Homework 2 - More ANN Fundamentals

NOTE: This assignment, like others in this class, is due at the beginning of the class period. This means that if you are even a minute late, you lose 20%. If you are worried about potentially being late, turn in your homework ahead of time. Do this by submitting them to me during office hours or by sliding it under my office door. Do not send assignments to me through email or leave them in my departmental mail box.

As discussed in class, artificial neurons (ANs) can be combined in many ways to compute more complex functions than could be computed by a single AN. The most fundamental way is by combining ANs into a layered, feedforward neural network (FFNN). Likewise, FFNNs can learn in many ways but the most fundamental way is supervised learning. Moreover, FFNNs may be used for many tasks but the two most fundamental are classification and function approximation, of which classification is the easier to visualize. These ANN fundamentals — FFNNs, supervised learning, and classification — are the topics of this homework.

Part 1 — FFNN Representation

1. Consider a two-layer FFNN — that is, one with two layers of computational elements (ANs) — used for classification in a 2D space with augmented vectors. The ANs in this FFNN are all SUs and their activation functions are identical to f_AN as given in 1.1. There are four ANs in the hidden layer and one in the output layer. Given the weights v_1,1=−0.3, v_2,1=0.1, v_3,1=0.4, v_4,1=−0.6, v_1,2=0.2, v_2,2=−0.3, v_3,2=0.9, v_4,2=0.5, v_1,3=−0.2, v_2,3=−0.9, v_3,3=0.2, v_4,3=−0.5, w₁=0.7, w₂=1.0, w₃=−0.6, and w₄=0.0, draw this FFNN.
2. Draw the decision region encoded by this FFNN. Be sure to indicate the γ₁ side of the region.
3. Add the following points on the graph you just drew and label the class of each according to the AN.
   • (0.7, 0.7)
   • (0, 0)
   • (0.1, 0.7)
   • (−0.8, −0.3)
   • (0.2, −0.1)
4. Explain how the decision region for this AN would change if γ₂ were changed to −1, rather than 0 and explain which points, if any, from those above would be classified differently and which would be classified the same.

Part 2 — FFNN Learning

1. Consider the FFNN given above in 1.1 but with sigmoidal activation functions for each AN and η = 0.5. Explain how its weights would be updated, using the backpropagation algorithm we covered in class, if presented with the data item (1.0, 0.2) γ₁. Show your work. Keep track of four significant digits.
2. Calculate the output value of the FFNN above if, after learning on (1.0, 0.2) γ₁, you were to present this data item to the FFNN again. Show your work. Keep track of four significant digits.
3. Explain whether the error value for the input (1.0, 0.2) γ₁ increased or decreased due to learning.

Turn in a neatly handwritten copy of your answers to the exercises for this assignment. The diagrams should be drawn on engineering or graph paper. You may also turn in a scanned electronic copy of this assignment as a backup in case your paper copy is misplaced.