# Special Topics: Rules of Thumb

Training neural networks is said to be more of an art than a science. While this claim could be made of many fields of science, it is true that there is no clear formula for successful network modeling. A good modeler is able to diagnose problems with the network and understand which changes can be made to try to resolve those problems. But it is also important that your initial parameter choices are in the right ballpark or the process of making incremental improvements may never succeed.

The purpose of this guide is to convey some of the things I have learned about training networks over the years. However, it is important to remember that there are no absolutes. What may be an appropriate parameter choice for most networks could be terribly wrong for your network. I have found many of the traditional rules of thumb, like using very small initial weights, to depend on certain assumptions that are not always valid. You may find the same to be true of my suggestions.

### Weight Update Algorithms

Lens includes four main weight update algorithms: steepest descent, momentum descent, Doug's momentum, and delta-bar-delta. Doug's momentum is probably the best overall method, which is why it is the default. The problem with steepest and momentum descent is that they are too aggressive at the outset of training and one must use a special initial learning rate and momentum to prevent outrageous weight changes. The reason is as follows.

The philosophy behind the backpropagation algorithm is that a small step should be taken in weight space in the direction of steepest error gradient. That is, if one were reverse hill-climbing, one would take a step down the steepest part of the slope. But, in practice, the steepest descent algorithm doesn't just take a step in the direction of steepest descent, it takes a step whose horizontal distance is proportional to the slope. It is as if, when the mountain is steeper, you reach your leg out even farther. This doesn't really make sense. If the mountain is steeper, a constant-size step will already take us farther downhill. To be safe, we might even want to take smaller steps when the gradient is steep.

This results in particular problems early in training. A randomized network usually has a huge error and a very steep gradient. Taking a step whose size is proportion to that gradient, unless it is scaled by a very small learning rate, will take you over the valley and probably the next ten mountains as well. This is even worse when momentum is used because the momentum from those huge first steps is hard to counteract. Therefore, when training with steepest or momentum descent, it is often necessary to start with a small learning rate and no momentum, and then increase these parameters once learning stabilizes.

The Doug's momentum method makes a simple change that helps alleviate these problems. Rather than having a weight step whose length is proportional to the product of the gradient and the learning rate, the length of the gradient is factored out. As long as the length of the gradient is greater than 1, the size of the weight step (prior to momentum) will just be equal to the learning rate. If the gradient is less than 1, which usually only occurs once the network is nearing a minimum, the size of the step is scaled by the gradient, as in normal momentum descent. Otherwise the network will never settle into a minimum and will perpetually bounce around.

Because Doug's momentum bounds the weight step, there is no need to use a special learning rate or momentum early on. However, you may still need to gradually reduce the learning rate as training goes on.

The delta-bar-delta algorithm maintains an individual learning rate for each link. As long as the link weight keeps heading in the same direction from one change to the next, its rate increases. If the direction of the weight change flips, the learning rate is cut back. Delta-bar-delta can be very effective in problems with smooth error surfaces. However, if you are not using batch learning or you are trying to solve a complex problem, delta-bar-delta can be quite bad as the weight changes are constantly switching directions.

When using DBD, I generally find a rate increment of 0.1 and a decrement of 0.9 to be pretty good. The main learning rate, which scales the links' values, should be set to the high end for the type of error measure you have (0.1-1.0 for squared error, 0.001 for cross-entropy or divergence). Again, train for the first little bit with no momentum and then turn it up to 0.9-0.95.

In general, though, I seem to get the best results with Doug's Momentum, especially with large networks on complicated tasks.

### Batch Size

Unless you have some principled bias against batch learning, it is probably best to train in full batch mode when you have 500 or fewer examples. Full batch mode is nice because the error surface is consistent from batch to batch. You can therefore use complex update algorithms like delta-bar-delta and it is easier to assess the progress of the network. If the error starts jumping around during batch training, you know it is because of aggressive training and not because of a changing environment.

However, batches that are too large can be inefficient. Too much time is spent gathering derivatives between weight updates. At the other extreme is online learning, which changes the weights after every example. In this case, too much time is generally spent performing weight updates and the error surface will be changing radically from one update to the next.

Somewhere between these extremes is the optimal batch size. I find that good batch sizes are often between 10 and 200 examples. This depends largely on the diversity of your example set. Batches should be large enough that their statistics are fairly representative of one another. That is, the network's ability on one batch should be a good predictor of its performance on another batch. A simple and repetitive training set may deserve smaller batches than a complex and diverse one.

One problem when not using full batches is that the error naturally jumps around from one batch to the next. This can be good because it is a natural sort of noise that can help get the network through local minima. On the other hand, it may also be hard to distinguish this normal variation from spikes in the error curve caused by overly aggressive training. You may need to rely on periodic testing to determine if the network is continuing to make progress.

When not using full batches, be sure to put the training set in PERMUTED, PROBABILISTIC, or RANDOMIZED mode so you don't train on the exact same batches each time.

### Momentum

Momentum causes a portion of the previous weight change to be added into the current step. When using a momentum of 0.9, the effective learning rate can grow to 10 times the actual rate if the direction of weight changes is consistent. In general, momentum is quite useful. As mentioned earlier, using momentum is dangerous at the start of training with ordinary momentum descent, but not with Doug's Momentum.

Momentum is particularly important when using small batches because it allows derivatives to be integrated across batches. The smaller the batch size, the greater the momentum you may want to use. Remember that the effect of momentum is basically proportional to the inverse of the difference between 1 and the momentum value. So a change from 0.9 to 0.95 is as significant as a change from 0.8 to 0.9 or 0.6 to 0.8.

The useful range of momentums is about 0.3 to 0.95. Below 0.3 the momentum plays an insignificant role. Going above 0.95 is probably not a good idea. When you have a choice between using momentum and a higher learning rate, I generally find it is better to use momentum, as long as it doesn't get too high. Overall, I have found a momentum of 0.9 to be ideal under a wide variety of circumstances, so that is definitely a good initial choice.

### Learning Rate and Error Functions

The learning rate is perhaps the most obvious parameter, and is indeed the one that you will need to pay the most attention to. Although a simple network can be trained effectively with a constant learning rate, most networks can only be trained well by adjusting the rate as the network trains.

The magnitude of the initial learning rate depends on the error function and weight update algorithm. If you are using sum squared error and momentum, steepest, or delta-bar-delta, typical learning rates are in the range 0.001 to 1.0. 0.01 is a good starting point. With Doug's momentum, you can use much larger learning rates, typically starting at 0.1.

Cross entropy is generally better than sum-squared error for networks with logistic output units. The related divergence function should be used for groups with normalized soft-max outputs. Because the error can grow exponentially large with these functions, networks that seriously miss their targets are punished severely. However, these functions should not be used with linear output units or when targets are not in the range [0,1].

When using cross entropy or divergence with one of the three non-normalized weight update algorithms, you will need a very small learning rate because of the large derivatives that can result. I find the useful range to be about 0.0001 to 0.001. 0.0005 is a good place to start and is generally what I have found to be the best on a wide range of problems. Again, Doug's momentum is not affected by the magnitude of the derivative so you can start with a learning rate around 0.1.

You are generally going to need to decrease the rate as training progresses. There was some old conventional wisdom that the learning rate should be increased as training progresses. I think that was because of the early instability of momentum descent. When using momentum descent, you will want to start with a low rate (or just no momentum) for a short time. From then on you will gradually decrease the rate. With Doug's momentum, you can just start with the highest rate and then gradually lower it.

If you want to train aggressively, you might compare several initial learning rates and select the optimal one. Perform about 1/10 of the intended training and choose the rate/momentum combination that leads to the lowest error. Then re-train with that rate and monitor the error. If the error levels out, begins to increase, or has detectable spikes, the rate is too large, so cut it in half. This will usually continue until the rate is roughly 1/20 of its initial value. If training is only making very slow, steady progress and the Gradient Linearity is high, your rate may be too low.

With some networks, especially those using small batches, I have found that there is a big improvement if you do a little training at the end with a very low learning rate, about 1/10 of the previous rate. So you might start with a rate of 0.1, eventually you will decrease it to 0.05, 0.02, 0.01, and 0.005. Finally, cut the rate to 0.0005 and do a little more training. With many networks this will help it nestle into a narrow minimum and thus results in a relatively large improvement in performance. However, you only want to do that at the end of training.

### Initial Weights

The magnitude of the initial random weights is not something people think about too much, but it can have a critical effect on the success of the network. The conventional wisdom has always been that very small initial weights (+/-0.001 or smaller) are best because logistic units will be at the most sensitive parts of their sigmoid. It was feared that larger weights would pin units on the flat part of their sigmoids where derivatives are very small.

However, there is a cost involved in using small initial weights. Error backprogated across a link is scaled by the weight of that link. Thus, small initial weights can also lead to very small derivatives reaching early links in a multi-layer network. Conceptually, if all hidden units have roughly the same activation on all examples, information will not propagate effectively from the input layer to the output layer. A change in the input pattern may have very little effect on the output unit activations. As a result, error cannot be attributed to early weights and learning will be slow.

Thus, you must choose an initial weight range that balances the problem of pinned units with the problem of inactive units due to tiny weights. A good rule of thumb for groups with sigmoidal units is that the distribution of activation values across units and across examples should be evenly distributed in the range [0,1]. Or, perhaps a bit better, the distribution should be slightly rounded in the middle, with fewer extreme values.

Therefore, the size of the initial weights into a unit should depend on the number of links into the unit. The more inputs, the smaller the weights on those inputs should be. Thus, you may want different randRange values for each group. The randRange value should be based on how many links there are into a group and the expected activation level of the units feeding those links. If we assume that the activation on all units feeding the group is fixed at 1.0 and the units have a logistic activation function with a gain of 1.0, the following table gives the appropriate range of initial weights:

Number of Inputs | Init Rand Range
5 | 1.2
10 | 0.90
20 | 0.60
40 | 0.45
60 | 0.35
80 | 0.30
100 | 0.25
150 | 0.20
200 | 0.18
500 | 0.12

As you can see from the table, the optimal weight randomization range is still fairly high, even with 200 inputs. The magnitude does not decrease much when jumping from 100 to 200 inputs. Note that these values are based on having sending units with activations of 1.0. In practice, the sending units will either be from an input group or a hidden layer. If they are from an input group, then you should estimate the number of inputs that will be active, on average, when choosing the randRange. If there will be about 10 inputs active, you might choose a randRange of 0.9 for the input-to-hidden projection. If the projection is coming from a hidden layer, you might estimate that the activations of units in the hidden layer will be evenly distributed in the range [0,1]. Therefore, this is effectively like having half as many units with activations of 1.0. So if there are 80 hidden units feeding the output layer, the randRange of the hidden-to-output projection should be around 0.45.

If you are not too interested in finely tuning the initial random weights, here is a simple rule of thumb. Use a value of 1.0 for networks with fewer than 50 units, 0.2 for networks with more than 200 units, and 0.5 otherwise.

### Improving Generalization

When given a finite training set, neural networks are particularly subject to the problem of over-training. Once the network seems able to learn the training set adequately, you will need to start worrying about improving its generalization. It is usually suggested that this can be done by either reducing the size of the network by eliminating hidden units or by using weight decay to encourage small weights. Actually, these techniques are conflicting and are only appropriate for different types of networks. This point is rarely understood, which can lead to confusion.

One class of network is used to solve logical, rule based problems. These include tasks such as solving boolean formulas, decision-tree based classification, or learning finite state machines. I will call these class A networks. Essentially, whenever the optimal behavior can be explained by a few simple rules, you have a class A task. Ideally, the network will learn just those rules necessary to solve the task.

In this case, you can usually encourage this by giving the network just enough hidden units to solve the task, but allow the weights to grow as large as they please. Such a task might require that the activation of one unit completely override the behavior of the rest of the network, which can only be done with large weights. You may also want to encourage the hidden units to use binary (close to 0 or 1) activations. This is discussed further below.

Class B tasks, on the other hand, cannot be characterized by a short rule. These are fuzzy problems that involve basing decisions on a range of accumulated weak evidence. They are the sort of natural task that people and networks are pretty good at, but symbolic machine learning methods are not. The handwritten digits problem is a good example of this.

With a class B task, you will probably obtain better generalization by using more hidden units but by trying to keep the weights small with weight decay. This will discourage the network from solving the training examples by activating just a few hidden units for each example that drive the outputs to their correct value. If the network did that, it would probably result in poor generalization. You may also want to introduce noise in either the input unit activations or in all unit activations. This prevents the network from memorizing irrelevant details of particular training examples.

When using weight decay, remember that a little goes a long way. A typical value is about 0.0001. You will need to experiment with weight decay to find the value that leads to the best validation-set performance. Of course, if you do that you should have another testing set to assess the true generalization performance. I'm still not certain if it is better to use weight decay right from the start or to introduce it later in training. You may want to use it right from the start just because it is simpler to do so.

In practice, most real-world problems probably have a mixture of class A and B properties. Language-related tasks are generally of this form. In this case, it isn't clear how you will obtain the best generalization. But hopefully you now have a better idea of how to go about experimenting.

### Encouraging Binary Representations

There are several reasons why you might want to encourage hidden units to use binary (close to 0 or 1) representations. If you have a class A task, binary representations can lead to greater generalization and can make the operation of the network easier to understand. You may be using the hidden representation as targets for another network, as in a recursive auto-associative machine (RAAM), in which case having binary targets is often easier. Also, binary activations make the units more resistant to noise.

There are several ways to encourage binary representations. The most direct is to use a unit output cost function, like LOGISTIC_COST or COSINE_COST. These add a term to the error function that penalizes non-binary units. That is certainly one option, but is not always the best one. If applied too early or too strongly, this can interfere with the ability of the network to develop good representations by preventing it from gradually shifting hidden units from an off to an on state during training.

Increasing the gain of the units is often a better option. A higher gain results in a sharper sigmoid. Thus, a smaller input can drive the unit's activation farther from 0.5. Gain is often more effective than a cost function. Again, you may want to increase the gain only later in training. Remember that a small change in gain can have a big effect on the network. Late in training, the gain should only be changed in small increments and the network allowed to re-train between changes.

Finally, an indirect way to encourage binary representations is by using noise. If noise is added to a unit's input, it will be hard for the unit to maintain a consistent activation in its sensitive middle range. The network will not be able to find good solutions that involve non-binary units, because those units are more sensitive to the noise. Thus, the network will tend to prefer more stable, binary patterns.

You can measure the degree to which the units in a group are adopting binary values using the polarity command.

Last modified: Tue Nov 28.31:14:00 EST 2000