Value iteration is a dynamic programming algorithm that computes the optimal value function for a Markov Decision Process (MDP). Each step applies the Bellman optimality equation: for every state, consider all possible actions, compute the expected value of each action using the transition probabilities, and take the maximum.
Given the current value estimates, transition probabilities, rewards, and a discount factor, perform one step of value iteration and return the updated values.
For each state s, compute:
V′(s)=amax[R(s,a)+γs′∑T(s,a,s′)⋅V(s′)]Where:
Input:
values = [0, 0], transitions = [[[0.8, 0.2], [0.3, 0.7]], [[0.5, 0.5], [0.1, 0.9]]], rewards = [[1, 2], [-1, 0]], gamma = 0.9
Output:
[2.0, 0.0]
State 0: Q(0,a0) = 1 + 0.9*(0.80 + 0.20) = 1, Q(0,a1) = 2 + 0.9*(0.30 + 0.70) = 2. V'(0) = max(1, 2) = 2. State 1: Q(1,a0) = -1 + 0 = -1, Q(1,a1) = 0 + 0 = 0. V'(1) = max(-1, 0) = 0.
Input:
values = [0, 0, 0], transitions = [[[0,1,0],[0,0,1]], [[1,0,0],[0,0,1]], [[0,1,0],[1,0,0]]], rewards = [[1,2],[3,0],[-1,5]], gamma = 0.9
Output:
[2.0, 3.0, 5.0]
With all values at 0, the future value terms vanish and V'(s) = max over immediate rewards. State 0: max(1, 2) = 2. State 1: max(3, 0) = 3. State 2: max(-1, 5) = 5.
Use three nested loops: outer loop over states s, middle loop over actions a, inner loop over next states s_next. For each (s, a), compute q = rewards[s][a] + gamma * sum(transitions[s][a][s_next] * values[s_next] for all s_next). Track the max q across actions.
Initialize best = negative infinity for each state before looping over actions. After the action loop, append best to the new values list.
Sign in to take notes on this problem
Accepts: array
Accepts: array
Accepts: array
Accepts: number
Value iteration is a dynamic programming algorithm that computes the optimal value function for a Markov Decision Process (MDP). Each step applies the Bellman optimality equation: for every state, consider all possible actions, compute the expected value of each action using the transition probabilities, and take the maximum.
Given the current value estimates, transition probabilities, rewards, and a discount factor, perform one step of value iteration and return the updated values.
For each state s, compute:
V′(s)=amax[R(s,a)+γs′∑T(s,a,s′)⋅V(s′)]Where:
Input:
values = [0, 0], transitions = [[[0.8, 0.2], [0.3, 0.7]], [[0.5, 0.5], [0.1, 0.9]]], rewards = [[1, 2], [-1, 0]], gamma = 0.9
Output:
[2.0, 0.0]
State 0: Q(0,a0) = 1 + 0.9*(0.80 + 0.20) = 1, Q(0,a1) = 2 + 0.9*(0.30 + 0.70) = 2. V'(0) = max(1, 2) = 2. State 1: Q(1,a0) = -1 + 0 = -1, Q(1,a1) = 0 + 0 = 0. V'(1) = max(-1, 0) = 0.
Input:
values = [0, 0, 0], transitions = [[[0,1,0],[0,0,1]], [[1,0,0],[0,0,1]], [[0,1,0],[1,0,0]]], rewards = [[1,2],[3,0],[-1,5]], gamma = 0.9
Output:
[2.0, 3.0, 5.0]
With all values at 0, the future value terms vanish and V'(s) = max over immediate rewards. State 0: max(1, 2) = 2. State 1: max(3, 0) = 3. State 2: max(-1, 5) = 5.
Use three nested loops: outer loop over states s, middle loop over actions a, inner loop over next states s_next. For each (s, a), compute q = rewards[s][a] + gamma * sum(transitions[s][a][s_next] * values[s_next] for all s_next). Track the max q across actions.
Initialize best = negative infinity for each state before looping over actions. After the action loop, append best to the new values list.
Sign in to take notes on this problem
Accepts: array
Accepts: array
Accepts: array
Accepts: number