Dynamically δ-dominant values

The two previous categories of dominance were exact: once found, a dominant value is certain to be the optimal value. We now present an approximative dominance: dominant values that allow for an error margin. They are computed in a very similar way with Equation7.4:

v_i^δ∗(t) =argmin_vj iδ

J OINi(t)^±,agent view, Xi=v_i^j (7.7)

The valuev^δ∗_i (t)computed like in Equation7.7has the smallest guaranteed distance to the optimal solution, given the currently available information. It is obvious that as time progresses and more and moreUTIL/VALUEmessages arrive, the bounds become tighter and tighter, thus offering the possibility for increasingly accurate decisions.

Ifδ(t, v_i^δ∗)issmall enough, then we say thatv^δ∗_i (t)is adynamicallyδ-dominant value, and we can safely assign it toXiand start theVALUEpropagation fromXi.

AnyPOPalso exhibits some built-in fault tolerance. If messages are lost, there is a graceful degra-dation of performance: theδs will not be updated anymore, and in case that would have meant changing a current assignment, solution quality degrades. However, the algorithm still provides the best solution it can infer based on the information thatwassent/received successfully.

7.2.6 Iterative A-DPOP for anytime behaviour

Another alternative for anytime solving is obtained by a straightforward iterative execution of A-DPOP with increasing boundsk, as described in Algorithm15. Iterative A-DPOP starts with low values for k, which means that the UTIL messages sent are small, and can be quickly computed and sent over the network. This means that a (relatively) good solution can be obtained very fast. As time goes by, executions of A-DPOP(k) proceed, with increasing values ofk, which mean that the approximate UTIL messages get larger andand more accurate, offering better bounds and better solutions. Like this we simulate an anytime behaviour with A-DPOP.

Remark 9 (Iterative A-DPOP can reuse computation between iterations.) Notice that as soon as the thresholdkexceeds the size of a nodeXi’s separator, and of all descendants ofXi, the UTIL mes-sage computed and sent byXi isexact(contains no approximations anymore). Afterwards, for sub-sequent executions of A-DPOP with larger values for k,Xi’s parentPican simply reuse theU T IL^P_iⁱ

Algorithm 14AnyPOP - Anytime approximate Distributed Pseudotree Optimization AnyPOP(X,D,R, k, δ): each agentXidoes:

UTILpropagation protocol

1 get all newUTILmessages (Xk,U T ILⁱ_k)

2 buildJ OINi(t)as in Equation7.6

3 ifXiis rootthenstartVALUEpropagation

4 else

message it has previously received fromXi. Like this,Xiand its whole subtree have no more com-putation or message passing to do until the end of the algorithm. This effectively means that Iterative A-DPOP explores easy (low width) parts of the problem very fast in the beginning, and then most of the work is concentrated in the difficult parts of the problem.

Algorithm 15Iterative A-DPOP: Anytime based on iterative A-DPOP Iterative A-DPOP(X,D,R):

1 Construct DFS treeusing Algorithm3

2 run Algorithm8; eachXi∈ XknowsSepi 3 w=argmaxX_i|Sepi|(the induced width)

4 fork= 1. . . wdo

5 run A-DPOP(k) as follows:

6 if|Sepi|< kandU T IL^P_iⁱin A-DPOP(k−1) was exactthen

7 Xireusesits UTIL message from A-DPOP(k−1) in A-DPOP(k).

8 set temporary solution according to A-DPOP(k)

7.2.7 Experimental evaluation

Our experiments were performed in the distributed meeting scheduling scenario described in [127]

and in Section2.3.1. In this context, the experiments were ran with an especially difficult problem

k Maxδ/msg % Avgδ/msg % δ/overall % Total UTIL payload Max msg size Utility

1 44.83 13.55 2.90 2104 16 2278

2 36.00 4.54 2.69 10032 128 2283

3 17.14 1.27 2.43 39600 1024 2289

4 13.11 0.57 0.81 130912 8192 2327

5 10.00 0.19 0.43 498200 65536 2336

6 1.36 0.04 0.30 1808904 524288 2339

7 0.00 0.00 0.00 3614064 2097152 2346

Table 7.3: Max. dimensions vs. solution accuracy: problem with 140 vars, 204 con-straints,width=7

Snapshot # Maxδ/var % Avgδ/var % Utility δ/overall % Assig changes

1 94.44 80.77 1555 33.72 0

2 66.07 16.7 1625 30.73 99

3 42.42 3.92 2036 13.21 73

4 13.51 1 2254 3.92 19

5 13.51 0.94 2289 2.43 1

Table 7.4:AnyPOPdynamic evolution: problem with 140 vars, 204 constraints,width=7

containing 70 agents, 140 variables and 204 binary constraints. The induced width is 7, meaning that the largest message holds over two million values. We executed A-DPOP with increasing bounds on the maximal dimensionality (k). We present in Table7.3the results in this order: maximal dimensionality, maximalδfor allUTILmessages (as in equation7.3), the averageδper message, the distance of the approximate solution to the true optimum, the total amount ofUTILinformation transmitted (computed as the sum of the sizes of the individualUTILmessages), the maximal message size, and the utility of the solutions found. The accuracy of the solutions increases with the increase ofk, culminating with the optimal value fork= 7, in which case A-DPOP(7) is equivalent to DPOP. However, there is also a dramatic increase in computation effort and network load. If we compare the first and the last lines of the table, we see that we can achieve a solution which is within 3% of the optimum with 3 orders of magnitude less effort (2k values sent over the network v.s. 3M). Therefore, if absolute optimality is not required, it might actually pay off to settle for a suboptimal solution obtained with much less effort.

We performed the second test with the same difficult instance from the previous test. This time, we wanted to test simultaneously both the anytime performance ofAnyPOP, and its ability to deal with low resources. Therefore, we imposedk= 3, and started AnyPOP. We took a number of 5 snapshots of

the assignments of the variables during the execution. The first snapshot is taken immediately after the initialδs are computed,before sending/receiving any message. Subsequent snapshots are taken after each node has received another message. The last snapshot is taken after all messages are sent/received.

The assignments discovered by each of the snapshots are used to compute the overall utility. We notice a steady progress of the algorithm towards a solution, culminating with the best solution found by A-DPOPon the same test problem, with the same boundk = 3. At the same time, there is a steady decrease of the error bounds, and of the assignment changes from one snapshot to the next.

7.2.8 Summary

We propose in this chapter an approximate algorithm for distributed optimization, which allows the de-sired tradeoff between solution quality and computational complexity. The algorithms can be extended with heuristics for selecting ”intelligently” the dimensions to be dropped out when exceeding maximal message size. In the second part of the chapter we present an anytime version of the first algorithm, which provides increasingly accurate solutions while the propagation is still in progress. This makes it suitable for very large, distributed problems, where propagations may take a long time to complete.

The anytime algorithm also exhibits some built-in fault-tolerance features, by graceful degradation of performance upon message loss. Experimental results show that these algorithms are a viable approach to real world, loose, possibly unbounded optimization problems.

PC-DPOP: Tradeoffs between Memory/Message Size and Centralization

“Congrego et impera.”

— Anonymous

In this chapter we discuss the idea of trading full decentralization for computational and communi-cation efficiency. We introduce PC-DPOP, a new hybrid algorithm that is controlled by a parameter kwhich upper bounds the size of the largest message, and the amount of available memory. PC-DPOP(k) operates in the framework of Section6.2for detecting high-width clusters, where it is not possible to perform full inference as in DPOP. Such clusters are centralized in therootof the cluster, and solved by the root in a centralized fashion, using an algorithm of its choice. Communi-cation requirements over any link in the network are limited thus toexp(k). The linear number of messages is preserved.

In high width clusters, PC-DPOP offers a tradeoff between the fully decentralized solving process of DPOP for polynomial memory and message size. The overall behavior of PC-DPOP(k) is as follows: ifwis the induced width of the problem given by the chosen DFS ordering, depending on the value chosen fork, we have 3 cases:

Fully decentralized algorithms for DCOP like DPOP or ADOPT often require excessive amounts of communication when applied to complex problems. Mailler and Lesser have introduced APO (Asyn-chronous Partial Overlay) [128], an algorithm which uses a strategy of partial centralization to mitigate this problem. While such a tradeoff is probably unfeasible in competitive settings where the agents

141

are non-cooperative (see the discussion from Section11.6), in settings where the agents are perfectly cooperative, this approach can offer communication and computation savings.

In this chapter we introduce PC-DPOP, a new hybrid algorithm that is controlled by a parameterk which upper bounds the size of the largest message, and the amount of available memory. PC-DPOP(k) operates in the framework of Section6.2for detecting high-width clusters, where it is not possible to perform full inference as in DPOP because the memory requirements would exceed the bound imposed byk. In low-width areas, PC-DPOP proceeds as normal DPOP, using a linear number of messages and memory at mostO(exp(k)). Clusters of high width are detected as in Section6.2.1, and centralized in therootof the cluster. The cluster root then solves the subproblem in a centralized fashion, using an algorithm of its choice. Communication requirements over any link in the network are limited thus to O(d^k). The linear number of messages is preserved.

Therefore, in these high width clusters, PC-DPOP offers a tradeoff between the fully decentralized solving process of DPOP for polynomial memory and message size. The overall behavior of PC-DPOP(k) is as follows: ifwis the induced width of the problem given by the chosen DFS ordering, depending on the value chosen fork, we have 3 cases:

1. Ifk= 1, only linear size messages and memory are used.

2. Ifk < w,P C−DP OP(k)performs full inference in areas of width lower than k, and centraliza-tion in areas of width higher than k. Memory and communicacentraliza-tion requirements areO(exp(k)).

3. Ifk ≥w, full inference is done throughout the problem; PC-DPOP(k) is then equivalent with DPOP (i.e. full inference everywhere). Memory requirements areO(exp(w)).

Partial results within each cluster are cached ( [42,8,132]) by the respective cluster root and then integrated as messages into the overall DPOP-type propagation. This avoids the need for any recom-putation during the final VALUE propagation phase.

Compared to OptAPO, PC-DPOP provides better control over what parts of the problem are cen-tralized and allows this centralization to be optimal with respect to the chosen communication structure.

PC-DPOP also allows for a priori, exact predictions about privacy loss, communication, memory and computational requirements on all nodes and links in the network. We also report strong efficiency gains over OptAPO in experiments on three problem domains.

The rest of this section is structured as follows: Section8.1introduces the PC-DPOP hybrid algo-rithm, which is evaluated empirically in Section8.2. Section8.3relates PC-DPOP to existing work.

Section8.4briefly discusses privacy, and Section8.5concludes.

Figure 8.1:A problem graph (a) and a DFS tree (b). In low-width areas the normal UTIL propagation is performed. In high width areas (shaded clustersC1,C2andC3in (b)) bounded UTIL propagation is used. All messages are of size at mostd^k. Clusters are centralized and solved in the cluster roots (the bold nodesX2, X9, X14).

8.1 PC-DPOP(k) - partial centralization hybrid

To overcome the space problems of DPOP, we introduce the control parameterkthat bounds the mes-sage dimensionality. This parameter should be chosen s.t. the available memory at each node and the capacity of its link with its parent is greater thand^k, where d is the maximal domain size.

As with DPOP, PC-DPOP also has 3 phases: aDFS constructionphase, anUTILphase, and a VALUEphase. The DFS construction is simply done using Algorithm3. Subsequently, on the es-tablished DFS structure, we run the LABEL-DFS algorithm from Section6.2.1(Algorithm8). This algorithm identifies clusters of high width and labels the nodes as eithernormal node,cluster-nodeor cluster-root node. The subsequent UTIL phase assumes this labeling is in place.