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
2160220160
We introduce the general and powerful scheme of predicting information re-use in optimization algorithms. This allows us to devise a computationally efficient algorithm for bandit convex optimization with new state-of-the-art guarantees for both Lipschitz loss functions and loss functions with Lipschitz gradients. This is the first algorithm admitting both a polynomial time complexity and a regret that is polynomial in the dimension of the action space that improves upon the original regret bound for Lipschitz loss functions, achieving a regret of @math . Our algorithm further improves upon the best existing polynomial-in-dimension bound (both computationally and in terms of regret) for loss functions with Lipschitz gradients, achieving a regret of @math . 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. Consider a sequence of bits where we are trying to predict the next bit from the previous bits. Assume we are allowed to say 'predict 0' or 'predict 1', and our payoff is +1 if the prediction is correct and -1 otherwise. We will say that at each point in time the loss of an algorithm is the number of wrong predictions minus the number of right predictions so far. In this paper we are interested in algorithms that have essentially zero (expected) loss over any string at any point in time and yet have small regret with respect to always predicting 0 or always predicting 1. For a sequence of length @math our algorithm has regret @math and loss @math in expectation for all strings. We show that the tradeoff between loss and regret is optimal up to constant factors. Our techniques extend to the general setting of @math experts, where the related problem of trading off regret to the best expert for regret to the special' expert has been studied by Even- (COLT'07). We obtain essentially zero loss with respect to the special expert and optimal loss regret tradeoff, improving upon the results of Even- and settling the main question left open in their paper. The strong loss bounds of the algorithm have some surprising consequences. A simple iterative application of our algorithm gives essentially optimal regret bounds at multiple time scales, bounds with respect to @math -shifting optima as well as regret bounds with respect to higher norms of the input sequence. Bandit convex optimization is a special case of online convex optimization with partial information. In this setting, a player attempts to minimize a sequence of adversarially generated convex loss functions, while only observing the value of each function at a single point. In some cases, the minimax regret of these problems is known to be strictly worse than the minimax regret in the corresponding full information setting. We introduce the multi-point bandit setting, in which the player can query each loss function at multiple points. When the player is allowed to query each function at two points, we prove regret bounds that closely resemble bounds for the full information case. This suggests that knowing the value of each loss function at two points is almost as useful as knowing the value of each function everywhere. When the player is allowed to query each function at d+1 points (d being the dimension of the space), we prove regret bounds that are exactly equivalent to full information bounds for smooth functions.