## ICM UGM Demo

The last demo ends the first series of demos coverings exact methods for decoding/inference/sampling. We now turn to the case of approximate methods. These methods can be applied to models with general graph structures and edge potentials, but don't necessarily perform these operations exactly. This first demo discusses approximate decoding with local search, generalizing the iterated conditional mode (ICM) algorithm mentioned in the previous demo.

### Iterated Conditional Modes

The ICM algorithm is one of the simplest methods for optimal decoding. In the ICM algorithm, we initialize the nodes to some starting state values (by default, UGM uses the states that maximize the node potentials), and we then start cycling through the nodes in order. When we get to node i, we consider all states that node i could take, and replace its current state with the state that maximizes the joint potential. We keep cycling through the nodes in order until we complete a full cycle without changing any nodes. At this point, we have reached a local optima of the joint potential that can not be improved by changing the state of any single node. The ICM algorithm is described in Besag's paper on analyzing "dirty pictures":
• J.E. Besag. On the Statistical Analysis of Dirty Pictures. Journal of the Royal Statistical Society Series B, 1986.
In UGM, we can apply ICM using:
```getNoisyX
ICMDecoding = UGM_Decode_ICM(nodePot,edgePot,edgeStruct);
figure;
imagesc(reshape(ICMDecoding,nRows,nCols));
colormap gray
```
This function UGM_Decode_ICM can also take an optional fourth argument that gives the intial configuration.

### Greedy Local Search Decoding

The ICM algorithm is a specific instance of a local search discrete optimization algorithm, sometimes described as a first improvement greedy algorithm. Instead of this particular local search method, we could instead consider other local search methods.

As one example, we could consider a best improvement greedy algorithm, where we search for the single state change that improves the joint potential by the largest amount. This is slightly more expensive than ICM since we only make a single state change after cycling through the nodes, but it ensures that we always take the 'best' move at each iteration. In UGM we can apply this greedy algorithm using:

```greedyDecoding = UGM_Decode_Greedy(nodePot,edgePot,edgeStruct);
figure;
imagesc(reshape(greedyDecoding,nRows,nCols));
colormap gray
```
This function can also take an optional fourth argument that gives the initial configuration.

### ICM with Restarts

A much more effective enhancement of the ICM algorithm is to restart it (with a different initial configuration) whenever it reaches a local optima. By doing this, we may be able to explore different local optima. Further, if we restart it enough times (and guarantee that our restart mechanism eventually generates all configurations), this procedure will eventually find the global optima. In UGM we can run the ICM algorithm with 1000 restarts using:
```nRestarts = 1000;
ICMrestartDecoding = UGM_Decode_ICMrestart(nodePot,edgePot,edgeStruct,nRestarts);
figure;
imagesc(reshape(ICMrestartDecoding,nRows,nCols));
colormap gray
```

### Comparison

Applying the the three methods to a sample of the noisy X problem yields the following results:

In this example, the best-improvement greedy decoding did slightly better than the ICM algorithm, but the (stochastic) ICM with restarts algorithm did better than both of the deterministic strategies.

### Notes

These are just three of the many possible local search methods that could be applied. For example, you could use any of the algorithms from the inside cover of "Stochastic Local Search: Foundations and Applications" by Hoos and Stutzle. This includes things like local search methods with stochastic moves, dynamic local search methods, iterated local search methods, genetic algorithms, ant colony optimization, etc.

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Mark Schmidt > Software > UGM > ICM Demo