Learning Neural Networks

Transcription

1 Learning Neural Networks Neural Networks can represent complex decision boundaries Variable size. Any boolean function can be represented. Hidden units can be interpreted as new features Deterministic Continuous Parameters Learning Algorithms for neural networks Local Search. The same algorithm as for sigmoid threshold units Eager Batch or Online

2 Neural Network Hypothesis Space ŷ W 9 a 6 a 7 a 8 W 6 W 7 W 8 x 1 x 2 x 3 x 4 Each unit a 6, a 7, a 8, and ŷ computes a sigmoid function of its inputs: a 6 = σ(w 6 X) a 7 = σ(w 7 X) a 8 = σ(w 8 X) ŷ = σ(w 9 A) where A = [1, a 6, a 7, a 8 ] is called the vector of hidden unit activitations Original motivation: Differentiable approximation to multi-layer layer LTUs

3 Representational Power Any Boolean Formula Consider a formula in disjunctive rmal form: (x 1 x 2 ) (x 2 x 4 ) ( x 3 x 5 ) Each AND can be represented by a hidden unit and the OR can be represented by the output unit. Arbitrary boolean functions require exponentially-many hidden units, however. Bounded functions Suppose we make the output linear: ŷ = W 9 A of hidden units. It can be proved that any bounded continuous function can be approximated to arbitrary accuracy with eugh hidden units. Arbitrary Functions Any function can be approximated to arbitrary accuracy with two hidden layers of sigmoid units and a linear output unit.

4 Fixed versus Variable Size In principle, a network has a fixed number of parameters and therefore can only represent a fixed hypothesis space (if the number of hidden units is fixed). However, we will initialize the weights to values near zero and use gradient descent. The more steps of gradient descent we take, the more functions can be reached from the starting weights. So it turns out to be more accurate to treat networks as having a variable hypothesis space that depends on the number of steps of gradient descent

5 Backpropagation: Gradient Descent for Multi-Layer Networks It is traditional to train neural networks to minimize the squared error. This is really a mistake they they should be trained to maximize the log likelihood instead. But we will study the MSE first. ŷ = σ(w 9 [1, σ(w 6 X), σ(w 7 X), σ(w 9 X)]) J i (W )= 1 2 (ŷ i y i ) 2 We must apply the chain rule many times to compute the gradient We will number the units from 0 to U and index them by u and v. w v,u will be the weight connecting unit u to unit v. (Note: This seems backwards. It is the uth input to de v.)

6 Derivation: Output Unit Suppose w 9,6 is a component of W 9, the output weight vector, connecting it from a 6. J i (W) w 9,6 = w 9,6 1 2 (ŷ i y i ) 2 = (ŷ i y i ) w 9,6 (σ(w 9 A i ) y i ) = (ŷ i y i ) σ(w 9 A i )(1 σ(w 9 A i )) = (ŷ i y i )ŷ i (1 ŷ i ) a 6 w 9,6 W 9 A i

7 The Delta Rule Define then δ 9 =(ŷ i y i )ŷ i (1 ŷ i ) J i (W) w 9,6 = (ŷ i y i )ŷ i (1 ŷ i ) a 6 = δ 9 a 6

8 Derivation: Hidden Units J i (W) w 6,2 = (ŷ i y i ) σ(w 9 A i )(1 σ(w 9 A i )) = δ 9 w 9,6 w 6,2 σ(w 6 X) = δ 9 w 9,6 σ(w 6 X)(1 σ(w 6 X)) = δ 9 w 9,6 a 6 (1 a 6 ) x 2 w 6,2 W 9 A i w 6,2 (W 6 X) Define δ 6 = δ 9 w 9,6 a 6 (1 a 6 ) and rewrite as J i (W ) w 6,2 = δ 6 x 2.

9 Networks with Multiple Output Units ŷ 1 ŷ 2 1 a 6 a 7 a 8 W 6 W 7 W 8 1 We get a separate contribution to the gradient from each output unit. Hence, for input-to to-hidden weights, we must sum up the contributions: δ 6 = a 6 (1 a 6 ) x 1 x 2 x 3 x 4 10 X w u,6 δu u=9

10 The Backpropagation Algorithm Forward Pass.. Compute a u and ŷ v for hidden units u and output units v. Compute Errors.. Compute ε v = (ŷ( v y v ) for each output unit v Compute Output Deltas.. Compute δ u = a u (1 a u ) v w v,u Compute Gradient. J i = δux ij w u,j Compute for input-to to-hidden weights. J i Compute = δ v a iu for hidden-to to-output output weights. wv,u Take Gradient Step. W := W η W J(x i ) v,u δ v

11 Proper Initialization Start in the linear regions keep all weights near zero, so that all sigmoid units are in their linear regions. This makes the whole net the equivalent of one linear threshold unit a a very simple function. Break symmetry. Ensure that each hidden unit has different input weights so that the hidden units move in different directions. Set each weight to a random number in the range [ 1, +1] 1 fan-in. where the fan-in of weight w v,u is the number of inputs to unit v.

12 Batch, Online, and Online with Momentum Batch.. Sum the W J(x i ) for each example i. Then take a gradient descent step. Online.. Take a gradient descent step with each W J(x i ) as it is computed. Momentum.. Maintain an exponentially-weighted moved sum of recent W (t+1) := µ W (t) + W J(x i ) W (t+1) := W (t) η W (t+1) Typical values of µ are in the range [0.7, 0.95]

13 Softmax Output Layer Let a 9 and a 10 be the output activations: a 9 = W 9 A, a 10 = W 10 A. Then define ŷ 1 = exp a 9 exp a 9 + exp a 10 ŷ 2 = The objective function is the negative log likelihood: J(W )= X i where I[expr] is 1 if expr is true and 0 otherwise X k ŷ 1 ŷ 2 softmax exp a 10 exp a 9 + exp a 10 I[y i = k]logŷ k W 9 W 10 1 a 6 a 7 a 8 W 6 W 7 W 8 1 x 1 x 2 x 3 x 4

14 Neural Network Evaluation Criterion Perc Logistic LDA Trees Nets Mixed data Missing values Outliers Motone transformations somewhat Scalability Irrelevant inputs somewhat Linear combinations Interpretable Accurate