4-Local search Artificial Intelligent Presentation

4-Local search Artificial Intelligent Presentation

• 1.
  Local search algorithms Localsearch algorithms • In many optimization problems, the state space is the y p p , p space of all possible complete solutions • We have an objective function that tells us how “good” i t t i d t t fi d th l ti ( l) a given state is, and we want to find the solution (goal) by minimizing or maximizing the value of this function
• 2.
  Example: n-queens problem Example:n queens problem • Put n queens on an n × n board with no two queens on q q the same row, column, or diagonal • State space: all possible n-queen configurations • What’s the objective function? – Number of pairwise conflicts
• 3.
  Example: Traveling salesman problem •Find the shortest tour connecting a given set of cities g g • State space: all possible tours • Objective function: length of tour
• 4.
  Local search algorithms Localsearch algorithms • In many optimization problems, the state space is the y p p , p space of all possible complete solutions • We have an objective function that tells us how “good” a i t t i d t t fi d th l ti ( l) b given state is, and we want to find the solution (goal) by minimizing or maximizing the value of this function • The start state may not be specified The start state may not be specified • The path to the goal doesn’t matter • In such cases, we can use local search algorithms that keep a single “current” state and gradually try to improve it
• 5.
  Example: n-queens problem Example:n queens problem • Put n queens on an n × n board with no two queens on q q the same row, column, or diagonal • State space: all possible n-queen configurations • Objective function: number of pairwise conflicts • What’s a possible local improvement strategy? Move one queen within its column to reduce conflicts – Move one queen within its column to reduce conflicts
• 6.
  Example: n-queens problem Example:n queens problem • Put n queens on an n × n board with no two queens on q q the same row, column, or diagonal • State space: all possible n-queen configurations • Objective function: number of pairwise conflicts • What’s a possible local improvement strategy? Move one queen within its column to reduce conflicts – Move one queen within its column to reduce conflicts h = 17
• 7.
  Example: Traveling Salesman Problem •Find the shortest tour connecting n cities g • State space: all possible tours • Objective function: length of tour • What’s a possible local improvement strategy? – Start with any complete tour, perform pairwise exchanges
• 8.
  Hill-climbing search Hill climbingsearch • Initialize current to starting state Initialize current to starting state • Loop: – Let next = highest-valued successor of current Let next = highest-valued successor of current – If value(next) < value(current) return current Else let current = next – Else let current = next • Variants: choose first better successor randomly • Variants: choose first better successor, randomly choose among better successors • “Like climbing mount Everest in thick fog with • Like climbing mount Everest in thick fog with amnesia”
• 9.
  Hill-climbing search Hill climbingsearch • Is it complete/optimal? Is it complete/optimal? – No – can get stuck in local optima – Example: local optimum for the 8-queens problem p p q p h = 1
• 10.
  The state space“landscape” • How to escape local maxima? – Random restart hill-climbing • What about “shoulders”? • What about “plateaux”?
• 11.
  Simulated annealing search •Idea: escape local maxima by allowing some "bad" moves but gradually decrease their bad moves but gradually decrease their frequency – Probability of taking downhill move decreases with Probability of taking downhill move decreases with number of iterations, steepness of downhill move – Controlled by annealing schedule • Inspired by tempering of glass, metal
• 12.
  Simulated annealing search •Initialize current to starting state • For i = 1 to ∞ • For i = 1 to ∞ – If T(i) = 0 return current – Let next = random successor of current Let next random successor of current – Let Δ = value(next) – value(current) – If Δ > 0 then let current = next – Else let current = next with probability exp(Δ/T(i))
• 13.
  Effect of temperature Effectof temperature exp(Δ/T) Δ Δ
• 14.
  Simulated annealing search •One can prove: If temperature decreases slowly h th i l t d li h ill fi d enough, then simulated annealing search will find a global optimum with probability approaching one H • However: – This usually takes impractically long The more downhill steps you need to escape a local – The more downhill steps you need to escape a local optimum, the less likely you are to make all of them in a row • More modern techniques: general family of Markov Chain Monte Carlo (MCMC) algorithms f l i li t d t t for exploring complicated state spaces
• 15.
  Local beam search •Start with k randomly generated states • At each iteration, all the successors of all k states are generated • If any one is a goal state, stop; else select the k best successors from the complete list and repeat successors from the complete list and repeat • Is this the same as running k greedy searches in ll l? parallel? Greedy search Beam search