Machine Learning CS 4900/5900. Lecture 03. Razvan C. Bunescu School of Electrical Engineering and Computer Science

Transcription

1 Machine Learning CS 4900/5900 Razvan C. Bunescu School of Electrical Engineering and Computer Science

2 Machine Learning is Optimization Parametric ML involves minimizing an objective function J(w): Also called cost function, loss function, or error function. Want to find w" = argmin w J(w) Numerical optimization procedure: 1. Start with some guess for w 0, set τ = Update w τ to w τ+1 such that J(w τ+1 ) J(w τ ). 3. Increment τ = τ Repeat from 2 until J cannot be improved anymore. 2

3 Gradient-based Optimization How to update w τ to w τ+1 such that J(w τ+1 ) J(w τ )? Move w in the direction of steepest descent: w /01 = w / + ηg g is the direction of steepest descent, i.e. direction along which J decreases the most. η is the learning rate and controls the magnitude of the change. 3

4 Gradient-based Optimization Move w in the direction of steepest descent: w /01 = w / + ηg What is the direction of steepest descent of J(w) at w τ? The gradient J(w) is in the direction of steepest ascent. Set g = J(w) => the gradient descent update: w /01 = w / η J(w / ) 4

5 Gradient Descent Algorithm Want to minimize a function J : R n R. J is differentiable and convex. compute gradient of J i.e. direction of steepest increase: J w = J J J,,, w 1 w ; w = 1. Set learning rate η = (or other small value). 2. Start with some guess for w 0, set τ = Repeat for epochs E or until J does not improve: 4. τ = τ w /01 = w / η J w / 5

6 Gradient Descent: Large Updates 6

7 Gradient Descent: Small Updates 7

8 The Learning Rate 1. Set learning rate η = (or other small value). 2. Start with some guess for w 0, set τ = Repeat for epochs E or until J does not improve: 4. τ = τ w /01 = w / η J w / How big should the learning rate be? o o If learning rate too small => slow convergence. If learning rate too big => oscillating behavior => may not even converge. 8

9 Learning Rate too Small 9

10 Learning Rate too Large 10

11 The Learning Rate How big should the learning rate be? If learning rate too big => oscillating behavior. If learning rate too small => hinders convergence. o Use line search (backtracking line search, conjugate gradient, ). o Use second order methods (Newton s method, L-BFGS,...). o o Requires computing or estimating the Hessian. Use a simple learning rate annealing schedule: Start with a relatively large value for the learning rate. Decrease the learning rate as a function of the number of epochs or as a function of the improvement in the objective. Use adaptive learning rates: Adagrad, Adadelta, RMSProp, Adam. 11

12 Gradient Descent: Nonconvex Objective Saddle point 12

13 Convex Multivariate Objective w 0 w 1 13

14 Gradient Step and Contour Lines w 0 w 1 14

15 Gradient Descent: Nonconvex Objectives 15

16 Gradient Descent & Plateaus 16

17 Gradient Descent & Saddle Points 17

18 Gradient Descent & Ravines 18

19 Gradient Descent & Ravines Ravines are areas where the surface curves much more steeply in one dimension than another. Common around local optima. GD oscillates across the slopes of the ravines, making slow progress towards the local optimum along the bottom. Use momentum to help accelerate GD in the relevant directions and dampen oscillations: Add a fraction of the past update vector to the current update vector. The momentum term increases for dimensions whose previous gradients point in the same direction. It reduces updates for dimensions whose gradients change sign. Also reduces the risk of getting stuck in local minima. 19

20 Gradient Descent & Momentum Vanilla Gradient Descent: v /01 = η J(w / ) w /01 = w / v /01 Gradient Descent w/ Momentum: v /01 = γv / + η J(w / ) w /01 = w / v /01 γ is usually set to 0.9 or similar. 20

21 Momentum & Nesterov Accelerated Gradient GD with Momentum: v /01 = γv / + η J(w / ) w /01 = w / v /01 Nesterov Accelerated Gradient: v /01 = γv / + η J(w / γv / ) w /01 = w / v /01 By making an anticipatory update, NAGs prevents GD from going too fast => significant improvements when training RNNs. 21

22 Gradient Descent Optimization Algorithms Momentum. Nesterov Accelerated Gradient (NAG). Adaptive learning rates methods: Idea is to perform larger updates for infrequent params and smaller updates for frequent params, by accumulating previous gradient values for each parameter. Adagrad. Adadelta. RMSProp. Adaptive Moment Estimation (Adam) 22

23 Visualization Adagrad, RMSprop, Adadelta, and Adam are very similar algorithms that do well in similar circumstances. Insofar, Adam might be the best overall choice. 23

24 Variants of Gradient Descent w /01 = w / η J w / Depeding on how much data is used to compute the gradient at each step: Batch gradient descent: Use all the training examples. Stochastic gradient descent (SGD). Use one training example, update after each. Minibatch gradient descent. Use a constant number of training examples (minibatch). 24

25 Batch Gradient Descent Sum-of-squares error: H J w = 1 2N C h w(x (=) ; ) t n =I1 w /01 = w / η J w / H w /01 = w / η 1 N C =I1 h w (x (=) ) t n x (=) 25

26 Stochastic Gradient Descent Sum-of-squares error: H J w = 1 2N C h w(x (=) ; ) t n =I1 H = 1 2N C J w/, x (=) =I1 w /01 = w / η J w /, x (=) w /01 = w / η h w (x (=) ) t n x (=) Update parameters w after each example, sequentially: => the least-mean-square (LMS) algorithm. 26

27 Batch GD vs. Stochastic GD Accuracy: Time complexity: Memory complexity: Online learning: 27

28 Batch GD vs. Stochastic GD 28

29 Pre-processing Features Features may have very different scales, e.g. x 1 = rooms vs. x 2 = size in sq ft. Right (different scales): GD goes first towards the bottom of the bowl, then slowly along an almost flat valley. Left (scaled features): GD goes straight towards the minimum. 29

30 Feature Scaling Scaling between [0, 1] or [ 1, +1]: For each feature x j, compute min j and max j over the training examples. Scale x (n) j as follows: Scaling to standard normal distribution: For each feature x j, compute sample μ j and sample σ j over the training examples. Scale x (n) j as follows: 30

31 Implementation: Gradient Checking Want to minimize J(θ), where θ is a scalar. Mathematical definition of derivative: d J(θ +ε) J(θ ε) J(θ) = lim dθ ε 2ε Numerical approximation of derivative: d J(θ +ε) J(θ ε) J(θ) dθ 2ε where ε =

32 Implementation: Gradient Checking If θ is a vector of parameters θ i, Compute numerical derivative with respect to each θ i. Aggregate all derivatives into numerical gradient G num (θ). Compare numerical gradient G num (θ) with implementation of gradient G imp (θ): G num (θ) G imp (θ) G num (θ)+ G imp (θ)

33 Gradient Descent vs. Normal Equations Gradient Descent: Need to select learning rate η. May need many iterations: Can do Early Stopping on validation data for regularization. Scalable when number of training examples N is large. Normal Equations: No iterations => easy to code. Computing (X M X) -1 has cubic time complexity => slow for large N. X M X may be singular: 1. Redundant (linearly dependent) features. 2. #features > #examples => do feature selection or regularization. 33