Chapter ML:VI (continued)

Transcription

1 Chapter ML:VI (continued) VI Neural Networks Perceptron Learning Gradient Descent Multilayer Perceptron Radial asis Functions ML:VI-64 Neural Networks STEIN

2 Definition 1 (Linear Separability) Two sets of feature vectors, X 0, X 1, of a p-dimensional feature space are called linearly separable, if p + 1 real numbers, w 0, w 1,, w p, exist such that holds: 1 x X 0 : p j0 w jx j < 0 2 x X 1 : p j0 w jx j 0 ML:VI-65 Neural Networks STEIN

3 Definition 1 (Linear Separability) Two sets of feature vectors, X 0, X 1, of a p-dimensional feature space are called linearly separable, if p + 1 real numbers, w 0, w 1,, w p, exist such that holds: 1 x X 0 : p j0 w jx j < 0 2 x X 1 : p j0 w jx j 0 x 2 x 1 x 2 x 1 linearly separable not linearly separable ML:VI-66 Neural Networks STEIN

4 Separability The XOR function defines the smallest example for two not linearly separable sets: x 2 1 x 1 x 2 XOR Class x 2 0 x 1 0 x 1 1 ML:VI-67 Neural Networks STEIN

5 Separability (continued) The XOR function defines the smallest example for two not linearly separable sets: x 2 1 x 1 x 2 XOR Class x 2 0 x 1 0 x 1 1 specification of several hyperplanes combination of several perceptrons ML:VI-68 Neural Networks STEIN

6 Separability (continued) Layered combination of several perceptrons: the multilayer perceptron Minimum multilayer perceptron that is able to handle the XOR problem: x 0 1 x {, } x 2 0 ML:VI-69 Neural Networks STEIN

7 Remarks: The multilayer perceptron was presented by Rumelhart and McClelland in 1986 Earlier, but unnoticed, was a similar research work of Werbos and Parker [1974, 1982] Compared to a single perceptron the multilayer perceptron poses a significantly more challenging training ( learning) problem, which requires continuous (and non-linear) threshold functions along wtih sophisticated learning strategies Marvin Minsky and Seymour Papert showed 1969 with the XOR problem the limitations of single perceptrons Moreover, they assumed that extensions of the perceptron architecture (such as the multilayer perceptron) would be similarly limited as a single perceptron fatal mistake In fact, they brought the research in this field to a halt that lasted 17 years [erkeley] [Marvin Minsky: MIT Media Lab, Wikipedia] ML:VI-70 Neural Networks STEIN

8 Computation in the Network [ Heaviside] perceptron with a continuous and non-linear threshold function: x 0 1 w 0 θ w 0 x 1 w 1 x 2 w 2 w p 0θ y x p Inputs Output The sigmoid function σ(z) as threshold function: σ(z) 1 where dσ(z) 1 + e z dz σ(z) (1 σ(z)) ML:VI-71 Neural Networks STEIN

9 Computation in the Network (continued) Computation of the perceptron output y(x) via the sigmoid function σ: y(x) σ(w T x) e wt x 1 0 w j x j j0 p n alternative to the sigmoid function is the tanh function: 1 tanh(x) ex e x e x + e e2x 1 x e 2x w j x j j0 p ML:VI-72 Neural Networks STEIN

10 Computation in the Network (continued) Distinguish units (nodes, perceptrons) of type input, hidden, and output: U I x 1 y x p U I, U H, U O w jk, w jk x j k y k, δ k w k x y H, y O Sets with units of type input, hidden, and output Weight and weight adaptation for the edge connecting the units j and k Input value (single incoming edge) for unit k, provided at the output of unit j Output value and classification error of unit k Weight vector (all incoming edges) of unit k Input vector for a unit of the hidden layer Output vector of the hidden layer and the output layer respectively ML:VI-73 Neural Networks STEIN

11 Computation in the Network (continued) Distinguish units (nodes, perceptrons) of type input, hidden, and output: U I U H x 1 x p y U I, U H, U O w jk, w jk x j k y k, δ k w k x y H, y O Sets with units of type input, hidden, and output Weight and weight adaptation for the edge connecting the units j and k Input value (single incoming edge) for unit k, provided at the output of unit j Output value and classification error of unit k Weight vector (all incoming edges) of unit k Input vector for a unit of the hidden layer Output vector of the hidden layer and the output layer respectively ML:VI-74 Neural Networks STEIN

12 Computation in the Network (continued) Distinguish units (nodes, perceptrons) of type input, hidden, and output: U I U H U O y 1 x 1 x p y 2 y k U I, U H, U O w jk, w jk x j k y k, δ k w k x y H, y O Sets with units of type input, hidden, and output Weight and weight adaptation for the edge connecting the units j and k Input value (single incoming edge) for unit k, provided at the output of unit j Output value and classification error of unit k Weight vector (all incoming edges) of unit k Input vector for a unit of the hidden layer Output vector of the hidden layer and the output layer respectively ML:VI-75 Neural Networks STEIN

13 Computation in the Network (continued) Distinguish units (nodes, perceptrons) of type input, hidden, and output: U I U H U O x 0 1 y H0 1 y 1 x 1 x p y 2 y k U I, U H, U O w jk, w jk x j k y k, δ k w k x y H, y O Sets with units of type input, hidden, and output Weight and weight adaptation for the edge connecting the units j and k Input value (single incoming edge) for unit k, provided at the output of unit j Output value and classification error of unit k Weight vector (all incoming edges) of unit k Input vector for a unit of the hidden layer Output vector of the hidden layer and the output layer respectively ML:VI-76 Neural Networks STEIN

14 Computation in the Network (continued) Distinguish units (nodes, perceptrons) of type input, hidden, and output: U I U H U O x 0 1 y H0 1 y 1 x 1 x p y 2 y k U I, U H, U O w jk, w jk x j k y k, δ k w k x y H, y O Sets with units of type input, hidden, and output Weight and weight adaptation for the edge connecting the units j and k Input value (single incoming edge) for unit k, provided at the output of unit j Output value and classification error of unit k Weight vector (all incoming edges) of unit k Input vector for a unit of the hidden layer Output vector of the hidden layer and the output layer respectively ML:VI-77 Neural Networks STEIN

15 Remarks: The units of the input layer, U I, perform no computations at all They distribute the input values to the next layer The network topology corresponds to a complete, bipartite graph between the units in U I and U H as well as between the units in U H and U O The non-linear characteristic of the sigmoid function allows for networks that approximate every (computable) function For this capability only three active layers are required, ie, two layers with hidden units and one layer with output units Keyword: universal approximator [Kolmogorov Theorem, 1957] Multilayer perceptrons are also called multilayer networks or (artificial) neural networks, NN for short ML:VI-78 Neural Networks STEIN

16 Classification Error The classification error Err(w) is computed as sum over the U O k network outputs: Err(w) 1 (c v (x) y v (x)) 2 2 v U O (x,c(x)) D Due its complex form, Err(w) may contain various local minima: Err(w) ML:VI-79 Neural Networks STEIN

17 Weight daptation: Incremental Gradient Descent [network] lgorithm: MPT Multilayer Perceptron Training Input: D Training examples (x, c(x)) with x p + 1, c(x) {0, 1} k (c(x) { 1, 1} k ) η Learning rate, a small positive constant Output: w Weights of the units in U I, U H, U O 1 initialize_random_weights(u I, U H, U O ), t 0 2 REPET 3 t t FORECH (x, c(x)) D DO 5 FORECH u U H DO y u σ(wu T x) // compute output of layer 1 6 FORECH v U O DO y v σ(wv T y H ) // compute output of layer 2 7 FORECH v U O DO δ v y v (1 y v ) (c v (x) y v ) // backpropagate layer 2 8 FORECH u U H DO δ u y u (1 y u ) w uv δ v // backpropagate layer 1 v U o 9 FORECH w jk, (j, k) (U I U H ) (U H U O ) DO 10 w jk η δ k x j k 11 w jk w jk + w jk 12 ENDDO 13 ENDDO 14 UNTIL(convergence(D, y O (D)) OR t > t max ) 15 return(w) ML:VI-80 Neural Networks STEIN

18 Weight daptation: Incremental Gradient Descent [network] lgorithm: MPT Multilayer Perceptron Training Input: D Training examples (x, c(x)) with x p + 1, c(x) {0, 1} k (c(x) { 1, 1} k ) η Learning rate, a small positive constant Output: w Weights of the units in U I, U H, U O 1 initialize_random_weights(u I, U H, U O ), t 0 2 REPET 3 t t FORECH (x, c(x)) D DO 5 FORECH u U H DO y u σ(wu T x) // compute output of layer 1 6 FORECH v U O DO y v σ(wv T y H ) // compute output of layer 2 7 FORECH v U O DO δ v y v (1 y v ) (c v (x) y v ) // backpropagate layer 2 8 FORECH u U H DO δ u y u (1 y u ) w uv δ v // backpropagate layer 1 v U o 9 FORECH w jk, (j, k) (U I U H ) (U H U O ) DO 10 w jk η δ k x j k 11 w jk w jk + w jk 12 ENDDO 13 ENDDO 14 UNTIL(convergence(D, y O (D)) OR t > t max ) 15 return(w) ML:VI-81 Neural Networks STEIN

19 Weight daptation: Incremental Gradient Descent [network] lgorithm: MPT Multilayer Perceptron Training Input: D Training examples (x, c(x)) with x p + 1, c(x) {0, 1} k (c(x) { 1, 1} k ) η Learning rate, a small positive constant Output: w Weights of the units in U I, U H, U O 1 initialize_random_weights(u I, U H, U O ), t 0 2 REPET 3 t t FORECH (x, c(x)) D DO 5 FORECH u U H DO y u σ(wu T x) // compute output of layer 1 6 FORECH v U O DO y v σ(wv T y H ) // compute output of layer 2 7 FORECH v U O DO δ v y v (1 y v ) (c v (x) y v ) // backpropagate layer 2 8 FORECH u U H DO δ u y u (1 y u ) w uv δ v // backpropagate layer 1 v U o 9 FORECH w jk, (j, k) (U I U H ) (U H U O ) DO 10 w jk η δ k x j k 11 w jk w jk + w jk 12 ENDDO 13 ENDDO 14 UNTIL(convergence(D, y O (D)) OR t > t max ) 15 return(w) ML:VI-82 Neural Networks STEIN

20 Remarks: The generic delta rule (Lines 7 and 8 of the MPT algorithm) allows for a backpropagation of the classification error and hence the training of multi-layered networks Gradient descent is based on the classification error of the entire network and hence considers the entire network weight vector ML:VI-83 Neural Networks STEIN

21 Weight daptation: Momentum Term Momentum idea: a weight adaptation in iteration t considers the adaptation in iteration t 1 : w uv (t) η δ v x u v + α w uv (t 1) The term α, 0 α < 1, is called momentum ML:VI-84 Neural Networks STEIN

22 Weight daptation: Momentum Term Momentum idea: a weight adaptation in iteration t considers the adaptation in iteration t 1 : w uv (t) η δ v x u v + α w uv (t 1) The term α, 0 α < 1, is called momentum Effects: due the adaptation inertia local minima can be overcome if the direction of the descent does not change, the adaptation increment and, as a consequence, the speed of convergence is increased ML:VI-85 Neural Networks STEIN