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A classification learning example Predicting when Rusell will wait for a table --similar to book preferences, predicting credit card fraud, predicting when people are likely to respond to junk mail Inductive Learning (Classification Learning) • Given a set of labeled examples, • Main variations: and a space of hypotheses • Bias: the “sort” of rule are you – Find the rule that underlies looking for? the labeling – If you are looking for only conjunctive hypotheses, • (so you can use it to there are just 3n predict future unlabeled – Search: examples) – Greedy search – Tabularasa, fully supervised – Decision tree learner – Systematic search • Idea: – Version space learner – Loop through all hypotheses – Iterative search • Rank each hypothesis in – Neural net learner terms of its match to data • Pick the best hypothesis It can be shown that sample complexity of PAC The main problem is that learning is proportional to the space of hypotheses is too large Given examples described in terms of n boolean variables 1/e, 1/d AND log |H| n There are 2 2 different hypotheses For 6 features, there are 18,446,744,073,709,551,616 hypotheses 5/5 Bias & Learning Accuracy • Having weak bias Fraction incorectly classified (large hypothesis Test (prediction) error space) – Allows us to capture Training error more concepts – ..increases learning cost – May lead to over- fitting Also the goal of a compression algorithm is to drive down the training error But the goal of a learning algorithm is to drive down the test error Uses different biases in predicting Russel’s waiting habbits Decision Trees --Examples are used to --Learn topology If patrons=full and day=Friday K-nearest --Order of questions then wait (0.3/0.7) neighbors If wait>60 and Reservation=no then wait (0.4/0.9) Association rules --Examples are used to --Learn support and confidence of association rules SVMs Neural Nets --Examples are used to --Learn topology --Learn edge weights Naïve bayes (bayesnet learning) --Examples are used to Russell waits --Learn topology RW None some full --Learn CPTs T 0.3 0.2 0.5 F 0.4 0.3 0.3 Wait time? Patrons? Friday? Learning Decision Trees---How? Basic Idea: --Pick an attribute --Split examples in terms of that attribute --If all examples are +ve label Yes. Terminate --If all examples are –ve label No. Terminate --If some are +ve, some are –ve continue splitting recursively (Special case: Decision Stumps If you don’t feel like splitting any further, return the majority label ) Which one to pick? Depending on the order we pick, we can get smaller or bigger trees Which tree is better? Why do you think so?? Basic Idea: --Pick an attribute --Split examples in terms of that attribute --If all examples are +ve label Yes. Terminate --If all examples are –ve label No. Terminate --If some are +ve, some are –ve continue splitting recursively --if no attributes left to split? (label with majority element) Would you split on patrons or Type? The Information Gain P+ : N+ /(N++N-) # expected comparisons needed to tell whether a Computation P- : N- /(N++N-) given example is +ve or -ve I(P+ ,, P-) = -P+ log(P+) - P- log(P- ) N+ N- Splitting on The difference is the information feature fk gain N1+ N2+ Nk+ So, pick the feature N1- N2- Nk- with the largest Info Gain I(P1+ ,, P1-) I(P2+ ,, P2-) I(Pk+ ,, Pk-) I.e. smallest residual info k Given k mutually exclusive and exhaustive events E1….Ek whose probabilities are S [Ni+ + Ni- ]/[N+ + N-] I(Pi+ ,, Pi-) p1….pk i=1 The “information” content (entropy) is defined as S i -pi log2 pi A split is good if it reduces the entropy.. A simple example Ex Masochistic Anxious Nerdy HATES EXAM V(M) = 2/4 * I(1/2,1/2) + 2/4 * I(1/2,1/2) 1 F T F Y = 1 2 F F T N V(A) = 2/4 * I(1,0) + 2/4 * I(0,1) 3 T F F N = 0 4 T T T Y V(N) = 2/4 * I(1/2,1/2) + 2/4 * I(1/2,1/2) = 1 So Anxious is the best attribute to split on Once you split on Anxious, the problem is solved m-fold cross-validation Split N examples into Evaluating the Decision Trees m equal sized parts for i=1..m train with all parts except ith Lesson: test with the ith part Every bias makes some concepts easier to learn and others harder to learn… “Majority” function Russell Domain (say yes if majority of attributes are yes) Learning curves… Given N examples, partition them into Ntr the training set and Ntest the test instances Loop for i=1 to |Ntr| Loop for Ns in subsets of Ntr of size I Train the learner over Ns Test the learned pattern over Ntest and compute the accuracy (%correct) Problems with Info. Gain. Heuristics • Feature correlation: We are splitting on one feature at a time • The Costanza party problem – No obvious easy solution… • Overfitting: We may look too hard for patterns where there are none – E.g. Coin tosses classified by the day of the week, the shirt I was wearing, the time of the day etc. – Solution: Don’t consider splitting if the information gain given by the best feature is below a minimum threshold • Can use the c2 test for statistical significance – Will also help when we have noisy samples… • We may prefer features with very high branching – e.g. Branch on the “universal time string” for Russell restaurant example – Branch on social security number to look for patterns on who will get A – Solution: “gain ratio” --ratio of information gain with the attribute A to the information content of answering the question “What is the value of A?” • The denominator is smaller for attributes with smaller domains. Decision Stumps • Decision stumps are decision trees where the leaf nodes do not necessarily have all +ve or all – N+ ve training examples N- – Could happen either because Splitting on examples are noisy and mis- feature fk classified or because you want to stop before reaching pure leafs N1+ N2+ Nk+ • When you reach that node, you N1- N2- Nk- return the majority label as the decision. • (We can associate a confidence P+= N1+ / N1++N1- with that decision using the P+ and P-) Sometimes, the best decision tree for a problem could be a decision stump (see coin toss example next) Bayes Network Learning • Bias: The relation between the class label and class attributes is specified by a Bayes Network. • Approach RW None some full Russell waits – Guess Topology T 0.3 0.2 0.5 – Estimate CPTs F 0.4 0.3 0.3 • Simplest case: Naïve Bayes Wait time? Patrons? Friday? – Topology of the network is “class label” causes all the attribute values independently – So, all we need to do is estimate CPTs P(attrib|Class) • In Russell domain, P(Patrons|willwait) – P(Patrons=full|willwait=yes)= #training examples where patrons=full and will wait=yes #training examples where will wait=yes – Given a new case, we use bayes rule to compute the class label Class label is the disease; attributes are symptoms Naïve Bayesian Classification • Problem: Classify a given example E into one of the classes among [C1, C2 ,…, Cn] – E has k attributes A1, A2 ,…, Ak and each Ai can take d different values • Bayes Classification: Assign E to class Ci that maximizes P(Ci | E) P(Ci| E) = P(E| Ci) P(Ci) / P(E) • P(Ci) and P(E) are a priori knowledge (or can be easily extracted from the set of data) • Estimating P(E|Ci) is harder – Requires P(A1=v1 A2=v2….Ak=vk|Ci) • Assuming d values per attribute, we will need ndk probabilities • Naïve Bayes Assumption: Assume all attributes are independent P(E| Ci) = P P(Ai=vj | Ci ) – The assumption is BOGUS, but it seems to WORK (and needs only n*d*k probabilities NBC in terms of BAYES networks.. NBC assumption More realistic assumption Estimating the probabilities for NBC Given an example E described as A1=v1 A2=v2….Ak=vk we want to compute the class of E – Calculate P(Ci | A1=v1 A2=v2….Ak=vk) for all classes Ci and say that the class of E is the one for which P(.) is maximum – P(Ci | A1=v1 A2=v2….Ak=vk) Common factor = P P(vj | Ci ) P(Ci) / P(A1=v1 A2=v2….Ak=vk) Given a set of training N examples that have already been classified into n classes Ci Let #(Ci) be the number of examples that are labeled as Ci Let #(Ci, Ai=vi) be the number of examples labeled as Ci that have attribute Ai set to value vj P(Ci) = #(Ci)/N P(Ai=vj | Ci) = #(Ci, Ai=vi) / #(Ci) USER PROFILE Example P(willwait=yes) = 6/12 = .5 Similarly we can show that P(Patrons=“full”|willwait=yes) = 2/6=0.333 P(Patrons=“full”|willwait=no) P(Patrons=“some”|willwait=yes)= 4/6=0.666 =0.6666 P(willwait=yes|Patrons=full) = P(patrons=full|willwait=yes) * P(willwait=yes) ----------------------------------------------------------- P(Patrons=full) = k* .333*.5 P(willwait=no|Patrons=full) = k* 0.666*.5 Using M-estimates to improve probablity estimates • The simple frequency based estimation of P(Ai=vj|Ck) can be inaccurate, especially when the true value is close to zero, and the number of training examples is small (so the probability that your examples don’t contain rare cases is quite high) • Solution: Use M-estimate P(Ai=vj | Ci) = [#(Ci, Ai=vi) + mp ] / [#(Ci) + m] – p is the prior probability of Ai taking the value vi • If we don’t have any background information, assume uniform probability (that is 1/d if Ai can take d values) – m is a constant—called ―equivalent sample size‖ • If we believe that our sample set is large enough, we can keep m small. Otherwise, keep it large. • Essentially we are augmenting the #(Ci) normal samples with m more virtual samples drawn according to the prior probability on how Ai takes values – Popular values p=1/|V| and m=|V| where V is the size of the vocabulary Also, to avoid overflow errors do addition of logarithms of probabilities (instead of multiplication of probabilities) How Well (and WHY) DOES NBC WORK? • Naïve bayes classifier is darned easy to implement – Good learning speed, classification speed – Modest space storage – Supports incrementality • It seems to work very well in many scenarios – Lots of recommender systems (e.g. Amazon books recommender) use it – Peter Norvig, the director of Machine Learning at GOOGLE said, when asked about what sort of technology they use “Naïve bayes” • But WHY? – NBC’s estimate of class probability is quite bad • BUT classification accuracy is different from probability estimate accuracy – [Domingoes/Pazzani; 1996] analyze this Uses different biases in predicting Russel’s waiting habbits Decision Trees --Examples are used to --Learn topology If patrons=full and day=Friday K-nearest --Order of questions then wait (0.3/0.7) neighbors If wait>60 and Reservation=no then wait (0.4/0.9) Association rules --Examples are used to --Learn support and confidence of association rules SVMs Neural Nets --Examples are used to --Learn topology --Learn edge weights Naïve bayes (bayesnet learning) --Examples are used to Russell waits --Learn topology RW None some full --Learn CPTs T 0.3 0.2 0.5 F 0.4 0.3 0.3 Wait time? Patrons? Friday? Decision Surface Learning (aka Neural Network Learning) • Idea: Since classification is really a question of finding a surface to separate the +ve examples from the -ve examples, why not directly search in the space of possible surfaces? • Mathematically, a surface is a function – Need a way of learning functions – “Threshold units” “Neural Net” is a collection of threshold units with interconnections I1 w1 = 1 if w1I1+w2I2 > k differentiable = 0 otherwise t=k I2 w2 Feed Forward Recurrent Uni-directional connections Bi-directional Single Layer Multi-Layer connections Any “continuous” Any linear decision decision surface Can act as surface can be represented (function) can be associative by a single layer neural net approximated to any memory degree of accuracy by some 2-layer neural net The “Brain” Connection A Threshold Unit Threshold Functions differentiable …is sort of like a neuron Perceptron Networks What happened to the “Threshold”? --Can model as an extra weight with static input I1 w1 t=k I2 w2 == I0=-1 w1 w0= k t=0 w2 Perceptron Learning • Perceptron learning algorithm Loop through training examples – If the activation level of the output unit is 1 when it should be 0, reduce the weight on the link to the jth input unit by a*Ij, where Ii is the ith input value and a a learning rate – If the activation level of the output unit is 0 when it should be 1, increase the weight on the link to the ith input unit by a*Ij – Otherwise, do nothing Until “convergence” A nice applet at: http://neuron.eng.wayne.edu/java/Perceptron/New38.html Perceptron Learning as Gradient Descent Search in the weight-space 1 E (T O) 2 2 i 2 1 E (W ) T g W j I j 2 i j 1 g ( x) ( sigmoid fn ) E 1 ex I j (T O) g W j I j W j j g ' ( x) g ( x)(1 g ( x)) Often a constant W j W j I j (T O) g W j I j learning rate parameter is used instead j Ij I Can Perceptrons Learn All Boolean Functions? --Are all boolean functions linearly separable? Comparing Perceptrons and Decision Trees in Majority Function and Russell Domain Majority function Russell Domain Majority function is linearly seperable.. Russell domain is apparently not.... Encoding: one input unit per attribute. The unit takes as many distinct real values as the size of attribute domain Max-Margin Classification & Support Vector Machines • Any line that separates the +ve & –ve examples is a solution • And perceptron learning finds one of them – But could we have a preference among these? – may want to get the line that provides maximum margin (equidistant from the nearest +ve/- ve) • The nereast +ve and –ve holding up the line are called support vectors • This changes the problem into an optimization one – Quadratic Programming can be used to directly find such a line Learning is Optimization after all! Lagrangian Dual Two ways to learn non-linear decision surfaces • First transform the data • Learn non-linear surfaces into higher dimensional directly (as multi-layer space neural nets) • Find a linear surface • Trick is to do training – Which is guaranteed to efficiently exist – Back Propagation to the • Transform it back to the rescue.. original space • TRICK is to do this without explicitly doing a transformation Linear Separability in High Dimensions “Kernels” allow us to consider separating surfaces in high-D without first converting all points to high-D Kernelized Support Vector Machines • Turns out that it is not always necessary to first map the data into high-D, and then do linear separation • The quadratic programming formulation for SVM winds up using only the pair-wise dot product of training vectors • Dot product is a form of similarity metric between points • If you replace that dot product by any non-linear function, you will, in essence, be transforming data into some high-dimensional space and then finding the max-margin linear classifier in that space – Which will correspond to some wiggly surface in the original dimension • The trick is to find the RIGHT similarity function – Which is a form of prior knowledge Kernelized Support Vector Machines • Turns out that it is not always necessary to first map the data into high-D, and then do linear separation • The quadratic programming formulation for SVM winds up using only the pair-wise dot product of training vectors • Dot product is a form of similarity metric between points • If you replace that dot product by any non-linear function, you will, in essence, be tranforming data into 0 some high-dimensional space and Polynomial Kernel: K (A ; A ) = (( 100 à 1)( 100 à 1) à 0:5) ï 0 A A 6 then finding the max-margin linear classifier in that space – Which will correspond to some wiggly surface in the original dimension • The trick is to find the RIGHT similarity function – Which is a form of prior knowledge Those who ignore easily available domain knowledge are doomed to re-learn it… Santayana’s brother Domain-knowledge & Learning • Classification learning is a problem addressed by both people from AI (machine learning) and Statistics • Statistics folks tend to ―distrust‖ domain-specific bias. – Let the data speak for itself… – ..but this is often futile. The very act of ―describing‖ the data points introduces bias (in terms of the features you decided to use to describe them..) • …but much human learning occurs because of strong domain- specific bias.. • Machine learning is torn by these competing influences.. – In most current state of the art algorithms, domain knowledge is allowed to influence learning only through relatively narrow avenues/formats (E.g. through ―kernels‖) • Okay in domains where there is very little (if any) prior knowledge (e.g. what part of proteins are doing what cellular function) • ..restrictive in domains where there already exists human expertise.. Multi-layer Neural Nets How come back-prop doesn’t get stuck in local minima? One answer: It is actually hard for local minimas to form in high-D, as the “trough” has to be closed in all dimensions Multi-Network Learning can learn Russell Domains Russell Domain …but does it slowly… Practical Issues in Multi-layer network learning • For multi-layer networks, we need to learn both the weights and the network topology – Topology is fixed for perceptrons • If we go with too many layers and connections, we can get over-fitting as well as sloooow convergence – Optimal brain damage • Start with more than needed hidden layers as well as connections; after a network is learned, remove the nodes and connections that have very low weights; retrain Humans make 0.2% Neumans (postmen) make 2% Other impressive applications: --no-hands across K-nearest-neighbor america The test example’s class is determined by the class of the majority of its k nearest --learning to speak neighbors Need to define an appropriate distance measure --sort of easy for real valued vectors --harder for categorical attributes True hypothesis eventually dominates… probability of indefinitely producing uncharacteristic data 0 Bayesian prediction is optimal (Given the hypothesis prior, all other predictions are less likely) Also, remember the Economist article that shows that humans have strong priors.. ..note that the Economist article says humans are able to learn from few examples only because of priors.. So, BN learning is just probability estimation! (as long as data is complete!) How Well (and WHY) DOES NBC WORK? • Naïve bayes classifier is darned easy to implement – Good learning speed, classification speed – Modest space storage – Supports incrementality • It seems to work very well in many scenarios – Lots of recommender systems (e.g. Amazon books recommender) use it – Peter Norvig, the director of Machine Learning at GOOGLE said, when asked about what sort of technology they use “Naïve bayes” • But WHY? – NBC’s estimate of class probability is quite bad • BUT classification accuracy is different from probability estimate accuracy – [Domingoes/Pazzani; 1996] analyze this Reinforcement Learning Based on slides from Bill Smart http://www.cse.wustl.edu/~wds/ What is RL? “a way of programming agents by reward and punishment without needing to specify how the task is to be achieved” [Kaelbling, Littman, & Moore, 96] Basic RL Model 1. Observe state, st World 2. Decide on an action, at 3. Perform action 4. Observe new state, st+1 S R A 5. Observe reward, rt+1 6. Learn from experience 7. Repeat Goal: Find a control policy that will maximize the observed rewards over the lifetime of the agent An Example: Gridworld Canonical RL domain +1 • States are grid cells • 4 actions: N, S, E, W • Reward for entering top right cell • -0.01 for every other move Minimizing sum of rewards Shortest path • In this instance The Promise of Learning The Promise of RL Specify what to do, but not how to do it • Through the reward function • Learning “fills in the details” Better final solutions • Based of actual experiences, not programmer assumptions Less (human) time needed for a good solution Learning Value Functions We still want to learn a value function • We’re forced to approximate it iteratively • Based on direct experience of the world Four main algorithms • Certainty equivalence • Temporal Difference (TD) learning • Q-learning • SARSA Certainty Equivalence Collect experience by moving through the world • s0, a0, r1, s1, a1, r2, s2, a2, r3, s3, a3, r4, s4, a4, r5, s5, ... Use these to estimate the underlying MDP • Transition function, T: SA → S • Reward function, R: SAS → Compute the optimal value function for this MDP • And then compute the optimal policy from it Temporal Difference (TD) [Sutton, 88] TD-learning estimates the value function directly • Don’t try to learn the underlying MDP Keep an estimate of Vp(s) in a table • Update these estimates as we gather more experience • Estimates depend on exploration policy, p • TD is an on-policy method TD-Learning Algorithm Initialize Vp(s) to 0, s Observe state, s Perform action, p(s) Observe new state, s’, and reward, r Vp(s) ← (1-a)Vp(s) + a(r + gVp(s’)) Go to 2 0 ≤ a ≤ 1 is the learning rate • How much attention do we pay to new experiences TD-Learning Vp(s) is guaranteed to converge to V*(s) • After an infinite number of experiences • If we decay the learning rate a a 2 t t t 0 t 0 c • at will work ct In practice, we often don’t need value convergence • Policy convergence generally happens sooner Actor-Critic Methods [Barto, Sutton, & Anderson, 83] TD only evaluates a particular policy Policy • Does not learn a better policy (actor) a V Value We can change the policy as we learn V Function • Policy is the actor (critic) • Value-function estimate is the critic s r World Success is generally dependent on the starting policy being “good enough” Q-Learning [Watkins & Dayan, 92] Q-learning iteratively approximates the state-action value function, Q • Again, we’re not going to estimate the MDP directly • Learns the value function and policy simultaneously Keep an estimate of Q(s, a) in a table • Update these estimates as we gather more experience • Estimates do not depend on exploration policy • Q-learning is an off-policy method Q-Learning Algorithm Initialize Q(s, a) to small random values, s, a Observe state, s Pick an action, a, and do it Observe next state, s’, and reward, r Q(s, a) ← (1 - a)Q(s, a) + a(r + gmaxa’Q(s’, a’)) Go to 2 0 ≤ a ≤ 1 is the learning rate • We need to decay this, just like TD Picking Actions We want to pick good actions most of the time, but also do some exploration • Exploring means that we can learn better policies • But, we want to balance known good actions with exploratory ones • This is called the exploration/exploitation problem Picking Actions e-greedy • Pick best (greedy) action with probability e • Otherwise, pick a random action Boltzmann (Soft-Max) • Pick an action based on its Q-value Q(s, a) t e • P(a | s) Q(s, a' ) , where t is the “temperature” e a' t SARSA SARSA iteratively approximates the state-action value function, Q • Like Q-learning, SARSA learns the policy and the value function simultaneously Keep an estimate of Q(s, a) in a table • Update these estimates based on experiences • Estimates depend on the exploration policy • SARSA is an on-policy method • Policy is derived from current value estimates SARSA Algorithm Initialize Q(s, a) to small random values, s, a Observe state, s Pick an action, a, and do it (just like Q-learning) Observe next state, s’, and reward, r Q(s, a) ← (1-a)Q(s, a) + a(r + gQ(s’, p(s’))) Go to 2 0 ≤ a ≤ 1 is the learning rate • We need to decay this, just like TD On-Policy vs. Off Policy On-policy algorithms • Final policy is influenced by the exploration policy • Generally, the exploration policy needs to be “close” to the final policy • Can get stuck in local maxima Off-policy algorithms • Final policy is independent of exploration policy • Can use arbitrary exploration policies • Will not get stuck in local maxima Convergence Guarantees The convergence guarantees for RL are “in the limit” • The word “infinite” crops up several times Don’t let this put you off • Value convergence is different than policy convergence • We’re more interested in policy convergence • If one action is really better than the others, policy convergence will happen relatively quickly Rewards Rewards measure how well the policy is doing • Often correspond to events in the world • Current load on a machine • Reaching the coffee machine • Program crashing • Everything else gets a 0 reward Things work better if the rewards are incremental • For example, distance to goal at each step • These reward functions are often hard to design The Markov Property RL needs a set of states that are Markov • Everything you need to know to make a decision is included in the state • Not allowed to consult the past Not holding key Holding key Rule-of-thumb K • If you can calculate the reward function from the state without any additional information, you’re OK S G But, What’s the Catch? RL will solve all of your problems, but • We need lots of experience to train from • Taking random actions can be dangerous • It can take a long time to learn • Not all problems fit into the MDP framework Learning Policies Directly An alternative approach to RL is to reward whole policies, rather than individual actions • Run whole policy, then receive a single reward • Reward measures success of the whole policy If there are a small number of policies, we can exhaustively try them all • However, this is not possible in most interesting problems Policy Gradient Methods Assume that our policy, p, has a set of n real- valued parameters, q = {q1, q2, q3, ... , qn } • Running the policy with a particular q results in a reward, rq R • Estimate the reward gradient, , for each qi θ i R • θi θi a θi This is another learning rate Policy Gradient Methods This results in hill-climbing in policy space • So, it’s subject to all the problems of hill-climbing • But, we can also use tricks from search, like random restarts and momentum terms This is a good approach if you have a parameterized policy • Typically faster than value-based methods • “Safe” exploration, if you have a good policy • Learns locally-best parameters for that policy An Example: Learning to Walk [Kohl & Stone, 04] RoboCup legged league • Walking quickly is a big advantage Robots have a parameterized gait controller • 11 parameters • Controls step length, height, etc. Robots walk across soccer pitch and are timed • Reward is a function of the time taken An Example: Learning to Walk Basic idea 1. Pick an initial q = {q1, q2, ... , q11} 2. Generate N testing parameter settings by perturbing q qj = {q1 + d1, q2 + d2, ... , q11 + d11}, di {-e, 0, e} 3. Test each setting, and observe rewards qj → rj 4. For each qi q d if θi largest Calculate q1+, q10, q1- and set θ'i θi 0 0 if θi largest d 5. Set q ← q’, and go to 2 if θi largest Average reward when qni = qi - di An Example: Learning to Walk Initial Final Video: Nate Kohl & Peter Stone, UT Austin Value Function or Policy Gradient? When should I use policy gradient? • When there’s a parameterized policy • When there’s a high-dimensional state space • When we expect the gradient to be smooth When should I use a value-based method? • When there is no parameterized policy • When we have no idea how to solve the problem Summary for Part I Background • MDPs, and how to solve them • Solving MDPs with dynamic programming • How RL is different from DP Algorithms • Certainty equivalence • TD • Q-learning • SARSA • Policy gradient