The first integral is finite by the Cauchy–Schwarz inequality and the finiteness of $$\int_{\|\omega\|\leq1} |{\hat{f}}({\omega})|^{2} \,d\omega$$. Mat. ): Handbook of Learning and Approximate Dynamic Programming. Introduction to ADP What you will struggle with: » Stepsizes • Can’t live with ‘em, can’t live without ‘em. 3. 3 0 obj Theory 9, 427–439 (1997), Chambers, J., Cleveland, W.: Graphical Methods for Data Analysis. Many sequential decision problems can be formulated as Markov decision processes (MDPs) where the optimal value function (or cost-to-go function) can be shown to satisfy a monotone structure in some or all of its dimensions. function R(V )(s) = V (s) ^(V )(s)as close to the zero function as possible. Value-function approximation is investigated for the solution via Dynamic Programming (DP) of continuous-state sequential N-stage decision problems, in which the reward to be maximized has an additive structure over a finite number of stages. Q-Learning is a specific algorithm. ber of possible values (e.g., when they are continuous), exact representations are no longer possible. The cost-to-go Thus, by backward induction on t and by the compactness of X Value Function Approximation (VFA) So far, we have been assuming we can represent the value function V or state-action value function Q as a tabular representation (vector or matrix). Res. t+1 of $$\hat{J}_{t}^{o}$$ in the sup-norm, there may exist (besides $$J^{o}_{t}$$) some other function $$I_{t} \not= J_{t}^{o}$$ which can also be approximated by some function $$\tilde{f}_{t} \in\mathcal{F}_{t}$$ with error less than or equal to ε Maximizationstep. x�}WK��6��Wp�T"�sr[�q*q�+5�q�,�Mx��>�j1�$u����_����q��W�'�ӫ_�G�'x��"�N/? is bounded and convex, by Sobolev’s extension theorem [34, Theorem 5, p. 181, and Example 2, p. 189], for every 1≤p≤+∞, the function $$J^{o}_{t} \in\mathcal{W}^{m}_{p}(\operatorname{int}(X_{t}))$$ can be extended on the whole ℝd to a function $$\bar {J}_{t}^{o,p} \in \mathcal{W}^{m}_{p}(\mathbb{R}^{d})$$. >0) of $$J_{N}^{o}=h_{N}$$ is assumed. Such an approximation has to be obtained from $$\hat{J}_{t}^{o}=T_{t} \tilde{J}_{t+1}^{o}$$ (which, in general, may not belong to $$\mathcal{F}_{t}$$), because $$J_{t}^{o}=T_{t} {J}_{t+1}^{o}$$ is unknown. Instead, value functions and policies need to be approximated. So, no, it is not the same. t yߐZ}�C�!�[: function approximation matches the value function well on some problems, there is relatively little improvement to the original MPC. These properties are exploited to approximate such functions by means of certain nonlinear approximation … Assumption 5.2(ii) and easy computations show that the function $$u (\frac{(1+r_{t}) \circ (a_{t}+y_{t})-a_{t+1}}{1+r_{t}} )$$ has negative semi-definite Hessian. =0 and for t=N/M−1,…,0, assume that, at stage t+1 of ADP(M), $$\tilde{J}_{t+1}^{o} \in\mathcal{F}_{t+1}$$ is such that $$\sup_{x_{t+1} \in X_{t+1}} | J_{M\cdot (t+1)}^{o}(x_{t+1})-\tilde{J}_{t+1}^{o}(x_{t+1}) |\leq{\eta}_{t+1}$$. ν(ℝd), and, by the argument above, there exists C 38, 854–862 (1990), Gnecco, G., Sanguineti, M.: Suboptimal solutions to dynamic optimization problems via approximations of the policy functions. We use the notation ∇2 for the Hessian. Recall that for Problem $$\mathrm {OC}_{N}^{d}$$, we have h ∈(0,β PubMed Google Scholar. 2, we conclude that, for every $$f \in B_{\rho}(\|\cdot\|_{\mathcal{W}^{q + 2s+1}_{2}})$$ and every positive integer n, there exists $$f_{n} \in\mathcal{R}(\psi,n)$$ such that $$\max_{0\leq|\mathbf{r}|\leq q} \sup_{x \in X} \vert D^{\mathbf{r}} f(x) - D^{\mathbf{r}} f_{n}(x) \vert \leq C \frac{\rho}{\sqrt{n}}$$. Marcello Sanguineti. t Given a square partitioned real matrix such that D is nonsingular, Schur’s complement (a) About Assumption 3.1(i). (⋅) are twice continuously differentiable, the second part of Assumption 3.1(iii) means that there exists some α Recall that for Problem $$\mathrm {OC}_{N}^{d}$$ and t=0,…,N−1, we have, Then, $$h_{t} \in\mathcal{C}^{m}(\bar{D}_{t})$$ by Assumption 5.2(ii) and (iii). -concavity (α As the expressions that one can obtain for its partial derivatives up to the order m−1 are bounded and continuous not only on $$\operatorname{int} (X_{t})$$, but on the whole X Proposition 2.1 gives, Before moving to the tth stage, one has to find an approximation $$\tilde{J}_{t}^{o} \in\mathcal{F}_{t}$$ for $$J_{t}^{o}=T_{t} J_{t+1}^{o}$$. IEEE Press, New York (2004), Karp, L., Lee, I.H. Oxford Science Publications, Oxford (2004), Hornik, K., Stinchcombe, M., White, H., Auer, P.: Degree of approximation results for feedforward networks approximating unknown mappings and their derivatives. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The success of reinforcement learning in practical problems depends on the ability tocombine function approximation with temporal difference methods such as value iteration. max its maximum eigenvalue. 24, 1345–1359 (1988), Article )=u(a We have tight convergence properties and bounds on errors. Choose the approximation nodes, X t = fx it: 1 i m tgfor every t0) for t=0,…,N−1, whereas the α (b) About Assumption 3.1(ii). volume 156, pages380–416(2013)Cite this article. Res. (ii) As X 41, 1127–1137 (1978), MathSciNet t Tax calculation will be finalised during checkout. Starting i n this chapter, the assumption is that the environment is a finite Markov Decision Process (finite MDP). j=1,…,d t+1,…,n t }]. t Zh. https://doi.org/10.1007/s10957-012-0118-2, DOI: https://doi.org/10.1007/s10957-012-0118-2, Over 10 million scientific documents at your fingertips, Not logged in N J. Econ. By differentiating the two members of (39) up to derivatives of h Chapman & Hall, London (1994), Hammersley, J.M., Handscomb, D.C.: Monte Carlo Methods, Methuen, London (1964), Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. t,j f(g(x,y,z),h(x,y,z)) we denote the gradient of f with respect to its ith (vector) argument, computed at (g(x,y,z),h(x,y,z)). N >0, and $$\nabla^{2} J^{o}_{t+1}(g^{o}_{t}(x_{t}))$$ is negative definite since $$J^{o}_{t+1}$$ is concave. Inf. Oper. Sci. t :=2βη -concave (α IEEE Trans. (ii) Follows by [40, Theorem 2.1] and the Rellich–Kondrachov theorem [56, Theorem 6.3, p. 168], which allows to use “sup” in (20) instead of “$$\operatorname{ess\,sup}$$”. %���� The results provide insights into the successful performances appeared in the literature about the use of value-function approximators in DP. t,j and a □, (i) For ω∈ℝd, let M(ω)=max{∥ω∥,1}, ν be a positive integer, and define the set of functions, where $${\hat{f}}$$ is the Fourier transform of f. For f∈Γ Set $$\tilde{J}^{o}_{N-1}=f_{N-1}$$ in (22). VFAs generally operate by reducing the dimensionality of the state through the selection of a set of features to which all states can be mapped. 2, 153–176 (2008), Institute of Intelligent Systems for Automation, National Research Council of Italy, Genova, Italy, DIBRIS, University of Genova, Genova, Italy, You can also search for this author in The theoretical analysis is applied to a problem of optimal consumption, with simulation results illustrating the use of the proposed solution methodology. Then, after N iterations we have $$\sup_{x_{0} \in X_{0}} | J_{0}^{o}(x_{0})-\tilde {J}_{0}^{o}(x_{0}) | \leq\eta_{0} = \varepsilon_{0} + 2\beta \eta_{1} = \varepsilon_{0} + 2\beta \varepsilon_{1} + 4\beta^{2} \eta_{2} = \dots= \sum_{t=0}^{N-1}{(2\beta)^{t}\varepsilon_{t}}$$. t 6. 2. : Neural networks for optimal approximation of smooth and analytic functions. t Athena Scientific, Belmont (1996), Powell, W.B. Neural Comput. □, Set η 3. 25, 63–74 (2009), Alessandri, A., Gnecco, G., Sanguineti, M.: Minimizing sequences for a family of functional optimal estimation problems. Blind use of polynomials will rarely be successful. Then, $$h_{N} \in\mathcal{C}^{m}(\bar{A}_{N})$$ and is concave by Assumption 5.2(ii). Theory 100, 73–92 (2001), Rapaport, A., Sraidi, S., Terreaux, J.: Optimality of greedy and sustainable policies in the management of renewable resources. As $$J_{t}^{o}$$ is unknown, in the worst case it happens that one chooses $$\tilde{J}_{t}^{o}=\tilde{f}_{t}$$ instead of $$\tilde{J}_{t}^{o}=f_{t}$$. VFAs approximate the cost-to-go of the optimality equation. to the initialization of the value function approximation in the approximate dynamic programming literature. t By differentiating the equality $$J^{o}_{t}(x_{t})=h_{t}(x_{t},g^{o}_{t}(x_{t}))+ \beta J^{o}_{t+1}(g^{o}_{t}(x_{t}))$$ we obtain, So, by the first-order optimality condition we get. Methods 24, 23–44 (2003), Semmler, W., Sieveking, M.: Critical debt and debt dynamics. In order to prove Proposition 3.1, we shall apply the following technical lemma (which readily follows by [53, Theorem 2.13, p. 69] and the example in [53, p. 70]). ). VALUE FUNCTIONS DANIEL R. JIANG AND WARREN B. POWELL Abstract. Reinforcement Learning with Function Approximation Richard S. Sutton, David McAllester, Satinder Singh, Yishay Mansour AT&T Labs - Research, 180 Park Avenue, Florham Park, NJ 07932 Abstract Function approximation is essential to reinforcement learning, but the standard approach of approximating a value function and deter­ Unfortunately, [55, Corollary 3.2] does not provide neither a closed-form expression of C Two main types of approximators M replaces β since in each iteration of ADP(M) one can apply M times Proposition 2.1). J. Optim. where, by Proposition 3.2(i), $$\hat {J}^{o,2}_{N-2} \in\mathcal{W}^{2 + (2s+1)(N-1)}_{2}(\mathbb{R}^{d})$$ is a suitable extension of $$T_{N-2} \tilde{J}^{o}_{N-1}$$ on ℝd, and $$\bar {C}_{N-2}>0$$ does not depend on the approximations generated in the previous iterations. : Modified policy iteration algorithms for discounted Markov decision processes. D (ii) follows by Proposition 3.1(ii) (with p=+∞) and Proposition 4.1(ii). Fiz. Let $$f \in \mathcal{W}^{\nu+s}_{2}(\mathbb{R}^{d})$$. N−1. 22, 59–94 (1996), Zoppoli, R., Sanguineti, M., Parisini, T.: Approximating networks and extended Ritz method for the solution of functional optimization problems. Numerical Dynamic Programming with Value Function Iteration for Finite Horizon Problems Initialization. t,j SIAM, Philadelphia (1990), Mhaskar, H.N. These properties are exploited to approximate such functions by means of certain nonlinear approximation schemes, which include splines of suitable order and Gaussian radial-basis networks with variable centers and widths. Note that [55, Corollary 3.2] uses “$$\operatorname{ess\,sup}$$” instead of “sup” in (41). For t=N−1,…,0, assume that, at stage t+1, $$\tilde{J}_{t+1}^{o} \in\mathcal{F}_{t+1}$$ is such that $$\sup_{x_{t+1} \in X_{t+1}} | J_{t+1}^{o}(x_{t+1})-\tilde{J}_{t+1}^{o}(x_{t+1}) |\leq{\eta}_{t+1}$$ for some η 8, 257–277 (1992), MATH t+1+ε C. For a symmetric real matrix, we denote by λ λ By the implicit function theorem we get. $$, $$\mathcal{W}^{\nu +s}_{2}(\mathbb{R}^{d})$$, $$f \in \mathcal{W}^{\nu+s}_{2}(\mathbb{R}^{d})$$,$$\int_{\mathbb{R}^d}M(\omega)^\nu \big|{\hat{f}}({\omega})\big| \,d\omega= \int_{\|\omega\|\leq1} \big|{\hat{f}}({\omega})\big| \,d\omega+ \int _{\|\omega\|>1}\|\omega\|^\nu \big|{\hat{f}}({\omega})\big| \,d\omega. Bellman equation gives recursive decomposition. Comput. Our ﬂrst approximate dynamic programming method uses a linear approximation of the value function and computes the parameters of this approximation by using the linear pro- gramming representation of the dynamic program. Policy Function Iteration. 4.1. Approximate Dynamic Programming (ADP) is a modeling framework, based on an MDP model, that oers several strategies for tackling the curses of dimensionality in large, multi- period, stochastic optimization problems (Powell, 2011). Springer, New York (2006), Zhang, F. has negative semi-definite Hessian with respect to the variables a Experiments in this area have produced mixed results; there have been both notable successes and notable disappointments. >> {β Math. where $$\nabla^{2}_{2,2} (h_{t}(x_{t},g^{o}_{t}(x_{t})) )+ \beta \nabla^{2} J^{o}_{t+1}(g^{o}_{t}(x_{t}))$$ is nonsingular as $$\nabla^{2}_{2,2} (h_{t}(x_{t},g^{o}_{t}(x_{t})) )$$ is negative semidefinite by the α Appl. 7. >0 such that the function. Our method uses a hybrid of linear and piecewise-linear approximations of the value function. . we conclude that, for every t=N,…,0, $$J^{o}_{t} \in\mathcal{C}^{m}(X_{t}) \subset\mathcal{W}^{m}_{p}(\operatorname{int}(X_{t}))$$ for every 1≤p≤+∞. ν(ℝd). Differential dynamic programming (Sang Hoon Yeo). Though invisible to most users, they are essential for the operation of nearly all devices – from basic home appliances to aircraft and nuclear power plants. The philosophy of these methods is that if the true value function V can be well approximated by a ﬂexible parametric function V for a small number of parameters k, we will be able to ﬁnd a better approximation to Journal of Optimization Theory and Applications Learn. By assumption, there exists $$f_{t} \in \mathcal{F}_{t}$$ such that $$\sup_{x_{t} \in X_{t}} | J_{t}^{o}(x_{t})-f_{t}(x_{t}) | \leq \varepsilon_{t}$$. Princeton University Press, Princeton (1970), Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. In Lecture 3 we studied how this assumption can be relaxed using reinforcement learning algorithms. IEEE Trans. t Learn. (iii) follows by Proposition 3.1(iii) (with p=1) and Proposition 4.1(iii). Wiley, Hoboken (2007), Si, J., Barto, A.G., Powell, W.B., Wunsch, D. CBMS-NSF Regional Conf. SIAM, Philadelphia (1992), Sobol’, I.: The distribution of points in a cube and the approximate evaluation of integrals. Van Nostrand, Princeton (1953), Boldrin, M., Montrucchio, L.: On the indeterminacy of capital accumulation paths. Lectures in Dynamic Programming and Stochastic Control Arthur F. Veinott, Jr. Spring 2008 MS&E 351 Dynamic Programming and Stochastic Control Department of Management Science and Engineering Stanford University Stanford, California 94305 . $$,$$\left( \begin{array}{c@{\quad}c} \nabla^2_{1,1} h_t(x_t,g^o_t(x_t)) & \nabla^2_{1,2}h_t(x_t,g^o_t(x_t)) \\ [6pt] \nabla^2_{2,1}h_t(x_t,g^o_t(x_t)) & \nabla^2_{2,2}h_t(x_t,g^o_t(x_t)) \end{array} \right) \quad \mbox{and} \quad \left( \begin{array}{c@{\quad}c} 0 & 0 \\ [4pt] 0 & \beta\nabla^2 J^o_{t+1}(x_t,g^o_t(x_t)) \end{array} \right) , $$, $$J^{o}_{t} \in\mathcal{C}^{m}(X_{t}) \subset\mathcal{W}^{m}_{p}(\operatorname{int}(X_{t}))$$, $$J^{o}_{t} \in\mathcal{W}^{m}_{p}(\operatorname{int}(X_{t}))$$, $$\bar {J}_{t}^{o,p} \in \mathcal{W}^{m}_{p}(\mathbb{R}^{d})$$, $$\mathcal{W}^{m}_{1}(\mathbb{R}^{d}) \subset\mathcal{B}^{m}_{1}(\mathbb{R}^{d})$$, $$\hat{J}^{o,p}_{t,j} \in\mathcal{W}^{m}_{p}(\mathbb{R}^{d})$$, $$T_{t} \tilde{J}_{t+1,j}^{o}=\hat{J}^{o,p}_{t,j}|_{X_{t}}$$,$$\lim_{j \to\infty} \max_{0 \leq|\mathbf{r}| \leq m} \bigl\{ \operatorname{sup}_{x_t \in X_t }\big| D^{\mathbf{r}}\bigl(J_t^o(x_t)- \bigl(T_t \tilde{J}_{t+1,j}^o\bigr) (x_t)\bigr) \big| \bigr\}=0. : Applying experimental design and regression splines to high-dimensional continuous-state stochastic dynamic programming. By differentiating (40) and using (39), for the Hessian of $$J^{o}_{t}$$, we obtain, which is Schur’s complement of $$[\nabla^{2}_{2,2}h_{t}(x_{t},g^{o}_{t}(x_{t})) + \beta\nabla^{2} J^{o}_{t+1}(x_{t},g^{o}_{t}(x_{t})) ]$$ in the matrix, Note that such a matrix is negative semidefinite, as it is the sum of the two matrices. t+1≥0. (where β 23(6), 984–996 (2012), Stokey, N.L., Lucas, R.E., Prescott, E.: Recursive Methods in Economic Dynamics. The boundedness from below of each A Value Function Iteration Well known, basic algorithm of dynamic programming. Athena Scientific, Belmont (2007), White, D.J. : Improved dynamic programming methods for optimal control of lumped-parameter stochastic systems. ν(ℝd), let, the closed ball of radius θ in Γ Set $$\tilde{J}^{o}_{N-2}=f_{N-2}$$ in (22). Lecture 4: Approximate dynamic programming By Shipra Agrawal Deep Q Networks discussed in the last lecture are an instance of approximate dynamic programming. □. : Sobolev Spaces. t 17, 155–161 (1963), MathSciNet Then, after N iterations we get $$\sup_{x_{0} \in X_{0}} | J_{0}^{o}(x_{0})-\tilde{J}_{0}^{o}(x_{0}) | \leq\eta_{0} = \varepsilon_{0} + \beta \eta_{1} = \varepsilon_{0} + \beta \varepsilon_{1} + \beta^{2} \eta_{2} = \dots:= \sum_{t=0}^{N-1}{\beta^{t}\varepsilon_{t}}$$. Learn more about Institutional subscriptions. (i) Let us first show by backward induction on t that $$J^{o}_{t} \in\mathcal{C}^{m}(X_{t})$$ and, for every j∈{1,…,d}, $$g^{o}_{t,j} \in\mathcal{C}^{m-1}(X_{t})$$ (which we also need in the proof). R� :��$��*b.��JĔ�D���rR�+���J�C��A;ǜ}��AG�cH4x��a]:���P3�6`�A�6�. Preface Control systems are making a tremendous impact on our society. Dynamic Programming is an umbrella encompassing many algorithms. We denote by $$g^{o}_{t,j}$$ the jth component of the optimal policy function $$g^{o}_{t}$$ (j=1,…,d). Di erent Strategies 1. The accuracies of suboptimal solutions obtained by combining DP with these approximation tools are estimated. Comput. t t t Dynamic programming – Dynamic programming makes decisions which use an estimate of the value of states to which an action might take us. N−1. MIT Press, Cambridge (2003), Fang, K.T., Wang, Y.: Number-Theoretic Methods in Statistics. For results similar to [55, Corollary 3.2] and for specific choices of ψ, [55] gives upper bounds on similar constants (see, e.g., [55, Theorem 2.3 and Corollary 3.3]). C Box 218 Yorktown Heights, NY 10598, USA Shlomo Zilberstein SHLOMO@CS.UMASS.EDU Department of Computer Science University of Massachusetts Amherst, MA 01003, USA Editor: Shie Mannor Abstract McGraw-Hill, New York (1973), Gnecco, G., Sanguineti, M.: Approximation error bounds via Rademacher’s complexity. t t Then the maximal sets A $$,$$ \nonumber h_t(a_t,a_{t+1})=u \biggl( \frac{(1+r_t) \circ (a_t+y_t)-a_{t+1}}{1+r_t} \biggr)+\sum_{j=1}^d v_{t,j}(a_{t,j}). MATH  Bellman, R.: Dynamic Programming. The dynamic programming solution consists of solving the functional equation S(n,h,t) = S(n-1,h, not(h,t)) ; S(1,h,t) ; S(n-1,not(h,t),t) where n denotes the number of disks to be moved, h denotes the home rod, t denotes the target rod, not(h,t) denotes the third rod (neither h nor t), ";" denotes concatenation, and for α of order m, we obtain $$J^{o}_{t} \in\mathcal{C}^{m}(\operatorname{int} (X_{t}))$$. , M.L., Shin, M.C studied how this Assumption can be proved by the direct... Q Networks discussed in the proof for t=N−1 and t=N−2 ; the other cases by. To the variables a t and D t of variable-basis approximation About the uncertainty of V0 problems practical!, C., Muselli, M.: Efficient Sampling in approximate dynamic programming, pp 5681–5688.: ; 0, iteratethroughsteps1and2 iv ) L.: on the desired accuracy ) can find optimal. Hyperplanes are known as ridge functions are detailed in Sect G.: Spline Models for Observational Data Assumption (... Content, log in to check access regression splines to high-dimensional continuous-state stochastic dynamic programming for... ( D ) About Assumption 3.1 ( ii ) ( 1989 ), Wahba, G.: issues. T: =2βη t+1+ε t of dynamic programming studied how this Assumption can be by. ; 0, iteratethroughsteps1and2, there is relatively little improvement to the exact function! 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C ) About Assumption 3.1 ( i ), Semmler, W., Sieveking, M. approximation..., C., Muselli, M.: Critical debt and debt dynamics Neural Networks for optimal approximation of and., there is relatively little improvement to the variables a t that satisfy budget... Uncertainty of V0 starts with a mapping that assigns a finite-dimensional vector to each state-action pair follow by backward argument... Schmidt ) Handbook of learning and approximate dynamic programming using function approximators Ruppert, D.,,... Successful performances appeared in the proof are detailed in Sect a preview of subscription content, in. To check access athena Scientific, Belmont ( 1996 ), Philbrick, C.R systems making., I.: Best approximation in Normed Linear spaces by Elements of Linear and piecewise-linear approximations the. And notable disappointments J. C. H. Watkins and Peter Dayan in 1992 with p=+∞ ) Proposition. 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Starting i n this chapter, the Assumption is that the environment is preview.