Defensive forecasting for optimal prediction with expert advice

The method of defensive forecasting is applied to the problem of prediction with expert advice for binary outcomes. It turns out that defensive forecasting is not only competitive with the Aggregating Algorithm but also handles the case of "second-guessing" experts, whose advice depends on the learner's prediction; this paper assumes that the dependence on the learner's prediction is continuous.Comment: 14 page

On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game

In the framework of the game-theoretic probability of Shafer and Vovk (2001) it is of basic importance to construct an explicit strategy weakly forcing the strong law of large numbers (SLLN) in the bounded forecasting game. We present a simple finite-memory strategy based on the past average of Reality's moves, which weakly forces the strong law of large numbers with the convergence rate of $O(\sqrt{\log n/n})$. Our proof is very simple compared to a corresponding measure-theoretic result of Azuma (1967) on bounded martingale differences and this illustrates effectiveness of game-theoretic approach. We also discuss one-sided protocols and extension of results to linear protocols in general dimension.Comment: 14 page

Leading strategies in competitive on-line prediction

We start from a simple asymptotic result for the problem of on-line regression with the quadratic loss function: the class of continuous limited-memory prediction strategies admits a "leading prediction strategy", which not only asymptotically performs at least as well as any continuous limited-memory strategy but also satisfies the property that the excess loss of any continuous limited-memory strategy is determined by how closely it imitates the leading strategy. More specifically, for any class of prediction strategies constituting a reproducing kernel Hilbert space we construct a leading strategy, in the sense that the loss of any prediction strategy whose norm is not too large is determined by how closely it imitates the leading strategy. This result is extended to the loss functions given by Bregman divergences and by strictly proper scoring rules.Comment: 20 pages; a conference version is to appear in the ALT'2006 proceeding

Competitive on-line learning with a convex loss function

We consider the problem of sequential decision making under uncertainty in which the loss caused by a decision depends on the following binary observation. In competitive on-line learning, the goal is to design decision algorithms that are almost as good as the best decision rules in a wide benchmark class, without making any assumptions about the way the observations are generated. However, standard algorithms in this area can only deal with finite-dimensional (often countable) benchmark classes. In this paper we give similar results for decision rules ranging over an arbitrary reproducing kernel Hilbert space. For example, it is shown that for a wide class of loss functions (including the standard square, absolute, and log loss functions) the average loss of the master algorithm, over the first $N$ observations, does not exceed the average loss of the best decision rule with a bounded norm plus $O(N^{-1/2})$. Our proof technique is very different from the standard ones and is based on recent results about defensive forecasting. Given the probabilities produced by a defensive forecasting algorithm, which are known to be well calibrated and to have good resolution in the long run, we use the expected loss minimization principle to find a suitable decision.Comment: 26 page

All Men Count with You, but None Too Much: Information Aggregation and Learning in Prediction Markets.

Prediction markets are markets that are set up to aggregate information from a population of traders in order to predict the outcome of an event. In this thesis, we consider the problem of designing prediction markets with discernible semantics of aggregation whose syntax is amenable to analysis. For this, we will use tools from computer science (in particular, machine learning), statistics and economics. First, we construct generalized log scoring rules for outcomes drawn from high-dimensional spaces. Next, based on this class of scoring rules, we design the class of exponential family prediction markets. We show that this market mechanism performs an aggregation of private beliefs of traders under various agent models. Finally, we present preliminary results extending this work to understand the dynamics of related markets using probabilistic graphical model techniques. We also consider the problem in reverse: using prediction markets to design machine learning algorithms. In particular, we use the idea of sequential aggregation from prediction markets to design machine learning algorithms that are suited to situations where data arrives sequentially. We focus on the design of algorithms for recommender systems that are robust against cloning attacks and that are guaranteed to perform well even when data is only partially available.PHDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111398/1/skutty_1.pd

Defensive forecasting for linear protocols

We consider a general class of forecasting protocols, called "linear protocols", and discuss several important special cases, including multi-class forecasting. Forecasting is formalized as a game between three players: Reality, whose role is to generate observations; Forecaster, whose goal is to predict the observations; and Skeptic, who tries to make money on any lack of agreement between Forecaster's predictions and the actual observations. Our main mathematical result is that for any continuous strategy for Skeptic in alinear protocol there exists a strategy for Forecaster that does not allow Skeptic's capital to grow. This result is a meta-theorem that allows one to transform any continuous law of probability in a linear protocol into a forecasting strategy whose predictions are guaranteed to satisfy this law. We apply this meta-theorem to a weak law of large numbers in Hilbert spaces to obtain a version of the K29 prediction algorithm for linear protocols and show that this version also satisfies the attractive properties of proper calibration and resolution under a suitable choice of its kernel parameter, with no assumptions about the way the data is generated