580.691/491 Learning Theory
Course Instructor: Reza Shadmehr

Spring semester 2017

Overview: This course introduces the probabilistic foundations of learning theory. We will discuss topics in regression, estimation, optimal control, system identification, Bayesian learning, and classification. Our aim is to first derive some of the important mathematical results in these topics, and then apply the framework to problems in biology, particularly animal learning and control of action.  The lectures are recorded and are available freely with the links below.


Lecture times: Mondays and Wednesdays, 3:00 – 4:15 PM, Shaffer 303

Teaching Assistant: Tehrim Yoon

Exams: Midterm on March 15. Final project due 5PM, May 13.

Textbook:  Biological Learning and Control, MIT Press, 2012


Useful mathematical identities


Course Outline:


Lecture 1: (intro.ppt, Lecture) Introduction.
Homework: predicting which movie to suggest  data
Extra credit: handwritten digit classification

Lecture 2: (Review of probability theory, Lecture).  Bayes rule, expected value and variance of random variables and sum of random variables, expected value of random variables raised to a power, Binomial distribution, Poisson distribution, Normal distribution.
Homework: probability theory


Regression, generalization, and maximum likelihood

Lecture 3: (LMS_1.ppt, Lecture) Loss function as mean squared error; batch learning and the normal equation; Cross validation, batch vs. online learning, steepest descent algorithm, LMS, convergence of LMS.
Homework: (simulation) classify using regressionData set.

Lecture 4: (LMS_2.ppt, Lecture) Newton-Raphson, LMS and steepest descent with Newton-Raphson, weighted least-squares, regression with basis functions, estimating the loss function for learning in humans.
Homework: moving centers of Gaussian bases.

Lecture 5: (generalization_2.ppt, Lecture) sensitivity to error, modulation of error sensitivity, generalization function, estimation of generalization function from sequence of errors
Herzfeld et al. (2014) A memory of errors in sensorimotor learning. Science 345:1349-1353.
Homework: (simulation) estimate generalization function from record of errorsData set.

Lecture 6: (ML_1.ppt, Lecture) Maximum likelihood estimation; likelihood of data given a distribution; ML estimate of model weights and model noise, integration of multiple sensory data.
Reading: chapters 4.1-4.5.
Homework: derive online estimates of model weights and model noise.


State estimation and the Kalman filter

Lecture 7:  (state_estimation1.ppt, Lecture) Optimal parameter estimation, parameter uncertainty, state noise and measurement noise, adjusting learning rates to minimize model uncertainty.  Derivation of the Kalman filter algorithm. 
Reading: chapters 4.6 and 4.7 of Shadmehr and Mussa-Ivaldi.
Homework: Convergence of the Kalman gain and uncertainty.

Lecture 8:  (Lecture) Estimation with multiple sensors, estimation with signal-dependent noise.
Reading: chapters 4.9 and 4.10.
Homework  data


Bayesian integration

Lecture 9:  (Bayes_2.ppt, Lecture) Kalman filter and Bayesian estimation; factorization of joint distribution of Gaussian variables.
Reading: chapter 5.1.
Homework: posterior distribution with two observed data points; maximizing the posterior directly.

Lecture 10:  (Lecture) Causal inference and the problem of deciding between two generative models; the influence of priors in how we make movements and perceive motion; the influence of priors in cognitive decision making.
Reading: chapters 5.2-5.3.

Lecture 11:  (Lecture) Use of the Kalman gain to account for learning in animals, classical conditioning, Kamin blocking, and backward blocking, with examples of adaptation in people. 
Reading: chapters 5.5, 6.1-6.4.


Sensorimotor adaptation and state-space models

Lecture 12:  (Lecture) A generative model of sensorimotor adaptation experiments; accounting for sensory illusions during adaptation; effect of statistics of prior actions on patterns of learning. 
Reading: chapters 6.5-6.7.

Lecture 13: (Lecture) Modulating sensitivity to error through manipulation of state and measurement noises; modulating forgetting rates.  Modulating sensitivity to error through memory of errors.
Reading: chapter 7.
Reading: Herzfeld et al. (2014) A memory of errors in sensorimotor learning. Science 345:1349-1353.
Homework: adaptive error-sensitivity (pdf) Data set

Lecture 14:  (Lecture) Multiple timescales of memory, savings, spontaneous recovery.
Reading: chapter 8.


Expectation Maximization and system identification

Lecture 15: Unsupervised classification.  Mixture models, K-means algorithm, and Expectation-Maximization (EM). 
Homework: image segmentationImagedata 

Lecture 16:  (Lecture) Identification of the learner, Expectation maximization as an algorithm for system identification.
Reading: chapters 9.8-9.9

Lecture 17: Generalized Expectation Maximization.

Homework: state-space model of sensorimotor adaptation


Optimal control and the Bellman equation

Lecture 18:  (Lecture) Motor costs and rewards. Movement vigor and encoding of reward.  Muscle tuning functions as a signature of motor costs.  Minimizing costs while meeting a constraint (Lagrange multipliers).
Reading: Chapter 10.

Lecture 19:  Lecture notes.  Open loop optimal control with signal-dependent noise to minimize endpoint variance with the constraint that the endpoint state should be at the goal location. 
Reading: Harris and Wolpert (1998) Signal-dependent noise determines motor planning. Nature 394:780-784.

Lecture 20:  (Lecture) Open loop optimal control with cost of time.  Temporal discounting of reward.  Optimizing movement duration with motor and accuracy costs.  Control of saccades as an example of a movement in which cost of time appears to be hyperbolic. 
Reading: Chapter 11.

Lecture 21:  (Lecture) Introduction to optimal feedback control. Bellman equation.
Reading: Chapter 12.1-12.3.
Example of Bellman’s equation

      Homework: noise free optimal feedback control in a grid-world

Lecture 22: (Lecture) Optimal feedback control of linear dynamical systems with and without additive noise.

Lecture 23: (Lecture) Optimal feedback control with signal dependent noise.
Reading: Chapter 12.4-12.6.


Classification via Bayesian estimation

Lecture 24: (Lecture) Introduction to classification; Fisher linear discriminant, classification using posterior probabilities with explicit models of densities, confidence and error bounds of the Bayes classifier, Chernoff error bounds. 
Homework: Bayesian classification of a binary decision

Lecture 25:  Linear and quadratic decision boundaries.  Equal-variance Gaussian densities (linear discriminant analysis), unequal-variance Gaussian densities (quadratic discriminant analysis), Kernel estimates of density.
Homework: Classification using assumptions of equal and unequal Gaussian distributions; classification using kernel density estimates.

Lecture 26:  Logistic regression as a method to model posterior probability of class membership as a function of state variables; batch algorithm: Iterative Re-weighted Least Squares; on-line algorithm.
Homework: logistic regression with multiple classes of unequal variance.


Subspace identification

(SubSpace.ppt, Lecture) Introduction to subspace analysis; projection of row vectors of matrices, singular value decomposition, system identification of deterministic systems using subspace methods.
Homework: system identification of a deterministic system
Data set
Reading: chapters 9.1-9.6
Overschee and De Moor (1996) Subspace identification for linear systems: theory, implementation, applications.  Kluwer Academic, The Netherlands


Reinforcement learning

Introduction to reinforcement learning; value functions and Bellman equations; generalized policy iteration
Homework: rat maze problem. Mazedata .

Temporal difference learning; policy improvement theorem; addiction and reinforcement learning.
Homework.  Randomwalkdata  Schultzpaper

TD-lambda and eligibility trace.