Supervised Learning in Neural Networks

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1 The Norwegian University of Science and Technology (NTNU Trondheim, Norway March 7, 2011

2 Supervised Learning Constant feedback from an instructor, indicating not only right/wrong, but also the correct answer for each training case. Many cases (i.e., input-output pairs to be learned. Weights are modified by a complex procedure (back-propagation based on output error. Feed-forward networks with back-propagation learning are the standard implementation. 99% of neural network applications use this. Typical usage: problems with a lots of input-output training data, and b goal of a mapping (function from inputs to outputs. Not biologically plausible, although the cerebellum appears to exhibit some aspects. But, the result of backprop, a trained ANN to perform some function, can be very useful to neuroscientists as a sufficiency proof.

3 Backpropagation Overview Training/Test Cases: {(d1, r1 (d2, r2 (d3, r3...} d3 r3 Encoder Decoder r* E = r3 - r* de/dw Feed-Forward Phase - Inputs sent through the ANN to compute outputs. Feedback Phase - passed back from output to input layers and used to update weights along the way.

4 Training -vs- Testing Cases Training N times, with learning Neural Net Test 1 time, without learning Generalization - correctly handling test cases (that ANN has not been trained on. Over-Training - weights become so fine-tuned to the training cases that generalization suffers: failure on many test cases.

5 Widrow-Hoff (a.k.a. Delta Rule X w T = target output value δ = error 1 Y Node N 0 0 S Δw = ηδx Y δ = T - Y Delta (δ = error; Eta (η = learning rate Goal: change w so as to reduce δ. Intuitive: If δ > 0, then we want to decrease it, so we must increase Y. Thus, we must increase the sum of weighted inputs to N, and we do that by increasing (decreasing w if X is positive (negative. Similar for δ < 0 Assumes derivative of N s transfer function is everywhere non-negative.

6 Gradient Descent Goal = minimize total error across all output nodes Method = modify weights throughout the network (i.e., at all levels to follow the route of steepest descent in error space. min ΔE (E Weight Vector (W w ij = η E i

7 Computing E i 1 x 1d w i1 f T t id sum id i o id n x nd w in E id Sum of Squared s (SSE E i = 1 2 (t id o id 2 d D E = 1 2 2(t id o id (t id o id = w d D ij (t id o id ( o id w d D ij

8 Computing ( o id 1 t id x 1d w i1 f T sum id i o id n x nd w in E id Since output = f(sum weighted inputs where E = (t id o id ( f T (sum id w d D ij n sum id = w ik x kd k=1 Using Chain Rule: f (g(x = f x g(x g(x x (f T (sum id = f T (sum id sum id = f T (sum id x sum id sum jd id

9 Computing sum id - Easy!! sum id = ( n k=1 w ik x kd = ( wi1 x 1d + w i2 x 2d w ij x jd w in x nd = (w i1x 1d + (w i2x 2d (w ijx jd (w inx nd = x jd = x jd

10 Computing f T (sum id sum id - Harder for some f T f T = Identity function: f T (sum id = sum id Thus: f T (sum id sum id = 1 (f T (sum id = f T (sum id sum id sum id = 1 x jd = x jd f T = Sigmoid: f T (sum id = 1 1+e sum id Thus: (f T (sum id = f T (sum id sum id f T (sum id sum id = o id (1 o id sum id = o id (1 o id x jd = o id (1 o id x jd

11 The only non-trivial calculation f T (sum id sum id But notice that: = ( (1 + e sum id 1 = ( 1 (1 + e sumid (1 + e sum id 2 sum id sum id = ( 1( 1e sum id (1 + e sum id 2 = e sum id (1 + e sum id 2 e sum id (1 + e sum id 2 = f T (sum id (1 f T (sum id = o id (1 o id

12 Putting it all together E i = (t id o id ( f ( T (sum id = w d D ij d D (t id o id f T (sum id sum id sum id So for f T = Identity: E i = (t id o id x jd d D and for f T = Sigmoid: E i = (t id o id o id (1 o id x jd d D

13 Weight Updates (f T = Sigmoid Batch: update weights after each training epoch w ij = η E i = η (t id o id o id (1 o id x jd d D The weight changes are actually computed after each training case, but w ij is not updated until the epoch s end. Incremental: update weights after each training case w ij = η E i = η(t id o id o id (1 o id x jd A lower learning rate (η recommended here than for batch method. Can be dependent upon case-presentation order. So randomly sort the cases after each epoch.

14 Backpropagation in Multi-Layered Neural Networks d(e d d(sum 1d 1 d(sum 1d d(o jd d(o jd d(sum jd w 1j sum jd j o jd E d w nj d(sum nd d(o jd n d(e d d(sum nd For each node (j and each training case (d, backpropagation computes an error term: δ jd = E d sum jd by calculating the influence of sum jd along each connection from node j to the next downstream layer.

15 Computing δ jd d(e d d(sum 1d 1 d(sum 1d d(o jd d(o jd d(sum jd w 1j sum jd j o jd E d w nj d(sum nd d(o jd n d(e d d(sum nd Along the upper path, the contribution to So summing along all paths: E d sum jd is: o jd sum jd sum 1d o jd E d sum 1d E d sum jd = o jd sum jd n sum kd o k=1 jd E d sum kd

16 Computing δ jd Just as before, most terms are 0 in the derivative of the sum, so: sum kd o jd = w kj Assuming f T = a sigmoid: Thus: o jd = f T (sum jd = o sum jd sum jd (1 o jd jd δ jd = E d = o n jd sum sum jd sum jd kd E d o k=1 jd sum kd n n = o jd (1 o jd w kj ( δ kd = o jd (1 o jd w kj δ kd k=1 k=1

17 Computing δ jd Note that δ jd is defined recursively in terms of the δ values in the next downstream layer: δ jd = o jd (1 o jd n k=1 w kj δ kd So all δ values in the network can be computed by moving backwards, one layer at a time.

18 Computing E d from δ jd - Easy!! 1 w i1 d(e d d(sum id j o jd w ij sum id i The only effect of w ij upon the error is via its effect upon sum id, which is: sum id = o jd So: E d = sum id E d = sum id ( δ sum id w id = o jd δ id ij

19 Computing w ij Given an error term, δ id (for node i on training case d, the update of w ij for all nodes j that feed into i is: w ij = η E d = η( o jd δ id = ηδ id o jd So given δ i, you can easily calculate w ij for all incoming arcs.

20 Learning XOR 1.2 Sum-Squared- 1.2 Sum-Squared- 1.2 Sum-Squared Epoch Epoch Epoch Epoch = All 4 entries of the XOR truth table. 2 (inputs X 2 (hidden X 1 (output network Random init of all weights in [-1 1]. Not linearly separable, so it takes awhile! Each run is different due to random weight init.

21 Learning to Classify Wines Class Properties Wine Properties Wine Class Hidden Layer

22 Wine Runs Sum-Squared Sum-Squared Epoch Epoch 13x5x1 (lrate = x5x1 (lrate = 0.1 Sum-Squared Sum-Squared Epoch Epoch 13 x 10 x 1 (lrate = x 25 x 1 (lrate = 0.3

23 Momentum: Combatting Local Minima w(t-1 w(t w ij (t = η E i + α w ij (t 1

24 Practical Tips 1 Only add as many hidden layers and hidden nodes as necessary. Too many more weights to learn + increased chance of over-specialization. 2 Scale all input values to the same range, typically [0 1] or [-1 1]. 3 Use target values of 0.1 (for zero and 0.9 (for 1 to avoid saturation effects of sigmoids. 4 Beware of tricky encodings of input (and decodings of output values. Don t combine too much info into a single node s activation value (even though it s fun to try, since this can make proper weights difficult (or impossible to learn. 5 For discrete (e.g. nominal values, one (input or output node per value is often most effective. E.g. car model and city of residence -vs- income and education for assessing car-insurance risk. 6 All initial weights should be relatively small: [ ] 7 Bias nodes can be helpful for complicated data sets. 8 Check that all your layer sizes, activation functions, activation ranges, weight ranges, learning rates, etc. make sense in terms of each other and your goals for the ANN. One improper choice can ruin the results.

25 Bias Nodes 1 w 1 1 bias node w w w Input Output Constant output = 1 All outgoing weights are independent and modifiable by backprop. The negative of each such weight functions like a threshold for the downstream node.

26 Supervised Learning in the Cerebellum Parallel Fibers Granular Cells Golgi Cells Mossy Fibers Climbing Fibers Purkinje Cells Inferior Olive Sensory + Cortical Inputs Somatosensory (touch, pain, body position + Cortical Inputs Efference Copy Inhibition of deep cerebellar To neurons Spinal Cord To Cerebral Cortex Granular cells detect contexts. Parallel fibers and Purkinje cells map contexts to actions Climbing fibers from Inferior Olive provide (supervisory feedback signals for LTD on Parallel-Purkinje synapses

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