COSC343: Artificial Intelligence

Transcription

1 COSC343: Artificial Intelligence Lecture 8 : Non-linear regression and Optimisation Lech Szymanski Dept. of Computer Science, University of Otago Lech Szymanski COSC343 Lecture 8

2 In today s lecture Error surface Iterative parameter search Non-linear regression and multiple minima Optimisation techniques: Random walk Steepest gradient Simulated annealing

3 Recall: Least squares and linear regression The least squares parameters that minimise, where, can be found by solving: Linear regression is consistent only when appropriate choice of base functions is made For models that are linear in parameters,, where, there is a closed form solution: and Matrix inversion is computationally intensive, where This matrix is sometimes singular

4 Recall: Least squares and regression The least squares parameters that minimise, where, can be found by solving: Can appropriate parameters be found without the closed form equation?

5 Space of possible solutions Let s think about the space of possible values that can take, and the corresponding cost for some hypothesis. The space of possible solutions is a U-dimensional state-space (where U is the number of parameters in the model); A cost function maps each point in the state space to a real number; The evaluations of all points in the state space can be visualised as a state-space landscape (in U+1 dimensions)

6 An example: fitting a line 3 Training data Cost surface 1 Cost surface 25 2 y = 2:1x! 2:99 y =!:5x + 8: ~y x J w w 2 w w 1 y i = w 1 x i + w 2 J = 1 2 Pi (y i! ~y i ) 2 w = (FF T )!1 F~y T w 1 = 2:1; w 2 =!2:99 Closed form solution Epoch : w 1 =!:5; w 2 = 8: Initial guess

7 Parameter search 1. Make an initial guess of the parameter values (often random) and compute the cost 2. Update the parameters and compute the new cost Learning rate parameter Change in parameter values a. Keep the update if 3. Go back to step 2

8 Random walk Pick the weight update at random, where Parameter update is a random variable (in this case with a normal distribution) keep the update COSC343 Lecture 8 if Lech Szymanski

9 An example: fitting a line with random walk

10 Learning rate parameter The learning rate parameter controls how big the update steps are when parameters are updated. Small results in smaller steps: more ordered and direct path towards the minimum, but it takes longer to get there Large results in bigger steps: faster convergence, but less direct path and more chance of overshooting the minimum

11 An example: random walk with smaller α

12 An example: random walk with larger α

13 Steepest gradient descent The weight update is the negative gradient of the cost function with respect to the parameters and Gradient is the positive slope (going up) of the cost surface at w t Also referred to as steepest gradient ascent or hill climbing if the objective function is being maximised

14 An example: fitting a line with steepest gradient descent Since, Therefore: true for any linear model then. Given: Output derivatives are:, Cost derivatives are: Since, then. Since then., j - index over number of weights i index over number of training samples

15 An example: fitting a line with steepest gradient descent

16 An example: non-linear regression y = sin(2:5x 1 +!1:5x 2 ) 1.5 Training data 25 Cost surface 2 1 Cost surface ~y J w x x w w w 1 y i = sin(w 1 x 1 + w 2 x 2 ) J = 1 2 Pi (y i! ~y i ) 2 Epoch : w 1 = 2:5; w 2 =!1:5 J = 89:23

17 An example: non-linear regression

18 An example: non-linear regression

19 An example: non-linear regression

20 Local minima Cost functions in non-linear optimisation are not necessarily convex The cost surface may not correspond to one giant valley, but a series of valleys separated by high cost ridges Global minimum the state of the system that gives lowest possible cost Best solution for the chosen model Local minimum the minimum closest to the current state May not be the best solution Steepest gradient always points towards the local minimum, and as a result, the outcome of the training is highly dependent on the starting value of the parameters

21 Local minima Global minimum

22 Local minima Global minimum Local minima

23 Simulated annealing Pick the weight update at random, where keep the update probability with, where COSC343 Lecture 8

24 Simulated annealing Pick the weight update at random, where keep the update probability with, where Reduction in cost COSC343 Lecture 8 Temperature

25 Simulated annealing: probability of accepting the update High temperature increases the chance of accepting an update that results in higher cost Low temperature reduces the change of accepting an update that results in higher cost Start training with high temperature More energy to jump out of the current minimum End training with low temperature No energy to jump out of the current minimum COSC343 Lecture

26 Simulated annealing: cooling schedule Starting temperature Rate of cooling

27 An example: non-linear regression with simulated annealing

28 Summary and reading Learning as the parameter search over the cost surface Models that are non-linear in parameters tend to have multiple minima Random walk slow, can t guarantee where it goes Gradient descent very fast, finds local minima Simulated annealing theoretical promise of finding the global minimum (but slow and hard to get it to work in practice) Reading for the lecture: AIMA Chapter 18 Section 4 Reading for next lecture: AIMA Chapter 18 Sections 7.1,7.2 Lech Szymanski COSC343 Lecture 6