## Monday, September 7, 2009

### KEYS and KEYS2

Within a model, there are chains of reasons that link inputs to the desired goals. As one might imagine, some of the links in the chain clash with others. Some of those clashes are most upstream; they are not dependent on other clashes. In the following chains of reasoning, the clashes are {e, -e}, {g, -g}, {j, -j}; the most upstream clashes are {e, -e}, {g, -g}

In order to optimize decision making about this model, we must first decide about these most upstream clashing reasons (the "keys"). Most of this model is completely irrelevant. In the context of reaching the goal, the only important discussions are the clashes {g,-g,j, -j}. Further, since {j, -j} are dependent on {g, -g}, then the core decision must be about variable g with two disputed values: true and false.

Setting the keys reduces the number of reachable states within the model. Formally, the reachable states reduce to the cross-product of all of the ranges of the collars. We call this the clumping effect. Only a small fraction of the possible states are actually reachable. This notion of $keys$ has been discovered and rediscovered
many times by many researchers. Historically, finding the keys has seen to be a very hard task. For example, finding the keys is analogous to finding the minimal environments of DeKleer's ATMS algorithm. Formally, this is logical abduction, which is an NP-hard task.

Our method for finding the keys uses a Bayesian sampling method. If a model contains keys then, by definition, those variables must appear in all solutions to that model. If model outputs are scored by some oracle, then the key variables are those with ranges that occur with very different frequencies in high/low scored model outputs. Therefore, we need not search for the keys- rather, we just need to keep frequency counts on how often ranges appear in best or rest outputs.

KEYS implements this Bayesian sampling methods. It has two main components - a greedy search and the BORE ranking heuristic. The greedy search explores a space of M mitigations over the course of M "eras". Initially, the entire set of mitigations is set randomly. During each era, one more mitigation is set to M_i=X_j, X_j being either true or false. In the original version of KEYS, the greedy search fixes one variable per era. Recent experiments use a newer version, called KEYS2, that fixes an increasing number of variables as the search progresses
(see below for details).

KEYS (and KEYS2), each era e generates a set as follows:
• [1:] MaxTries times repeat:
• Selected[1{\ldots}(e-1)] are settings from previous eras.
• Guessed are randomly selected values for unfixed mitigations.
• Input = selected from guessed.
• Call model to compute score=ddp(input);

• [2:] The MaxTries scores are divided into B "best" and the remainder become the "rest".
• [3:] The input mitigation values are then scored using BORE (described below).
• [4:] The top ranked mitigations (the default is one, but the user may fix multiple mitigations at once) are fixed and stored in selected[e].

The search moves to era e+1 and repeats steps 1,2,3,4. This process stops when every mitigation has a setting. The exact settings for MaxTries and B must be set via engineering judgment. After some experimentation, we used MaxTries=100 and B=10.

Pseudocode for the KEYS algorithm:
1. Procedure KEYS2. while FIXED_MITIGATIONS != TOTAL_MITIGATIONS3.    for I:=1 to 1004.       SELECTED[1...(I-1)] = best decisions up to this step5.       GUESSED = random settings to the remaining mitigations6.       INPUT = SELECTED + GUESSED7.       SCORES= SCORE(INPUT)8.    end for   9.    for J:=1 to NUM_MITIGATIONS_TO_SET 10.      TOP_MITIGATION = BORE(SCORES)       11.      SELECTED[FIXED_MITIGATIONS++] = TOP_MITIGATION 12.   end for  13. end while          14. return SELECTED \end{verbatim} }

KEYS ranks mitigations by combining a novel support-based Bayesian ranking measure. BORE (short for "best or rest") divides numeric scores seen over K runs and stores the top 10% in best and the remaining 90% scores in the set rest (the bes\$ set is computed by studying the delta of each score to the best score seen in any era). It then computes the probability that a value is found in best using Bayes theorem. The theorem uses evidence E and a prior probability P(H) for hypothesis H in{best, rest}, to calculate a posterior probability

When applying the theorem, likelihoods are computed from observed frequencies. These likelihoods (called "like" below) are then normalized to create probabilities. This normalization cancels out P(E) in Bayes theorem. For example, after K=10,000 runs are divided into 1,000 best solutions and 9,000 rest, the value mitigation31=false might appear 10 times in the best solutions, but only 5 times in the rest. Hence:

Previously, we have found that Bayes theorem is a poor ranking heuristic since it is easily distracted by low frequency evidence. For example, note how the probability of E belonging to the best class is moderately high even though its support is very low; i.e. P(best|E)=0.66 but freq(E|best) = 0.01.

To avoid the problem of unreliable low frequency evidence, we augment the equation with a support term. Support should increase as the frequency of a value increases, i.e. like(best|E) is a valid support measure. Hence, step 3 of our greedy search ranks values via

For each era, KEYS samples the model and fixes the top N=1 settings. Observations have suggested that N=1 is, perhaps, an overly conservative search policy. At least for the DDP models, we have observed that the improvement in costs and attainments generally increase for each era of KEYS. This lead to the speculation that we could jump further and faster into the solution space by fixing N=1 settings per era. Such a jump policy can be implemented as a small change to KEYS, which we call KEYS2.

• Standard KEYS assigns the value of one to NUM_MITIGATIONS_TO_SET (see the pseudocode above);
• Other variants of KEYS assigns larger values.

In era 1, KEYS2 behaves exactly the same as KEYS. In (say) era 3, KEYS2 will fix the top 3 ranked ranges. Since it sets more variables at each era, KEYS2 terminates earlier than KEYS.