Surrogate regret bounds for generalized classification performance metrics

We consider optimization of generalized performance metrics for binary classification by means of surrogate losses. We focus on a class of metrics, which are linear-fractional functions of the false positive and false negative rates (examples of which include $F_{\beta}$-measure, Jaccard similarity coefficient, AM measure, and many others). Our analysis concerns the following two-step procedure. First, a real-valued function $f$ is learned by minimizing a surrogate loss for binary classification on the training sample. It is assumed that the surrogate loss is a strongly proper composite loss function (examples of which include logistic loss, squared-error loss, exponential loss, etc.). Then, given $f$, a threshold $\widehat{\theta}$ is tuned on a separate validation sample, by direct optimization of the target performance metric. We show that the regret of the resulting classifier (obtained from thresholding $f$ on $\widehat{\theta}$) measured with respect to the target metric is upperbounded by the regret of $f$ measured with respect to the surrogate loss. We also extend our results to cover multilabel classification and provide regret bounds for micro- and macro-averaging measures. Our findings are further analyzed in a computational study on both synthetic and real data sets.Comment: 22 page

Online Isotonic Regression

We consider the online version of the isotonic regression problem. Given a set of linearly ordered points (e.g., on the real line), the learner must predict labels sequentially at adversarially chosen positions and is evaluated by her total squared loss compared against the best isotonic (non-decreasing) function in hindsight. We survey several standard online learning algorithms and show that none of them achieve the optimal regret exponent; in fact, most of them (including Online Gradient Descent, Follow the Leader and Exponential Weights) incur linear regret. We then prove that the Exponential Weights algorithm played over a covering net of isotonic functions has a regret bounded by $O\big(T^{1/3} \log^{2/3}(T)\big)$ and present a matching $\Omega(T^{1/3})$ lower bound on regret. We provide a computationally efficient version of this algorithm. We also analyze the noise-free case, in which the revealed labels are isotonic, and show that the bound can be improved to $O(\log T)$ or even to $O(1)$ (when the labels are revealed in isotonic order). Finally, we extend the analysis beyond squared loss and give bounds for entropic loss and absolute loss.Comment: 25 page

Generalized test utilities for long-tail performance in extreme multi-label classification

Extreme multi-label classification (XMLC) is the task of selecting a small subset of relevant labels from a very large set of possible labels. As such, it is characterized by long-tail labels, i.e., most labels have very few positive instances. With standard performance measures such as precision@k, a classifier can ignore tail labels and still report good performance. However, it is often argued that correct predictions in the tail are more "interesting" or "rewarding," but the community has not yet settled on a metric capturing this intuitive concept. The existing propensity-scored metrics fall short on this goal by confounding the problems of long-tail and missing labels. In this paper, we analyze generalized metrics budgeted "at k" as an alternative solution. To tackle the challenging problem of optimizing these metrics, we formulate it in the expected test utility (ETU) framework, which aims to optimize the expected performance on a fixed test set. We derive optimal prediction rules and construct computationally efficient approximations with provable regret guarantees and robustness against model misspecification. Our algorithm, based on block coordinate ascent, scales effortlessly to XMLC problems and obtains promising results in terms of long-tail performance.Comment: This is the authors' version of the work accepted to NeurIPS 2023; the final version of the paper, errors and typos corrected, and minor modifications to improve clarit

Random permutation online isotonic regression

We revisit isotonic regression on linear orders, the problem of fitting monotonic functions to best explain the data, in an online setting. It was previously shown that online isotonic regression is unlearnable in a fully adversarial model, which lead to its study in the fixed design model. Here, we instead develop the more practical random permutation model. We show that the regret is bounded above by the excess leave-one-out loss for which we develop efficient algorithms and matching lower bounds. We also analyze the class of simple and popular forward algorithms and recommend where to look for algorithms for online isotonic regression on partial orders

Robust Online Convex Optimization in the Presence of Outliers

We consider online convex optimization when a number k of data points are outliers that may be corrupted. We model this by introducing the notion of robust regret, which measures the regret only on rounds that are not outliers. The aim for the learner is to achieve small robust regret, without knowing where the outliers are. If the outliers are chosen adversarially, we show that a simple filtering strategy on extreme gradients incurs O(k) additive overhead compared to the usual regret bounds, and that this is unimprovable, which means that k needs to be sublinear in the number of rounds. We further ask which additional assumptions would allow for a linear number of outliers. It turns out that the usual benign cases of independently, identically distributed (i.i.d.) observations or strongly convex losses are not sufficient. However, combining i.i.d. observations with the assumption that outliers are those observations that are in an extreme quantile of the distribution, does lead to sublinear robust regret, even though the expected number of outliers is linear

Generalized test utilities for long-tail performance in extreme multi-label classification

Extreme multi-label classification (XMLC) is the task of selecting a small subset of relevant labels from a very large set of possible labels. As such, it is characterized by long-tail labels, i.e., most labels have very few positive instances. With standard performance measures such as precision@k, a classifier can ignore tail labels and still report good performance. However, it is often argued that correct predictions in the tail are more "interesting" or "rewarding," but the community has not yet settled on a metric capturing this intuitive concept. The existing propensity-scored metrics fall short on this goal by confounding the problems of long-tail and missing labels. In this paper, we analyze generalized metrics budgeted "at k" as an alternative solution. To tackle the challenging problem of optimizing these metrics, we formulate it in the expected test utility (ETU) framework, which aims to optimize the expected performance on a fixed test set. We derive optimal prediction rules and construct computationally efficient approximations with provable regret guarantees and robustness against model misspecification. Our algorithm, based on block coordinate ascent, scales effortlessly to XMLC problems and obtains promising results in terms of long-tail performance

Quantum learning: optimal classification of qubit states

Pattern recognition is a central topic in Learning Theory with numerous applications such as voice and text recognition, image analysis, computer diagnosis. The statistical set-up in classification is the following: we are given an i.i.d. training set $(X_{1},Y_{1}),... (X_{n},Y_{n})$ where $X_{i}$ represents a feature and $Y_{i}\in \{0,1\}$ is a label attached to that feature. The underlying joint distribution of $(X,Y)$ is unknown, but we can learn about it from the training set and we aim at devising low error classifiers $f:X\to Y$ used to predict the label of new incoming features. Here we solve a quantum analogue of this problem, namely the classification of two arbitrary unknown qubit states. Given a number of `training' copies from each of the states, we would like to `learn' about them by performing a measurement on the training set. The outcome is then used to design mesurements for the classification of future systems with unknown labels. We find the asymptotically optimal classification strategy and show that typically, it performs strictly better than a plug-in strategy based on state estimation. The figure of merit is the excess risk which is the difference between the probability of error and the probability of error of the optimal measurement when the states are known, that is the Helstrom measurement. We show that the excess risk has rate $n^{-1}$ and compute the exact constant of the rate.Comment: 24 pages, 4 figure