**580.691/491 Learning Theory
**

**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 9. Final project due 5PM, May 13.

**Textbook**:
Biological Learning and Control, MIT Press, 2012

**Useful mathematical identities**

**Course Outline**:

**Introduction**

**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 regression. Data
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
errors. Data 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.

Homework

**Expectation
Maximization and system identification**

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

Homework: image segmentation.
Imagedata

**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).

Homework.

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.

Homework

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.

Homework.

**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