CMU-Q 15-381 Lecture 8: Optimization I: Optimization for CSP - - PowerPoint PPT Presentation

CMU-Q 15-381

Lecture 8: Optimization I: Optimization for CSP Local Search

Teacher: Gianni A. Di Caro

LOCAL SEARCH FOR CSP

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§ Real-life CSPs can be very large and hard to solve … § Methods so far: construct a solution by assigning one variable at-a- time, if an assignment fails because of constraint violation, backtrack, and keep doing until all variables have been assigned feasible values § At any point of the construction process we have one partial solution (partial assignment of values to variables) § The states of the process are partial states of the problem

LOCAL SEARCH FOR CSP

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1. Start with some unfeasible assignment (i.e., featuring 𝑜 constraint violations) 2. LS operators reassign variable values (one or more at each search step)

• A. Variable selection (e.g., randomly select any variable

involved in constraints violation)

• B. Values selection (e.g., min-conflicts heuristic h, to

choose a value such that the new CSP assignment violates the fewest constraints) 3. Iterate 2 (A-B) until a feasible solution is found or only a few constraint violations survive or … Local search methods: work with complete states (i.e., all variables assigned, it can be an unfeasible assignment)

LOCAL SEARCH FOR CSP

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Neighbor states (one color change)

EXAMPLE N-QUEENS

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• States: 4 queens in 4 columns (44 = 256 states)
• LS Operator: move queen in column
• Goal test: no attacks
• Evaluation: h = number of attacks

LOCAL SEARCH

Local search algorithms at each step consider a single “current” state, and try to improve it by moving to one of its neighbors ➔ Iterative improvement algorithms § Pros and cons

• No complete (no optimal), except with random restarts
• Space complexity 𝒫(𝑐)
• Time complexity 𝒫(𝑒), 𝑒 can be ∞!
• Can perform well also in large (infinite, continuous)

spaces

• Relatively easy to implement

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HILL-CLIMBING SEARCH

§ Move in the direction of strictly increasing value (up to the hill) § Steepest ascent / Steepest descent § Terminate when no neighbor has higher value § Greedy (myopic) local search § We necessarily end into a local optimum or a plateau § Which optimum: depends on the starting point

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Like climbing Everest in thick fog with amnesia

HILL-CLIMBING SEARCH

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18 14 13 13 14 14 14 16 13 15 14 16 14 18 13 15 14 14 15 14 14 13 16 13 16 14 17 15 14 16 16 17 16 18 15 15 18 14 15 15 14 16 14 14 13 17 14 18 12 12 12 12 12 12 12 12 State with 17 conflicts, showing the #conflicts by moving a queen within its column, with best moves in red Local optimum: state that has only

• ne conflict, but every move leads to

larger #conflicts

HILL-CLIMBING SEARCH

§ Hill-climbing can solve large instances of 𝑜-queens (𝑜 = 106) in a few seconds § 8 queens statistics:

• State space of size ≈17 million
• Starting from random state, steepest-ascent hill

climbing solves 14% of problem instances

• It takes 4 steps on average when it succeeds, 3 when it

gets stuck

• When sideways moves are allowed, things change …
• When multiple restarts are allowed, things change even

more

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HILL-CLIMBING CAN GET STUCK!

10 state space Objective function global maximum shoulder “flat” local maximum current state neighborhood

Plateaux Local optima

VARIANTS OF HILL-CLIMBING

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§ Sideways moves: if no uphill moves, allow moving to a state with the same value as the current one (escape shoulders)

state space Objective function global maximum shoulder “flat” local maximum current state neighborhood

Plateaux Local optima

sideways moves (M): M=100 → 94% solved instances for the 8-queens! 21 steps avg. on success 64 steps avg. on “failure”

VARIANTS OF HILL-CLIMBING

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§ Sideways moves: if no uphill moves, allow moving to a state with the same value as the current one (escape shoulders) § Stochastic hill-climbing: selection among the available uphill moves is done randomly (uniform, proportional, soft-max, ε- greedy, …) to be “less” greedy § First-choice hill-climbing: successors are generated randomly,

• ne at a time, until one that is better than the current state is

found (deal with large neighborhoods) § Random-restart hill climbing: probabilistically complete (how do we select the next restart configuration?)

In general, these variants apply to all Local Search algorithms

HILL-CLIMBING CAN GET STUCK!

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Diagonal ridges: From each local maximum all the available actions point downhill, but there is an uphill path! Zig-zag motion, very long ascent time! Gradient ascent doesn’t have this issue: all state vector components are (potentially) changed when moving to a successor state, climbing can follow the direction of the ridge

LOCAL SEARCH + MIN-CONFLICTS HEURISTIC

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§ min-conflicts heuristic h chooses a value such that the new CSP assignment violates the fewest constraints § Given a random initial state, can solve 𝑜-queens in almost constant time for very large 𝑜

The same appears to be true for any randomly-generated CSP except in a narrow range of the ratio:

WALKSAT: LS FOR SAT

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§ Binary literals (true / false) § Clause: disjunction of literals § Conjunctive Normal Form (CNF) for a logical Formula: Conjunction of clauses 3-SAT (all clauses have 3 literals)

WALKSAT: LS FOR SAT

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§ Random 3-SAT

• sample uniformly from

space of all possible 3- clauses

• 𝑜 variables, 𝐷 clauses

§ Which are the hard instances?

• around

' ( = 4.26

WALKSAT: LS FOR SAT

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n Complexity peak is very

stable …

q across problem sizes q across solver types n

systematic

n

stochastic

WALKSAT: LS FOR MAX-SAT

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§ At each step, the randomly chosen clause is satisfied, but other clauses may become unsatisfied § The parameter 𝑞 is called the "mixing probability" and determined approximately by experiment for a given class of CNF formulas § For random, hard 3-SAT problems (those with the ratio of clauses to variables around 4.25) 𝑞 = 0.5 works well § For 3-SAT formulas with more structure, as generated in many applications, slightly more greediness, i.e. 𝑞 < 0.5, is often better § Empirically, restarting after 𝑃(𝑜4) flips, 𝑜 = number of variables, works well

GENERAL LOCAL SEARCH OPTIMIZER

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Function Search by Iterative Solution Modification() π = instance of optimization problem of class Π S = {set of all feasible solutions of π} N = neighborhood structure for Π, can be variable in ξ, t, m eval() = evaluation function for candidate solutions in N t = iteration, time ξt = search state at time t, current feasible solution mt = memory structure of search states and values t0 ← 0 m0 ← ∅ ξ0 ← initial feasible solution(π, S) while ¬ terminate(ξt, π, N(ξt, π), t, . . .) (ξ0, mt) ← step(N(ξt, π), mt, eval()) if accept(ξ0, ξt, t, mt) ξt+1 ← ξ0 mt+1 ← update solution best value(π, ξt+1, t) t ← t + 1 if at least one feasible solution has been generated(mt, S) return best solution found(m) else return “No feasible solution found!”

§ We only need to be able to compute the function … § No derivatives, analytical properties are needed