Title: Quantum Speedups for Minimax Optimization and Beyond
Venue: NeurIPS 2025
Accepted: yes
Keywords: minimax optimization

quantum computing
Abstract: This paper investigates convex-concave minimax optimization problems where only the function value access is allowed.
We introduce a class of Hessian-aware quantum zeroth-order methods that can find the $\epsilon$-saddle point within $\tilde{\mathcal{O}}(d^{2/3}\epsilon^{-2/3})$ function value oracle calls.
This represents an improvement of $d^{1/3}\epsilon^{-1/3}$ over the $\mathcal{O}(d\epsilon^{-1})$ upper bound of classical zeroth-order methods, where $d$ denotes the problem dimension.
We extend these results to $\mu$-strongly-convex $\mu$-strongly-concave minimax problems using a restart strategy, and show a speedup of $d^{1/3}\mu^{-1/3}$ compared to classical zeroth-order methods.
The acceleration achieved by our methods stems from the construction of efficient quantum estimators for the Hessian and the subsequent design of efficient Hessian-aware algorithms.
In addition, we apply such ideas to non-convex optimization, leading to a reduction in the query complexity compared to classical methods.

Quantum Speedups for Minimax Optimization and Beyond

Abstract

This paper investigates convex-concave minimax optimization problems where only the function value access is allowed. We introduce a class of Hessianaware quantum zeroth-order methods that can find the /epsilon1 -saddle point within ˜ O ( d 2 / 3 /epsilon1 -2 / 3 ) function value oracle calls. This represents an improvement of d 1 / 3 /epsilon1 -1 / 3 over the O ( d/epsilon1 -1 ) upper bound of classical zeroth-order methods, where d denotes the problem dimension. We extend these results to µ -stronglyconvex µ -strongly-concave minimax problems using a restart strategy, and show a speedup of d 1 / 3 µ -1 / 3 compared to classical zeroth-order methods. The acceleration achieved by our methods stems from the construction of efficient quantum estimators for the Hessian and the subsequent design of efficient Hessian-aware algorithms. In addition, we apply such ideas to non-convex optimization, leading to a reduction in the query complexity compared to classical methods.

1 Introduction

We consider the following unconstrained minimax problem:

where h ( · , · ) is convex in x and concave in y . Let d = m + n , z = ( x /latticetop , y /latticetop ) /latticetop ∈ R d , and define f ( z ) /defines h ( x , y ) . The above problem has received considerable attention in the field of machine learning due to its wide applications, including fairness-aware learning [52, 32], AUC maximization [51, 20], robust optimization [3], game theory [48], and reinforcement learning [15, 43].

There are numerous classical algorithms to solve the convex-concave minimax problem (1). Firstorder methods, such as the optimistic gradient descent ascent (OGDA) [44, 36], extra gradient method (EG) [27, 38], along with their variants [40, 39] find an /epsilon1 -saddle point within O ( /epsilon1 -1 ) gradient oracle queries when h ( · , · ) is smooth. When ∇ 2 h ( · , · ) is Lipschitz continuous, second-order

methods offer faster convergence rates. The Newton proximal extra gradient method (NPE) [37] and its cubic-regularized realization [31, 22] require O ( /epsilon1 -2 / 3 ) queries to the second-order oracle. Very recently, Chen et al. [11] further improved the second-order oracle complexity to ˜ O ( /epsilon1 -4 / 7 ) . These results have been generalized to the cases where the p -th order derivative of h ( · , · ) is Lipschitz, with query complexity of O ( /epsilon1 -p p +1 ) to the p -th order oracle [30, 23, 24].

However, access to the gradient oracle or higher-order oracles of h ( · , · ) is not always available, as the calculation of the exact gradient (or higher order) information can be expensive or even infeasible [42, 35, 25, 53]. This necessitates the design of efficient derivative-free algorithms to solve equation (1). Beznosikov et al. [4] proposed a gradient-free algorithm with O ( d/epsilon1 -2 ) query complexity to h ( · , · ) for the smooth convex-concave minimax problem (1). Subsequently, this rate was improved to O ( d/epsilon1 -1 ) by Sadiev et al. [45], whose dependence on /epsilon1 -1 matches the lower bound for first-order methods [54].

The aforementioned methods for minimax problems are all designed for classical computing machines. However, the advantage of quantum computing has been investigated for various optimization problems. Accessing quantum counterparts of the classical oracles always leads to better query complexities or even a breakthrough over the classical lower bounds, including convex [47, 8, 9, 55] or non-convex optimization [46, 19], semi-definite programming [5, 6], non-smooth optimization [8, 33, 28], stochastic optimization [41, 55, 46], and so on. There are also some previous studies on some specific minimax problems, including the zero-sum game [29, 16] and minimizing the maximal loss [49]. However, it remains an open question whether quantum speedup is available for general convex-concave minimax problems. Thus, the first question we aim to address is:

Can quantum zeroth-order methods be designed to surpass the O ( d/epsilon1 -1 ) query complexity of classical zeroth-order methods, thereby demonstrating a quantum speedup for general convex-concave minimax problems?

A natural approach to achieve this is to leverage the fast gradient estimator [26], which can approximate the gradient of a smooth objective function within ˜ O (1) queries to the quantum function value oracle, based on the quantum Fourier transform [50]. In convex optimization, employing the fast gradient estimator within cutting-plane algorithms and gradient descent leads to improved query complexities compared to classical zeroth-order methods. Specifically, the quantum cuttingplane method [8] and the quantum gradient descent method [2] achieve query complexities of ˜ O ( d ) and ˜ O ( /epsilon1 -1 ) , respectively, improving upon the ˜ O ( d 2 ) and ˜ O ( d/epsilon1 -1 ) complexities of classical zerothorder methods by a factor of d . For non-convex optimization, a similar strategy has been used in the design of quantum perturbed AGD [19], resulting in an improved query complexity of ˜ O ( /epsilon1 -7 / 4 ) compared to the ˜ O ( d/epsilon1 -7 / 4 ) complexity of the classical zeroth-order method [53]. These methods use the quantum gradient estimators to achieve the same query complexities as classical first-order methods by accessing only zeroth-order oracles, which reduces the dependency on d . On the other hand, the oracle complexity of classical second-order methods enjoys a better dependency on /epsilon1 -1 when compared to the classical first-order methods. This motivates us to ask:

Can we go beyond quantum estimation for the gradient and design efficient Hessian-aware quantum zeroth-order algorithms with better dependence on both /epsilon1 -1 and d ?

In this paper, we provide an affirmative answer to the above two questions. To this end, we develop a quantum estimator for the Hessian matrix using the finite difference method and design a novel Hessian-aware zeroth-order optimization framework. We summarize our contributions as follows.

For the convex-concave problem, we propose a Hessian-aware quantum zeroth-order method (HAQZO), with query complexity of ˜ O ( d/epsilon1 -2 / 3 ) to find the /epsilon1 -saddle point, which surpasses the classical zeroth-order method by a factor of /epsilon1 -1 / 3 . We further accelerate such query complexity to ˜ O ( d + d 2 / 3 /epsilon1 -2 / 3 ) by proposing a double-loop Hessian-aware quantum method (HAQZO + ) that can reuse the Hessian estimators. HAQZO + accelerates the classical algorithms in terms of /epsilon1 -1 and d . We compare HAQZO and HAQZO + with the existing method in Table 1. The detailed analysis of HAQZO and HAQZO + can be found in Sections 4.1 and 4.2, respectively.

For the strongly-convex-strongly-concave problem, we apply the restart strategy in HAQZO + and propose Restart-HAQZO + . We prove that Restart-HAQZO + finds the /epsilon1 -point with query complexity of ˜ O ( d + d 2 / 3 ( L 2 /µ ) 2 / 3 ) , outperforms the classical method by a factor d 1 / 3 µ -1 / 3 .

(

)

(

)

(

)

The comparison of the query complexities can be found in Table 2 and the detailed analysis is presented in Section 4.3.

We further generalize the design of Hessian-aware quantum methods to solve non-convex problems with Lipschitz continuous Hessian. We propose the quantum cubic regularized-Newton method (QCNM) with query complexity of ˜ O ( d + d 1 / 2 /epsilon1 -3 / 2 ) to find the /epsilon1 -stationary point, which is better than all classical zeroth-order algorithms. The proposed QCNM method also enjoys an improved quantum query complexity over the existing state-of-the-art quantum algorithm when d = O ( /epsilon1 -1 / 2 ) , demonstrating the power of designing Hessian-aware quantum algorithms. We compare QCNM with existing classical algorithms and quantum algorithms in Table 3 and present the results in Section 5.

2 Preliminaries

We make the following assumptions on f ( z ) /defines h ( x , y ) .

Assumption 2.1. We assume f ( z ) = h ( x , y ) is convex in x and concave in y .

Assumption 2.2. We assume the f ( · ) , ∇ f ( · ) , ∇ 2 f ( · ) are L 0 , L 1 , and L 2 -Lipschitz continuous, respectively, i.e. we have | f ( z ) -f ( z ′ ) | ≤ L 0 ‖ z -z ′ ‖ , ‖∇ f ( z ) -∇ f ( z ′ ) ‖ ≤ L 1 ‖ z -z ′ ‖ , and for any z , z ′ ∈ R d .

We aim to find the approximate saddle point [39, 31, 12], which is defined as follows.

be the saddle point of function f ( · ) . For a given point ˆ z /defines [ ˆ x ˆ y ] , we let β sufficiently large such that max {‖ ˆ x -x ∗ ‖ , ‖ ˆ y -y ∗ ‖} ≤ β holds, we define the restricted gap function as

Definition 1 (Nesterov [39]) . Let B β ( w ) be the ball centered at w with radius β . Let z ∗ /defines [ x ∗ y ∗ ]

We call ˆ z an /epsilon1 -saddle point if Gap(ˆ z ; β ) ≤ /epsilon1 and β = Ω( ‖ z 0 -z ∗ ‖ ) .

diag ( I m , -I n ) . The Jacobian of F ( · ) can be written as ∇ F ( x ) = J ∇ 2 F ( x ) . The following proposition shows that F ( · ) is monotone if f ( · ) satisfies Assumption 2.1.

In the following context, we define F ( · ) as F ( z ) /defines J ∇ f ( z ) = [ ∇ x h ( x , y ) -∇ y h ( x , y ) ] , where J =

Proposition 2.3 (Lemma 2.7 [30]) . If f satisfies Assumption 2.1, then for all z , z ′ ∈ R d it holds that 〈 F ( z ) -F ( z ′ ) , z -z ′ 〉 ≥ 0 . For a given ˆ z ∈ R d , its gap can be bounded by Gap(ˆ z ; β ) ≤ max z ∈ B √ 2 β ( z ∗ ) 〈 F (ˆ z ) , ˆ z -z 〉 .

We define the quantum evaluation oracle for a function f ( · ) .

Definition 2.4 (Quantum Function Evaluation Oracle) . A quantum evaluation oracle for a function f is defined as the following unitary transformation

Here ⊕ is the bit-wise XOR operation. We say that we have a quantum evaluation oracle for f with accuracy /epsilon1 0 if we have a quantum evaluation oracle for f , such that | f ( z ) -f ( z ) | ≤ /epsilon1 0 for all z .

˜ ˜ Remark 2.5 . The quantum advantages are stated in terms of query complexity on the function evaluation oracle. In many situations, query complexity dominates the computational complexity of the algorithm, which is a natural setting in both classical and quantum optimization. For example, considering the generalized linear model such that f ( x ) = h ( A T x ) where A ∈ R d × n . The circuit implementation of the oracle of f may involve dominating computational complexity if n /greatermuch d in this example, and our algorithm achieves meaningful quantum speedups under such a setting.

3 Gradient and Hessian Estimation via Quantum Function Evaluation Oracle

Before introducing our quantum algorithms, we first introduce the quantum estimators for the gradient and Hessian of the objective function by using the quantum evaluation oracle on f ( · ) , which are the critical components of our methods. These results are natural and direct extensions of Jordan's method [26] for the smooth objective. We do not consider them to be our primary technical contribution, but state them for completeness.

3.1 Quantum Gradient Estimator

Quantum gradient estimator is first proposed in [26] for the smooth objective, and its rigorous statement is given by Gilyén et al. [18], Chakrabarti et al. [8], van Apeldoorn et al. [47]. The following is one of the statements.

Lemma 3.1 (Lemma 2.2 [8]) . Let f be an L 0 -Lipschitz continuous and L 1 -smooth function. Given the access to a quantum evaluation oracle of f with /epsilon1 0 accuracy, then for /epsilon1 g ≥ /epsilon1 0 there is a quantum algorithm A ( f, /epsilon1 g , L 0 , L 1 , z ) which outputs an estimate ˜ ∇ f ( z ) of ∇ f ( z ) , satisfying that ∀ i ∈ [ d ] , Pr (∣ ∣ ∣ [ ˜ ∇ f ( z ) ] i -[ ∇ f ( z )] i ∣ ∣ ∣ ≥ 1500 √ L 1 d/epsilon1 g ) ≤ 1 3 . Moreover, the A algorithm uses O (1) queries to the quantum evaluation oracle and O ( d log L 0 dL 1 /epsilon1 g ) quantum gates.

The following lemma allows for an arbitrarily small failure probability δ ∈ (0 , 1) to the quantum gradient estimator, which generalizes the results above.

Lemma 3.2 (Quantum Gradient Estimator) . Let f ( · ) be a L 0 -Lipschitz continuous and L 1 -smooth function. Given the access to a quantum evaluation oracle of f with /epsilon1 0 accuracy, then for /epsilon1 g ≥ /epsilon1 0 , there exists a quantum algorithm QuantumGradient ( f, /epsilon1 g , L 0 , L 1 , z , δ ) which outputs an estimate ˜ ∇ f ( z ) of ∇ f ( z ) , satisfying

∥ ˜ ∥ Moreover, QuantumGradient uses O (log( d δ )) queries to U f and O ( d log( L 0 dL 1 /epsilon1 g ) log( d δ ) ) gates.

3.2 Quantum Hessian-vector Estimator and Quantum Hessian Estimator

In this section, we show that the Hessian vector product of a smooth object function can also be constructed within the ˜ O (1) quantum function evaluation oracle. Furthermore, since ∇ 2 f ( z ) = [ ∇ 2 f ( z ) e 1 , · · · , ∇ 2 f ( z ) e d ] , the Hessian of a smooth object function can be constructed within the ˜ O ( d ) quantum function evaluation oracle.

We formally present our construction of the quantum Hessian vector product estimator in Algorithm 1 and state its complexity in the following lemma.

Algorithm 1 QuantumHessianVector( f, /epsilon1 hv , L 0 , L 1 , L 2 , z , v , δ )

1: M = ‖ v ‖ 2

3: ˜ ∇ f ( z ) := QuantumGradient ( f, /epsilon1 hv , L 0 , L 1 , z , δ/ 2) 4: ∇ f ( z +∆ v ) := QuantumGradient ( f, /epsilon1 hv , L 0 , L 1 , z +∆ · v , δ/ 2)

2: ∆ = 20 √ 15 /epsilon1 1 / 4 hv M -1 / 2 L -1 / 2 2 L 1 / 4 1 d 1 / 2

˜ 5: Return 1 ∆ ( ˜ ∇ f ( x +∆ v ) -˜ ∇ f ( x ) )

Lemma 3.3 (Quantum Hessian Vector Estimator) . Suppose f satisfies Assumption 2.2. Given the access to U f with /epsilon1 0 accuracy, let hv = QuantumHessianVector ( f, /epsilon1 hv , L 0 , L 1 , L 2 , z , v , δ ) be the output of Algorithm 1 where /epsilon1 hv ≥ /epsilon1 0 , then it holds that

Remark 3.4 . He et al. [21] proposed a quantum estimator for a row of a Hessian, which can be viewed as a special case of our quantum Hessian vector estimator. Besides, they do not provide results on the query complexity and gate complexity.

∥ ∥ Moreover, Algorithm 1 uses O (log( d δ )) queries to U f and O ( d log( L 0 dL 1 /epsilon1 hv ) log( d δ ) ) gates.

Given the Hessian vector estimator, we are ready to construct the Hessian estimator by calculating the estimators of the Hessian vector set {∇ 2 f ( x ) e i } i ∈ [ d ] , which is formally given in Algorithm 2. The following results show how well the output of Algorithm 2 approximates ∇ 2 f ( z ) .

∥ ˜ ∥ Moreover, Algorithm 2 uses O ( d log( d δ )) queries to U f and O ( d 2 log( L 0 dL 1 /epsilon1 H ) log( d δ ) ) gates.

Lemma 3.5 (Quantum Hessian Estimator) . Suppose f satisfies Assumption 2.2. Given access to a quantum evaluation oracle of f with /epsilon1 0 accuracy and /epsilon1 H ≥ /epsilon1 0 , let ˜ ∇ 2 f ( z ) = QuantumHessian ( f, /epsilon1 H , L 0 , L 1 , L 2 , z , δ ) be the output of Algorithm 2, then it holds that

Algorithm 2 QuantumHessian( f, /epsilon1 H , L 0 , L 1 , L 2 , z , δ )

1: H = 0 d × d

3: H [ i, :] = QuantumHessianVector( f, /epsilon1 H , L 0 , L 1 , L 2 , z , e i , δ/d )

2: for i ∈ [ d ]

4: ˜ H = 1 2 ( H + H /latticetop )

5: Return ˜ H

Algorithm 3 HAQZO ( z 0 , T, L 0 , L 1 , L 2 , δ )

1: for t = 0 , · · · , T -1 do

2: Choose /epsilon1 1 ,t > 0 and /epsilon1 H ,t > 0

3: ˜ g t = QuantumGradient ( f, /epsilon1 1 ,t , L 0 , L 1 , z t , δ/ (3 T )) and g t = J ˜ g t

5: Compute the inexact cubic step i.e. find z t +1 / 2 that satisfies

4: ˜ H t = QuantumHessian ( f, /epsilon1 H ,t , L 0 , L 1 , L 2 , z t , δ/ (3 T )) and H t = J ˜ H t

6: λ t = 6 ( L 2 ‖ z t -z t +1 / 2 ‖ + √ 1500 d 1 / 2 L 1 / 4 1 /epsilon1 1 / 4 1 ,t + √ 1500 d 2 L 1 / 4 1 L 1 / 2 2 /epsilon1 1 / 4 H ,t ) ) . 7: Choose /epsilon1 2 ,t > 0

8: ˜ v t +1 / 2 = QuantumGradient ( f, /epsilon1 2 ,t , L 0 , L 1 , z t +1 / 2 , δ/ (3 T )) and v t = J ˜ v t

9: z t +1 = z t -λ -1 t v t .

10: end for

11: return ¯ z T = 1 ∑ T -1 t =0 λ -1 t ∑ T -1 t =0 λ -1 t z t +1 / 2 .

4 Quantum Speedups for Minimax Optimization

In this section, we introduce quantum algorithms to find the /epsilon1 -saddle point for general convexconcave minimax problems. In Section 4.1, we propose a Hessian-aware algorithm with ˜ O ( d/epsilon1 -2 / 3 ) queries to the quantum function evaluation oracle, which outperforms the classical state-of-the-art algorithm by a factor of /epsilon1 -1 / 3 . We further improve such query complexity to ˜ O ( d 2 / 3 /epsilon1 -2 / 3 ) , which outperforms the classical algorithm by a factor of d 1 / 3 /epsilon1 -1 / 3 , by proposing a double-loop algorithm that reuses the Hessian estimators in Section 4.2. In Section 4.3, we generalize our results to stronglyconvex-strongly-concave problems.

4.1 Hessian-Aware Quantum Algorithm with Better Dependency on /epsilon1 -1

Our idea is to use the quantum gradient estimator and the quantum Hessian estimator to obtain a close approximation of F ( z ) and ∇ F ( z ) and then apply the Newton proximal extragradient framework [37]. We present our Hessian-aware quantum zeroth-order method (HAQZO) in Algorithm 3.

To analyze Algorithm 3, we first consider the following generalized NPE update:

where g t , v t , and H t are some approximations to F ( z t ) , F ( z t +1 / 2 ) , and ∇ F ( z t ) , which satisfy

The following lemma shows that the update of (6) recovers the convergence rates of the NPE method if δ 1 ,t , δ 2 ,t , and δ H ,t are small enough.

Remark 4.2 . We note that some prior works have also studied the inexact NPE methods [31, 1]. However, these methods only consider the case where the Hessian is inexact, while our Lemma 4.1 allows inexactness from both the gradient and the Hessian.

Because the iteration rule of Algorithm 3 can be interpreted as the generalized NPE update in (6) with high probability, we can determine the query complexity of Algorithm 3 by incorporating the quantum gradient and Hessian estimators. This result is formally stated in the following theorem.

Theorem 4.3. Under Assumptions 2.1 and 2.2, let R = Ω( ‖ z 0 -z ∗ ‖ ) , given desired accuracy /epsilon1 > 0 , we run Algorithm 3 with

then with probability at 1 -δ , Algorithm 3 finds the /epsilon1 -saddle point of f ( · ) with ˜ O ( dL 2 / 3 2 R 2 /epsilon1 -2 / 3 ) queries to U f , where ˜ O ( · ) hides the polylogarithm dependency on d , L 0 , L 1 , L 2 , /epsilon1 -1 , δ -1 , and R .

4.2 Hessian-Aware Quantum Algorithm with Better Dependency on /epsilon1 -1 and d

In this section, we further improve the query complexity of HAQZO by proposing a double-loop Hessian-aware quantum method HAQZO + in Algorithm 4, which is inspired by the recent advance in lazy Hessian methods [14, 12, 10, 34].

The main difference between Algorithm 4 and Algorithm 3 is that we eliminate calling quantumHessian in every iteration, but only call it at the snapshot point in iterations t when t mod m = 0 , and reuse such Hessian estimator in the next m iterations. In addition, we replace L 2 with a larger parameter M ≥ L 2 on Line 9 and Line 10 and tune it to guarantee convergence. We

Algorithm 5 Restart-HAQZO + ( z 0 , T, L 0 , L 1 , L 2 , M, m, S, δ )

1: z (0) = z 0

2: for s = 0 , · · · , S -1

3: z ( s +1) = HAQZO + ( z ( s ) , T, L 0 , L 1 , L 2 , M, m, δ/S )

4: end for

5: return z ( S )

first consider the following iteration rule

where π ( t ) /defines t -( t mod m ) and g t , v t , H π ( t ) are some approximations to F ( z t ) , F ( z t +1 / 2 ) , ∇ F ( z π ( t ) ) such that

The following lemma shows that the update of (8) still enjoys the rate of T -3 / 2 if the regularization term λ t is chosen large enough and δ 1 ,t , δ 2 ,t , and δ H are small.

The iteration rule of Algorithm 4 can also be interpreted as (8) with high probability. At each iteration, the algorithm calls QuantumGradient with ˜ O (1) quantum function evaluation queries to obtain g t and v t . Every m iterations, the algorithm calls QuantumHessian with ˜ O ( d ) quantum function evaluation queries to obtain H t . The following theorem provides the query complexity of Algorithm 4 with a proper choice of m = d .

Theorem 4.5. Under Assumptions 2.1 and 2.2, let R = Ω( ‖ z 0 -z ∗ ‖ ) , given desired accuracy /epsilon1 > 0 , we run Algorithm 4 with

then Algorithm 4 finds the /epsilon1 -saddle point of f ( · ) with ˜ O ( d + d 2 / 3 L 2 / 3 2 R 2 /epsilon1 -2 / 3 ) queries to U f with probability at 1 -δ , where ˜ O ( · ) hides the polylogarithm dependency on d , L 0 , L 1 , L 2 , /epsilon1 -1 , δ -1 , and R .

4.3 Restarted Hessian-Aware Quantum Algorithm for Strongly-Convex Strongly-Concave Minimax Optimization

In this section, we generalize our results to solve strongly-convex-strongly-concave minimax problems. We make the following assumption on f ( · ) , which is stronger than Assumption 2.1.

Assumption 4.6. We assume f ( z ) = h ( x , y ) is µ -strongly-convex in x and µ -strongly-concave in y for some µ > 0 .

We apply the restart strategy which is widely used in minimization [17] and minimax optimization [22, 30, 12] on our HAQZO + and propose Restart-HAQZO + in Algorithm 5. To avoid confusion, we use the superscript ( s ) to denote the parameters in the HAQZO + subroutine in the s -th iteration of Algorithm 5. The following lemma shows that by properly choosing the parameter in HAQZO + , ‖ z ( s +1) -z ∗ ‖ 2 will descend linearly with high probability.

Lemma 4.7. Under Assumptions 2.2 and 4.6, set the parameter in subroutine HAQZO + in the s -th iteration of Algorithm 5 as follows:

Lemma 4.7 means it is enough to set S = /ceilingleft log(1 //epsilon1 ) /ceilingright to obtain some z ( S ) such that ‖ z ( S ) -z ∗ ‖ 2 ≤ /epsilon1 . Given this, we are ready to present the query complexity of Algorithm 5.

Theorem 4.8. Under Assumptions 2.2 and 4.6, set the parameter in subroutine HAQZO + in the s iteration of Algorithm 5 as in Lemma 4.7 with m = d , and set S = /ceilingleft log( ‖ z (0) -z ∗ ‖ 2 //epsilon1 ) /ceilingright , then with probability at least 1 -δ , the output of Algorithm 5 satisfies that ‖ z ( S ) -z ∗ ‖ 2 ≤ /epsilon1 with ˜ O ( d + d 2 / 3 L 2 / 3 2 µ -2 / 3 ) queries to U f , where ˜ O ( · ) hides the polylogarithm dependency on d , L 0 , L 1 , L 2 , /epsilon1 -1 , δ -1 .

5 Extension to Non-convex Optimization

In the previous section, we have shown that, using quantum Hessian estimators, it is possible to design fast quantum algorithms which outperform the classical algorithms in terms of accuracy /epsilon1 -1 and dimension d . Wehighlight that the quantum estimators designed in Section 3 are not restricted to convex-concave minimax problems. In this section, we extend the idea of designing Hessian-aware quantum zeroth-order methods to non-convex problems

Definition 5.1. We say z is an /epsilon1 -stationary point of the nonconvex minimization problem (10) if it holds that ‖∇ f ( z ) ‖ ≤ /epsilon1 .

where f ( · ) is smooth but possibly not convex. We aim to find the /epsilon1 -stationary point of (10).

Wepresent the quantum cubic-regularized Newton methods in Algorithm 6, which replace the classical gradient and Hessian estimators in the zeroth-order CNM method [13] by the quantum estimators designed in Section 3. The following theorem gives the query complexity of Algorithm 6 to find the /epsilon1 -stationary point of f ( · ) .

Theorem 5.2. Under Assumption 2.2 and suppose f ∗ /defines min z ∈ R d f ( z ) > -∞ , given desired accuracy /epsilon1 > 0 , we run Algorithm 6 with

then with probability at least 1 -δ , the output of Algorithm 6 finds the /epsilon1 -stationary point of problem 10 with ˜ O ( d + d 1 / 2 L 1 / 2 2 ( f ( z 0 ) -f ∗ ) /epsilon1 -3 / 2 ) queries to U f , where ˜ O ( · ) hides the polylogarithm dependency on d , L 0 , L 1 , L 2 , /epsilon1 -1 , δ -1 .

6 Conclusion

In this paper, we have proposed quantum algorithms to speed up training for minimax optimization problems. Our Hessian-aware quantum zeroth-order method reduces the query complexity of the function evaluation oracle of the classical methods by a factor of d 1 / 3 /epsilon1 -1 / 3 and d 1 / 3 µ -1 / 3 for convex-concave and strongly-convex-strongly-concave problems, respectively. Moreover, we find that the proposed quantum oracles for estimating the Hessian matrix can be used to solve other important optimization problems, i.e. non-convex optimization. However, the query complexity of the proposed Hessian-aware quantum zeroth-order methods still depends on the dimension, and the quantum lower bound for this question is still unknown. We leave this for future work.

Acknowledgment

We thank the anonymous reviewers for their helpful suggestions. Zongqi Wan, Jialin Zhang, and Xiaoming Sun are supported by the National Natural Science Foundation of China Grants No. 62325210 and 12447107. Chengchang Liu is supported by the National Natural Science Foundation of China (624B2125). John C.S. Lui is supported in part by the GRF-14207721 and SRFS21224S02.

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A Auxiliary Lemmas

Lemma A.1. Given a positive sequence { λ t } T -1 t =0 , if ∑ T -1 t =0 λ 2 t ≤ C , then we have ∑ T -1 t =0 1 λ t ≥ T 3 / 2 √ C .

Proof. By Holder's inequality, we have that

Therefore,

Lemma A.2 (Lemma 4.2, [12]) . For any sequence of positive numbers { r t } t ≥ 0 , it holds for any m ≥ 2 that ∑ m -1 t =1 ( ∑ t -1 i =0 r i ) 2 ≤ m 2 2 ∑ m -1 t =0 r 2 t .

Lemma A.3 (Lemma 4.2, [14]) . For any sequence of positive numbers { r t } t ≥ 0 , it holds for any m ≥ 1 that ∑ m -1 t =1 ( ∑ t -1 i =0 r i ) 3 ≤ m 3 3 ∑ m -1 t =0 r 3 t .

B The Proof of Section 3

B.1 The Proof of Lemma 3.2

Proof. Running A ( f, /epsilon1 g , L 0 , L 1 , x ) for M times. Let ˜ ∇ ( m ) f ( z ) denotes the output estimates for m -th running. Then for each coordination, take the median of the output estimates, and return the resulting vector as the output of QuantumGradient ( f, /epsilon1 g , L 0 , L 1 , z , δ ) , denoted as ˜ ∇ f ( z ) .

For any i ∈ [ d ] , m ∈ [ M ] , X i,m denotes the indicator random variable of the event

∣ ∣ By Lemma 3.1, we have Pr( X m,i ) ≥ 2 3 and { X m,i } M m =1 are independent random variables. By Chernoff's bound,

Let Y i denote the event that ∑ M m =1 X i,m ≥ 5 9 M , then Pr( Y i ) ≥ 1 -e -13 M 500 . If Y i happens, we have ∣ ∣ ∣ [ ˜ ∇ f ( vz )] i -[ ∇ f ( z )] i ∣ ∣ ∣ ≤ 1500 √ L 1 d/epsilon1 g . By union bound, Pr( ∩ d i =1 Y i ) ≥ 1 -d · e -13 M 500 . Under event ∩ d i =1 Y i , we have ‖ ˜ ∇ f ( z ) - ∇ f ( z ) ‖ 2 ≤ √ 1500 2 d · L 1 d/epsilon1 g = 1500 d √ L 1 /epsilon1 g . Set M := O (log( d δ )) , we have Pr( ∩ d i =1 Y i ) ≥ 1 -δ .

Since we have invoked A for M times, the total query complexity is O (log( d δ )) and the total gate complexity is O ( d log L 0 dL 1 /epsilon1 g log d δ ) by Lemma 3.1.

B.2 The Proof of Lemma 3.3

Proof. Let ˜ ∇ f ( x ) and ˜ ∇ f ( x + ∆) be the quantum gradient estimates with probability at least (1 -δ/ 2) . By Lemma 3.2 and union bound, we have

‖ ˜ ∇ f ( x ) -∇ f ( x ) ‖ 2 ≤ 1500 d √ L 1 /epsilon1 hv and ‖ ˜ ∇ f ( x +∆) -∇ f ( x +∆) ‖ 2 ≤ 1500 d √ L 1 /epsilon1 hv hold with probability at least 1 -δ . Condition on this good event, we have

Since we have choose ∆ = 20 √ 15 /epsilon1 1 / 4 hv M -1 / 2 L -1 / 2 2 L 1 / 4 1 d 1 / 2 , it holds that

B.3 The Proof of Lemma 3.5

Proof. It holds that H [ i, :] estimates ∇ 2 f ( z ) e i with failure probability δ/d , then by Lemma 3.3, holds with probability 1 -δ . Under this event, we have

Therefore, we have

∥ ∥ ∥ -∇ ∥ ∥ ∥ 2 ≤ ∥ ∥ -∇ ∥ ∥ ≤ √ ∥ ∥ -∇ ∥ ∥ ≤ √ holds with probability at least 1 -δ .

C The Proof of Section 4

C.1 The Proof of Lemma 4.1

Proof. The iteration rule (6) means

Thus, we have

where the last inequality is due to the choice of λ t . We also have that

Since ‖ z t +1 / 2 -( z t -1 λ t v t ) ‖ = ‖ z t +1 / 2 -z t +1 ‖ , plugging the bound of (14) into (15) and using the fact that ‖ v t -F ( z t +1 / 2 ) ‖ ≤ δ 2 ,t , we have

We let R ≥ 10 ‖ z 0 -z ∗ ‖ and the choice of δ 1 ,t and δ 2 ,t means that we have

Let z = z ∗ in (16) and due to 〈 F ( z k +1 / 2 ) , z k +1 / 2 -z ∗ 〉 ≥ 0 for all k , we have

and

Thus, we have

Summing up the (16) from t = 0 to t = T -1 , for all z ∈ B √ 6 R ( z ∗ ) , we have

We then bound the regularization term ∑ T -1 t =0 λ 2 t by

Thus, we have

where the first inequality is from Proposition 2.3, ∗ is due to the monotone of F ( · ) , and the last inequality is due to ∑ T -1 t =0 1 λ t ≥ T 3 / 2 √ (864 L 2 2 +4) R 2 according to Lemma A.1.

C.2 The Proof of Theorem 4.3

Proof. According to Lemmas 3.2 and 3.5, we know that g t , H t , v t can be constructed within ˜ O (1) , ˜ O ( d ) , and ˜ O (1) quantum function evaluation oracle, respectively. The following statements

and

hold with probability at least ( 1 -δ T ) .

Let δ 1 ,t = R 2 10 T , δ H ,t = R √ T , and δ 2 ,t = min { λ t R 2 10 T ( ‖ z t +1 / 2 -z 0 ‖ + R ) , δ 1 ,t 2 } , we know that the condition of Lemma 4.1 holds with probability at least (1 -δ ) . Thus, the output ¯ z T of Algorithm 3 holds that with probability at least (1 -δ ) . The total query of quantum evaluation oracle can be bounded by

C.3 The Proof of Lemma 4.4

Proof. The iteration of (8) means

Thus, we have

where the last inequality is due to the choice of λ t . On the other hand, it holds that

We denote r t /defines ‖ z t +1 -z t ‖ , since r 2 t ≤ 2 ‖ z t +1 / 2 -z t ‖ 2 +2 ‖ z t +1 -z t ‖ 2 and that then it holds that

Summing up the above inequality from i = π ( t ) to π ( t ) + s -1 where 1 ≤ s ≤ m and take M ≥ mL 2 √ 3 , we have

where the last inequality is due to Lemma A.2. Combining the error from v t , we have that

Similar to the proof in Lemma 4.1, we let R ≥ 10 ‖ z 0 -z ∗ ‖ and the choice of δ 1 ,t and δ 2 ,t means

Let z = z ∗ in (25), we have

and

Thus, we have

Summing up the batch of (25), for all z ∈ B ( z ∗ , 3 √ 2 R ) , we have

We then bound the regularization term T -1 t =0 λ 2 t can be bounded by

Finally, we have

where the last inequality is due to Lemma A.1.

C.4 The Proof of Theorem 4.5

Proof. According to Lemmas 3.2 and 3.5, we know that g t , H , v t can be constructed within ˜ O (1) , ˜ O ( d ) , and ˜ O (1) quantum function evaluation oracle, respectively and

and hold with probability at least ( 1 -δ T ) .

Let δ 1 ,t = R 2 10 T , δ H ,t = R √ T , and δ 2 ,t = min { λ t R 2 10 T ( ‖ z t +1 / 2 -z 0 ‖ + R ) , δ 1 ,t 2 } , we know that the condition of Lemma 4.4 holds with probability at least (1 -δ ) . Thus, the output ¯ z T of Algorithm 4 satisfies that with probability at least (1 -δ ) . The total number of queries to the quantum evaluation oracle can be bounded by

C.5 The Proof of Lemma 4.7

Proof. We can obtain a good approximation of F ( z ( s ) i +1 / 2 ) , F ( z ( s ) i ) , and ∇ F ( z ( s ) π ( t ) ) with probability at least (1 -δ/S ) for all i ∈ [ T -1] . Recalling the proof of Lemma 4.4 in Appendix C.3, from (25), we have holds with probability at least (1 -δ/S ) . Assumption 4.6 means that

Let z = z ∗ in (30), we have

Summing up above inequality from i = 0 to i = T -1 , we have

The choice of δ ( s ) 1 ,t and δ ( s ) 2 ,t guarantees that

Thus we have

inequality, we have

Since z ( s +1) = ¯ z ( s ) T = ∑ T -1 i =0 z ( s ) i λ ( s ) i ∑ T -1 i =0 1 λ ( s ) i and z ( s ) = z ( s ) 0 , using the convexity of ‖ · ‖ 2 and by Jensen's

We then bound the term ∑ T -1 i =0 ( λ ( s ) i ) 2 by

it enough to guarantee the linear decent on ‖ z ( s ) -z ∗ ‖ such that

C.6 The Proof of Theorem 4.8

Proof. Using Lemma 4.7, we have that ‖ z ( s +1) -z ∗ ‖ 2 ≤ 1 2 ‖ z ( s ) -z ∗ ‖ 2 holds for all s ∈ [ S -1] with probability at least 1 -δ . Then we have ‖ z ( S ) -z ∗ ‖ 2 ≤ ( 1 2 ) S ‖ z (0) -z ∗ ‖ 2 ≤ /epsilon1 . For each call of the subroutine HAQZO + , the query complexity can be bounded by

Thus, the total query complexity can be bounded by

D The Proof of Section 5

We first present some useful results for one step of lazy CRN [14, 13]:

where g t and H π ( t ) are good estimations to ∇ f ( z t ) and ∇ 2 f ( z π ( t ) ) , respectively, such that

Lemma D.1 (Theorem 2.4 [13]) . Consider the cubic regularization step in (33) where g t and H π ( t ) satisfy (34) , then it holds that

Then we know that by properly choosing δ g , δ H , the CRN step can make the gradient small with a rate of O ( T -3 / 2 ) .

Lemma D.2. Under Assumptions 2.2, let ¯ z be uniformly chosen from { z i } T i =1 , generated by (33) , then it holds that

Proof. Summing up (35) from k = π ( t ) to t , then it holds that

where the last inequality is due to Lemma A.3 and M = 30 mL 2

Summing up (36) from 0 to T -1 , we have and

Now we are ready to prove Theorem 5.2.

D.1 The Proof of Theorem 5.2

Proof. The choice of /epsilon1 g and /epsilon1 H means that

hold with probability at least (1 -δ ) for all t ∈ [ T ] . Using Lemma D.2, we have

due to the convexity of x 3 / 2 , we have

We require using ˜ O (1) query complexity of function evaluation oracle to construct the gradient estimator for all T iterations and using ˜ O ( d ) query complexity to construct the Hessian estimator for T m +1 iterations at the snapshot point. Thus, the total query complexity of Algorithm 6 can be bounded by

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper tackles convex–concave minimax optimization with only zeroth-order (function-value) access by introducing Hessian-aware quantum zeroth-order algorithms that use fast quantum estimators requiring $\\tilde O(1)$ gradient and $\\tilde{O}(d)$ Hessian queries. The main scheme, HAQZO, achieves query complexity $\\tilde{O}(d\\epsilon^{-2/3})$, while a double-loop variant, HAQZO+, improves this to $\\tilde{O}(d^{2/3}\\epsilon^{-2/3})$, surpassing the classical $\\tilde{O}(d\\epsilon^{-1})$ rate. For $\\mu$-strongly-convex–strongly-concave games, a restarted method reaches $\\tilde{O}(d^{2/3}(L_2/\\mu)^{2/3})$ queries. Extending to non-convex problems, a Quantum Cubic-Regularised Newton Method (QCNM) finds $\\epsilon$-stationary points in $\\tilde{O}(d^{1/2}\\epsilon^{-3/2})$ oracle calls. These complexities establish quantum speedups for gradient-free optimization convex, strongly-convex, and non-convex settings.\n\nSTRENGTHS AND WEAKNESSES:\nFrom my perspective, this is a technically solid work providing quantum algorithms with provably better performance for minimax optimization over best-known classical counterparts given queries to only zero-th order queries to the function. As far as I know, this is the first work that provides quantum algorithms that outperform classical algorithms in all settings of minimax optimization given queries to either function value or gradients. In addition, the result is not a direct application of the well-known quantum gradient estimation algorithm. Instead, improved quantum Hessian-vector estimator and Hessian estimator are proposed with better performance than the best-known classical counterparts. Therefore, I am happy to vote for acceptance of this paper at NeurIPS 2025.\n\nThe weakness of this paper, if I have to provide one, is that this work provides no understanding of whether the query complexity achieved by HAQZO+ or restart-HAOZO+ is optimal in certain parameter regimes. This is also mentioned by the authors in the conclusion. However, I believe the contribution of this work by proposing the quantum algorithm with a provable quantum advantage is already significant enough.\n\nBesides, a minor comment is that I believe the overall presentation of the work can be significantly improved by adding a more detailed literature review regarding classical minimax optimization in different settings, including (strong) convex-(strong) concave functions, non-convex-(strong) concave functions, zeroth-order queries, first-order queries, etc.\n\nQUESTIONS:\nI hope the authors can respond and address my minor comment in the Strengths And Weaknesses part.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors provide a method for solving $\\epsilon$ min-max problems using a zeroth order quantum method that approximates Hessians.  Their algorithm claims a lower query complexity than previous work, improving upon previously known classical methods by a factor of $d^{1/3} \\epsilon^{1/3}$ for convex-concave problems, and $d^{1/3} \\mu^{1/3}$ for strongly convex-concave problems, where $d$  is the dimension of the domain, $\\epsilon$ is the error in the saddle point estimate, and $\\mu$ is the coefficient determining the strength of the strongly concave-convex problems.  The method relies on an oracle which computes function evaluations for a function $f$, which is convex in the first variable and concave in the second variable, as well as Lipschitz continuity in the 0th, 1st, and 2nd derivatives.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is very technically solid, and the appendices go into strong detail about many of the derivations.  They do a good job of listing the core assumptions of their method as well as providing very detailed appendices outlining the derivation of their key lemmas.  That said, the overall readability of the main text is somewhat difficult to follow - most likely due to the page limit.  I would have liked it if they referred readers to the appendices for proofs of their key points in the main text to make it easier to follow where some of the bounds were coming from.  It was also somewhat unclear how viable this method would be to implement on a fault tolerant computer, as hidden overheads in the oracle embedding or mapping onto hardware for general functions is not clear.  That said, this was a theory paper, and those types of concerns are beyond the scope of this work.  There was nothing overall wrong technically with their paper, as far far as I could tell, and it provides a good theoretical foundation for further work in the future.  Therefore I believe that the paper should be accepted.\n\nQUESTIONS:\nHow well does this method generalize to functions with degenerate saddle points? \n\nHow will the oracle implementation effect the performance of this algorithm?  It strikes me that the algorithm scales with $d^{2/3}$, but just loading the problem would take at least $d$, which could limit your performance.  Is this a limitation?\n\n$x$ and $y$ are said to be real valued, but it is not obvious to me how you would embed real-valued, rather than simply natural valued inputs (i.e. a linear combination over the computational basis).  Does this restriction on your input $x$ and $y$ values pose a problem for your algorithm or is there an amplitude embedding, etc. that avoids this issue (or is it a non-issue)?\n\nLIMITATIONS:\nIt is unclear how generally applicable this method is to real world problem instances.  They claim that these types of problems are important for \"fairness aware learning, AUC maximization, robust optimization, game theory, and reinforcement learning,\" but it is unclear to me what problem scales of these problems become relevant.  This is important, since as was shown in M. Beverland, 2022 (https://arxiv.org/abs/2211.07629), scaling advantages that are sub-quadratic are unlikely to be practically realizable on currently considered quantum hardware.  Again, this is a theory paper, so this limitation is not particularly major in the context of this work, but it is a practical limitation nonetheless.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-21"}

comments: SUMMARY:
This work studies quantum algorithms for zeroth order optimization problems, and specifically, for the minimax problem. The authors introduce a class of Hessian-aware quantum zeroth-order methods that achieve speedups over classical methods in terms of query complexity to the function value oracle.

STRENGTHS AND WEAKNESSES:
Strength:

- The paper has proposed new quantum zeroth order optimization algorithms, with provable speedups in query complexity than classical algorithms.
- The paper has given new quantum algorithms for estimating the Hessian matrix.

Weakness:

- The acceleration in query complexity of the methods are modest, less than quadratic.
- In the quantum gradient and Hessian estimation, the gate complexity is similar to the one in the classical case. Therefore, I guess the reason why focusing on the query complexity should be clearly explained with examples.
- The work fails to give lower bounds for query complexity of quantum algorithms in the problem setting.

QUESTIONS:
Typos:
- Line 136: exits -> exists;

The writing could be improved. Many theorem statements has grammar mistakes.

Could the authors give more explanations about the novelty of their paper? As far as I can see, the quantum Hessian estimation is a direct application of the quantum gradient estimation method. Whether the method is optimal or not still needs discussion.

LIMITATIONS:
The work fails to give lower bounds for query complexity of quantum algorithms in the problem setting, leaving the optimality of their algorithms as an open question.
The method of estimating the Hessian matrix seems to be a direct application of quantum gradient estimation, which might not be so novel.
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates quantum algorithms for convex-concave minimax optimization problems with access to a quantum zeroth-order oracle only. By combining a novel quantum Hessian vector estimator (based on the quantum gradient estimation subroutine) with existing classical (stochastic) second-order methods, such as Newton proximal extragradient, the proposed quantum algorithm achieves an improvement of $d^{1/3}\\epsilon^{-1/3}$ over best-known classical counterparts. Application of the quantum Hessian estimator to non-convex optimization problems also achieves polynomial improvement in the query complexity compared to classical methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The Hessian-vector estimation subroutine is a novel technical contribution that can potentially accelerate other classical (second-order) optimization/minimax algorithms.\n- The theoretical analysis appears complete and solid, with a comprehensive comparison with prior art in Tables 1-3. \n- Extension to non-convex optimization problems is discussed, complementing the discussion on minimax optimization and showcasing the usefulness of the quantum Hessian estimation.\n\nWeaknesses:\n- While the Hessian-vector and Hessian estimation subroutines (Algorithms 1 & 2) appear original and have not been formally analyzed in previous work, they are quite straightforward extensions of the quantum gradient estimators (Lemma 3.2) relying on a standard 2-point estimation technique (e.g., finite difference). \n- While the quantum Hessian estimation subroutine seems to achieve an improvement of $d$ over classical algorithms, the achieved speedups for minimax optimization in terms of quantum queries (Theorem 4.2, 4.4) are quite limited. \n- The quantum advantage is formulated using query complexity. However, in Algorithms 3 and 4, it is still required to find an inexact cubic step using classical methods (e.g., solving linear systems). It is unclear if this algorithm achieves meaningful end-to-end quantum speedups.\n- Besides, there are several obvious typos/formatting issues in this submission, see the questions below.\n\nQUESTIONS:\nTechnical questions:\n- How is the inexact cubic step in Algorithm 3 computed? If this step is performed purely classically, how can we expect any **genuine** quantum speedup, given that representing the Hessian as a classical array may already negate the quantum advantage gained from the Hessian estimation?\n- The classical algorithms for convex-concave minimax optimization problems are already very good (with a provable query upper bound that is sublinear in both $d$ and $1/\\epsilon$). In what parameter regime would we expect practical quantum advantage? Is it possible to have large (i.e., super-quadratic) quantum speedups?\n- Lemma 4.1: Is the Gap() defined anywhere? I do not find it near the lemma, and it is not directly clear to me how this gap is defined.\n\nMinor typos/formatting issues:\n- Line 24: \"When ... is Lipschitz continuous, second-order offers faster convergence rates.\" A word like \"method\" or \"algorithm\" is missing after \"second-order\".\n- Line 29: the h(\\cdot, \\cdot) is not in the mathematical italic format.\n- Line 57: There should be a space between the big-O notation and \"complexity\". \n- Theorem 5.2: what does the $\\Delta$ symbol mean in the in-line math equation: $f^* \\Delta \\min_{x\\in \\mathbb{R}^d}$?\n\nLIMITATIONS:\n– The main technical contribution (quantum Hessian estimation via a zeroth-order oracle) is relatively straightforward and appears to offer only limited benefit when integrated into a classical algorithm.\n– The quantum speedups achieved are modest compared to classical baselines, and it remains unclear whether these would translate into a meaningful practical advantage once classical computation and post-processing overheads are accounted for.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies dimension-dependent quantum algorithms for convex-concave minimax optimization, assuming access to a quantum function value oracle. Classically, the problem can be solved using $O(d/\\epsilon)$ queries to a classical oracle. In contrast, the proposed quantum algorithm achieves a query complexity of $O(d + d^{2/3}\\epsilon^{-2/3})$, yielding an improvement of order $d^{1/3}\\epsilon^{-1/3}$. Additionally, for the strongly convex--strongly concave case, the authors present an algorithm with a query complexity improvement of order $d^{1/3}\\mu^{-1/3}$. The techniques are further extended to the non-convex setting, where the objective is to find an $\\epsilon$-stationary point.\n\nTechnically, this paper introduces quantum algorithms for estimating the Hessian-vector product and the Hessian of the objective function. These techniques are used as subroutines in the main algorithm, which maintains an estimate of the Hessian and performs approximate Newton updates. In particular, the Hessian estimate is updated using a lazy updating framework. The algorithm for finding stationary points of a non-convex function follows a similar high-level approach.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n1. This paper is the first to discuss quantum speedups for convex-concave minimax optimization problem with function value access.\n2. The paper is well-written, and the proofs are presented clearly.\n3. The analysis of the lazy Hessian algorithm (Algorithm 4) is interesting and may be of independent interest\n\nWeaknesses:\n1. The query complexity bounds are not written in a very precise way, it should be $d+\\cdots$, since at the beginning of the algorithm you need $d$ queries to compute the full Hessian. Omitting it is inaccurate.\n2. The algorithms for estimating the Hessian-vector product and the full Hessian appear to be, to some extent, relatively straightforward applications of Jordan’s gradient estimation algorithm.\n\nQUESTIONS:\n1. For every quantum algorithm in this paper, can we have a corresponding classical algorithm with first-order input that achieves the same query complexity, using basically the same algorithms? \n2. Can we have dimension-independent quantum algorithms using an evaluation oracle? If so, can we design algorithms whose complexity interpolate between this dimension-independent bound and the bounds obtained in this paper?\n3. For the dimension dependent non-convex optimization setting, there is a more recent paper \"Improved Complexity for Smooth Nonconvex Optimization: A Two-Level Online Learning Approach with Quasi-Newton Methods\" that might be worth discussing.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}