  The overarching goal of this paper is to derive excess risk bounds for
learning from exp-concave loss functions in passive and sequential learning
settings. Exp-concave loss functions encompass several fundamental problems in
machine learning such as squared loss in linear regression, logistic loss in
classification, and negative logarithm loss in portfolio management. In batch
setting, we obtain sharp bounds on the performance of empirical risk
minimization performed in a linear hypothesis space and with respect to the
exp-concave loss functions. We also extend the results to the online setting
where the learner receives the training examples in a sequential manner. We
propose an online learning algorithm that is a properly modified version of
online Newton method to obtain sharp risk bounds. Under an additional mild
assumption on the loss function, we show that in both settings we are able to
achieve an excess risk bound of $O(d\log n/n)$ that holds with a high
probability.
