Deriving and Analyzing Learning Algorithms

Project Summary There is a large variety of learning problems across all disciplines waiting for the right algorithms. Many of these are on-line problems, where the learning algorithm continually makes predictions and updates its hypothesis after getting each “correct” outcome. Ef£cient algorithms may be unable to keep the entire history, and thus must compress their experience into hypotheses. This leads to a tension when the algorithm predicts incorrectly: it must correct its hypothesis in case the same instance is seen again, yet the algorithm must move cautiously to preserve its previously acquired knowledge. One way to quantify this tradeoff is to put a distance measure on the space of possible hypotheses and optimize the improvement of the prediction on the last example versus the distance moved. For the simple linear regression setting, Kivinen and Warmuth showed how two different distances lead to two radically different families of algorithms. One of these families makes additive updates to its hypothesis and includes the standard gradient descent methods. The other family makes multiplicative updates and has radically different performance. Amortized analysis techniques are used to prove relative loss bounds (similar to competitive ratios) on the algorithms, and these relative loss bounds provide a yardstick to measure the effectiveness of each learning family. Although neither family is better all of the time, the new multiplicative family performs exponentially better in many natural settings. The proposed work will extend the framework of Kivinen and Warmuth in a variety of ways. The existing setup requires a £xed learning rate that must be carefully tuned, and an important proposed direction is to analyze annealed and self-tuned learning rates. The Boosting setting is different from, but closely related to, the on-line learning setting, and the second proposed direction is to modify the framework to cover boosting problems. Most current bounds compare the loss of the algorithm against the best £xed predictor, and the third main direction of the proposal is to extend the framework so that algorithms can be compared against shifting predictors that can change over time.