Stochastic $L^\natural$-convex Function Minimization

1 Introduction . As an important branch of optimization , discrete stochastic optimization problems ( Futschik & Pflug , 1995 ; Gutjahr & Pflug , 1996 ; Futschik & Pflug , 1997 ; Kleywegt et al. , 2002 ; Semelhago et al. , 2020 ) appear in a wide range of applications such as image processing ( Jegelka & Bilmes , 2011 ) , pricing optimization ( Ito & Fujimaki , 2016 ) and simulation-based optimization ( Altman et al. , 2003 ; Freund et al. , 2017 ) . Since a discrete optimization problem is generallyNP-hard to solve , specific structures are necessary to guarantee the existence of polynomial-time algorithms . In the special case when the feasible set is { 0 , 1 } d for some positive integer d , submodular functions are viewed as the discrete analogy of convex functions in the field of combinatorial optimization . The submodular function minimization ( SFM ) problem has wide applications in computer vision , economics , game theory and is well-studied in literature ( Lee et al. , 2015 ; Axelrod et al. , 2020 ; Zhang et al. , 2020 ) . However , for certain problems , because of the stochastic structure , the exact value of the objective function is not available but only noisy samples of the objective function value are available . These scenarios , for example , include the optimization of expected performances for complex stochastic systems . In contrast to the deterministic problem , the stochastic SFM problem is less understood ( Ito , 2019 ; Chakrabarty et al. , 2017 ; Blais et al. , 2019 ; Axelrod et al. , 2020 ) . In this work , we consider the stochastic minimization problem of a class of generalized submodular functions , named the L\-convex functions ( Murota , 2003 ) . More concretely , a function 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . f : { 1 , . . . , N } d 7→ R is called an L\-convex function if it satisfies the discrete midpoint convexity : f ( x ) + f ( y ) ≥ f ( d ( x+ y ) /2e ) + f ( b ( x+ y ) /2c ) , ∀x , y ∈ { 1 , . . . , N } d , where N , d are positive integers and d·e , b·c are the ceiling and flooring functions applied componentwisely to vectors . We note that L\-convex functions are reduced to submodular functions if N = 2 . As an extension to the stochastic SFM problem , the stochastic L\-convex minimization problem also appears in a variety of decision making problems . For example , the user satisfaction function of a bike-sharing system ( Freund et al. , 2017 ) and the expected total waiting time of a service network ( Altman et al. , 2003 ) are proved to be multimodular functions , which is equivalent to an L\-convex function under a linear transformation . Since the evaluation of both functions includes the simulation of random user behaviors , we only have access to noisy objective function value oracles and thus both problems can be formulated as stochastic L\-convex minimization problems . We refer the readers to the appendix and Murota ( 2003 ) for more concrete examples of L\-convex minimization problems . The contributions of this work are two-fold . For the algorithm design , we provide the first polynomialtime algorithms to achieve high-probability bounds for the stochastic L\-convex function minimization problem . We note that we only consider the number of objective function value evaluations in the computational complexity , since the function value evaluation oracle is usually the major contributor to the running time of algorithms . The first algorithm is based on the observation that the Lovász extension of an L\-convex function transforms the original problem to an equivalent continuous convex optimization problem . Therefore , we can apply the stochastic subgradient method to solve the continuous convex problem and then round the solution to an integral solution . The idea of combining the Lovász extension and the stochastic subgradient method has already been considered in Hazan & Kale ( 2011 ) ; Ito ( 2019 ) for stochastic online convex optimization and stochastic submodular minimization , respectively . However , the direct application of the stochastic subgradient method will lead to a large computational complexity . We add an extra truncation step to the stochastic subgradient and the computational complexity is reduced by O ( d ) . We also implement numerical experiments to show the significance of the truncation step . In the special case when N = 2 , our problem reduces to the stochastic SFM problem and the computational complexity of the truncated subgradient method is O ( d2 ) smaller than the stochastic cutting-plane method in Blais et al . ( 2019 ) . In addition , we propose the dimension reduction method and get a strongly polynomial approximate algorithm ( see the definition in Section 4.2 ) with computational complexity O ( d3N2 ( d + log ( N ) ) −2 log ( 1/δ ) ) . Finally , the uniform sampling method in Zhang et al . ( 2021 ) is extended to the multi-dimensional case and the computational complexity is independent of N . Table 1 summarizes the upper bounds on the computational complexities to find a solution whose objective value is at most larger than the optimum with probability at least 1− δ . On the other hand , we derive lower bounds on the computational complexity for achieving a highprobability error bound . We construct a set of L\-convex functions whose landscapes are close to each other while their optima are different with each other . Intuitively , the algorithm requires sufficiently many evaluations to differentiate those “ similar ” L\-convex functions , where the similarity between functions is characterized by the KL-divergence . More rigorously , we utilize the information inequality in Kaufmann et al . ( 2016 ) and get the lower bound O ( d −2 log ( 1/δ ) ) on the complexity . Recently , Ito ( 2019 ) proved a lower bound for achieving an -expected error . There is a difference in terms of d and δ regarding the lower bound results between ours and Ito ’ s . We note that these two lower bounds are established under different settings , i.e. , the high-probability setting and the expectation setting . Hence , a directly comparison of the two lower bounds is unavailable ; see the discussion in Section 5 . 1.1 Notation . For N ∈ N , we define [ N ] : = { 1 , 2 , . . . , N } . For a given set S and an integer d ∈ N , the product set Sd is defined as { ( x1 , x2 . . . , xd ) : xi ∈ S , i ∈ [ d ] } . For two vectors x , y ∈ Rd , we use ( x ∧ y ) i : = min { xi , yi } and ( x ∨ y ) i : = max { xi , yi } to denote the component-wise minimum and maximum . Similarly , the ceiling function d·e and the flooring function b·c round each component to an integer when applied to vectors . The failing probability of algorithms is denoted as δ . The notation f = O ( g ) ( resp . f = Θ ( g ) ) means that there exist absolute constants c1 , c2 > 0 such that f ≤ c1g ( resp . c1g ≤ f ≤ c2g ) . Similarly , the notation f = Õ ( g ) ( resp . f = Θ̃ ( g ) ) means that there exist absolute constants c1 , c2 > 0 and constant c3 > 0 independent of δ such that f ≤ c1g + c3 ( resp . c1g ≤ f ≤ c2g + c3 ) . 2 Problem Formulation . We have in mind a complex stochastic system and consider the following discrete stochastic optimization problem min x f ( x ) : = E [ F ( x , ξx ) ] s.t . x ∈ X , ( 1 ) in which X is the discrete feasible set , ξx summarizes all the randomness supported on some proper probability space ( Y , BY ) and F : X×Y 7→ R is a deterministic function . Specifically , ξx summarizes all the random variables and stochastic processes involved in the system for the decision variable x . The function F maps the decision and associated randomness into a random system performance . For example , in service systems such as the bike-sharing service , this function F captures the full operational logic regarding how services are arranged . We write ξx,1 , ξx,2 , . . . , ξx , n as independent and identically distributed ( i.i.d . ) copies of ξx . We use F̂n ( x ) : = 1n ∑n j=1 F ( x , ξx , j ) to denote the empirical mean of the n independent evaluations for the choice of decision variables x . The triad of the set X , the space of randomness ( Y , BY ) and the function F ( · , · ) is called the model of problem ( 1 ) . In this work , we assume the L\-convexity of the objective function . For the ease of presentation , we assume the feasible set X = [ N ] d , where d denotes the dimension and N represents the number of choices on each dimension for the decision variable x . Assumption 1 . The objective function f ( · ) is an L\-convex function on X = [ N ] d , i.e. , f ( x ) + f ( y ) ≥ f ( d ( x+ y ) /2e ) + f ( b ( x+ y ) /2c ) , ∀x , y ∈ X , Remark 1 . In general , the definition of L\-convexity also requires the feasible set X to be an L\convex set , namely , the indicator function of X should be L\-convex on Zd . In our case , the feasible set X = [ N ] d is an L\-convex set . Hence , we only need the discrete mid-point convexity property to define L\-convex functions on X . Moreover , we assume that the distribution of the stochastic output F ( x , ξx ) is sub-Gaussian . Assumption 2 . The distribution of F ( x , · ) is sub-Gaussian with a known parameter σ2 for all x ∈ X . The knowledge of σ2 represents the maximum level of stochasticity involved in the system performance . For scenarios where this σ2 is not known , it has to be adaptively learned and adds additional complexity to the problem , but is not the focus of this work . We further note that our algorithm design and analysis can be generalized to any probability distribution with provable concentration inequalities . The set of all models satisfying both Assumption 1 and Assumption 2 is denoted as MC ( X ) , or simplyMC . The main goal is to design algorithms that guarantee the selection of a near-optimal solution with high probability . Formally , this criterion is called the ( , δ ) -Probably Approximately Correct ( PAC ) guarantee ; see Even-Dar et al . ( 2002 ) ; Kaufmann et al . ( 2016 ) ; Ma & Henderson ( 2017 ) . We say that a solution x̂ , as the output of an algorithm , satisfies the ( , δ ) -PAC guarantee if P [ f ( x̂ ) −min x∈X f ( x ) ≥ ] ≤ δ , where the probability is calculated over the distribution of x̂ . An algorithm is called an [ ( , δ ) -PAC , MC ] -algorithm if it always returns an ( , δ ) -PAC solution for any model inMC . The computational complexity of an algorithm is defined as the number of evaluations of F ( · , · ) before termination . The difficulty of satisfying the PAC guarantee for any model inMC is characterized by the worst-case performance of an optimal algorithm . Definition 1 . The computational complexity of the ( , δ ) -PAC guarantee and classMC is defined as T ( , δ , MC ) : = inf A is ( ( , δ ) -PAC , MC ) sup M∈MC E [ τA ] , where the random variable τA is the computational complexity for each implementation of A . The reminder of this paper is focused on providing upper bounds on T ( , δ , MC ) by designing algorithms and providing lower bounds by an information-theoretical analysis . In this paper , we consider the behavior of algorithms in the high-probability regime , namely , in the setting when the failing probability δ is small enough . Assumption 3 . The failing probability δ is sufficiently small . In general , the sub-Gaussian randomness leads to T ( , δ , M ) = C ( , M ) · log ( 1/δ ) 0 < δ 1 , where C ( , M ) is some function only depending on the precision and the model classM . For example , it is known that the sample complexity is Θ ( Nd −2 ) log ( 1/δ ) for the general best-arm selection problem ( Even-Dar et al. , 2002 ) when no convex structure is in presence .

Summary: This paper focuses on the problem of minimizing $f(x) = \mathbb{E}_{\xi}[F(x,\xi_x)]$ where $f:[N]^d\to\mathbb{R}$ is an $L^\natural$ function. This class of functions can be interpreted as a discrete analog of convex functions. The goal is to design polynomial time algorithms that guarantee a solution whose value is $\epsilon$ away from the optimal value with probability at least $1-\delta$ (in the paper referred as $(\epsilon,\delta)$-PAC guarantee).  The authors provide upper bounds and a lower bound for the computational complexity of the problem, which is defined as the worst-case expected time over all possible $(\epsilon,\delta)$-PAC algorithms. First, the authors present the standard Lovasz extension for $L^\natural$ functions. This extension is convex so common techniques in convex optimization can be used. First, they present a truncated stochastic subgradient algorithm that has an appropriate rounding procedure as a final step. They show this method has a complexity of $\tilde{O}(\frac{d^2N^2}{\epsilon^2}\log(1/\delta))$ when $f$ is $L$-Lipschitz. Later, they show that under the assumption that the subgradients of the function are uniformly bounded with probability 1, the same algorithm without truncation has an improved complexity. The authors also study the performance of the algorithms when the suboptimality gap is known. Then, they give two methods for the case when the Lipschitz constant is not known: multidimensional uniform sampling and dimension reduction. Finally, they give an information-theoretical lower bound of $\Theta(\frac{d}{\epsilon^2}\log(1/\delta))$.