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Introduction to Training and Learning in Neural Networks

Introduction to Training and Learning in Neural Networks. CS/PY 399 Lab Presentation # 4 February 1, 2001 Mount Union College. More Realistic Models. So far, our perceptron activation function is quite simplistic f (x 1 , x 2 ) = 1, if  x k ·w k >  , or

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Introduction to Training and Learning in Neural Networks

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  1. Introduction to Training and Learning in Neural Networks • CS/PY 399 Lab Presentation # 4 • February 1, 2001 • Mount Union College

  2. More Realistic Models • So far, our perceptron activation function is quite simplistic • f (x1 , x2 ) = 1, if  xk·wk> , or • = 0, if  xk·wk <  • To more closely mimic actual neuronal function, our model needs to become more complex

  3. Problem # 1: Need more than 2 input connections • addressed last time: activation function becomes f (x1 , x2 , x3 , ..., xn ) • vector and summation notation help with writing and describing the calculation being performed

  4. Problem # 2: Output too Simplistic • Perceptron output only changes when an input, weight or theta changes • Neurons don’t send a steady signal (a 1 output) until input stimulus changes, and keep the signal flowing constantly • Action potential is generated quickly when threshold is reached, and then charge dissipates rapidly

  5. Problem # 2: Output too Simplistic • when a stimulus is present for a long time, the neuron fires again and again at a rapid rate • when little or no stimulus is present, few if any signals are sent • over a fixed amount of time, neuronal activity is more of a firing frequency than a 1 or 0 value (a lot of firing or a little)

  6. Problem # 2: Output too Simplistic • to model this, we allow our artificial neurons to produce a graded activity level as output (some real number) • doesn’t affect the validity of the model (we could construct an equivalent network of 0/1 perceptrons) • advantage of this approach: same results with smaller network

  7. Output Graph for 0/1 Perceptron 1 output 0 θ Σ xk · wk

  8. LIMIT function: More Realism • Define a function with absolute minimum and maximum output values (say 0 and 1) • Establish two thresholds: lower and upper • f (x1 , x2 , ..., xn ) = 1, if  xk·wk>upper, • = 0, if  xk·wk<lower, or • some linear function between 0 and 1, otherwise

  9. Output Graph for LIMIT function 1 output 0 θlower θupper Σ xk · wk

  10. Sigmoid Ftns: Most Realistic • Actual neuronal activity patterns (observed by experiment) give rise to non-linear behavior between max & min • example: logistic function • f (x1 , x2 , ..., xn ) = 1 / (1 + e-  xk·wk), where e  2.71828... • example: arctangent function • f (x1 , x2 , ..., xn ) = arctan( xk·wk) / ( / 2)

  11. Output Graph for Sigmoid ftn 1 output 0 0 Σ xk · wk

  12. TLearn Activation Function • The software simulator we will use in this course is called TLearn • Each artificial neuron (node) in our networks will use the logistic function as its activation function • gives realistic network performance over a wide range of possible inputs

  13. TLearn Activation Function • Table, p. 9 (Plunkett & Elman) InputActivationInputActivation -2.00 0.119 0.50 0.622 -1.50 0.182 1.00 0.731 -1.00 0.269 1.50 0.818 -0.50 0.378 2.00 0.881 0.00 0.500

  14. TLearn Activation Function • output will almost never be exactly 0 or exactly 1 • reason: logistic function approaches, but never quite reaches these maximum and minimum values, for any input from -  to  • limited precision of computer memory will enable us to reach 0 and 1 sometimes

  15. Automatic Training in Networks • We’ve seen: manually adjusting weights to obtain desired outputs is difficult • What do biological systems do? • if output is unacceptable (wrong), some adjustment is made in the system • how do we know it is wrong? Feedback • pain, bad taste, discordant sound, observing that desired results were not obtained, etc.

  16. Learning via Feedback • Weights (connection strengths) are modified so that next time the same input is encountered, better results may be obtained • How much adjustment should be made? • different approaches yield various results • goal: automatic (simple) rule that is applied during weight adjustment phase

  17. Rosenblatt’s Training Algorithm • Developed for Perceptrons (1958) • illustrative of other training rules; simple • Consider a single perceptron, with 0/1 output • We will work with a training set • a set of inputs for which we know the correct output • weights will be adjusted based on correctness of obtained output

  18. Rosenblatt’s Training Algorithm • for each input pattern in the training set, do the following: • obtain output from perceptron • if output is correct: (strengthen) • if output is 1, set w = w + x • if output is 0, set w = w - x • but if output is incorrect: (weaken) • if output is 1, set w = w - x • if output is 0, set w = w + x

  19. Example of Rosenblatt’s Training Algorithm • Training data: x1x2out 0 1 1 1 1 1 1 0 0 • Pick random values as starting weights and θ: w1 = 0.5, w2 = -0.4, θ = 0.0

  20. Example of Rosenblatt’s Training Algorithm • Step 1: run first training case through a perceptron x1x2out 0 1 1 • (0, 1) should give answer 1 (from table), but perceptron produces 0 • do we strengthen or weaken? • do we add or subtract? • based on answer produced by perceptron!

  21. Example of Rosenblatt’s Training Algorithm • obtained answer is wrong, and is 0: we must ADD input vector to weight vector • new weight vector: (0.5, 0.6) • w1 = 0.5 + 0 = 0.5 • w2 = -0.4 + 1 = 0.6 • Adjust weights in perceptron now, and try next entry in training data set

  22. Example of Rosenblatt’s Training Algorithm • Step 2: run second training case through a perceptron x1x2out 1 1 1 • (1, 1) should give answer 1 (from table), and it does! • do we strengthen or weaken? • do we + or -?

  23. Example of Rosenblatt’s Training Algorithm • obtained answer is correct, and is 1: we must ADD input vector to weight vector • new weight vector: (1.5, 1.6) • w1 = 0.5 + 1 = 1.5 • w2 = 0.6 + 1 = 1.6 • Adjust weights, then on to training case # 3

  24. Example of Rosenblatt’s Training Algorithm • Step 3: run last training case through the perceptron x1x2out 1 0 0 • (1, 0) should give answer 0 (from table); does it? • do we strengthen or weaken? • do we + or -?

  25. Example of Rosenblatt’s Training Algorithm • determine what to do, and calculate a new weight vector • should have SUBTRACTED • new weight vector: (0.5, 1.6) • w1 = 1.5 - 1 = 0.5 • w2 = 1.6 - 0 = 1.6 • Adjust weights, then try all three training cases again

  26. Ending Training • This training process continues until: • perceptron gives correct answers for all training cases, or • a maximum number of training passes has been carried out • some training sets may be impossible for a perceptron to calculate (e.g., XOR ftn.) • In actual practice, we train until the error is less than an acceptable level

  27. Introduction to Training and Learning in Neural Networks • CS/PY 399 Lab Presentation # 4 • February 1, 2001 • Mount Union College

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