In this paper we address the problem of predicting a time series using the ARMA (autoregressive moving average) model, under minimal assumptions on the noise terms. Using regret minimization techniques, we develop effective online learning algorithms for the prediction problem, without assuming that the noise terms are Gaussian, identically distributed or even independent. Furthermore, we show that our algorithm's performances asymptotically approaches the performance of the best ARMA model in hindsight.
Furthermore, an online approach that relies on regret minimization techniques was never considered for ARMA prediction, and hence regret bounds of the type we are interested simply do not exist. Yet, results on the convergence rate of the coefficient vectors do exist, and regret bounds can be derived from these results. E.g., in @cite_2 such results are presented, and a regret bound of @math can be derived for the squared loss. We are not familiar of these kind of results for general convex loss functions.
@cite_2
2138682935 2547508527 2163735605 2186453173
This paper examines the generalization properties of online convex programming algorithms when the loss function is Lipschitz and strongly convex. Our main result is a sharp bound, that holds with high probability, on the excess risk of the output of an online algorithm in terms of the average regret. This allows one to use recent algorithms with logarithmic cumulative regret guarantees to achieve fast convergence rates for the excess risk with high probability. As a corollary, we characterize the convergence rate of PEGASOS (with high probability), a recently proposed method for solving the SVM optimization problem. We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. We study online aggregation of the predictions of experts, and first show new second-order regret bounds in the standard setting, which are obtained via a version of the Prod algorithm (and also a version of the polynomially weighted average algorithm) with multiple learning rates. These bounds are in terms of excess losses, the differences between the instantaneous losses suffered by the algorithm and the ones of a given expert. We then demonstrate the interest of these bounds in the context of experts that report their confidences as a number in the interval [0; 1] using a generic reduction to the standard setting. We conclude by two other applications in the standard setting, which improve the known bounds in case of small excess losses and show a bounded regret against i.i.d. sequences of losses. We study the online convex optimization problem, in which an online algorithm has to make repeated decisions with convex loss functions and hopes to achieve a small regret. We consider a natural restriction of this problem in which the loss functions have a small deviation, measured by the sum of the distances between every two consecutive loss functions, according to some distance metrics. We show that for the linear and general smooth convex loss functions, an online algorithm modified from the gradient descend algorithm can achieve a regret which only scales as the square root of the deviation. For the closely related problem of prediction with expert advice, we show that an online algorithm modified from the multiplicative update algorithm can also achieve a similar regret bound for a different measure of deviation. Finally, for loss functions which are strictly convex, we show that an online algorithm modified from the online Newton step algorithm can achieve a regret which is only logarithmic in terms of the deviation, and as an application, we can also have such a logarithmic regret for the portfolio management problem.