proofs for Section 3

A.1 Proof of Theorem 3.2

Proof. Since each individual's allocation x A i depends on all shadow prices ( p (1) , . . . , p ( T ) ) on the training trajectory and the individual's own data, by Billboard Lemma 2.3, it suffices to verify the whole sequence ( p (1) , . . . , p ( T ) ) is ( ε, δ ) -DP. We characterize the DP guarantee through Rényi Differential Privacy [Mir17]. Below are some useful lemmas on Rényi DP.

Definition A.1 ( ( α, ε ) -Rényi Differential Privacy, [Mir17]) . Let M : D → P be a randomized mechanism. For any neighboring datasets D ∼ D ′ , let P , P ′ be the distribution of M ( D ) and M ( D ′ ) , respectively. Then, M is said to be ε -Rényi differentially private of order α , or ( α, ε ) -RDP for short, if it holds that

where D α ( P || Q ) := 1 α -1 ln ( E Q [( P ( x ) Q ( x ) ) α ]) is the Rényi divergence between P and Q .

Lemma A.2 (Gaussian Mechanism is RDP, Proposition 7 in [Mir17]) . Let f : D → Z be a vectorvalued function whose global sensitivity is GS := sup D∼D ′ ∥ f ( D ) -f ( D ′ ) ∥ 2 , then the Gaussian Mechanism M ( D ) := f ( D ) + N ( 0 , σ 2 I ) is ( α, α · GS 2 2 σ 2 ) -RDP, for any α > 1 .

Lemma A.3 (Composition for RDP, Proposition 1 in [Mir17]) . Let M 1 : D → P 1 be ( α, ε 1 ) -RDP and M 2 : D × P 1 → P 2 be ( α, ε 2 ) -RDP. Let X := M 1 ( D ) be the output of M 1 and Y := M 2 ( D , X ) be the output of M 2 , then the adaptive composition ( X,Y ) satisfies ( α, ε 1 + ε 2 ) -RDP.

Lemma A.4 (From RDP to DP, Proposition 3 in [Mir17]) . If M is an ( α, ε ) -RDP mechanism, it also satisfies ( ε + ln (1 /δ ) α -1 , δ ) -DP, for any δ ∈ (0 , 1) .

For iteration t , we notice that data only plays a role in the gradient g ( p ( t ) ; D ) := nγ b -a ( x ( p ( t ) ; D )) through intermediate allocation decision x ( p ( t ) ; D ) := arg max x ∈X F ( x ) -〈 p ( t ) , a ( x ) 〉 . It is immediate to show that the Global Sensitivity of gradient function is

where the last inequality is by our assumption that serving one individual consumes at most b resource (see assumption 2.5). Then, injecting a zero-mean Gaussian noise n ( t ) ∼ N ( 0 , σ 2 I ) into gradients g ( t ) := g ( p ( t ) ; D ) can guarantee ( α, α ·∥ b ∥ 2 2 2 σ 2 ) -RDP by Lemma A.2.

Moreover, the algorithm comprises T iterations, and each iteration takes as input the shadow price in previous iteration; therefore, T -fold iterations form a sequential composition. Applying Composition Lemma A.3, we immediately conclude that ( g (1) + n (1) , . . . , g ( T ) + n ( T ) ) satisfies ( α, Tα ·∥ b ∥ 2 2 2 σ 2 ) -RDP. As the noisy mirror descent update step in Eq.(6) is merely post-processing after obtaining ( g (1) + n (1) , . . . , g ( T ) + n ( T ) ) , the sequence ( p (1) , . . . , p ( T ) ) is thus also ( α, Tα ·∥ b ∥ 2 2 2 σ 2 ) -RDP. Lastly, by Lemma A.4, ( α, Tα ·∥ b ∥ 2 2 2 σ 2 ) -RDP implies ( Tα ·∥ b ∥ 2 2 2 σ 2 + ln (1 /δ ) α -1 , δ ) -DP ∀ δ ∈ (0 , 1) . Thus, for a given pair of ( ε, δ ) , if we assign privacy budget in the following way:

which is the desired variance level.

we can get

A.2 Proof of Lemma 3.4

Proof. For any gradient ˜ g ( t ) used for the mirror descent update step Eq.(6), shadow prices p ( t ) , and updated shadow prices p ( t +1) , there exists an inequality linking them together (Lemma 9.13, [Bec17]):

Summing up the above inequality in a telescoping way over t = 1 , . . . , T gives

When stepsizes η ( t ) = η are the same for all t , the inequality becomes

A.3 Proof of Theorem 3.5

Proof. Let us first fix a sequence of noises { n ( t ) } T t =1 . Recall that the final allocation decision x A is the average over all decisions along the trajectory and the objective function is concave; therefore, by Jensen's inequality, F ( x A ) ≥ 1 T ∑ T t =1 F ( x ( t ) ) . Moreover, by weak duality, F ( x ∗ ) ≤ D ( p ( t ) ) , ∀ t . With the above results, the optimality gap F ( x ∗ ) -F ( x A ) can be upper controlled as:

Taking expectation with respect to noises { n ( t ) } T t =1 on both sides, we can remove the second term on the right hand side in preceding inequality, because n ( t ) is zero-mean and is independent of p ( t ) . This leads to

The inner summation can be upper bounded by applying Lemma 3.4 with p = 0 and η ( t ) = η > 0 , ∀ t ; we therefore obtain

Here are some helpful inequalities:

Denote G := γ 2 n 2 ∥ b ∥ 2 q , and G A := 1 T ∑ T t =1 ∥ ∥ n ( t ) ∥ ∥ 2 q . Using above helpful inequalities, we can upper bound 1 T ∑ T t =1 ∥ ∥ ˜ g ( t ) ∥ ∥ 2 q as

Plugging (11) back into (10), we further reach an upper bound:

Because G A = 1 T ∑ T t =1 ∥ ∥ n ( t ) ∥ ∥ 2 q and n ( t ) ∼ N ( 0 , σ 2 I m × m ) with σ 2 = Tc ε,δ , ∀ t , direct calculation gives E A [ G A ] = σ 2 E [ ∥ z ∥ 2 q ] with z ∼ N ( 0 , I m × m ) being a standard Gaussian random vector. Therefore,

Let C Φ ( p (1) ) := √ B Φ ( 0 , p (1) ) /α . Then, plugging in the stepsize η = √ αB Φ ( 0 , p (1) ) T · ( G + σ 2 E [ ∥ z ∥ 2 q ]) leads to

Lastly, if we run sufficiently many iterations T ≥ G c ε,δ · E [ ∥ z ∥ 2 q ] , we get the stated result in the theorem.

A.4 Proof of Theorem 3.6

Proof. We first briefly introduce the idea for the proof. We will find a lower bound and an upper bound for the cumulative stationarity gap ∑ T t =1 〈 ˜ g ( t ) , p ( t ) -p 〉 in Lemma 3.4. With a properly chosen p ∈ R m + ∩ W as a bridge, comparing these two bounds leads to the desired conclusion.

For a given final allocation decision x A , we denote the resource underage level by

where the positive part operator ( · ) + applies element-wise. It is self-evident that v A ≥ 0 by definition, and v A = 0 implies no constraint violation. Let j A := arg max j ∈ [ m ] ∪{ 0 } { u A j } be the index of the most-severely violated constraint. Specially, if j A = 0 , then we know v A = 0 , and we are done. Otherwise, we let e j A be an indicator vector with 1 on its ( j A ) 's position and 0 on others.

Choose p := C e j A ∈ R m + ∩ W with a constant C to be determined later, we reformulate the cumulative stationarity gap ∑ T t =1 〈 ˜ g ( t ) , p ( t ) -C e j A 〉 into

and control the three terms one by one.

(*): First, rewrite ( ∗ ) into a function of dual form:

where the last inequality is from D ( p ( t ) ) ≥ D ( p ∗ ) and applying Jensen's inequality to F ( · ) . Next, we turn to control F ( x A ) . We observe that if more resource v A becomes available for the original allocation problem, then x A will be a feasible allocation. Hence, we can upper bound the value of F ( x A ) with another problem having more resources. Denote the set of feasible allocations when b resource available by X ( b ) := { x := ( x 1 , . . . , x n ) ∈ ⊗ n i =1 X i : a ( x ) ≤ b } . We have

Let p in (14) be p := p ∗ , and plug (14) back into (13), we obtain

(**): The second term ( ∗∗ ) helps characterize constraint violations:

(***): The third term ( ∗ ∗ ∗ ) involves zero-mean Gaussian noise vectors { n ( t ) } t that are independent of p ( t ) . While e j A is dependent on noise vectors, the dependence is not a big issue here, since e j A ∈ E 1 := { e ∈ { 0 , 1 } m : ⟨ e , 1 ⟩ = 1 } , and the Gaussian Complexity G ( E 1 ) := E z ∼N ( 0 , I m × m ) [ sup e ∈ E 1 ⟨ e , z ⟩ ] of the set E 1 can be well controlled:

Therefore, the term ( ∗ ∗ ∗ ) admits a lower bound in expectation w.r.t. A as shown below:

Finally, replacing three terms back and taking expectation w.r.t. A , we obtain a lower bound on the cumulative stationarity gap:

Now, we turn to another side and bound the cumulative stationarity gap from above. Suppose we can choose a proper C > 0 such that C e j A ∈ R m + ∩ W almost surely, the expected cumulative stationarity gap thus can be bounded from above by employing Lemma 3.4:

Similarly, denote G := γ 2 n 2 ∥ b ∥ 2 q and G A := ∑ T t =1 ∥ ∥ n ( t ) ∥ ∥ 2 q /T . Then E A [ ∑ T t =1 ∥ ∥ ˜ g ( t ) ∥ ∥ 2 q ] ≤ 2 T · ( G + σ 2 E [ ∥ z ∥ 2 q ]) with z being a standard Gaussian vector. Plugging these values back, we get

Comparing the lower bound (15) with the upper bound (16) and using σ 2 = Tc ε,δ , we immediately obtain

Rearranging the above inequality yields (assume C > ∥ p ∗ ∥ 1 )

When T ≥ G c ε,δ · E [ ∥ z ∥ 2 q ] , (17) implies

The inequality (18) gives an upper bound on the expected maximal constraint violation.

However, the above analysis holds only when constant C satisfies (i) C > ∥ p ∗ ∥ 1 and (ii) C e j A ∈ R m + ∩ W for any j A ∈ [ m ] . To this end, we can simply let C = 2 ∥ p ∗ ∥ 1 , and replacing C with 2 ∥ p ∗ ∥ 1 gives the desired result:

A.5 Proof of Lemma 3.8

Proof. For any agent i , the objective value of its individual dual problem is always positive, i.e., max x i ∈X i { u i ( x i ) -⟨ p ∗ , a i ( x i ) ⟩} ≥ 0 by Assumption 2.5 part 4. With this inequality, we are ready to show bounded shadow prices:

Dividing both sides by nγ results in ⟨ b , p ∗ ⟩ ≤ u γ . Similarly,

which implies ⟨ 1 , p ∗ ⟩ ≤ u γ min j b j .

A.6 Proof of Lemma 3.9

Proof. By the definition of strong convexity, it suffices to show

According to the definition of parameterized negative entropy function, its gradient is ∇ Φ( p ; θ ) j = θ j ln ( p j θ j ) + θ j , ∀ j ∈ [ m ] . By the well celebrated Hermite-Hadamard inequality f ( a + b 2 ) ≤ 1 b -a ∫ b a f ( x ) dx , if we use convex function f ( t ) = 1 t , ∀ t > 0 , we immediately have 2 a + b ≤ ln b -ln a b -a , which implies

where the case a = 0 or b = 0 , or both are obtained by continuity and the conventions that 0 0 = 0 and 0 ln 0 = 0 . If we set b = p j θ j and a = q j θ j for (21), we can get

Summing over j ∈ [ m ] , we have

It remains to further lower bound the summation on the r.h.s. We follow the proof idea of Lemma 2 in [BLM23] to complete our proof:

A.7 Proof of Theorem 3.10

Proof. The proof follows the general proof of Theorem 3.5 and 3.6 for any potential function Φ . It remains to properly modify results for the Φ chosen. When the potential function is the parameterized negative entropy Φ := Φ( w ; b ) = ∑ m j =1 w j b j ln ( w j b j ) defined on ∆ int K ( b ) := { w > 0 : ⟨ b , w ⟩ ≤ K } with K = 2 u γ , the conjugate index q = ∞ , strong convexity parameter α = b 2 /K . Moreover, direct calculation gives the Bregman divergence B Φ ( 0 , p (1) ) = 〈 b , p (1) 〉 . Because we choose p (1) j = K/ ( mb j ) , ∀ j ∈ [ m ] , we thus have B Φ ( 0 , p (1) ) = K .

Utility Guarantee: It is evident that C Φ ( p (1) ) := √ B Φ ( 0 , p (1) ) /α = √ K/ ( b 2 /K ) = K/b . Therefore, the utility upper bound (7) becomes

Feasibility Guarantee: We can work from (19) and first figure out C Φ , 1 ( p (1) ) . Direct calculation gives B Φ ( C e j , p (1) ) = B Φ ( 0 , p (1) ) + Cb j · ( ln( C/p (1) j ) -1 ) , ∀ j ∈ [ m ] . Plugging C = 2 ∥ p ∗ ∥ 1 , we can upper bound C Φ , 1 ( p (1) ) as:

Substitute C Φ , 1 ( p (1) ) back into (19) with K = 2 u γ , we get

B Proofs for Section 4

B.1 Proof of self post-processing Lemma 4.1 for JDP

Proof. The proof is for a deterministic function f . The result on randomized functions follows immediately as any randomized mapping can be decomposed into a convex combination of deterministic functions. The proof here largely follows the same idea for the proof of Billboard Lemma. For any x ∈ X n , let f ( x -k , D -k ) := ( f ( x 1 , z 1 ) , . . . , f ( x k -1 , z k -1 ) , f ( x k +1 , z k +1 ) , . . . , f ( x n , z n )) . For any subset S ⊆ Y n -1 , let f -1 ( S ; D -k ) := { x -k ∈ X n -1 : f ( x -k , D -k ) ∈ S } be the preimage of S with respect to the mapping f ( · , D -k ) . Then, we can show that

which confirms JDP guarantee after self post-processing.

B.2 Proof of post-processing Lemma 4.2 for JDP

Proof. Similarly, the proof is for a deterministic function f . Let f -1 ( S ) be the preimage of S . For any S ⊆ Y :

B.3 Proof of Theorem 4.3

The proof consists of several steps: (i) construct a 'hard' distribution of datasets; (ii) identify necessary conditions on the JDP algorithm A with the help of Lemma 4.1 and 4.2; (iii) reformulate the minimax lower bound into a better form that aligns these necessary conditions; (iv) combine all steps together. Without loss of generality, we assume b = 1 .

Step 1: Construct a 'hard' distribution of datasets.

Let Id( · ) be the identity function, and let 1 and 0 be vectors of 1 's and 0 's whose lengths are m . Define three types of requests:

All three types of requests are the same in their utility and consumption functions, both taking linear forms. It is also easy to verify that all three types of requests satisfy Assumptions 2.5 (part 1 and 2). In what follows, we construct datasets that satisfy Assumption 2.5 (part 3 and 4) and Assumption 3.7 as well.

Based on these requests, we construct a pair of neighboring datasets that differ in agent n only:

It is easy to verify that part 3 of Assumption 2.5 is also met, i.e., the primal problem is always feasible, and the optimal shadow price is p ∗ = 1 , which are non-zeros. The result p ∗ = 1 is by noticing that one unit increase in available resource will increase the objective value of the primal problem by 1 . Since p ∗ = 1 , it immediate to see that part 4 of Assumption 2.5 also holds, because now the function u i ( x i ) + ⟨ p ∗ , -a i ( x i ) ⟩ becomes ⟨ 1 , x i ⟩ + ⟨ 1 , -x i ⟩ , which is always non-negative for any x i ∈ X i . Moreover, the constraint to be dualized is a linear constraint; thus Assumption 3.7 strong duality holds.

Lastly, let dataset D be drawn from Unif {D 0 , D 1 } , i.e. uniformly distributed over these two datasets.

Step 2: Identify necessary conditions on JDP algorithm A

Let x A ( D ) := ( x A 1 ( D ) , . . . , x A n ( D )) be the output of ( ε, δ ) -JDP algorithm A . The self postprocessing Lemma 4.1 says that if we process each x A i with i 's own data, the resulting output is still JDP. That implies (Id( x A 1 ) , . . . , Id( x A n )) should also be ( ε, δ ) -JDP. Moreover, the postprocessing Lemma 4.2 indicates that the aggregated consumption, excluding agent n 's, should be indistinguishable even if agent n 's data is changed, i.e.,

̸

̸

Instantiating the above inequality with D = D 0 , D ′ = D 1 , and S = ( nγ 1 -c 1 , nγ 1 ] with c := min { 1 , nγ } leads to

̸

̸

We can simplify the preceding inequality by changing decision variables from bold format x i to normal format x i , because by construction of requests z, z 1 , z 0 , each decision vector x i := 1 x i is fully determined by a single scalar x i . This leads to the following equivalent inequality,

̸

̸

̸

Moreover, when dataset is D 1 , agent n should always be assigned c unit, forcing resource available for others to be at most nγ -c . Thus, if A is an ( ε, δ ) -JDP algorithm outputting a feasible allocation with probability 1, then ∑ i = n x A i ( D 1 ) ≤ nγ -c w.p. 1, and (23) becomes:

̸

Therefore, we get a necessary condition (24) for any A being ( ε, δ ) -JDP: when dataset is D 0 , Algorithm A should not assign many resources to other agents, and only with a small probability δ , other agents can get at least nγ -c resources.

Step 3: Reformulate the minimax bound

Let ∆ ∈ [0 , nγ ] be a constant to be determined later, and let S + ∆ := ( nγ -∆ , nγ ] , S -∆ := [0 , nγ -∆] be two disjoint regions partitioning the interval [0 , nγ ] . Let I ∈ { 0 , 1 } m be an indicator so that 1 in j 's position implies ( ∑ n i =1 x A i ) j ∈ S + ∆ ; otherwise, in S -∆ . Let event E I := { ( x 1 , . . . , x n ) : ✶ { ( ∑ n i =1 x i ) j ∈ S + ∆ } = I j , ∀ j ∈ [ m ] } . With these notations and the law of total probability, we first rewrite the expression of Algorithm A 's utility when dataset is D 0 as:

We observe that when event E I happens, the resource consumed (∑ n i =1 x A i ( D 0 ) ) j is no more than either nγ -∆ or nγ , depending on I j . Hence, we can reach an upper bound expressed by I j :

Due to the construction of D 0 that x A i := 1 x A i , the probability terms in the preceding inequality should be the same among all j = 1 , . . . , m ; thus, we have

Moreover, with D 0 , the allocation to agent n should be x A n ( D 0 ) = 0 , modifying the preceding inequality into

̸

We note that with either dataset D 0 or D 1 , the optimal utility obtained in the non-private setting is always mnγ , i.e., F ( x ∗ ( D 0 )) = F ( x ∗ ( D 1 )) = mnγ . As a result, the minimax regret is lower bounded as follows:

̸

where the second inequality is by considering D 0 and plugging in (25).

Step 4: Combine all together

̸

Suppose ε ≥ max { 1 , 1 / ( nγ ) } . If we set ∆ = 1 ε , the r.h.s. of (26) becomes m 2 ε · inf A Pr [ ∑ i = n x A i ( D 0 ) ∈ [0 , nγ -1 /ε ] ] . Since 1 /ε ≤ min { 1 , nγ } = c , it follows that

̸

̸

̸

Plugging the preceding inequality back into (26) immediately leads to

which completes the proof.

B.4 Discussion on lower bounds for general ε

The lower bound in Theorem 4.3 only holds for ε ≥ max { 1 , 1 / ( nγ ) } . Intuitively, this is because we only use feasibility of A to prove the lower bound: 'we reserve some resource exclusively for agent k .' The optimality condition of A in the minimax expression inf A sup D is not used. Noticing this oversight, we provide a lower bounding problem that fills this gap.

We start with expressing DP in terms of hockey-stick divergence. Suppose µ and µ ′ are two probability measures that are absolutely continuous with respect to each other. Let H e ε ( X,Y ) :=

̸

Let M be an ( ε, δ ) -DP mechanism, and let M ( D ) -k := Π -k M ( D ) be the random vector of n -1 elements projected to others than agent k . Here are some relationships among indistinguishability, DP, and JDP.

( ε, δ ) -indistinguishability : if H e ε ( X,Y ) ≤ δ , then X is said to be ( ε, δ ) -indistinguishable from Y ;

( ε, δ ) -DP : for a mechanism M , if sup D∼D ′ H e ε ( M ( D ) , M ( D ′ )) ≤ δ , we say M is ( ε, δ ) -DP [BBG18];

( ε, δ ) -JDP : for a mechanism M , if sup D∼D ′ ,k H e ε ( M ( D ) -k , M ( D ′ ) -k ) ≤ δ , we say M is ( ε, δ ) -JDP.

It is self-evident from these relationships that DP requires indistinguishability between outputs for any pair of neighboring datasets, while JDP requires indistinguishability between projected outputs without agent k for any pair of neighboring datasets.

To obtain another lower bound exploiting optimality of A , we can follow the idea and notation in the proof of Theorem 4.3, but consider different neighboring datasets:

̸

It is obvious D 1 -∼ D 1+ , and D 1+ ∼ D 1+ , 2+ are pairs of neighboring datasets. Let S -k ( x A ) := ∑ i = k x A i be the total consumption excluding agent k . By the definition of ( ε, δ ) -JDP, any JDP algorithm A should satisfy following necessary constraints:

The variable x A ( D 1+ , 2+ ) in both constraints (28) and (29) should be the same, and thus links both constraints. Moreover, the variable x A ( D 1 -) in constraints (27) and (30) are the same. Therefore, the four constraints form a loop restricting each other.

The four constraints (27)-(30) are necessary conditions for any JDP algorithm. Furthermore, the dataset D 1 -here is essentially the same as D 0 in the proof of Theorem 4.3; thus all analysis for D 0 can be borrowed. Consequently, following the same analysis for (26), we get a lower bound expressed by an inf problem:

We believe analyzing the inf problem with tools in [BBG18], such as couplings, is promising. After this base case, one can easily extend the set of constraints (27)-(30) to take account of more loops.

C Additional results of numerical experiments

All experiments were run on a PC with an AMD 3700X CPU, 16GB memory; no GPU was used. The algorithm was implemented in Python 3.11, and optimization was solved by scipy.optimize.minimize method. Source code is available in supplementary materials. Datasets are publicly available as discussed in the main text.

C.1 Workforce scheduling

When running the algorithm, the only parameter different from Theorem 3.10 is K := 1 . 1 u/ ( γb ) , where Theorem 3.10 suggests K := 2 u/ ( γb ) . The constant factor is modified from 2 to 1.1, as a result of hyperparameter tuning. We discuss the tuning and its impact later.

We first report optimality gaps in Table 4. It is clear that our algorithm, equipped with either potential

Notes. Werun our algorithm A Noisy Dual MD with two potential functions: negative entropy (abbr. ne) and squared ℓ 2 -norm function (abbr. ℓ 2 ). Settings are the same as in Figure 1. bold =better functions, always outperform the existing method. However, the performance of ' A w. neg.entr' seems independent of ε and the performance of [HHRW16] seems worse when ε increases, both of which are strange. The observations here are because the optimality gap itself does not reflect the whole picture, for example, constraint violations are not reflected. We thus report constraint violations in Table 5 below. It should be clear from Table 5 that constraint violations are reduced

when ε increases, which indicates that our algorithm can maintain a good optimality performance while better reducing constraint violations. While [HHRW16] has the lowest constraint violations, our algorithms' constraint violations are only slightly worse.

Runtime For this workforce scheduling problem, the runtime of our algorithm varies from potential function to potential function, see Table 6.

Impact of 'K_constant' From our experiments, we found that the constant factor of K significantly impacts algorithms' performance. To avoid confusion, we want to point out that the algorithm by [HHRW16] set K_constant to be 2, a fixed value. But for all results in Appendix here, we let their K_constant change as well, in order to gain more understandings of its impact. We find that, when the constant is close to 1, the final allocations are more likely to achieve a smaller optimality gap, but violate constraints more severely. On the other side, when the constant increases, the final allocations will violate fewer constraints, but end with a larger optimality gap. In other words, the constant of K trade-offs between optimality and constraint violations. To see the trade-off more clearly, we draw Figure 3. The x-axis represents the constant of K , the left y-axis is optimality gap, and the right y-axis is total constraint violation; and for all curves, the lower the better. It is evident that when K_constant is small, we get smaller gaps but higher constraint violations. Additionally, when ε takes large values (say, ε = 5 , 10 ), constraint violations of all algorithm are at almost the same levels, but our algorithms have better optimality performance. One may notice that when K_constant=1, the optimality gap is negative, which means F ( x A ) ≥ F ( x ∗ ) . Considering x A may violate some constraints, this phenomenon is possible.





















ε

=1













ε

=2













ε

=5













ε

=10

























C.2 Assignment Problems

We repeat Table 3 below for discussion convenience. When reading the table, we should keep in mind

Notes. Algorithm parameters: K = 1 . 1 u/ ( γb ) , δ = . 01 , T = 10 4 . Other settings follow Theorem 3.10. MD_l2 and MD_ne means potential function is squared ℓ 2 and negative entropy, respectively. For three cases n = 800 , 1500 , 3000 , we set resource level γ = 0 . 1 , 0 . 05 , 0 . 02 , respectively. For all values in the table, the lower the better. Bold =better.

that optimality gaps should be understood together with constraint violations. Some observations from the table are:

When ε is small ( ε = 1 , 2 ), while [HHRW16] achieves the lowest optimality gaps in most instances, it achieves them at severe constraint violations, which might be not acceptable in practice. In contrast, our algorithms have reasonably satisfying optimality performances and much smaller constraint violations.

When ε gradually grows, our algorithms have a clear pattern of convergence, i.e., smaller optimality gap and fewer constraint violations; but the pattern for [HHRW16] is not clear enough. We conjecture that this is because of high-variance noises injected.

If constraint violation matters, MD_ne is most preferred.

Runtime Runtimes are reported in Table 8. Compared to existing methods, our algorithms do not observe a significant increase in runtime. Moreover, as we set T = 10 4 , we can in fact complete training within one hour, even for the largest case with n = 3000 .

Dual convergence Figure 4 shows the progress of dual variables convergence. The y-axis is the gap between dual variables by our algorithms and the optimal dual variables, while x-axis is the training progress. Shadow areas indicate standard deviations. It is evident that when ε is larger, dual variables converge faster to optimal values, and stay around the optimal values in remaining iterations. When ε is small, say ε = 1 , dual variables still show a converging tendency. Moreover, dual variables by MD_ne converge much faster, which further justifies its better performance observed earlier.















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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies a class of convex resource allocation problems, in which the utilities and constraints are private (and bounded).\nThe paper proposes a simple algorithm that applies mirror descent to the dual problem (while the update of the primal variables is assumed to be exact).\nThe main technical result is an improvement of the utility bound (improves dependency on number of constraints)\n\nSTRENGTHS:\n- The paper is well-written, and the technical results are interesting  (though I did not carefully review proofs)\n- This class of convex problems (with private utilities and constraints) is well motivated and seems relevant in practice.\n- There is a concrete improvement by a factor of m in the utility upper bound for this class, under strong duality assumptions.\n\nWEAKNESSES:\n- Algorithm 1 assumes that the maximization problem (line 3) is solved exactly. This might be realistic for a subclass of problems (for instance when utilty and constraints are linear -- and I suppose this is true in experiments), but the analysis considers a much more general class, for which exact updates are generally not possible (so I find Table 1 to be a bit misleading about this). For the analysis to be complete, this would require allowing inexact updates in line 3. Can the authors work out the details in this case (please explain in the rebuttal)? Otherwise this should be clearly stated as an assumption and reflected in Table 1.\n- Near optimality: The lower bound assumes that the solution is feasible, while the utility upper bound is for infeasible solutions. I don't think you can reasonably conclude that \"noisy mirror descent is near optimal\" (line 258). It should be made clear that optimality is still very much open. The title should be revised too.\n- Dependence on $\\underline{b}$ in Theorem 3.10: it seems one is trading-off a $\\sqrt m$ utility improvement for a $1/\\underline b^2$ term (both in utility and feasibility) which could be arbitrarily bad depending on the problem instance. Is this unavoidable? Can the authors comment?\n\nMinor:\n- I am not sure how meaningful the \"compulsory requests\" relaxation is, as this requires some restriction on $p^*$, and comes at the expense of simplicity/presentation.\n- the introduction mentions personalized recommendation as a motivating example. Can the authors provide more details or references? How is this problem modeled as a constrained resource allocation problem?\n- Theorem 3.5 and 3.6 are written in terms of $p^{(1)}$. Presumably the bounds in Table 1 use a specific choice of $p^{(1)}$. It would be good to comment on that.\n- Using $u$ both for utility functions and \"violation levels\" (line 183) is confusing. Please choose a different notation.\n- In Condition 3.1, I believe you technically want to require differentiability on relint W, and not on W (otherwise your negative entropy examples would not satisfy this -- please comment on whether this is problematic in your proofs).\n- Some quantities are not defined (even if standard, please make sure to define them to make the paper self-contained), for example the $-i$ notation and strong convexity wrt $\\|\\cdot\\|_p$.\n- Assumption 2.4.3 should be rephrased. Maybe \"the constant $\\gamma$ in (4) is assumed to be ...\"\n\nQUESTIONS:\nPlease see weaknesses section.\n\nLIMITATIONS:\nAdditional limitations should be discussed:\n- The near optimality claim is not fully justified (given the feasibility assumption in the lower bound).\n- The algorithm and analysis assume primal updates can be done exactly.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses the allocation problem under Joint Differential Privacy within the framework of a convex consumption function, a concave utility function, and a convex 'personal' domain.\n\nThe contributions of this work are threefold. Firstly, it derives results similar to those in previous research, but notably, it does so using only weak duality. Secondly, the paper improves the previously established optimality gap by leveraging strong duality. Lastly, it establishes a lower bound for the Joint Differential Privacy allocation problem, specifically for algorithms whose outputs are feasible allocations, within a certain regime of the epsilon parameter.\n\nSTRENGTHS:\nThe work effectively distinguishes between the capabilities of weak and strong duality. In the case of strong duality, it achieves a significant improvement by reducing the dependency on the quantity of resources $m$ by a square root factor. This improvement, however, comes with the introduction of polylogarithmic dependencies in both the duality gap and the constraint violation.\nThe authors also present a lower bound for the Joint Differential Privacy (JDP) allocation problem under the assumption of strong duality.\n\nWEAKNESSES:\nThe text could benefit from increased clarity in certain areas. For example, it is not immediately clear in the abstract that the lower bound is derived using the strong duality assumption. While this is clarified later in the paper, it would be beneficial for the abstract to be more specific on this point.\nAdditionally, it is not clear to me in what sense the lower bound is near matching the upper bound. It seems to me that there is an extra $m$ factor in the lower bound, which makes the difference not merely polylog. Am I missing something?\n\nQUESTIONS:\nIt caught my attention that the upper bound is for an algorithm that can violate the constraints, but the lower bound is for one that always output feasible allocations. Is it significantly more difficult to analyze a lower bound for the first type of algorithm (fixing a range of constraint violation)?\n\nLIMITATIONS:\nYes, the authors have adressed the limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe submission studies jointly differentially private algorithms for resource allocation problems, which are a broad generalization of packing linear programs. The work addresses this challenge by considering a primal-dual formulation of the problem, and running a noisy mirror descent algorithm on the dual. Standard analyses of the mirror descent method and the impact of noise addition then provide bounds on the suboptimality and approximate feasibility of the method. The most interesting result is that under strong duality, the approach yields bounds which are polylogarithmic on the number of resources. The intuition behind this result is that strong duality provides bounds on the magnitude of an optimal dual solution. This suffices to more accurately locate the dual fesible region, which allows the use of entropy regularization for the noisy mirror descent method. The paper also provides lower bounds, which yield some insight on the possible near optimality of the method.\n\nSTRENGTHS:\n1. The generality of the class of allocation problems considered.\n2. The insights derived from duality and the mirror descent algorithm.\n3. Numerical results are encouraging.\n\nWEAKNESSES:\n1. I am confused about how the lower bound of Theorem 4.3 compares to the upper bounds. The upper bounds of the paper provide in-expectation guarantees for both suboptimality and infeasibility; moreover, the latter is quantified in an $\\ell_{\\infty}$-sense (which I think it's the right approach in this case). By contrast, the lower bound is expressed for algorithms which are almost surely feasible, leading to the expression that only involves the objective. Since the upper bounds do not contain linear factors in m, I don't see how this lower bound is optimal (or that it gives insight on the efficiency of the upper bound).\n\n2. I also left some more specific comments in the questions section. I believe clarifying these it is very important for the submission to be publishable (and I also think these should be easy fixes).\n\nQUESTIONS:\nMAJOR QUESTIONS:\n1. About point 1 in weaknesses, it would be more consistent if the authors try to:\na) Provide high-probability results for the constraint violation upper bounds (this should be easy, using the high probability tail bounds of Gaussian noise addition).\nb) Provide lower bounds for joint DP resource allocation problems.\n2. About the upper bounds without strong duality, it is unclear to me how the set $\\mathcal{W}$ should be chosen in this case. I think that under quadratic potential they can just let $\\mathcal{W}$ to be the whole space, but please clarify.\n\n\nMINOR QUESTIONS:\n1. In page 3, I don't understand why is it claimed that the dual may not be convex. The dual objective is written as a maximum of functions which are affine in p, so it should always be convex.\n2. The paper does not accurately represent the work done on differentially private versions of mirror descent. The authors should make a more thorough search about this topic, in order to put their work in context.\n3. If I understand correctly, the vector $b$ represents the resource capacities, which it is reasonable to expect them to be public information. Either way, I believe the comment right before theorem 3.10 comes way too late, and the discussion about what is private and what is public it should be brought up much earlier.\n4. The paragraph right after Theorem 3.10 about \"purchasing resources\" I also found it very confusing. Please make efforts to clarify it.\n5. The last comment at the proof ot Theorem 3.6 should not be in the appendix (neither in the proof). This is a more practical consideration that does not belong to the proof, and it should be in the main file instead.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-27"}