| ### Chapter 8 - The Error Surface
Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks
Russell D. Reed and Robert J. Marks II
Copyright © 1999 Massachusetts Institute of Technology

## 8.4 Remarks

The efficiency of any optimization method depends on having a good fit between the basic assumptions of the algorithm and the actual characteristics of the function being minimized. Many advanced optimization methods assume the error surface is locally quadratic, for example, and may not do well on the "cliffs-and-plateaus" surface common in neural network classifiers. In this case, the quadratic assumption is not reasonable on a large scale so these optimizers may be no more efficient than simpler methods in finding a good approximate solution. The assumption, however, is usually reasonable near a minimum, in which case these methods may be very efficient for final tuning of a near-solution found by other methods.

For back-propagation, a large learning rate is needed to make progress across the large flat regions. But near the "cliffs" where ||∂ E/∂w || is large, a small learning rate is necessary to prevent huge weight changes in essentially random directions. If a fixed learning rate is used, the value will have to be a compromise. One of the advantages of the common technique of initializing with small random weights is that the system starts in the area near the origin where the error surface is smoother and it has a better chance of finding the right trough.

Before ending this discussion, it should be noted that the error surfaces illustrated in the figures are for classification problems with small training sets. The error surfaces may be very different for regression problems with large sample sizes. Based on figure 8.1, it is reasonable to expect it to be smoother.

For regression problems where the target is a continuous function of its inputs, smooth input-output functions are usually preferred. If there is sufficient data that the system cannot fit every point exactly, then it must approximate multiple points by fitting a surface "close" to them. For many cost functions, this surface can be thought of as the local average of nearby points and will tend to be smooth because of the smoothing effects of averaging. Because smoother functions generally correspond to smaller weights, the good minima will usually be in the interior of the weight space rather than at infinity. Similarly, in classification problems with many samples in overlapping clusters, it may be better to form gradual transitions between classes in regions where they overlap. This again corresponds to smaller weights and moves the minima in from infinity.

Of course, if there is so little data and the network is so powerful that it can fit every point exactly, then there is so reason to expect it to form a smooth function, and minima at infinity may survive. Even when a smooth function would be preferable, local minima at infinity may survive corresponding to fitting a few of the points exactly while ignoring the rest; these are likely to be shallow and narrow, however, and will probably be shadowed by better minima closer to the origin. Although plateaus and cliffs will be apparent at large distances from the origin, this may be irrelevant because those regions will never be investigated by the learning algorithm.

The stair-step shape may survive in very underconstrained networks that can essentially classify each of the training points internally (e.g., by assigning a hidden "grandmother node" to each training sample). In this case, the global minima of the training set error would be at infinity and low generalization-error areas corresponding to smooth functions are unlikely to be minima.

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