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Probabilistic Modeling for Combinatorial Optimization Scott Davies School of Computer Science Carnegie Mellon University Joint work with Shumeet Baluja

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Combinatorial Optimization zMaximize “evaluation function” f(x) yinput: fixed-length bitstring x youtput: real value zx might represent: yjob shop schedules yTSP tours ydiscretized numeric values yetc. zOur focus: “Black Box” optimization yNo domain-dependent heuristics

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Most Commonly Used Approaches zHill-climbing, simulated annealing yGenerate candidate solutions “neighboring” single current working solution (e.g. differing by one bit) yTypically make no attempt to model how particular bits affect solution quality zGenetic algorithms yAttempt to implicitly capture dependency of solution quality on bit values by maintaining a population of candidate solutions yUse crossover and mutation operators on population members to generate new candidate solutions

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Using Explicit Probabilistic Models zMaintain an explicit probability distribution P from which we generate new candidate solutions yInitialize P to uniform distribution yUntil termination criteria met: xStochastically generate K candidate solutions from P xEvaluate them xUpdate P to make it more likely to generate solutions “similar” to the “good” solutions zSeveral different choices for what sorts of P to use and how to update it after candidate solution evaluation

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Probability Distributions Over Bitstrings zLet x = (x 1, x 2, …, x n ), where x i can take one of the values {0, 1} and n is the length of the bitstring. zCan factorize any distribution P(x 1 …x n ) bit by bit: P(x 1,…,x n ) = P(x 1 ) P(x 2 | x 1 ) P(x 3 | x 1, x 2 )…P(x n |x 1, …, x n-1 ) zIn general, the above formula is just another way of representing a big lookup table with one entry for each of the 2 n possible bitstrings. zObviously too many parameters to estimate from limited data!

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Representing Independencies with Bayesian Networks zGraphical representation of probability distributions yEach variable is a vertex yEach variable’s probability distribution is conditioned only on its parents in the directed acyclic graph (“dag”) D F IA H Wean on Fire Fire Alarm Activated Hear Bells Ice Cream Truck Nearby Office Door Open P(F,D,I,A,H) = P(F) * P(D) * P(I|F) * P(A|F) * P(H|D,I,A)

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“ Bayesian Networks” for Bitstring Optimization? P(x 1,…,x n ) = P(x 1 ) P(x 2 | x 1 ) P(x 3 | x 1, x 2 )…P(x n |x 1, …, x n-1 ) x1x1 x2x2 x3x3 xnxn Yuck. Let’s just assume all the bits are independent instead. (For now.) x1x1 x2x2 x3x3 xnxn Ahhh. Much better. P(x 1,…,x n ) = P(x 1 ) P(x 2 ) P(x 3 )…P(x n )

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Population-Based Incremental Learning zPopulation-Based Incremental Learning (PBIL) [Baluja, 1995] yMaintains a vector of probabilities: one independent probability P(x i ) for each bit x i. yUntil termination criteria met: xGenerate a population of K bitstrings from P xEvaluate them xUse the best M of the K to update P as follows: if x i is set to 1 if x i is set to 0or xOptionally, also update P similarly with the bitwise complement of the worst of the K bitstrings yReturn best bitstring ever evaluated

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PBIL vs. Discrete Learning Automata zEquivalent to a team of Discrete Learning Automata, one automata per bit. [Thathachar & Sastry, 1987] yLearning automata choose actions independently, but receive common reinforcement signal dependent on all their actions yPBIL update rule equivalent to linear reward-inaction algorithm [Hilgard & Bower, 1975] with “success” defined as “best in the bunch” zHowever, Discrete Learning Automata typically used previously in problems with few variables but noisy evaluation functions

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PBIL vs. Genetic Algorithms zPBIL originated as tool for understanding GA behavior zSimilar to Bit-Based Simulated Crossover (BSC) [Syswerda, 1993] yRegenerates P from scratch after every generation yAll K used to update P, weighted according to probabilities that a GA would have selected them for reproduction zWhy might normal GAs be better? yImplicitly capture inter-bit dependencies with population yHowever, because model is only implicit, crossover must be randomized. Also, limited population size often leads to premature convergence based on noise in samples.

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Four Peaks Problem zProblem used in [Baluja & Caruana, 1995] to test how well GAs maintain multiple solutions before converging. zGiven input vector X with N bits, and difficulty parameter T: yFourPeaks(T,X)=MAX(head(1,X), tail(0,X))+Bonus(T,X) xhead(b,X) = # of contiguous leading bits in X set to b xtail(b,X) = # of contiguous trailing bits in X set to b xBonus(T,X) = 100 if (head(1,X)>T) AND (tail(0,X) > T), or 0 otherwise zShould theoretically be easy for GA to handle with single-point crossover

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Four Peaks Problem Results zInsert nasty photocopy here

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Large-Scale PBIL Empirical Comparison zSee bug-ugly photocopied table

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Modeling Inter-Bit Dependencies zHow about automatically learning probability distributions in which at least some dependencies between variables are modeled? zProblem statement: given a dataset D and a set of allowable Bayesian networks {B i }, find the B i with the maximum posterior probability:

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Equations. (Mwuh hah hah hah!) zWhere: yd j is the j th datapoint yd i j is the value assigned to x i by d j. i is the set of x i ’s parents in B is the set of values assigned to i by d j yP’ is the empirical probability distribution exhibited by D

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Mmmm...Entropy. zFact: given that B has a network structure S, the optimal probabilities to use in B are just the probabilities in D, i.e. P’. So: Pick B to Maximize: Pick S to Maximize:

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Entropy Calculation Example x 17 x 20 x 32 x 20 ’s contribution to score: 80 total datapoints where H(p,q) = p log p + q log q.

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Single-Parent Tree-Shaped Networks zNow let’s allow each bit to be conditioned on at most one other bit. x 17 x5x5 x 22 x2x2 x6x6 x 19 x 13 x 14 x8x8 zAdding an arc from x j to x i increases network score by H(x i ) - H(x i |x j ) (x j ’s “information gain” with x i ) = H(x j ) - H(x j |x i ) (not necessarily obvious, but true) = I(x i, x j ) Mutual information between x i and x j

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Optimal Single-Parent Tree-Shaped Networks zTo find optimal single-parent tree-shaped network, just find maximum spanning tree using I(x i, x j ) as the weight for the edge between x i and x j. [Chow and Liu, 1968] yStart with an arbitrary root node x r. yUntil all n nodes have been added to the tree: xOf all pairs of nodes x in and x out, where x in has already been added to the tree but x out has not, find the pair with the largest I(x in, x out ). xAdd x out to the tree with x in as its parent. zCan be done in O(n 2 ) time (assuming D has already been reduced to sufficient statistics)

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Optimal Dependency Trees for Combinatorial Optimization [Baluja & Davies, 1997] zStart with a dataset D initialized from the uniform distribution zUntil termination criteria met: yBuild optimal dependency tree T with which to model D. yGenerate K bitstrings from probability distribution represented by T. Evaluate them. Add best M bitstrings to D after decaying the weight of all datapoints already in D by a factor between 0 and 1. zReturn best bitstring ever evaluated. zRunning time: O(K*n + n 2 ) per iteration

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Tree-based Optimization vs. MIMIC zTree-based optimization algorithm inspired by Mutual Information Maximization for Input Clustering (MIMIC) [De Bonet, et al., 1997] yLearned chain-shaped networks rather than tree-shaped networks yDataset: best N% of all bitstrings ever evaluated x+: dataset has simple, well-defined interpretation x-: have to remember bitstrings x-: seems to converge too quickly on some larger problems x4x4 x8x8 x 17 x5x5

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MIMIC dataset vs. Exp. Decay dataset zInsert skanky photocopy here

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Graph-Coloring Example zPhotocopy or PostScript inclusion

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Peaks problems results zPostscript/photcopy

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Tree-Max problem results zYet another dummy slide

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Checkerboard problem results zDummy slide

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Linear Equations Results zYucky photocopy maybe

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Modeling Higher-Order Dependencies zThe maximum spanning tree algorithm gives us the optimal Bayesian Network in which each node has at most one parent. zWhat about finding the best network in which each node has at most K parents for K>1? yNP-complete problem! [Chickering, et al., 1995] yHowever, can use search heuristics to look for “good” network structures (e.g., [Heckerman, et al., 1995]), e.g. hillclimbing.

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Scoring Function for Arbitrary Networks zRather than restricting K directly, add penalty term to scoring function to limit total network size yEquivalent to priors favoring simpler network structures yAlternatively, lends itself nicely to MDL interpretation ySize of penalty controls exploration/exploitation tradeoff |B| |B| penalty factor |B|: number of parameters in B

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Bayesian Network-Based Combinatorial Optimization zInitialize D with C bitstrings from uniform distribution, and Bayesian network B to empty network containing no edges zUntil termination criteria met: yPerform steepest-ascent hillclimbing from B to find locally optimal network B’. Repeat until no changes increase score: xEvaluate how each possible edge addition, removal or deletion would affect penalized log-likelihood score xPerform change that maximizes increase in score. ySet B to B’. yGenerate and evaluate K bitstrings from B. Decay weight of datapoints in D by . yAdd best M of the K recently generated datapoints to D. zReturn best bit string ever evaluated.

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Cutting Computational Costs zCan cache contingency tables for all possible one-arc changes to network structure yOnly have to recompute scores associated with at most two nodes after arc added, removed, or reversed. yPrevents having to slog through the entire dataset recomputing score changes for every possible arc change when dataset changes. yHowever, kiss memory goodbye zOnly a few network structure changes required after each iteration since dataset hasn’t changed much (one or two structural changes on average)

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Evolution of Network Complexity zPlaceholder for parity-based network complexity graph

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Summation Cancellation zMinimize magnitudes of cumulative sum of discretized numeric parameters (s 1, …, s n ) represented with standard binary encoding: Average value over 50 runs of best solution found in 2000 generations

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Bayesian Networks: Empirical Results Summary zDoes better than Tree-based optimization algorithm on some toy problems ySignificantly better on “Summation Cancellation” problem y10% reduction in error on System of Linear Equation problems zRoughly the same as Tree-based algorithm on others, e.g. small Knapsack problems zSignificantly more computation despite efficiency hacks, however. zWhy not much better performance? yToo much emphasis on exploitation rather than exploration? ySteepest-ascent hillclimbing over network structures not good enough?

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Using Probabilistic Models for Intelligent Restarts zTree-based algorithm’s O(n 2 ) execution time per generation very expensive for large problems zEven more so for more complicated Bayesian networks zOne possible approach: use probabilistic models to select good starting points for faster optimization algorithms, e.g. hillclimbing

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COMIT zCombining Optimizers with Mutual Information Trees [Baluja & Davies, 1997b]: yInitialize dataset D with bitstrings drawn from uniform distribution yUntil termination criteria met: xBuild optimal dependency tree T with which to model D. xGenerate K bitstrings from the distribution represented by T. Evaluate them. xExecute a hillclimbing run starting from single best bitstring of these K. xReplace up to M bitstrings in D with the best bitstrings found during the hillclimbing run. yReturn best bitstring ever evaluated

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COMIT, cont’d zEmpirical tests performed with stochastic hillclimbing algorithm that allows at most PATIENCE moves to points of equal value before restarting zCompare COMIT vs.: yHillclimbing with restarts from bitstrings chosen randomly from uniform distribution yHillclimbing with restarts from best bitstring out of K chosen randomly from uniform distribution yGenetic algorithms?

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COMIT: Example of Behavior zDummy slide for graphs showing all evaluations in TSP domain

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COMIT: Empirical Comparisons zBig ugly table goes here

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Summary zPBIL uses very simple probability distribution from which new solutions are generated, yet works surprisingly well zAlgorithm using tree-based distributions seems to work even better, though at significantly more computational expense zMore sophisticated networks: past the point of diminishing marginal returns? “Future research” zCOMIT makes tree-based algorithm applicable to much larger problems

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Future Work zMaking algorithm based on complex Bayesian Networks more practical yCombine w/simpler search algorithms, ala COMIT? zApplying COMIT to more interesting problems yWALKSAT? zOptimization in real-valued state spaces. What sorts of PDF representations might be useful? xGaussians? xKernel-based representations? xHierarchical representations?

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Acknowledgements zShumeet Baluja zDoug Baker, Justin Boyan, Lonnie Chrisman, Greg Cooper, Geoff Gordon, Andrew Moore, …?

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