local and stochastic search
DESCRIPTION
Local and Stochastic Search. Based on Russ Greiner’s notes. A Different Approach. So far: systematic exploration: Explore full search space (possibly) using principled pruning (A*, . . . ) Best such algorithms (IDA*) can handle - PowerPoint PPT PresentationTRANSCRIPT
Local and Stochastic Search
Based on Russ Greiner’s notes
A Different Approach
So far: systematic exploration: Explore full search space (possibly) using
principled pruning (A*, . . . )
Best such algorithms (IDA*) can handle 10100 states ≈ 500 binary-valued variables
(ballpark figures only!)
but. . . some real-world problem have 10,000 to 100,000 variables 1030,000 states We need a completely different approach: Local Search Methods or Iterative Improvement Methods
Local Search Methods
Applicable when seeking Goal State & don't care how to get there. E.g., N-queens, map coloring, . . . VLSI layout, planning, scheduling, TSP,
time-tabling, . . .
Many (most?) real Operations Research problems are solved using local search!
Random Search
1. Select (random) initial state (initial guess at solution)
2. Make local modification to improve current state
3. Repeat Step 2 until goal state found (or out of time)
Requirements: generate a random (probably-not-optimal)
guess evaluate quality of guess move to other states (well-defined
neighborhood function) . . . and do these operations quickly. . .
Example: 4 QueenStates: 4 queens in 4 columns (256 states) Operators: move queen in column Goal test: no attacks Evaluation: h(n) = number of attacks
Example: Graph Coloring
1. Start with random coloring of nodes
2. Change color of one node to reduce # of conflict
3. Repeat 2
Hill-Climbing
3-Coloring
Problems with Hill Climbing
Problems with Hill Climbing
Foothills / Local Optimal: No neighbor is better, but not at global optimum. (Maze: may have to move AWAY from
goal to find (best) solution)
Plateaus: All neighbors look the same. (8-puzzle: perhaps no action will change
# of tiles out of place)
Ridges: going through a sequence of local maxima
Ignorance of the peak: Am I done?
Hill-climbing search: 8-queens problem
A local minimum with h = 1
Issues
The Goal is to find GLOBAL optimum. 1. How to avoid LOCAL optima? 2. How long to do plateau walk? 3. When to stop? 4. Climb down hill? When?
Overcoming Local Optimum and Plateau
Random restarts Simulated annealing Mixed-in random walk Tabu search (prevent repeated states) Others (Genetic algorithms/programming, . . . )
Local Search Example: SAT (Russell & Norvig p221-225)
Many real-world problems can be translated into propositional logic (A v B v C) ^ (¬B v C v D) ^ (A v ¬C v D) . . . solved by finding truth assignment to variables (A, B, C, . . . ) that satisfies the formula Applications planning and scheduling circuit diagnosis and synthesis deductive reasoning software testing . . .
Satisfiability Testing
Best-known systematic method: Davis-Putnam Procedure (1960) Backtracking depth-first search (DFS)
through space of truth assignments (with unit-propagation)
Greedy Local Search: GSAT
GSAT: 1. Guess a random truth assignment 2. Flip value assigned to the variable that yields
the largest increase of the total # of satisfied clauses. (Note: Flip even if the increase is 0, but not negative)
3. Repeat until all clauses satisfied, or have performed “enough” flips
4. If no sat-assign found, repeat entire process, starting from a different initial random assignment.
Systematic vs. Stochastic
Systematic search: DP systematically checks all possible
assignments. Can determine if the formula is
unsatisfiable.
Stochastic search: Guided random search approach. Once we find it, we're done. Can't determine unsatisfiability.
GSAT vs. DP on Hard Random Instances
What Makes a SAT Problem Hard?
Suppose we have n variables to write m clauses with k variables each. We negate the resulting variables randomly
(flip an unbiased coin).
What is the number of possible sentences in terms of n, m, and k ?What if there are many variables and only a smaller number of clauses?What if there are many clauses and only a smaller number of variables?
Phase Transition
For 3-SAT m/n < 4.2, under constrained: nearly all
sentences sat. m/n > 4.3, over constrained: nearly all sentences
unsat. m/n ~ 4.26, critically constrained: need to search
Phase Transition
Under-constrained problems are easy, just guess an assignment. Over-constrained problems are easy, just say “unsatisfiable” (often easy to verify using Davis-Putnam). At a m/n ratio of around 4.26, there is a phase transition between these two different types of easy problems. This transition sharpens as n increases. For large n, hard problems are extremely rare (in some sense).
Improvements to Basic Local Search
Issue: How to move more quickly to
successively better plateaus? Avoid “getting stuck” / local maxima?
Idea: Introduce uphill moves (“noise”) to escape from plateaus/local maxima Noise strategies: 1. Simulated Annealing
Kirkpatrick et al. 1982; Metropolis et al. 1953
2. Mixed Random Walk Selman and Kautz 1993
Simulated Annealing
Pick a random variable If flip improves assignment: do it. Else flip with probability p = e-δ/T = 1/(eδ/T ) δ = # of additional clauses becoming unsatisfied T = “temperature”
Higher temperature = greater chance of wrong-way move
Slowly decrease T from high temperature to near 0. Q: What is p as T tends to infinity? . . . as T tends to 0?
For δ = 0? Thus, bad moves likely to be allowed at the start when T is high.
Simulated Annealing Algorithm
Notes on SA
Noise model based on statistical mechanics . . . introduced as analogue to physical process of
growing crystals Kirkpatrick et al. 1982; Metropolis et al. 1953
Convergence: 1. W/ exponential schedule, will converge to global
optimum 2. No more-precise convergence rate (Recent work on
rapidly mixing Markov chains)
Key aspect: upwards / sideways moves Expensive, but (if have enough time) can be best
Hundreds of papers/ year; Many applications: VLSI layout, factory scheduling, . . .
Properties of simulated annealing search
One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1
Widely used in VLSI layout, airline scheduling, etc
Local Beam Search
Keep track of k states rather than just one
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.
Pure WalkSat
PureWalkSat( formula ) Guess initial assignment While unsatisfied do
Select unsatisfied clause c = ±Xi v ±Xj v ±Xk
Select variable v in unsatisfied clause c Flip v
Example:
Mixing Random Walk with Greedy Local Search
Usual issues: Termination conditions Multiple restarts
Value of p determined empirically, by finding best setting for problem class
Finding the best value of p
WalkSat[p]: W/prob p, flip var in unsatisfied clause W/prob 1-p, make a greedy flip to minimize # of
unsatisfied clauses
Q: What value for p? Let: Q[p, c] be quality of using WalkSat[p] on problem c.
Q[p, c] = Time to return answer, or = 1 if WalkSat[p] return (correct) answer within 5
minutes and 0 otherwise, or = . . . perhaps some combination of both . . .
Then, find p that maximize the average performance of WalkSat[p] on a set of challenge problems.
Experimental Results: Hard Random 3CNF
Time in secondsEffectiveness: prob. that random initial assignment leads to a solution. Complete methods, such as DP, up to 400 variables
Mixed Walk better than Simulated Annealing better than Basic GSAT better than Davis-Putnam
Overcoming Local Optimum and Plateau
Random restarts Simulated annealing Mixed-in random walk Tabu search (prevent repeated states) Others (Genetic algorithms/programming, . . . )
Other Techniques
random restarts: restart at new random state after pre-defined # of local steps.
[Done by GSAT] tabu: prevent returning quickly to same state.
Implement: Keep fixed length queue (tabu list). Add most recent step to queue; drop oldest step. Never make step that's on current tabu list.
Example: without tabu:
flip v1, v2, v4, v2, v10, v11, v1, v10, v3, ... with tabu (length 5)—possible sequence:
flip v1, v2, v4, v10, v11, v1, v3, ...
Tabu very powerful; competitive w/ simulated annealing or random walk (depending on the domain)
Genetic Algorithms
A class of probabilistic optimization algorithms A genetic algorithm maintains a population of
candidate solutions for the problem at hand, and makes it evolve by iteratively applying a set of stochastic operators
Inspired by the biological evolution processUses concepts of “Natural Selection” and “Genetic Inheritance” (Darwin 1859)Originally developed by John Holland (1975)
The Algorithm
1. Randomly generate an initial population.
2. Select parents and “reproduce” the next generation
3. Evaluate the fitness of the new generation
4. Replace the old generation with the new generation
5. Repeat step 2 though 4 till iteration N
Stochastic Operators
Cross-over decomposes two distinct solutions
and then randomly mixes their parts to form novel solutions
Mutation randomly perturbs a candidate
solution
1 0 1 0 1 1 1
1 1 0 0 0 1 1
Parent 1
Parent 2
1 0 1 0 0 1 1
1 1 0 0 1 1 0
Child 1
Child 2 Mutation
Genetic Algorithm Genetic Algorithm OperatorsOperators
Mutation and CrossoverMutation and Crossover
Examples: Recipe
To find optimal quantity of three major ingredients (sugar, wine, sesame oil) denoting ounces. Use an alphabet of 1-9 denoting
ounces.. Solutions might be 1-1-1, 2-1-4, 3-3-1.
Examples
Mutation:The recipe example:
1-2-3 may be changed to 1-3-3 or 3-2-3.
Parameters to adjust How often? How many digits change? How big?
More examples:
CrossoverRecipe :
Parents 1-3-3 & 3-2-3. Crossover point after the first digit. Generate two offspring: 3-3-3 and 1-2-3.
Can have one or two point crossover.
Randomized Algorithms
A randomized algorithm is defined as an algorithm where at least one decision is based on a random choice.
Monte Carlo and Las Vegas
There are two kinds of randomized algorithms: Las Vegas: A Las Vegas algorithm always
produces the correct answer, but randomness only influence the resources used.
E.g. Randomized Quiksort with pivot randomly
chosen.
Monte Carlo: A Monte Carlo algorithm runs for a fixed number of steps for each input and produces an answer that is correct with a bounded probability
Local Search Summary
Surprisingly efficient search technique Wide range of applicationsFormal properties elusive Intuitive explanation: Search spaces are too large for
systematic search anyway. . .
Area will most likely continue to thrive