Stochastic Enforced Hill-Climbing
Jia-Hong Wu and Rajesh Kalyanam and Robert Givan Electrical and Computer Engineering, Purdue University, W. Lafayette, IN 47907 jw, rkalyana, givan @purdue.edu { }
Abstract enforced hill-climbing . (Hoffmann and Nebel 2001) The immediate goal of enforced hill-climbing is to nd Enforced hill-climbing is an effective deterministic hill- climbing technique that deals with local optima using a descendant that is strictly better than the current state in breadth-first search (a process called “basin flooding”). We heuristic value by performing a local search. The planner propose and evaluate a stochastic generalization of enforced then moves to that descendant and repeats this process. The hill-climbing for online use in goal-oriented probabilistic effectiveness of enforced hill-climbing is demonstrated by planning problems. We assume a provided heuristic func- the success of the planner FF (Hoffmann and Nebel 2001). tion estimating expected cost to the goal with flaws such as Enforced hill-climbing is not de ned for probabilistic local optima and plateaus that thwart straightforward greedy problems, due to the stochastic outcomes of actions. In the action choice. While breadth-first search is effective in ex- presence of stochastic outcomes, nding descendants of bet- ploring basins around local optima in deterministic problems, ter values no longer implies the existence of a policy that for stochastic problems we dynamically build and solve a lo- cal Markov-decision process model of the basin in order to reaches those descendants with high probability. One may find a good escape policy exiting the local optimum. argue that FF-Replan (Yoon, Fern, and Givan 2007) a cur- We evaluate our proposal in a wide range of recent proba- rent top performer for probabilistic planning benchmarks bilistic planning-competition benchmark domains. For eval- uses enforced hill-climbing during its call to FF; however, uation, we show that stochastic enforced hill-climbing pro- the enforced hill-climbing process is used on a determinized duces better policies than greedy action choice for value func- problem, and FF-Replan does not use any form of hill climb- tions derived in two very different ways. First, we propose a ing directly in the stochastic problem. In fact, FF-Replan novel heuristic function derived from the ideas in the effec- does not consider the outcome probabilities at all. tive re-planner FF-Replan. This new “controlled-randomness One problem to consider in generalizing enforced hill- FF heuristic” is the deterministic FF heuristic computed on climbing to stochastic domains is that the solution to a de- the simple determinization of the probabilistic problem that terministic problem is typically concise, a sequential plan; makes available a deterministic transition wherever a proba- bilistic transition was possible. Our results show that stochas- in contrast, the solution to a stochastic problem is a policy tic enforced hill-climbing with this heuristic significantly out- (action choice) for all possibly reached states. The essential performs simply greedily following the heuristic, and also motivation for hill-climbing is to avoid storing exponential substantially outperforms FF-Replan. We additionally eval- information during search, and even the explicit solution to uate our technique on automatically learned value functions a stochastic problem cannot be directly stored while respect- that on their own perform at the state-of-the-art when used ing this motivation. For this reason, we limit consideration to construct a greedy policy, and again show significant im- to the online setting, where the solution to the problem is a provement over greedy action selection. single action, for the current state. After this action is com- mitted to and executed, the planner then has a new online Introduction problem to solve (possibly retaining some information from the previous solution). Our approach generalizes directly to Heuristic estimates of distance-to-the-goal have long been the construction of of ine policies in situations where space used in deterministic search and deterministic planning. to store such policies is available. Such estimates typically have aws such as local extrema We propose a novel tool for stochastic planning by gener- and plateaus that limit their utility. Methods such as sim- alizing enforced hill-climbing to stochastic domains. Rather ulated annealing (Kirkpatrick, Gelatt, and Vecchi 1983; than seeking a sequence of actions deterministically leading Cerny 1985) and A* (Nilsson 1980) search have been de- to a better state, our method constructs a breadth- rst local veloped for handling aws in heuristics. More recently, ex- Markov Decision Process (MDP) around the current state cellent practical results have been obtained by ooding lo- and seeks a policy that expects to exit this MDP with a better cal optima using breadth- rst search; this method is called valued state. When such a policy is found, the method exe- Copyright c 2008, Association for the Advancement of Artificial cutes the policy until the local MDP is exited. Critical to this Intelligence (www.aaai.org). All rights reserved. process is the direct incorporation of the probabilistic model π∗ in nding the desired policy. Here, we nd this policy us- There is at least one optimal policy π∗ for which V (s), ab- π ing value iteration on the local MDP, where the only rewards breviated V ∗(s), is no less than V (s) at every state s, for are the heuristic values assigned at the exits. As in enforced any other policy π. The following Q function evaluates hill-climbing, breadth- rst expansion around a state occurs an action a with respect to a value function V by using V to only when there is no action available with Q-value (rela- estimate the value after action a is applied, tive to the heuristic) achieving the current state's heuristic Q(s, a, V ) = T (s, a, s0)[R(s, a, s0) + V (s0)]. value. Note that although stochastic enforced hill-climbing s S is an explicit statespace technique, it can be suitable for use X0∈ Recursive Bellman equations use Q() to describe V ∗ and in astronomically large statespaces if the heuristic used is in- π π π formative enough to limit the size of the local MDPs needed V as follows. First, V (s) = Q(s, π(s), V ). Then, V ∗(s) = maxa A Q(s, a, V ∗). Also using Q(), we can se- to nd expected heuristic improvement. ∈ Stochastic enforced hill-climbing can be applied to any lect an action greedily relative to any value function. The (V ) s heuristic function. We test the technique on a broad range of policy Greedy selects, at any state , a randomly selected action from arg maxa A Q(s, a, V ). domains from the rst two international probabilistic plan- ∈ ning competitions. We select two very different heuristic Value iteration iterates the operation V 0(s) = maxa A Q(s, a, V ), computing the Bellman update functions: First we use a novel heuristic function based ∈ on the determinization from the successful re-planner FF- V 0 from V , producing a sequence of value functions Replan. This new controlled-randomness FF heuristic is converging to V ∗, regardless of the initial V used, as shown the FF heuristic (Hoffmann and Nebel 2001) computed on by Bertsekas for a similar formulation in (Bertsekas 1995). the FF-Replan determinization of the probabilistic problem. While goal-based MDP problems can be directly spec- Our results show that stochastic enforced hill-climbing with i ed as above, they may also be speci ed exponentially this heuristic outperforms FF-Replan substantially, and also more compactly using planning languages such as PPDDL outperforms simply greedily following the heuristic. We (Younes et al. 2005), as used in our experiments. Our tech- additionally evaluate our technique on value functions pro- nique below avoids converting the PPDDL problem explic- duced by a relational feature-learning method in (Wu and itly into the above form, but constructs a sequence of smaller Givan 2007). These value functions have been shown to per- problems of this form modeling heuristic aws. form at the state-of-the-art when used to construct a greedy policy. We again show improvement over greedy selection Local Markov decision processes Given a heuristic func- when using these value functions as the heuristic. tion h : S R estimating V ∗ for a MDP M = (S, A, R, T ), i.e.,→ estimating negated distance to the ab- Technical Background and Related Work sorbing state, we de ne the sub-MDP induced by state set S0 S to be the MDP that equals M inside S0, but gives ⊆ Goal-oriented Markov decision processes We give a terminal reward according to h upon exiting S0. Formally, brief review of Markov decision processes (MDPs) special- the sub-MDP induced by S0 and h is given by (S, A, R0, T 0), ized to goal-region objectives. For more detail, see (Bert- where R0 and T 0 are given as follows. Let s be any state in sekas and Tsitsiklis 1996) and (Sutton and Barto 1998). S and a any action in A. For any state s0 S0, we take ∈ A stochastic shortest path Markov decision process R0(s0, a, s) = 0 and T 0(s0, a, s) = T (s0, a, s). For any state (MDP) M is a tuple (S, A, R, T ) with nite state and ac- x in S S0, R0(x, a, s) = h(x) and T 0(x, a, ) = 1. We − ⊥ tion spaces S and A, reward function R : S A S R, refer to the number of states reachable from S0 in zero or and a transition probability function T : S A× × (S→) that one steps as the size of the sub-MDP induced by S . × → P 0 maps (state, action) pairs to probability distributions over S. The rough motivation for setting action rewards to zero To avoid discounting while still assigning nite value to each for the analysis within S0 is that such actions are being con- state, we require that S contain a zero-reward absorbing state sidered by our method to remediate a awed heuristic; the , i.e., such that R( , a, s) = 0 and T ( , a, ) = 1 for all cumulative negative action reward paid to reach a state of ⊥ ⊥ ⊥ ⊥ s S and a A, and that there is some policy (as de ned higher heuristic value is a measure of the magnitude of the ∈ ∈ below) from any state leading to with probability 1. aw in the heuristic, but we remove this reward from the ⊥ Goal-oriented MDPs are MDPs where there is a subset analysis in order to directly express the subgoal of reach- G S of the statespace, containing , such that: (1) ing a state with higher heuristic value . Instead of diluting ⊆ ⊥ R(g, a, s0) is zero whenever g G and minus one other- this subgoal by adding in negative action rewards, our meth- wise, and (2) T (g, a, ) is one for∈ all a A and g G. ods seek the shortest path to a heuristic improvement by ⊥ ∈ ∈ Given policy π : S A for an MDP, the value func- enlarging S0 in a breadth rst manner. The resulting breadth- π → tion V (s) gives the expected cumulative reward obtained rst expansion of S0 generalizes if needed to uniform-cost from state s selecting action π(s) at each state encountered1. expansion to handle more general action costs.2
1Formally, we have restricted consideration to domains where We defer more formal algorithmic treatment of deadend domains some policy reaches a goal state with probability one from all for future work, but note that some domains evaluated in this paper states, to avoid discounting. In practice we can and do handle dead- do contain unavoidable deadends. end states by assigning them large negative values if they are recog- 2Also note that in this zero-reward undiscounted setting, greed- nized by simple reachability tests, and by adjusting the parameters ily acting based upon the converged value function is not guaran- for our algorithm as discussed in subsection “Deadend handling.” teed to achieve that value. It is necessary to remember the action Note: technically, we seek to climb in value , even we generalize the idea of searching for a single strictly im- though we think of heuristics as measuring distance to the proved successor state to nding a policy that expects to goal. We simply use the negative value of the distance-to-go improve on the heuristic value of the initial state inside a heuristic as our value function. dynamically constructed sub-MDP. As formalized in Local Markov decision processes above, the sub-MDP ignores Related work Stochastic enforced hill-climbing dynami- the negative action rewards paid to reach such improved cally constructs local MDPs to nd a local policy leading states; the secondary goal of minimizing these costs is em- to heuristically better state regions. The concept of form- bodied by constructing the smallest sub-MDP enabling suc- ing sub-MDPs, or envelopes , and using them to facilitate cess, as described formally below. Note that, in contrast to probabilistic planning has been used in previous research replanning techniques, with this approach the system adjusts such as (Bonet and Geffner 2006; Dean et al. 1995), which to the uncertainty in action outcomes without replanning. we brie y review here. We present pseudo-code for stochastic enforced hill- The envelope-based method in (Dean et al. 1995) starts climbing in Figure 1, and explain the terminology used in with a partial policy in a restricted area of the problem (the the pseudo-code next. The algorithm assumes a non-positive envelope ), then iteratively improves the solution quality heuristic function h : S R as input that assigns zero to by changing the envelope and recomputing the partial policy. all goal states. Stochastic→enforced hill-climbing iteratively The typical assumption of implementing this method is to builds and solves sub-MDPs and seeks improved policies have an initial trajectory from the starting state to the goal, inside such sub-MDPs3. Each sub-MDP is induced as de- generated by some stochastic planner. ned previously by a state set Γ together with the heuristic Another line of work (Barto, Bradtke, and Singh 1995; function h. We use two parameters k and α to control the Hansen and Zilberstein 2001; Bonet and Geffner 2006) starts aggressiveness with which the sub-MDPs are constructed in from only the initial state in an envelope, iteratively ex- an attempt to escape the local optimum or plateau. The hori- pands the envelope by expanding states according to state zon parameter k limits the number of steps from the entry values and using dynamic programming methods to backup state s0; the heuristic radius parameter α limits the heuristic state values, until some convergence criterion is reached. distance from h(s0) that states in the sub-MDP may have, Stochastic enforced hill-climbing can be viewed as repeat- representing the energy barrier to escape. We assume some edly deploying the envelope method with the goal, each schedule for selecting more and more aggressive values of time, of improving on the heuristic estimate of distance to k and α to construct increasingly large sub-MDP problems go. For a good h function, most invocations result in trivial seeking an exit. Our algorithm applies to any such schedule. one-step envelopes; however, when local optima or plateaus For any pair (k, α) in the schedule, we de ne a state set are encountered, the envelope may need to grow to locate a Γ . For any α, we de ne Γ (s ) = s . We de ne an kα 0α 0 { 0} stochastically reachable set of exits. operation Extend on a state set to be Extend(Γ) = s0 s { | ∃ ∈ All of the above referenced previous search methods have Γ, a A, P (s, a, s0) > 0 . We can then iteratively extend ∃ ∈ } constructed envelopes seeking a high quality policy to the Γ to be Γ (s ) = Γ s0 Extend(Γ ) kα (k+1)α 0 kα ∪ { ∈ kα | goal rather than our far more limited and relatively inexpen- h(s0) h(s ) α . | − 0 | ≤ } sive goal of basin escape. Preliminary experiments with one Thus, in line 6 of Figure 1 , value iteration is conducted such recent previous technique, LDFS (Bonet and Geffner for increasingly large sub-MDPs around s0, seeking a policy 2006), on the domains and heuristics used for evaluation in 4 improving h(s0) . Depending on the schedule chosen for k this paper gave poor results. Our results derive from on- and α, implementation choices may allow reuse of states- line greedy exploitation of the heuristic rather than the more pace and value information as larger MDPs are constructed, expensive of ine computation of converged values proving expanding on smaller ones. The implementation may also overall (near) optimality. LDFS, for example, will com- proceed down the schedule without value iteration when no pute/check values for at least all states reachable under the new states of high value are added to the sub-MDP. optimal policy (even if given V ∗ as input) and possibly vastly many others as well during the computation. Early termination The primary termination condition for repeated sub-MDP construction is the discovery of a policy Stochastic Enforced Hill Climbing improving on the heuristic estimate of the initial state. How- Deterministic enforced hill-climbing (Hoffmann and Nebel ever, for awed heuristic functions that overestimate state 2001) searches for a strictly improving successor state and value, especially in domains with unavoidable deadends, returns a path from the initial state to such a successor. This this may not be possible. For this reason, in practice, the al- path is an action sequence that guarantees reaching the de- gorithm stops enlarging sub-MDPs when user-speci ed re- sired successor. In a stochastic environment, there may be source limits are exceeded. no single improved descendant that can be reached with probability one, as the actions may have multiple stochas- 3Note that we must avoid building the full MDP explicitly as it tic outcomes. Thus, in stochastic enforced hill-climbing, is exponentially large. 4Given the negative-reward-per-step structure of a goal- which first achieved the converged value during value iteration, for oriented MDP, h() is an estimate of (negated) expected distance each state, and take that action in preference to others that may look to the goal, so this subgoal corresponds to seeking a policy that equal in greedy value due to zero-cost cycles. expects to move closer to the goal. Stochastic enforced hill-climbing yi is easily computed during sub-MDP construction using αi. Throughout our experiments, we terminate sub-MDP 1. Repeat construction (as in line 8 of table 1) when the size of the 2. // for some schedule of (k, α) pairs (ki, αi) sub-MDP exceeds 1.5 105 or the runtime to construct the 3. s0 current state ← sub-MDP exceeds 1 minute.∗ We set the limit on repeated 4. i = 1, Γ = Γk1α1 5. Repeat states during policy execution σ to 50 and the random walk 6. V Value iteration in sub-MDP(Γ,h) length ω to 9. The value τ assigned to recognized dead-end 7. i =←i + 1, Γ = Γ states is set to 1.0 105. We denote this implementation kiαi − ∗ 8. Until V (s0) > h(s0) or resource consumption exceeds running on heuristic h with SEH(h). user-set limits 9. Follow Greedy(V ) until exit Γ or a state visited σ times Goals of the evaluation Our empirical goal is to show 10. If h(new current state) < h(s0), take random walk of that stochastic enforced hill-climbing improves signi cantly ω steps upon greedy following of the same heuristic (using the pol- 11. Until goal achieved icy Greedy(h) as described in the technical background above. We will show this is true both for a simple new Figure 1: Pseudo-code for stochastic enforced hill-climbing. heuristic based on determinization, described below, and for learned heuristics from previously published research. A Once a sub-MDP is constructed assigning V (s0) > h(s0) secondary goal of our evaluation is to show that the result or exceeding the resource limits, the system executes the is strong in comparison to FF-Replan. greedy policy within the sub-MDP until the policy exits the sub-MDP. Again, in practice, we impose a bound σ on the The Controlled-randomness FF heuristic We de ne a number of times a state may be repeated during the execu- novel heuristic function, the controlled-randomness FF tion. If the greedy policy execution terminates in a state with heuristic (CR-FF), as the FF heuristic (Hoffmann and poor heuristic value (worse than h(s0)), our method adds a Nebel 2001) computed on the FF-Replan (Yoon, Fern, and random walk of length ω; this additional random walk al- Givan 2007) determinization of the probabilistic problem5. lows our method to retain some of the bene cial properties The determinization used in FF-Replan is constructed by of random exploration in domains where heuristic aws are creating a new deterministic action for each possible out- too large for MDP analysis. come of a stochastic action while ignoring the probability of such outcome happening. This effectively allows the planner Deadend handling The implementation tested in this pa- to control the randomness in executing actions, making this per adds ad-hoc handling of dead-end states, which can be determinization a kind of relaxation of the problem. We use incompletely identi ed by the heuristics we use. Recog- the CR-FF heuristic in each domain as one kind of heuristic nized dead-ends are assigned a large negative value τ. Also, function h in our experiments. once a dead-end state is detected in a problem, we set a min- imum sub-MDP size of 500 states for every sub-MDP con- For our experiments with the CR-FF heuristic, we com- pute the heuristic value h(s) as h(s) = h (s) where h (s) structed for that problem (to reduce the chance of greedily − d d falling into a dead-end). Also, because dead-ends break our is the FF distance estimate (Hoffmann and Nebel 2001) of s computed on the above determinization. We denote the termination guarantee based on improving on h(s0), for any sub-MDP that contains a known dead-end state, we termi- resulting heuristic F . nate sub-MDP growth once a good exit is discovered (with Learned heuristics We also test stochastic enforced h h(s ) greater than 0 ) and the sub-MDP size is at least 2000. hill-climbing on automatically generated value functions from (Wu and Givan 2007), which on their own perform at Setup for Empirical Evaluation the state-of-the-art when used to construct a greedy policy. Here, we prepare for our results section by describing the We scale these value functions to t the non-positive range parameters used for our method, the heuristics we will test requirement of h discussed previously. These learned value the method on, the domains in which the tests will be con- functions are currently available for only ve of our test do- ducted, and issues arising in interpreting the results. mains, and so are only tested in those domains. We note that these heuristics were learned for a dis- When implementing Parameterizing the algorithm counted setting without action costs and so are not a direct t stochastic enforced hill-climbing, we use a schedule of to the negated distance-to-go formalization adopted here. choices for the horizon k and heuristic radius α parameters We are still able to get signi cant improvements from apply- that explores a sequence of more and more aggressive ing our technique. We denote the resulting heuristic L. heuristic radius values α1, α2, . . .. For each radius parame- ter αi, a sequence of horizons k is considered 0, 1, . . . , 10i, Domains considered We evaluate stochastic enforced before extending the radius. We take α1 to be zero and hill-climbing on a wide range of recent probabilistic de ne subsequent αi+1 as follows. Let yi be the smallest 5 heuristic radius value such that Γkyi properly contains Γkαi The deterministic FF heuristic, described in (Hoffmann for any k in 0, 1, . . . , 10i. We then choose α = 1.5 y . i+1 ∗ i and Nebel 2001), from FF planner version 2.3 available If there is no such yi, stochastic enforced hill-climbing can- at http://members.deri.at/˜joergh/ff.html, computes a sequential not proceed further and the schedule terminates. The value relaxed-plan length using a sequence of ordered subgoal steps. SEH(F ) Greedy(F ) SEH(L) Greedy(L) FF-R SEH(F ) Greedy(F ) SEH(L) Greedy(L) FF-R Avg1 0.947 0.364 0.857 0.760 0.828 Avg1 1.598 2.60 1.525 8.180 2.494 Avg2 0.882 0.417 – – 0.785 Avg2 1.465 2.069 – – 1.832
Figure 2: Average of average success ratios across testing domains. Figure 3: Average of “normalized average plan length” across do- Avg1 represents the average across the five domains that all plan- mains, where plan length is normalized by dividing by the best av- ners address, and Avg2 represents the average across all nine do- erage plan length achieved by any planner in the domain in ques- mains for the indicated planners. FF-R abbreviates “FF-Replan.” tion. We omit two domains (tireworld and exploding blocksworld) where large variations in success rate makes successful length par- ticularly misleading. Avg1 represents the average across the three planning-competition benchmark domains. For each prob- domains that all planners address, and Avg2 represents the aver- lem size tested in each domain, 30 problems are generated age across all seven domains for the indicated planners. FF-R from the given problem generator and results are averaged abbreviates “FF-Replan.” over 30 attempts per problem. We use a length cutoff of 2000 for each attempt. Each attempt is given a time limit and our stochastic-enforced-hillclimbing algorithm SEH(h) of 30 minutes. We run our experiments on Intel Pentium 4 to determine whether stochastic enforced hill-climbing im- Xeon 2.8GHz machines. proves performance. We also show, on every plot, the per- In (Wu and Givan 2007), value functions are learned for formance of a recent stochastic re-planner, FF-Replan, that ve domains: performs quite well in these benchmark domains. blocksworld and boxworld from the rst international • In the rst set of experiments, we evaluate the novel probabilistic planning competition (IPPC); heuristic F de ned above (in the section titled The tireworld, zenotravel, and exploding blocksworld from controlled-randomness FF heuristic ). In the results in the • the second IPPC. next section, we show that SEH(F ) signi cantly outper- We use these ve domains when evaluating both the CR- forms Greedy(F ) across seven of nine benchmark planning FF heuristic and the learned value-function heuristic. We domains, roughly equalling Greedy(F ) in the other two do- additionally evaluate the following domains from the sec- mains (tireworld and schedule). ond IPPC using CR-FF heuristic: Random, schedule, ele- In addition to nearly always signi cantly improving on vators, and pitchcatch. For these four domains we take the greedy hill-climbing, SEH(F ) manages very similar perfor- problems from the competition rather than use the problem mance to FF-Replan (with small variations either way) in generators to generate new ones6. three of the nine domains tested, and more dominant perfor- For each problem in each domain we show three types of mance in either success ratio or successful plan length in the measurement: success ratio (SR), average plan length over remaining six, making it a state-of-the-art approach to these only the successful attempts (Slen), and average runtime for competition benchmark planning domains. only the successful attempts (Stime). In blocksworld, box- In a second set of experiments, we consider the learned world, tireworld, zenotravel, and exploding blocksworld, heuristic functions. Here, again, SEH(L) signi cantly these per-problem results are aggregated by problem size to improves upon Greedy(L) in four of ve domains, and compress the data. In addition, averaging across all prob- slightly improves Greedy(L) in the other domain (tire- lems in the domain, we also plot the overall success ratio, world). SEH(L) also outperforms FF-Replan in three overall average successful plan length, and overall average of ve domains, but loses in zenotravel and exploding runtime of successful attempts (labeled A on the x-axis). blocksworld, where L itself is poor. Because there is a lot of data shown across the 480 prob- Notes on interpreting the data It is very important to note that successful plan length and successful runtime are lem instances considered, we attempt in Figs. 2 and 3 a only appropriate to compare between planners when those very rough aggregation for concise comparison purposes. planners exhibit very similar success ratios (ideally near We report the average success ratio for each of the ve 1.0). Otherwise, differences in plan lengths may be due approaches averaged across across all domains considered, to differences in the problems being solved rather than plan giving equal weight to each domain (combining all prob- quality. Plan lengths are shown in these cases only for very lem sizes within each domain). Likewise, we report the av- erage across domains of normalized average plan length , coarse comparison. Some advantages for the SEH algorithm occur in plan length and some in success ratio, mandating where plan length is normalized by dividing by the best av- careful data interpretation. Note: all runtime and plan length erage plan length achieved by any planner in the domain results are presented on a logarithmic scale. in question. For the plan length average, we omit two do- mains (tireworld and exploding blocksworld) where large variations in success rate, even on single problems, makes Empirical Results successful length particularly misleading. Results summary For each heuristic function h evaluated, On these averages, SEH(F ) is substantially better than we compare the greedy hill-climbing algorithm h Greedy( ) FF-Replan in both success ratio and average length. In ad- 6We do this because for three of these “domains”, the problem dition, SEH(F ) has a dramatically higher success rate than generators change the action set between instances, so reusing the Greedy(F ). For learned heuristics, SEH(L) gives a marked actual instances aids in consistency with the competition. improvement over Greedy(L) in overall average success ra- L S-EFH(F ) G-rFeedy(F ) FFFR-Replan L S-EFH(F ) GG-rFeedy(F ) FFFFR-Replan 1 1 R R S 0.5 S 0.5 0 0 15 20 25 30 A 20 30 40 A
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o 10,2,2 5 10 A l ( 10 Zenotravel EX-BW, # blocks e m i
t 1 Figure 7: Zenotravel and exploding blocksworld results for the S 10,5 10,10 10,15 15,5 A CR-FF heuristic. The problem size for zenotravel is 10 cities, 2 Boxworld, (# boxes, # cities) people, 2 aircraft. Figure 5: Boxworld results for the CR-FF heuristic. In Figs. 8 to 11, we present detailed results for the CR- tio, and a dramatic improvement in the average length mea- FF heuristic for the four remaining domains, and we discuss sure. SEH(L) also shows similar success ratio and much each brie y here. For these plots, each label on the x-axis shorter average length than FF-Replan. indicates the problem number from the second IPPC. Detailed results for the controlled FF heuristic In When compared to Greedy(F ), SEH(F ) has signi cantly Figs. 4 to 7, we present detailed results for the CR-FF heuris- higher success ratio in random, elevators, and pitchcatch, tic for the ve domains handled by all planners. though the advantage in random is present on only one of When compared to Greedy(F ), SEH(F ) has signi cantly fteen problems. In schedule, SEH(F ) and Greedy(F ) ex- higher success ratio in blocksworld, boxworld, zenotravel, hibit similar success ratios. and exploding blocksworld. In tireworld SEH(F ) and When compared to FF-Replan, SEH(F ) performs sim- Greedy(F ) are equal in success ratio. ilarly in success ratio across two of the four domains When compared to FF-Replan, SEH(F ) has signi cantly (random and elevators), and improves on FF-Replan in the higher success ratio in blocksworld, tireworld, and explod- two others (schedule and pitchcatch). Also, SEH(F ) nds ing blocksworld. Note that these differences in success rate signi cantly shorter plans than FF-Replan in pitchcatch. make successful length comparisons misleading in these do- mains. In boxworld, SEH(F ) and FF-Replan have simi- lar success ratio, but SEH(F ) yields a signi cantly shorter Detailed results for the learned heuristics In Figs. 12 average successful plan length than FF-Replan. In zeno- to 15, we present detailed results for the learned heuristics travel, both planners are equally successful in nding the taken from (Wu and Givan 2007) for the ve domains where goals while FF-Replan is slightly shorter in plan length. learned heuristics are available, and we discuss each brie y L S-EFH(F ) G-rFeedy(F ) FFFFR-Replan L SS-EFH(F ) GG-rFeedy(F ) FFFFR-Replan 1 1 R R S 0.5 S 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 1000
) 1000 ) g g o 100 l o ( l
(
n 100 n e 10 e L L S
1 S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A )
) 1000 c c e e 1000 s s 100
g g 100 o o l l 10 ( (
10 e e 1 m m 1 i i t t
0.1 S S 0.1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A Random, Prob # Schedule, Prob # Figure 8: Random results for the CR-FF heuristic. Figure 10: Schedule results for the CR-FF heuristic. L SS-EFH(F ) GG-rFeedy(F ) FFFFR-Replan L S-EFH(F ) GG-rFeedy(F ) FFFR-Replan 1 1 R
S 0.5 R
S 0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 1000 )
g 1000 ) o g l 100 ( o
l ( n
e
10 n 100 L e S L
1 S
) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 10 c
1000 ) e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A c s
e s g 100 100
o g l ( 10 o
l 10 e (
1 e m
i 1 t m i
S 0.1 t
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A S 0.1 Elevators, Prob # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A Pitchcatch, Prob # Figure 9: Elevators results for the CR-FF heuristic. Figure 11: Pitchcatch results for the CR-FF heuristic. here7. In blocksworld and boxworld, SEH(L) preserves the Summary high success ratio of Greedy(L) and signi cantly improves the average successful plan length. SEH(L) improves slightly on the success ratio of Greedy(L) in tireworld. Note We have proposed and evaluated stochastic enforced hill- that in these three domains SEH(L) also signi cantly outper- climbing, a novel generalization of the deterministic en- forms FF-Replan as Greedy(L) already has excellent perfor- forced hill-climbing from FF (Hoffmann and Nebel 2001). mance in these domains. Instead of nding a descendant that is strictly better than the In zenotravel, SEH(L) again has signi cant improve- current state in heuristic value, we construct a local MDP ment on average successful plan length over Greedy(L). In around any local optimum or plateau reached and seek a pol- exploding blocksworld, the base heuristic Greedy(L) per- icy that expects to exit this MDP with a better valued state. forms so poorly as to provide inadequate guidance to SEH, We have demonstrated that this approach provides substan- although SEH(L) is able to improve on Greedy(L) in 5 tial improvement over greedy hill-climbing for fourteen dif- blocks problems. In both of these domains, FF-Replan out- ferent heuristics in nine different domains, created using two performs either hill-climbing method. different styles for heuristic de nition. We have also demon- strated the resulting planners are a substantial improvement 7We take the best-performing value functions used for compar- over FF-Replan (Yoon, Fern, and Givan 2007) in our exper- ison purpose in that paper. iments. L SS-ELH(L ) G-rLeedy(L ) FFFFR-Replan L SS-EFH(L ) GG-rFeedy(L ) FFFFR-Replan 1 1 R R S 0.5 S 0.5
0 0 15 20 25 30 A 20 30 40 A 10 ) ) 1000 g g o o l l ( (
100 n n e e L L S S 10 1 15 20 25 30 A 20 30 40 A
) 1000
c Tireworld, # nodes e s
g Figure 14: Tireworld results for learned value functions. We omit
o 100 l (
runtime results as on average each planner uses less than 1 second e per attempt. m
i 10 t
S 15 20 25 30 A L S-F G-F FFFR-Replan Blocksworld, # blocks SEH(L ) Greedy(L ) 1 1 Figure 12: Blocksworld results for learned value functions. R R S S 0.5 0.5 L S-EFH(L ) G-rFeedy(L ) FFFR-Replan 1 0 0 10,2,2 5 10 A R 1000 100 S 0.5 ) ) g g o o l l ( (
0 10 n n e
10,5 10,10 10,15 15,5 A e L
L 100
) 1000 S S 1 g o ) ) l 10,2,2 5 10 A ( c c
e 100 100 e 10 n s s
e g g L o o l l S 1 (
10 10 (
e 10,5 10,10 10,15 15,5 A e ) m m c i i e t 100 1 t 0.1 s
S S
g 10,2,2 5 10 A o l
( Zenotravel EX-BW, # blocks
10 e m
i Figure 15: Zenotravel and exploding blocksworld results for t 1 S 10,5 10,10 10,15 15,5 A learned value functions. The problem size for zenotravel is 10 Boxworld, (# boxes, # cities) cities, 2 people, 2 aircraft. Figure 13: Boxworld results for learned value functions. Hansen, E., and Zilberstein, S. 2001. LAO*: A heuris- tic search algorithm that nds solutions with loops. AIJ References 129:35 62. Barto, A. G.; Bradtke, S. J.; and Singh, S. P. 1995. Learn- Hoffmann, J., and Nebel, B. 2001. The FF planning sys- ing to act using real-time dynamic programming. AIJ tem: Fast plan generation through heuristic search. JAIR 72:81 138. 14:253 302. Bertsekas, D. P., and Tsitsiklis, J. N. 1996. Neuro-Dynamic Kirkpatrick, S.; Gelatt, Jr, C.; and Vecchi, M. 1983. Opti- Programming. Athena Scienti c. mization by simulated annealing. Science 220:671 680. Bertsekas, D. P. 1995. Dynamic Programming and Optimal Nilsson, N. 1980. Principles of Arti cial Intelligence. Control. Athena Scienti c. Tioga Publishing, Palo Alto, CA. Bonet, B., and Geffner, H. 2006. Learning depth- rst Sutton, R. S., and Barto, A. G. 1998. Reinforcement Learn- search: A uni ed approach to heuristic search in determin- ing: An Introduction. MIT Press. istic and non-deterministic settings, and its application to Wu, J., and Givan, R. 2007. Discovering relational domain MDPs. In ICAPS. features for probabilistic planning. In ICAPS. Cerny, V. 1985. Thermodynamical approach to the travel- Yoon, S.; Fern, A.; and Givan, R. 2007. FF-Replan: A ing salesman problem: An ef cient simulation algorithm. baseline for probabilistic planning. In ICAPS. J. Optim. Theory Appl. 45:41 51. Younes, H.; Littman, M.; Weissman, D.; and Asmuth, J. Dean, T.; Kaelbling, L. P.; Kirman, J.; and Nicholson, A. 2005. The rst probabilistic track of the international plan- 1995. Planning under time constraints in stochastic do- ning competition. JAIR 24:851 887. mains. AIJ 76:35 74.