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.
In standard time series analysis, the squared loss is usually considered and the noise terms are assumed to be independent with bounded variance and zero-mean. In this specific setting, one can assume without loss of generality that the noise terms have identical Gaussian distribution (see @cite_12 @cite_3 @cite_9 for more information). This allows the use of statistical methods, such as least squares and maximum likelihood based methods, for the tasks of analysis and prediction. However, when different loss functions are considered these assumptions do not hold in general, and the aforementioned methods are not applicable. We are not aware of a previous approach that tries to relax these assumptions for general convex loss functions. We note that there has been previous work which tries to relax such assumptions for the squared loss, usually under additional modelling assumptions such as -distribution of the noise (e.g., @cite_11 @cite_4 ). We emphasize that the independence assumption is rather strict and previous works that relax this assumption usually offer specific dependency model, e.g., as proposed by @cite_8 for the ARCH model.
@cite_4 @cite_8 @cite_9 @cite_3 @cite_12 @cite_11
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We prove that the empirical risk of most well-known loss functions factors into a linear term aggregating all labels with a term that is label free, and can further be expressed by sums of the same loss. This holds true even for non-smooth, non-convex losses and in any RKHS. The first term is a (kernel) mean operator -- the focal quantity of this work -- which we characterize as the sufficient statistic for the labels. The result tightens known generalization bounds and sheds new light on their interpretation. Factorization has a direct application on weakly supervised learning. In particular, we demonstrate that algorithms like SGD and proximal methods can be adapted with minimal effort to handle weak supervision, once the mean operator has been estimated. We apply this idea to learning with asymmetric noisy labels, connecting and extending prior work. Furthermore, we show that most losses enjoy a data-dependent (by the mean operator) form of noise robustness, in contrast with known negative results. We study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function. We characterise when margin losses can be proper composite losses, explicitly show how to determine a symmetric loss in full from half of one of its partial losses, introduce an intrinsic parametrisation of composite binary losses and give a complete characterisation of the relationship between proper losses and "classification calibrated" losses. We also consider the question of the "best" surrogate binary loss. We introduce a precise notion of "best" and show there exist situations where two convex surrogate losses are incommensurable. We provide a complete explicit characterisation of the convexity of composite binary losses in terms of the link function and the weight function associated with the proper loss which make up the composite loss. This characterisation suggests new ways of "surrogate tuning" as well as providing an explicit characterisation of when Bregman divergences on the unit interval are convex in their second argument. Finally, in an appendix we present some new algorithm-independent results on the relationship between properness, convexity and robustness to misclassification noise for binary losses and show that all convex proper losses are non-robust to misclassification noise. We study binary classification in the presence of class-conditional random noise, where the learner gets to see labels that are flipped independently with some probability, and where the flip probability depends on the class. Our goal is to devise learning algorithms that are efficient and statistically consistent with respect to commonly used utility measures. In particular, we look at a family of measures motivated by their application in domains where cost-sensitive learning is necessary (for example, when there is class imbalance). In contrast to most of the existing literature on consistent classification that are limited to the classical 0-1 loss, our analysis includes more general utility measures such as the AM measure (arithmetic mean of True Positive Rate and True Negative Rate). For this problem of cost-sensitive learning under class-conditional random noise, we develop two approaches that are based on suitably modifying surrogate losses. First, we provide a simple unbiased estimator of any loss, and obtain performance bounds for empirical utility maximization in the presence of i.i.d. data with noisy labels. If the loss function satis_es a simple symmetry condition, we show that using unbiased estimator leads to an efficient algorithm for empirical maximization. Second, by leveraging a reduction of risk minimization under noisy labels to classification with weighted 0-1 loss, we suggest the use of a simple weighted surrogate loss, for which we are able to obtain strong utility bounds. This approach implies that methods already used in practice, such as biased SVM and weighted logistic regression, are provably noise-tolerant. For two practically important measures in our family, we show that the proposed methods are competitive with respect to recently proposed methods for dealing with label noise in several benchmark data sets. A general approach for estimating an unknown signal x0 ∈ ℝn from noisy, linear measurements y = Ax0 + z ∈ ℝm is via solving a so called regularized M-estimator: x := arg minx ℒ(y−Ax)+λf(x). Here, ℒ is a convex loss function, f is a convex (typically, non-smooth) regularizer, and, λ > 0 a regularizer parameter. We analyze the squared error performance ∥x − x0∥22 of such estimators in the high-dimensional proportional regime where m, n → ∞ and m n → δ. We let the design matrix A have entries iid Gaussian, and, impose minimal and rather mild regularity conditions on the loss function, on the regularizer, and, on the distributions of the noise and of the unknown signal. Under such a generic setting, we show that the squared error converges in probability to a nontrivial limit that is computed by solving four nonlinear equations on four scalar unknowns. We identify a new summary parameter, termed the expected Moreau envelope, which determines how the choice of the loss function and of the regularizer affects the error performance. The result opens the way for answering optimality questions regarding the choice of the loss function, the regularizer, the penalty parameter, etc.