Discrete Stochastic Approximation with Application to Resource Allocation


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1 Discrete Stochastic Aroximation with Alication to Resource Allocation Stacy D. Hill An otimization roblem involves fi nding the best value of an obective function or fi gure of merit the value that otimizes the function. If the set of otions is fi nite in number, then the roblem is discrete. If the value of the obective function is uncertain because of measurement noise or some other source of random variation, the roblem is stochastic. Mathematically, the discrete otimization roblem is a nonlinear otimization roblem involving integer variables, and its solution will require some iterative rocedure. This article discusses one such rocedure for solving diffi cult otimization roblems. The rocedure is a discretevariables version of the Simultaneous Perturbation Stochastic Aroximation algorithm, develoed at APL, for solving otimization roblems involving continuous variables. The discretevariables algorithm shares some of the comutational effi ciency of its continuous counterart. INTRODUCTION Discrete otimization roblems occur in a wide variety of ractical alications. One imortant class of such roblems is the resource allocation roblem: There is a fi nite quantity of some resource that can be distributed in discrete amounts to users or to erform a set of tass; the roblem is to distribute the resource so as to otimize some figure of merit or obective function. This article discusses a discrete otimization algorithm for solving such roblems. The algorithm,, develoed in collaboration with László Gerencsér (Comuter and Automation Res. Inst., Hungarian Academy of Sciences, Budaest) and Zsuzsanna Vágó (Pázmány Péter Catholic University, Budaest), relies on the method of stochastic aroximation (SA). 3 It is a discretevariables version of an SA method, also develoed at APL, called the Simultaneous Perturbation Stochastic Aroximation (SPSA) algorithm, 4,5 which is used for solving otimization roblems involving continuous variables. The goal in develoing the discrete version of the SPSA, which we will sometimes call discrete SPSA, was to design an algorithm that, lie its continuous counterart, is comutationally efficient and solves roblems in which the obective function is analytically unavailable or difficult to comute. Before resenting the discrete algorithm, several examle roblems are given to illustrate the variety of discrete resource allocation roblems and the need for discrete SPSA. (For other examles, see Ref. 6.) JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005) 5
2 S. D. HILL The fi rst examle is the weaons assignment roblem. 7 There are multile weaons systems that differ in number, yield, and accuracy, and there are multile targets that differ in hardness and tye. The roblem is to assign weaons to targets in some otimal fashion. The resources are weaons assets, and the tass are the targets to be attaced. The obective function might reflect the cost of deloying assets, target hardness, and the strategic value of each target; it might also reflect the goal to attac a certain minimum number of targets or the requirements to achieve a certain level of damage and minimize undesired collateral damage. Another examle is the facilities location roblem 8 : Facilities (e.g., manufacturing lants, military suly bases, schools, warehouses) can be built at a fi xed number of locations. There is a cost if the facility is underutilized or if it cannot ee u with the demand for its services. The roblem is to determine the best location and size of each facility. The last examle is the roblem of scheduling the transmission of messages in a radio networ. 9 A message is transmitted over the nodes in a networ as a set of frames, where the total number of frames a message requires deends on the length of the message. There is a fi xed number of time slots buffer sace that can be allocated to message frames. The roblem is to allocate buffer sace to the nodes to minimize average transmission delays or some other quantity such as the number of messages that are bloced or cannot be transmitted. In each of these examles, the resource can only be distributed in discrete amounts, i.e., the amount of a resource that can be allocated is an integer value. The obective function is some scalarvalued function; deending on its interretation, the otimal value and consequently the otimal allocation corresonds to its minimum or maximum value. For examle, if the obective function measures a loss such as the cost of an allocation, then an otimal allocation minimizes the loss. If, on the other hand, the obective function measures some gain such as rofit or reward, then an otimal allocation is one for which the obective function is at a maximum. In what follows, we will assume that the obective function is a loss function. This assumtion imoses no loss in generality, since a maximization roblem is easily transformed into one of minimization. More recisely, the roblem of fi nding the minimum of a loss function, L, say, is equivalent to the roblem of fi nding the maximum value of L. One feature of discrete otimization roblems that maes them otentially difficult to solve is the size or cardinality of their search saces, which can be large in roblems involving a relatively small number of users and resources. For examle, the number of ways of allocating, say, 0 units of a resource to 30 users exceeds 0 3. More generally, the size of the search sace for a constrained resource allocation roblem consisting of N users and K identical resources (i.e., the resources are indistinguishable) exceeds (K + N )!/(N )!K!. 0 Thus, the search sace is tyically too large to mae an exhaustive search a feasible aroach. Adding to the difficulty of dealing with large search saces is the roblem of oerating in a noisy or stochastic environment. An algorithm for fi nding the otimal value requires the ability to evaluate the loss function at estimated or candidate solutions. The comuted values of the loss will be noisy if the loss function deends on quantities having uncertain values or is corruted by measurement noise. In the weaons assignment roblem, for examle, some uncertainty may exist in the location or characteristics of the targets or in estimates of damage, and hence the loss may deend on damage assessments obtained by sensor devices that may contain measurement noise. In the facilities location roblem, the actual use at a location may vary unredictably, as will the gain and loss in locating there. In the roblem of transmitting messages in a radio networ, the loss will be random if users can request networ resources at random instants of time or hold them for random lengths of time. Any algorithm for fi nding the minimum must be alicable to noisy loss functions. Noisy loss functions and large search saces resent two difficult challenges to solving discrete otimization roblems and are the main motivation for the develoment of a discrete version of SPSA. PROBLEM FORMULATION The resource allocation roblem is easy to state. Consider the case involving a single tye of resource. Suose K units of the resource are to be distributed to users or tass. An allocation rule assigns a fi xed number of the units to each user and therefore determines a vector of dimension an allocation vector whose comonents are the quantities of the resource allocated to users. If denotes the allocation vector, then = (,..., ), where is the amount allocated to the th user. Since the amounts allocated are discrete, each is a nonnegative integer, and since the total quantity allocated is K, it follows that = K. = The total loss associated with the allocation is L( ) and deends on the loss in allocating resources to each of the users. If L ( ) is the loss associated with the th user, the total loss is L() = (). = L The roblem, then, is to fi nd the allocation that minimizes the total loss, i.e., minimize L ( ) = subect to = K, = () 6 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005)
3 where is a nonnegative integer, =,,...,. In general terms, this is a nonlinear integer roblem an otimization roblem with integer variables and a realvalued obective function. An allocation vector is a feasible solution if it satisfies the constraints, i.e., the allocations are nonnegative integers and their sum equals K. A feasible solution * is a solution if L( * ) L( ) for any other feasible solution. The otimization roblem as currently formulated is intractable; that is, any algorithm that solves it must enumerate a nontrivial art of the set of feasible solutions and is essentially equivalent to an exhaustive search. In the language of comutational comlexity theory, the roblem is NPcomlete. (See Ibarai and Katoh, and 4, for a discussion of comutational comlexity and how it relates to the resource allocation roblem.) For this reason, we consider a class of obective functions that lead to tractable roblems. In articular, we consider obective functions that are searable and integer convex. The notion of integer convexity will be defi ned later. The loss function L is searable if L = L() = ( ). () This form is a secial case of a loss function in which the th user loss deends only on the allocation to the th user. Algorithms for solving an otimization roblem are iterative rocedures that generate a sequence of estimates that converges to an otimum. These rocedures are tyically recursive, i.e., the next estimate deends on the revious ones. In the deterministic setting roblems in which the loss function can be evaluated at each there are a number of rocedures for fi nding the minimum. 6 For many ractical roblems, however, uncertainty or noise may exist that maes the loss function difficult or imossible to evaluate. Algorithms for discrete otimization roblems involving noisy loss functions are limited. 0 In such roblems, the loss values must be relaced by estimates, which may contain measurement noise. (For examle, in the facilities location roblem, loss deends on the difference between the lanned caacity and that which is required to meet user demand, and will therefore be unnown if the actual demand is unredictable.) The discrete SPSA algorithm, lie the SPSA algorithm, is a recursive algorithm in which the next estimate deends in a very simle way on the estimates of the loss function. Uncertainties that mae the loss difficult to evaluate can be viewed as random variables that influence the actual loss, and the (total) loss L( ) can be viewed as the average or exected value of the actual loss. More secifically, assume that the uncertain quantities are random variables denoted, and denote the actual loss by (, ); then L( ) = the exected value of (, ). Similarly, if the actual loss for the th user is (, ), then and L ( ) = the exected value of (, ) = (, ) = (, ). Another way of viewing the actual and exected losses is to thin of the actual loss as a measurement of the exected loss that is corruted by additive noise. In other words, if the measurement noise is (, ) and has zero mean, then (, ) = L ( ) + (, ). (3) Thus (, ) is a noisy measurement of L ( ). Liewise, the total actual loss, (, ), is a noisy measurement of L( ), the total exected loss. Since the loss function is not directly available, the otimization algorithm must rely on noisy estimates of the loss for fi nding the minimum. THE OPTIMIZATION ALGORITHM The discrete SPSA method is an analogue of the SPSA algorithm for continuousvariables roblems. Let us briefly review the SPSA algorithm to see how it is modified to obtain the discrete version. ContinuousVariables SPSA In the continuous setting, is a continuous vector arameter, i.e., its comonents are real numbers. The otimization roblem is minimize L( ), where is a real number, =,,...,. (4) The loss function L is assumed to be a differentiable realvalued function. Thus, if a oint * minimizes L, then L( * ) = 0. (5) JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005) 7
4 S. D. HILL Under additional assumtions, the root of this equation minimizes L. As in the discrete otimization roblem, the loss function is assumed to be unavailable. However, one can obtain noisy measurements of the loss, y( ) = L( ) + (, ), where is measurement noise and, as before, denotes uncertainty. Since L( ) is unavailable, its gradient, L/, is also unavailable and must be estimated. The SPSA algorithm for solving Eq. 5 uses a comutationally efficient estimate of the gradient, which is comuted in terms of the y( ) values, the noisy observations of the loss. The algorithm is ˆ ˆ ˆ(ˆ ). = = ag (6) For =,, 3,, the gain sequence a is an aroriately chosen sequence of ositive real numbers, and gˆ(ˆ ) is an estimate of the gradient g( ) = L( )/ of the loss function evaluated at ˆ defi ned as follows: Ste. Generate a vector = (,,..., ), the comonents of which are Bernoulli random variables taing the values with robability /. Ste. Tae a ositive real number c, the ste size, and consider the two erturbations about ˆ : and ˆ (+) ˆ( ) = ˆ c. = ˆ + c (Sall, , contains guidelines for choosing values for c and a.) Ste 3. Evaluate y at the erturbed values ˆ (+) ˆ ( ), to obtain y (+) ( + ) ( ) ( ) = y(ˆ ) and y = y(ˆ ). These are measurements of (+) ( ) L(ˆ ) and L(ˆ ), resectively. Ste 4. Form the estimate gˆ(ˆ ) by taing ( y ) ˆ(ˆ ( + ) y ( ) g ) =. (7) c M Under suitable conditions on the loss function, the estimates ˆ converge to a solution of Eq. 5. The gradient estimate requires, at each iteration, only ( + ) ( ) two measurements of the loss, namely, y and y. The standard method of estimating the gradient from observations, the method of fi nite differences, requires at least + measurements of the loss. The SPSA is comutationally efficient comared with an SA algorithm that uses fi nite difference, esecially if loss measurements are timeconsuming or costly to obtain. It is this tye of efficiency that is sought for the discrete algorithm. Discrete Parameter SPSA A Secial Case The discrete algorithm is similar to the continuous algorithm; however, in the discrete setting there is no derivative. Under suitable conditions, differences between the loss function at different oints behave lie derivatives in the continuousvariables setting. One condition that guarantees this behavior is a convexity condition for functions of a discrete argument 6,4,5 and leads to a discretevariables analogue of the SPSA algorithm. Before exloring the notion of discrete convexity, it may be helful to review convexity in the continuous setting. Geometrically, a function is convex if, at each oint on its grah, there is a line which asses through that oint which lies on or below the grah. Any such line is called a line of suort. For examle, in Fig., the solid curve (blue) is the grah of a convex function and the dashed line (red) is a line of suort. The integer convexity condition is not too difficult to describe in the onedimensional case, where the loss function is a function of a single integer variable. In this instance, the loss function is said to be integer convex or simly convex if, for each integer, L( + ) + L( ) L(). (8) This condition is similar to midoint convexity for functions of a real argument, where midoint convexity and convexity are equivalent. An equivalent form of the revious inequality is L( ) L( ) L( ) L( ). (9) It is this last inequality that motivates the use of differences as a relacement for derivatives. To see the similarity between differences and gradients, we need the following fact about integer convex functions. Let L( ) = L( ) L( ). A oint * minimizes an integer convex function L if L( * ) 0 L( * ). (0) 8 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005)
5 Figure. Continuous convex function L( + ) L( ) g() =, () is zero or close to zero. The quantity g( ) behaves very much lie a gradient. To see this, we need to extend the loss function to a function of a continuous variable. Consider the function L( ) obtained by linearly interolating L between and, =,, 3,... The extension L( ) is defi ned for each real number. This function is a continuous convex function, but is not everywhere differentiable. Furthermore, if the ste size c is small enough, where c > 0, then L( + c. ) L( c. ) = ( L( + ) L( )) c = ( L( +) L()) (3) + ( L( ) L( )) if is an integer and L( + c. ) L( c. ) = L([ ]+ ) L([ ]) (4) c if is not an integer, where [ ] denotes the integer art of. In either instance, denote the difference by g( ), so that 3 Figure. Integer convex function. Figure illustrates this roerty and also the connection between L( ) and the loss function. The blue dots, which are connected by the solid line, lot the values of the loss function, and the red dots, which are connected by the dashed line, lot the values of L( ). The minimum of the function occurs at * = 4. If we thin of differences L( ), with being an integer, as the discrete analogue of a gradient, then this last inequality imlies that to fi nd the minimum of L, we need only loo for the oint at which the gradients L( ) and L( ) are close to zero, or, equivalently, the oint at which their average ( L( ) L( ))/ is close to zero. Observe that L( + ) + L( ) L( + ) L( ) =. () So the roblem of minimizing L reduces to the roblem of fi nding the oint at which the discrete gradient, ( L( + ) L( )), = an integer g( ) = L([ ] + ) L([ ]), otherwise. (5) Then g( ) behaves, in some sense, lie a gradient for extended function L( ). In fact, g is a subgradient and serves as a gradient in the otimization of functions that are not necessarily differentiable. An estimate of the subgradient is y( + c ) y( c ) g ˆ( ) =, c (6) where y( ) is the (iecewise linear) extension of y( ), and is a Bernoulli random variable taing the values with equal robability. This estimate is a simultaneous erterbation estimate of the subgradient. The foregoing suggests an SA algorithm that is analogous to the continuous version (Eq. 6): ˆ = ˆ agˆ([ ˆ ]), ˆ = an integer. + (7) JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005) 9
6 S. D. HILL If the algorithm is stoed at iteration n, the estimate of the otimizing value is [ ˆ n ]. Note that this algorithm is similar to the discrete SA algorithm in Ref. 6. The General Algorithm Assume that the loss function is searable and convex, i.e., L() = L ( ), = (8) where each L is an integer convex function. Similar to Inequality 0, the oint * minimizes this loss function if L ( ) 0 L ( + ), =, K,. (9) * * Unlie the onedimensional case, this condition is sufficient but not necessary for a minimum. (Cassandras et al. 0 give sufficient conditions for a minimum.) Let L( ) = ( L ( ),..., L ( )). If we view L( ) and L( ) as gradients of L, then the minimum occurs at a oint where their average ( L( ) L( ))/ is close to zero. Let g( )= L( + ) + L( ). Again, the roblem of minimizing the loss reduces to fi nding a oint at which g is zero or close to zero. Since L is searable, g satisfies the following identity: g( )= L ( + ) L ( ) L ( + ) L ( ), K,. (0) This form of g and the discussion in the last section suggest the following simle gradient estimate similar to the estimate in SPSA. Let = (,,..., ), where the comonents of are indeendent Bernoulli random variables taing the values. Then ˆ ˆ ˆ y ) ˆ )) g( )= (( + ( y. () M This estimate satisfies an imortant roerty: it is an unbiased estimate of the subgradient. The subgradient estimate in Eq. leads to an algorithm similar to the one in Eq. 7: ˆ ˆ ˆ([ ˆ ]). + = ag () The initial value is ˆ = (ˆ, ˆ, K, ˆ ), where each ˆ is an integer, =,,, and [ ] is the vector obtained by taing the integer art of each comonent of. To solve the resource allocation roblem, this algorithm requires a slight modification that ensures that each iterate is a feasible solution. An easy way to guarantee this is by a roection if an iterate lands outside the feasible solution set, roect it onto the feasible solution set before generating the next iterate. The stes to imlement the discrete algorithm are similar to those for the continuous one: Ste. Generate a vector = (,,..., ), the comonents of which are Bernoulli random variables taing the values with robability /. Ste. Form the value [ ˆ ], the vector with integer comonents. Ste 3. Consider the two erturbations about [ ˆ ]: and ( + ) [ ˆ ] = [ˆ ] + ( ) [ ˆ ] = [ˆ ]. ± Ste 4. Evaluate y at the erturbed values [ ˆ ] ( ) to ( + ) ( + ) obtain y = y(ˆ ) and y( ) = y(ˆ ( ) ) (measurements of L(ˆ + ( ) ) and L(ˆ ( ) ), resectively) and form the estimate of gˆ(ˆ ). Ste 5. Udate the algorithm according to Eq.. DISCUSSION The erformance of the algorithm is an imortant ractical issue. Its numerical erformance has been studied reviously.,7 It has also been comared (see Ref. 8) with the method of simulated annealing, an algorithm that alies to functions of a discrete argument. Numerical studies rovide insight into secific alications and erformance in secial roblems when comared with existing algorithms. However, numerical studies do not rove convergence, which is a requirement of any algorithm. Convergence is a theoretical roerty and must be established by a roof of convergence, as has been done for the continuous algorithm. 4,9 With resect to convergence, there are two imortant questions: Does the algorithm converge? How fast does it converge? The discrete SPSA algorithm (Eq. ) converges under some restrictions on the loss function. The latter question about the rate of convergence remains oen and is currently under investigation. 0 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005)
7 REFERENCES Gerencsér, L., Hill, S. D., and Vágó, Z., Otimization over Discrete Sets via SPSA, in Proc. 38th IEEE Conf. on Decision and Control, Phoenix, AZ, (999). Hill, S. D., Gerencsér, L., and Vágó, Z., Stochastic Aroximation on Discrete Sets Using Simultaneous Difference Aroximations, in Proc. 004 Am. Control Conf., Boston, MA, (004). 3 Kushner, H. J., and Clar, D. S., Stochastic Aroximation Methods for Constrained and Unconstrained Systems, SringerVerlag (978). 4 Sall, J. C., Multivariate Stochastic Aroximation Using a Simultaneous Perturbation Gradient Aroximation, IEEE Trans. Auto. Contr. 37, (99). 5 Sall, J. C., An Overview of the Simultaneous Perturbation Method for Effi cient Otimization, Johns Hoins APL Tech. Dig. 9(4), (998). 6 Ibarai, T., and Katoh, N., Resource Allocation Problems: Algorithmic Aroaches, MIT Press (988). 7 Cullenbine, C. A., Gallagher, M. A., and Moore, J. T., Assigning Nuclear Weaons with Reactive Tabu Search, Mil. Oerations Res. 8, (003). 8 Ermoliev, Y., Facility Location Problem, in Numerical Techniques for Stochastic Otimization, Y. Ermoliev and R. JB. Wets (eds.), Sringer, New Yor, (988). 9 Wieselthier, J. E., Barnhart, C. M., and Ehremides, A., Otimal Admission Control in CircuitSwitched Multiho Networs, in Proc. 3st IEEE Conf. on Decision and Control, Tucson, AZ, (99). 0 Cassandras, C. G., Dai, L., and Panayiotou, C. G., Ordinal Otimization for a Class of Deterministic and Stochastic Discrete Resource Allocation Problems, IEEE Trans. Auto. Contr. 43(7), (998). Gelfand, B., and Mitter, S. K., Simulated Annealing with Noisy or Imrecise Energy Measurements, J. Otimiz. Theory A. 6, 49 6 (989). Kleywegt, A., HomemdeMello, T., and Shairo, A., The Samle Average Aroximation Method for Stochastic Discrete Otimization, SIAM J. Otimiz. (), (00). 3 Sall, J. C., Introduction to Stochastic Search and Otimization, Wiley, NJ (003). 4 Favati, P., and Tardella, F., Convexity in Nonlinear Integer Programming, Ric. Oerat. 53, 3 34 (990). 5 Miller, B. L., On Minimizing Nonsearable Functions Defi ned on the Integers with an Inventory Alication, SIAM J. A. Math., (97). 6 Duac, V., and Herenrath, U., Stochastic Aroximation on a Discrete Set and the MultiArmed Bandit Problem, Comm. Stat. Sequential Anal., 5 (98). 7 Whitney, J. E. II, Hill, S. D., and Solomon, L. I., Constrained Otimization over Discrete Sets via SPSA with Alication to NonSearable Resource Allocation, in Proc. 00 Winter Simulation Conf., Arlington, VA, (00). 8 Whitney, J. E. II, Hill, S. D., Wairia, D., and Bahari, F., Comarison of the SPSA and Simulated Annealing Algorithms for the Constrained Otimization of Discrete NonSearable Functions, in Proc. 003 Am. Control Conf., Denver, CO, (003). 9 Gerencsér, L., Rate of Convergence of Moments for a Simultaneous Perturbation Stochastic Aroximation Method for Function Minimization, IEEE Trans. Auto. Contr. 44, (999). ACKNOWLEDGMENT: This wor was artially suorted by the APL IR&D rogram. THE AUTHOR Stacy D. Hill oined APL in 983 and is a member of the Strategic Systems Deartment. He received a B.S. in sychology and an M.S. in mathematics in 975 and 977, resectively, both from Howard University, and a D.Sc. in engineering (systems science and mathematics) in 98 from Washington University in St. Louis. Dr. Hill has led a variety of systems analysis and modeling roects, including develoing techniques for analyzing weaons systems accuracy erformance. He has ublished articles on diverse toics, including simulation, otimization, and arameter estimation. His article, Otimization of Discrete Event Dynamic Systems via Simultaneous Perturbation Stochastic Aroximation, was ublished in a secial issue of IIE Transactions and received a Best Paer award. His current research interest is stochastic discrete otimization. Dr. Hill can be reached via at Stacy Hill JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005)
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