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 studies the minimization of a stochastic L$^\\natural$-convex function. - The authors analyze the convergence of three algorithms, truncated subgradient, multi-dim uniform sampling, and stochastic cutting-plane. The time complexity of these algorithms is typically $O(d^2N^2/\\epsilon^2)$ or something higher, where $d$ is the dimension, $N$ is the range of the feasible region, and $\\epsilon$ is the error-accuracy. - The authors also proved a lower bound that any algorithm takes $\\Omega(d/\\epsilon^2)$-time to find an $\\epsilon$-optimal solution. - In numerical experiments, the truncated subgradient method rapidly found an $\\epsilon$-optimal solution.