Neural Networks, the Top Quark, and Brain Computation
Denise Sawicki
A neural network is a computational device which, in certain ways, functions similarly to the human brain. In a computerized neural network, a mathematical function is modeled as a set of connected nodes. Each input node is connected to a certain number of internal nodes, these connections having characteristic weights that provide a constant by which the input is multiplied. The internal nodes perform certain mathematical manipulations upon the sum of the received inputs, thus producing an output, which in turn goes to other nodes as input. There can be more than one level of internal nodes, connected similarly, and eventually the last level of internal nodes are connected to an output node or nodes. The neural network is given flexibility by the fact that its weights can be adjusted to optimize the function to fit a set of data. In the brain, neurons serve as nodes. Neurons are known to fire based upon the sum of their inputs, and information is stored by changing the strength of connections between neurons, which is the brain's analog of changing the weights of connections. In the brain, neural network algorithms may be at work without our conscious knowledge. This would be a good way of explaining the human learning process. Once we have solved a certain kind of math problem a number of times, for instance, our brain may have developed a neural net function that helps us to solve similar problems more efficiently in the future.
Neural networks have been found very useful in a variety of practical applications. When we examine some of these applications, it becomes clear that the hardware in the brain is more than sufficient for performing those operations that we would normally classify as "computations". In fact, the massive computational power of neural networks raises the question of why the brain is so inadequate at performing complex mathematical computations. In this paper, I will describe neural networks as they function in a computer in a certain practical application, and I will attempt to explain the reasons why the human brain cannot achieve similar tasks through the use of its neurons.
One practical field in which neural networks have been used with great success is particle physics. An experiment in particle physics involves beams of particles being smashed into each other at extremely high velocities. This sometimes results in a reaction in which new particles go flying outwards, to be detected in the form of energy deposits in calorimeters surrounding the collision point. The problem with analyzing data from such experiments is that one cannot interpret the observed energy deposits such as to state unambiguously the nature of the reaction that occurred. The recent discovery of the top quark, which took place at Fermilab, was the culmination of much analysis, including extensive use of neural networks. Theorists had already made models to predict the patterns of energy deposits that would indicate the presence of a top quark. These quarks are created in a top-antitop pair, whose most likely decay mode is into six other quarks, which are perceived in the detector as jets or sprays of particles. The term "signal" refers to the events for which one is searching, in this case, events involving top-antitop production and decay, while "background" refers to events which may appear to be signal events but which are not. The major background for an event in which two tops decay into six jets is simply the case in which six jets are produced from some other source. Neural networks proved to be an excellent tool for distinguishing between signal and background events in this experiment.
A neural network helps to classify events as signal and background by providing a function that takes an input of a number of variables dealing with the energy deposits characterizing an event, and gives an output estimating the probability that the event is signal as opposed to background. To compute this function, the network undergoes a "training" period, in which one feeds in a sample of background events and a sample of signal events. Using the training samples, the neural network adjusts the weights of its various connections, trying to find the weights that correspond to the best fit between the function and the data. The data consists of ones for signal events and zeros for background events, and the neural net output consists of numbers between zero and one depicting the likelihood that an event with the given parameters is a signal event. With each iteration, the neural net weights are modified such as to cause the function to better model the data; this is done by moving through parameter space in the direction opposite the gradient of the mean squared difference between the function and the data. If one allows the neural network training algorithm to run for an extremely long time, there can be a danger of over-training. This means that instead of reporting general trends, the network focuses in on incidental aberrations in the training sample that do not have physical significance and will not be helpful when one goes to test the neural net function. For this reason, it is a good idea to use different samples for training as opposed to testing. This way, if the neural net function performs well on both samples, one knows that it is not over-trained.
Once an optimal neural net function has been found, the experimenter must then make a decision as to what number to cut on. Cutting on a value of 0.95, for instance, means that all events that the neural network estimates to be at least 95% likely to be signal will be considered signal events. In Bayesian notation, then if, one denotes an event being signal as S and event passing the cut as C, P(S|C)>=0.95. Choosing a neural net cut automatically selects the signal efficiency (The probability that an event passes the cut if it is signal, P(C|S)) and the background efficiency (The probability that the event passes the cut if it is background, P(C|~S)). Ideally, the background efficiency should be very low, so any events passing the cut are most likely to be signal. Making the background efficiency low enough generally requires that the signal efficiency be quite low as well. The higher the signal efficiency, the more illuminating the results of the experiment can be.
An example of a variable that is good to distinguish between signal and background for top decay processes deals with the invariant masses of groups of jets. In a top-antitop decay process, the six jets should be divisible into two sets of three, each of which have the same combined mass, this being the mass of a top quark. In analysis of the data at Fermilab, various groups chose between four and thirteen such variables to distinguish between signal and background. Before applying neural network cuts, there were at least 1000 times as many background events as signal events in the data sample. Using neural net cuts involving the chosen variables, however, it was possible to determine with near certainty that top quarks were in fact being produced.
The crux of the matter is as follows: even in this extremely complicated process of sorting out interesting events from a huge background of uninteresting events that are very similar, the total number of network nodes required was always considerably less than 100. Network nodes are computational models for neurons, and there are 1011 neurons in the human brain: the number of nodes required for this computation is thus miniscule compared to the number of neurons in the brain. Why, then, do we even need a computer to perform these calculations? It might seem that the brain should be able to perform this computation easily, as well as solve problems almost inconceivably more complex. Yet it was shown that any criteria that a human could come up with by inspection did not provide nearly as good results for distinguishing between signal and background as did the criteria found from the neural net algorithm.
One major reason that a computerized neural network performs better than the brain at certain tasks is speed. Due to physical limitations, the brain cannot process information nearly as quickly as a computer. Thus, a person could not concentrate long enough to perform a task comparable to the search for the top quark. Furthermore, the human brain does not have a programmer in any easily definable sense. It programs itself in response to input from the person's senses. So, though the hardware may be more than sufficient to make a neural network that could eventually solve this kind of problem, a person cannot simply tell his brain to work this way. In this particular case, the solution to the problem is not the kind of thing that a person could come up with based on intuition from the outside world. When there is a large number of input variables, the task becomes very difficult to visualize. Nothing in the physical world prepares a person for the task of creating a twelve-dimensional boundary that divides a thirteen-dimensional space into regions that are characterized by different likelihoods of events being signal and background. Our brains have learned from birth how to help us perform tasks in a three-dimensional environment, so anything more is a struggle to learn.
It is true in general, and not just in the particle physics example, that the brain is not very good at performing the kind of tasks that neural networks are used for in computing. Nonetheless, neural networks do conform very well to what we know of the functioning of neurons. Putting a neural network into a computer allows the computer to make seemingly intelligent judgments, but quickly and without the human errors that could be induced into a person's judgements by factors in the person's environment. In certain projects, computerized neural networks can function in ways that are a huge improvement upon the brain's own algorithms. It is easy to visualize how the huge number of neurons in the brain could provide the computational power that a person might need, but it is harder to explain why the brain is not even better at forming complex mathematical judgments. This paper has attempted to explain the kind of task that a computerized neural network excels at, and to give reasons why the brain is not better at such tasks.