USCO-Solver: Solving Undetermined Stochastic Combinatorial Optimization Problems

1 Introduction . Combinatorial optimization problems are not only of great theoretical interest but also central to enormous applications . Traditional research assumes that the problem settings ( i.e. , objective function ) are completely known [ 1 ] , but the reality is that the underlying system is often very complicated and only partial knowledge ( from historical data ) is provided . In a recommendation system , the item-user similarities could be unknown [ 2 ] , which makes it impossible to compute the optimal recommendation scheme . In the study of wireless communications , the backbone network depends on stochastic node connections [ 3 ] , incurring extra difficulties in designing management strategies . In the most general sense , we can formalize such scenarios by assuming that the objective function is governed by a configuration space associated with an unknown distribution , where our goal is to maximize the expected objective . For example , the configuration space may specify possible item-user similarities or candidate network structures . We call such problems as undetermined stochastic combinatorial optimization ( USCO ) problems . Since the distribution of the configuration space is unknown , one can adopt the learning-andoptimization strategy [ 4 ] , where the unknown distribution is first learned from data so that the subsequent optimization problem can be solved using existing methods . While such a strategy is natural and proved successful in several applications [ 5 ] , it may require a large amount of data to learn an excessive number of parameters – for example , the weight between each pair of nodes in a network or the similarity between each pair of user and item in a recommendation system . In the worst case , we may not even have access to such kind of data ( due to , for example , privacy issues 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . [ 6 ] ) . Furthermore , a more technical concern is that the learning process is often independent of the optimization process , causing the situation that optimizing the learned objective function does not produce a good approximation to the true objective function , which is theoretically possible [ 7 ] . These concerns motivate us to think of the settings that can eschew model learning and can address USCO problems aiming right at the approximation guarantees . To this end , we consider the regression between the input space and solution space , and wish to directly compute the solution for future inputs without learning the hidden objective function . USCO-Solver . In this paper , we present USCO-Solver , a general-purpose solver to USCO problems . Our framework is designed through two main techniques : randomized function approximation and approximate structured prediction . The key idea is to construct a hypothesis space composed of affine combinations of combinatorial kernels generated using random configurations , from which a large-margin machine is trained using input-solution pairs via approximate inference . The main advantage of USCO-Solver is that it can handle an abstract USCO problem as long as an oracle to solve its deterministic counterpart is given . Another significance of USCO-Solver lies in our use of combinatorial kernels , which suggests a novel and principled way for incorporating an approximation algorithm into a learning process . In doing so , we are able to handle combinatorial objects easily without worrying much about their indifferentiability or the inherited symmetries , which is often not possible for mainstream learning methods [ 8 ] . Theoretical analysis . For USCO-Solver , we present a learning-error analysis under the PACBayesian framework , where we introduce the multiplicative margin dedicated to bounding the approximation guarantees . In particular , we prove that the approximation ratio of the predictions is essentially bounded by O ( α2 ) , whenever the considered problem admits an α-approximation in the deterministic sense . Such a result is possible due to the fact the hypothesis space of USCO-Solver can approximate the configuration space arbitrarily well . To our knowledge , this is the first result of such kind . Empirical studies . We conduct experiments with USCO problems concerning three classic combinatorial objects : path , coverage , and matching . On various real-world datasets , we consistently observe that USCO-Solver not only works the way it is supposed to , but also outperforms other competitors by an evident margin in most cases . In many case studies , near-optimal solutions can be computed without consulting the true configuration distribution . Supplementary material . The proofs , together with more results and discussions on the experiments , can be found in the supplementary material . In addition , we release the experimental materials , including source code , datasets , and pretrain models , as well as their instructions . The experiment materials can be found online1 . 2 Preliminaries . 2.1 Undetermined stochastic combinatorial optimization problems . We are concerned with an abstract combinatorial optimization problem associated with three finite combinatorial spaces : the input space X , the output space Y , and a configuration space C ; in addition , we are given a bounded non-negative function f ( x , y , c ) : X ×Y ×C → R+ denoting the objective value to be maximized . Let Φ be the set of all distributions over C. We consider stochastic combinatorial optimization problem in the sense that we seek to maximize f in terms of a distribution φtrue ∈ Φ rather than a fixed configuration in C. This is desired because the system that defines the objective values is often essentially probabilistic , which may stem from random networks or functions with random parameters . Therefore , the objective function is given by F ( x , y , φtrue ) = ∫ c∈C φtrue ( c ) · f ( x , y , c ) dc . ( 1 ) Given x and φtrue , we are interested in the problem max y∈Y F ( x , y , φtrue ) . ( 2 ) We may wish to compute either the optimal solution H ( x , φtrue ) : = arg maxy∈Y F ( x , y , φtrue ) or its α-approximation Hα ( x , φtrue ) for some α ∈ ( 0 , 1 ) . Such problems are fundamental and also 1https : //github.com/cdslabamotong/USCO-Solver ubiquitous , ranging from the stochastic version of classic combinatorial problems [ 9 ] to applications subject to environmental uncertainties , such as commercial recommendation [ 10 ] , viral marketing [ 11 ] , and behavioral decision making in autonomous systems [ 12 ] . Taking the stochastic longest path problem [ 13 ] as an example , the length of each edge could follow a certain distribution , and therefore , each configuration c ∈ C corresponds to a weighted graph ; in such a case , x = ( u , v ) denotes a pair of source and destination , y denotes a path from u to v , and f ( x , y , c ) amounts to the length of y in c. While the above formulation is natural , we are always limited by our imperfect understanding of the underlying system , which could be formalized by assuming that the distribution φtrue over the configuration is not known to us . Since the true distribution φtrue is unknown , the objective function F ( x , y , φtrue ) can not be directly optimized . Essentially , a learning process is required . In such a sense , we call these problems undetermined stochastic combinatorial optimization ( USCO ) problems . Oracle . In general , USCO problems are made difficult by a ) the learning challenges in dealing with the unknown distribution φtrue and b ) the approximation hardness in solving Equation 2 . We focus on the learning challenges and thus assume the approximation hardness ( if any ) can be resolved by oracles . Notice that the objective function ( Equation 1 ) lies in the positive affine closure of its discretizations , which is ∪∞k=1 { k∑ i=1 wi · f ( x , y , ci ) : wi ∈ R+ , ci ∈ C } . ( 3 ) For some fixed α ∈ ( 0 , 1 ] , we assume the access to a polynomial oracle for computing an αapproximation to maximizing any function in the above class , which implies that Hα ( x , φ ) is obtainable for each x ∈ X and φ . We can think of α as the best ratio that one can obtain by a polynomial algorithm , with α = 1 denoting the scenario when the optimal solution can be efficiently computed . For a vector w = ( w1 , ... , wk ) ∈ Rk and a subset C = { c1 , ... , ck } ⊆ C of configurations , we denote the induced kernel function as KC ( x , y ) : = ( f ( x , y , c1 ) , ... , f ( x , y , ck ) ) and the associated affine combination as F̂w , C ( x , y ) : = k∑ i=1 wi · f ( x , y , ci ) = wTKC ( x , y ) . Finally , we denote the α-approximation to maxy∈Y F̂w , C ( x , y ) as hαw , C ( x ) . 2.2 Learning settings . We consider samples of input-solution pairs : Sm : = { ( xi , y α i ) } m i=1 ⊆ 2X ×Y where yαi : =Hα ( xi , φtrue ) is an α-approximation associated with the input xi . Such samples intuitively offer the historical experience in which we successfully obtained good solutions to some inputs . From a learning perspective , we have formulated a regression task between two combinatorial spaces X and Y . Some formal treatments are given as follows . Let us assume that the true distribution φtrue is unknown but fixed , and there is a distribution DX over X . Our goal is to design a learning framework AS : X → Y that can leverage a set S of samples to compute a prediction AS ( x ) ∈ Y for each future x ∈ X . For a prediction ŷ ∈ Y associated with an input x , let l ( x , ŷ ) ∈ [ 0 , 1 ] be a general loss function . Since we aim to examine if ŷ is a good approximation to Equation 2 , the loss l is determined jointly by ŷ and x. Consequently , we measure the performance of A by L ( AS , DX , l ) : =Ex∼DX [ l ( x , AS ( x ) ) ] 3 A learning framework . In this section , we first present the learning framework and then analyze its generalization bounds . Finally , we discuss training algorithms . 3.1 USCO-Solver . Following the standard idea of structured prediction [ 14 , 15 , 16 ] , we wish for a score function F̂ ( x , y ) such that , for each x ∈ X , the y ∈ Y that can maximize F̂ ( x , y ) is a good solution to maximizing F ( x , y , φtrue ) . Suppose that we are provided with a set Sm of m ∈ Z samples . USCO-Solver consists of the following steps : • Step 1 . Select a distribution φem ∈ Φ over C and decide a hyperparameter K ∈ Z . • Step 2 . Sample K configurations CK = { c1 , ... , cK } ⊆ C independently following φem . • Step 3 . Compute a seed vector w̃ = ( w̃1 , ... , w̃K ) ∈ RK using the training data . • Step 4 . Sample a vector w ∈ RK from Q ( w|β · w̃ , I ) , which is an isotropic Gaussian with identity convariance and a mean of β · w̃ , with β being β : = 4 minp |w̃p| · α2 √ 2 ln 2mK ‖w̃‖2 . ( 4 ) • Score function . With CK and w = ( w1 , ... , wK ) , we adopt the score function F̂w , CK ( x , y ) : = K∑ i=1 wi · f ( x , y , ci ) . • Inference . For each input x ∈ X , the prediction is given by hαw , CK ( x ) , which can be computed by the oracle . Given that the framework is randomized , the error of interest is L ( USCO-SolverSm , DX , l ) : =Ew∼Q , CK∼φem , x∼DX [ l ( x , hαw , CK ( x ) ) ] . ( 5 ) So far , we have not seen how to determine the distribution φem , the parameter K , and the seed vector w̃ . We will first analyze how they may affect the generalization bound ( Sec . 3.2 ) , and then discuss how to make selections towards minimizing the generalization bound ( Sec . 3.3 ) .

Summary: The paper presents USCO-Solver, a new solver that is able to handle abstract undetermined stochastic combinatorial optimisation problems. The main contribution is a learning framework that allows the solver to infer high quality solutions from samples of input-output pairs without having to learn the objective function. This approach seems to contrast the well known learning-and-optimisation strategy. An empirical evaluation on several classic combinatorial problem domains demonstrate the effectiveness of the proposed approach.