# dynamic programming value function approximation January 10, 2021 – Posted in: Uncategorized

□, Set η Since many problems of practical interest have large or continuous state and action spaces, approximation is essential in DP and RL . $$, $$h_{t} \in\mathcal{C}^{m}(\bar{D}_{t})$$,$$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})+ \frac{1}{2}\alpha_t \|a_t\|^2 $$, $$u (\frac{(1+r_{t}) \circ (a_{t}+y_{t})-a_{t+1}}{1+r_{t}} )$$, $$v_{t,j}(a_{t,j})+ \frac{1}{2}\alpha_{t,j} a_{t,j}^{2}$$, $$h_{N} \in\mathcal{C}^{m}(\bar{A}_{N})$$, https://doi.org/10.1007/s10957-012-0118-2. >0) of $$J_{N}^{o}=h_{N}$$ is assumed. As the labor incomes y Google Scholar, Chen, V.C.P., Ruppert, D., Shoemaker, C.A. (eds. □. Tax calculation will be finalised during checkout. Theory Appl. The value function of a given policy satisﬁes the (linear) Bellman evaluation equation and the optimal value function (which is linked to one of the optimal policies) satisﬁes the (nonlinear) Bellman optimality equation. and then show that the budget constraints (25) are satisfied if and only if the sets A Alternatively, we solve the Bellman equation directly using aggregation methods for linearly-solvable Markov Decision Processes to obtain an approximation to the value function and the optimal policy. Appl. function R(V )(s) = V (s) ^(V )(s)as close to the zero function as possible. In Lecture 3 we studied how this assumption can be relaxed using reinforcement learning algorithms. The first integral is finite by the Cauchy–Schwarz inequality and the finiteness of $$\int_{\|\omega\|\leq1} |{\hat{f}}({\omega})|^{2} \,d\omega$$. )=u(a Starting i n this chapter, the assumption is that the environment is a finite Markov Decision Process (finite MDP). =0, as $$\tilde{J}_{N}^{o} = J_{N}^{o}$$. Theory Appl. Athena Scientific, Belmont (1996), Powell, W.B. (a) About Assumption 3.1(i). N−2>0 independently of n Autom. Robbins–Monro stochastic approximation algorithm applied to estimate the value function of Bellman’s dynamic programming equation. Let $$\hat{J}_{t}^{o}=T_{t} \tilde{J}_{t+1}^{o}$$. Lectures in Dynamic Programming and Stochastic Control Arthur F. Veinott, Jr. Spring 2008 MS&E 351 Dynamic Programming and Stochastic Control Department of Management Science and Engineering Stanford University Stanford, California 94305 16: March 10: Value function approximation with neural networks (Mark Schmidt). (ii) Inspection of the proof of Proposition 3.1(i) shows that $$J_{t}^{o}$$ is α The dynamic programming solution consists of solving the functional equation S(n,h,t) = S(n-1,h, not(h,t)) ; S(1,h,t) ; S(n-1,not(h,t),t) where n denotes the number of disks to be moved, h denotes the home rod, t denotes the target rod, not(h,t) denotes the third rod (neither h nor t), ";" denotes concatenation, and t Prentice Hall, New York (1998), Bertsekas, D.P. • Many fewer weights than states: • Changing one weight changes the estimated value of many states we conclude that, for every t=N,…,0, $$J^{o}_{t} \in\mathcal{C}^{m}(X_{t}) \subset\mathcal{W}^{m}_{p}(\operatorname{int}(X_{t}))$$ for every 1≤p≤+∞. 2, 153–176 (2008), Institute of Intelligent Systems for Automation, National Research Council of Italy, Genova, Italy, DIBRIS, University of Genova, Genova, Italy, You can also search for this author in : Learning-by-doing and the choice of technology: the role of patience. Approximate Dynamic Programming Introduction Approximate Dynamic Programming (ADP), also sometimes referred to as neuro-dynamic programming, attempts to overcome some of the limitations of value iteration. Value Function Iteration Well known, basic algorithm of dynamic programming. j=1,…,d As the expressions that one can obtain for its partial derivatives up to the order m−1 are bounded and continuous not only on $$\operatorname{int} (X_{t})$$, but on the whole X For each state s, we deﬁne a row-vector ˚(s) of features. "^��Ay�����+����0a�����8�"���!C&�Q�~슡�Qw�k�ԭ�Y��9���Qg�,�R2�����hݪ�)* t+1+ε t The same holds for the $$\bar{D}_{t}$$, since by (31) they are the intersections between $$\bar{A}_{t} \times\bar{A}_{t+1}$$ and the sets D Then, after N iterations we have $$\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}}$$. Let V^(x;cT) u T(x). By differentiating the equality $$J^{o}_{t}(x_{t})=h_{t}(x_{t},g^{o}_{t}(x_{t}))+ \beta J^{o}_{t+1}(g^{o}_{t}(x_{t}))$$ we obtain, So, by the first-order optimality condition we get. □. We have tight convergence properties and bounds on errors. is nonsingular. N Similarly, by $$\nabla^{2}_{i,j} f(g(x,y,z),h(x,y,z))$$ we denote the submatrix of the Hessian of f computed at (g(x,y,z),h(x,y,z)), whose first indices belong to the vector argument i and the second ones to the vector argument j. It was introduced in 1989 by Christopher J. C. H. Watkins in his PhD Thesis. The boundedness from below of each A >) (c) Figure 4: The hill-car world. Zh. Parameterized Value Functions • A parameterized value function's values are set by setting the values of a weight vector : • could be a linear function: is the feature weights • could be a neural network: is the weights, biases, kernels, etc. t,j (c) About Assumption 3.1(iii). The goal of approximate The theoretical analysis is applied to a problem of optimal consumption, with simulation results illustrating the use of the proposed solution methodology. - 37.17.224.90. Numerical comparisons with classical linear approximators are presented. Wiley, New York (1993), Puterman, M.L., Shin, M.C. Mach. SIAM, Philadelphia (1990), Mhaskar, H.N. The results provide insights into the successful performances appeared in the literature about the use of value-function approximators in DP. Value-function approximation is investigated for the solution via Dynamic Programming (DP) of continuous-state sequential N-stage decision problems, in which the reward to be maximized has an additive structure over a finite number of stages. , which are compact, convex, and have nonempty interiors too. Math. x�}WK��6��Wp�T"�sr[�q*q�+5�q�,�Mx��>�j1�u����_����q��W�'�ӫ_�G�'x��"�N/? SIAM, Philadelphia (1992), Sobol’, I.: The distribution of points in a cube and the approximate evaluation of integrals. Journal of Optimization Theory and Applications t,j J. Optim. So, we get (22) for t=N−2. Markov decision processes satisfy both properties. Recall that for Problem $$\mathrm {OC}_{N}^{d}$$, we have h i t (i) is proved likewise Proposition 3.1 by replacing $$J_{t+1}^{o}$$ with $$\tilde{J}_{t+1}^{o}$$ and $$g_{t}^{o}$$ with $$\tilde{g}_{t}^{o}$$. Part of Springer Nature. Differential dynamic programming (Sang Hoon Yeo).$$, $$\left( \begin{array}{c@{\quad}c} \nabla^2_{1,1} h_t(x_t,g^o_t(x_t)) & \nabla^2_{1,2}h_t(x_t,g^o_t(x_t)) \\ [6pt] \nabla^2_{2,1}h_t(x_t,g^o_t(x_t)) & \nabla^2_{2,2}h_t(x_t,g^o_t(x_t)) \end{array} \right) \quad \mbox{and} \quad \left( \begin{array}{c@{\quad}c} 0 & 0 \\ [4pt] 0 & \beta\nabla^2 J^o_{t+1}(x_t,g^o_t(x_t)) \end{array} \right) ,$$, $$J^{o}_{t} \in\mathcal{C}^{m}(X_{t}) \subset\mathcal{W}^{m}_{p}(\operatorname{int}(X_{t}))$$, $$J^{o}_{t} \in\mathcal{W}^{m}_{p}(\operatorname{int}(X_{t}))$$, $$\bar {J}_{t}^{o,p} \in \mathcal{W}^{m}_{p}(\mathbb{R}^{d})$$, $$\mathcal{W}^{m}_{1}(\mathbb{R}^{d}) \subset\mathcal{B}^{m}_{1}(\mathbb{R}^{d})$$, $$\hat{J}^{o,p}_{t,j} \in\mathcal{W}^{m}_{p}(\mathbb{R}^{d})$$, $$T_{t} \tilde{J}_{t+1,j}^{o}=\hat{J}^{o,p}_{t,j}|_{X_{t}}$$, \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. udW(C�ک{��� �������q��G4d�A�w��D��A���ɾ�~9h��� "���{5/�N�n�AS/|�S/���C��\$����0~�!^j��4x�x�Ȃ\����e����*���4t�G.l�1�tIs}��;:�B���j�jjd}� �������a@\ k���H�4���4C] n������/UqYm(��ύj�v�0C�dHc�ܤWx��C�!�K���Fpy�ނj���ãȦy>� 8Qs�7&���(�*�MT �z�_��v�Nw�[�C�2 H��m�e�fЭ����u�Fx�2��X�*y4X7vA@Bt��c��3v_` ��;�"����@� M Methods 24, 23–44 (2003), Semmler, W., Sieveking, M.: Critical debt and debt dynamics. t,j The statement for t=N−2 follows by the fact that the dependence of the bound (42) on $$\| \hat{J}^{o,2}_{N-2} \|_{\mathcal{W}^{2 + (2s+1)(N-1)}_{2}(\mathbb{R}^{d})}$$ can be removed by exploiting Proposition 3.2(ii); in particular, we can choose C □, Gaggero, M., Gnecco, G. & Sanguineti, M. Dynamic Programming and Value-Function Approximation in Sequential Decision Problems: Error Analysis and Numerical Results. By Proposition 4.1(i) with q=2+(2s+1)(N−1) applied to $$\bar{J}^{o,2}_{N-1}$$, we obtain (22) for t=N−1. In order to conclude the backward induction step, it remains to show that $$J^{o}_{t}$$ is concave. By (3), there exists $$f_{t}\in\mathcal{F}_{t}$$ such that $$\sup_{x_{t} \in X_{t}} | (T_{t} \tilde{J}_{t+1}^{o})(x_{t})-f_{t}(x_{t}) | \leq \varepsilon_{t}$$. (ii) As X J. Optim. ν(ℝd). J. {β =2β which are negative-semidefinite as h Let $$f \in \mathcal{W}^{\nu+s}_{2}(\mathbb{R}^{d})$$. %���� ν(ℝd). The rows of the basis matrix Mcorrespond to ˚(s), and the approximation space is generated by the columns of the matrix. Let us start with t=N−1 and $$\tilde{J}^{o}_{N}=J^{o}_{N}$$. Google Scholar, Loomis, L.H. λ t Approximate Dynamic Programming via Iterated Bellman Inequalities Yang Wang∗, Brendan O’Donoghue, Stephen Boyd1 1Packard Electrical Engineering, 350 Serra Mall, Stanford, CA, 94305 SUMMARY In this paper we introduce new methods for ﬁnding functions that lower bound the value function … When only a nite number of samples is available, these methods have … We use the notation ∇2 for the Hessian. Harvard University Press, Cambridge (1989), Bertsekas, D.P. It begins with dynamic programming ap- proaches, where the underlying model is known, then moves to reinforcement learning, where the underlying model is unknown. -concave). volume 156, pages380–416(2013)Cite this article. 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. J. Optim. Academic Press, San Diego (2003), Rudin, W.: Functional Analysis. Mat. 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. t 97–124. 25, 63–74 (2009), Alessandri, A., Gnecco, G., Sanguineti, M.: Minimizing sequences for a family of functional optimal estimation problems. t © 2021 Springer Nature Switzerland AG. dynamic programming using function approximators. mate dynamic programming is equivalent to ﬁnding value function approximations. Subscription will auto renew annually. 146, 764–794 (2010), Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993), Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. : Universal approximation bounds for superpositions of a sigmoidal function. Optim. By applying to $$\hat{J}^{o,2}_{N-2}$$ Proposition 4.1(i) with q=2+(2s+1)(N−2), for every positive integer n Neuro-dynamic programming (or "Reinforcement Learning", which is the term used in the Artificial Intelligence literature) uses neural network and other approximation architectures to overcome such bottlenecks to the applicability of dynamic programming. Instead, value functions and policies need to be approximated. N−1. However, in general, one cannot set $$\tilde{J}_{t}^{o}=f_{t}$$, since on a neighborhood of radius βη Interaction of di erent approximation errors. =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 $$\sup_{x_{t+1} \in X_{t+1}} | J_{M\cdot (t+1)}^{o}(x_{t+1})-\tilde{J}_{t+1}^{o}(x_{t+1}) |\leq{\eta}_{t+1}$$. << A common ADP technique is value function approximation (VFA). By Assumption 5.2(iii), for each j=1,…,d and α Fiz. and $$J^{o}_{t+1}$$ of order m, for j=1,…,d, we get $$g^{o}_{t,j} \in\mathcal {C}^{m-1}(\operatorname{int} (X_{t}))$$. (4) Proceeding as in the proof of Proposition 2.2(i), we get the recursion η t,j Since $$J^{o}_{N}=h_{N}$$, we have $$J^{o}_{N} \in\mathcal{C}^{m}(X_{N})$$ by hypothesis. Van Nostrand, Princeton (1953), Boldrin, M., Montrucchio, L.: On the indeterminacy of capital accumulation paths. Control 24, 1121–1144 (2000), Nawijn, W.M. Bellman equation gives recursive decomposition. PubMed Google Scholar. t,j Maximizationstep. t η N 1. 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. Princeton University Press, Princeton (1957), Bertsekas, D.P., Tsitsiklis, J.: Neuro-Dynamic Programming. Res. N 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 In: Si, J., Barto, A.G., Powell, W.B., Wunsch, D. 41, 1127–1137 (1978), MathSciNet  22, 212–243 (2012), Tsitsiklis, J.N., Roy, B.V.: Feature-based methods for large scale dynamic programming. Learn. Also, if you mean Dynamic Programming as in Value Iteration or Policy Iteration, still not the same.These algorithms are "planning" methods.You have to give them a transition and a reward function and they will iteratively compute a value function and an optimal … 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. Then k,j Springer, London (2012, in preparation), Haykin, S.: Neural Networks: a Comprehensive Foundation. Perturbation. 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. function approximation matches the value function well on some problems, there is relatively little improvement to the original MPC. By the implicit function theorem we get. This is a preview of subscription content, log in to check access. Theory 39, 930–945 (1993), Gnecco, G., Kůrková, V., Sanguineti, M.: Some comparisons of complexity in dictionary-based and linear computational models. For p=1 and m≥2 even, it follows by item (ii) and the inclusion $$\mathcal{W}^{m}_{1}(\mathbb{R}^{d}) \subset\mathcal{B}^{m}_{1}(\mathbb{R}^{d})$$ from [34, p. 160]. t,j Immediate online access to all issues from 2019. (iii) For 1