Wojciech Kotłowski

Erik Schultheis, Marek Wydmuch, Wojciech Kotłowski et al.

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.

Gaetan De Waele, Marek Wydmuch, Krzysztof Dembczyński et al.

One of the central challenges in the computational analysis of liquid chromatography-tandem mass spectrometry (LC-MS/MS) data is to identify the compounds underlying the output spectra. In recent years, this problem is increasingly tackled using deep learning methods. A common strategy involves predicting a molecular fingerprint vector from an input mass spectrum, which is then used to search for matches in a chemical compound database. While various loss functions are employed in training these predictive models, their impact on model performance remains poorly understood. In this study, we investigate commonly used loss functions, deriving novel regret bounds that characterize when Bayes-optimal decisions for these objectives must diverge. Our results reveal a fundamental trade-off between the two objectives of (1) fingerprint similarity and (2) molecular retrieval. Optimizing for more accurate fingerprint predictions typically worsens retrieval results, and vice versa. Our theoretical analysis shows this trade-off depends on the similarity structure of candidate sets, providing guidance for loss function and fingerprint selection.

Erik Schultheis, Wojciech Kotłowski, Marek Wydmuch et al.

We consider the optimization of complex performance metrics in multi-label classification under the population utility framework. We mainly focus on metrics linearly decomposable into a sum of binary classification utilities applied separately to each label with an additional requirement of exactly $k$ labels predicted for each instance. These "macro-at-$k$" metrics possess desired properties for extreme classification problems with long tail labels. Unfortunately, the at-$k$ constraint couples the otherwise independent binary classification tasks, leading to a much more challenging optimization problem than standard macro-averages. We provide a statistical framework to study this problem, prove the existence and the form of the optimal classifier, and propose a statistically consistent and practical learning algorithm based on the Frank-Wolfe method. Interestingly, our main results concern even more general metrics being non-linear functions of label-wise confusion matrices. Empirical results provide evidence for the competitive performance of the proposed approach.

Eugenio Clerico, Hamish Flynn, Wojciech Kotłowski et al.

We develop a methodology for constructing confidence sets for parameters of statistical models via a reduction to sequential prediction. Our key observation is that for any generalized linear model (GLM), one can construct an associated game of sequential probability assignment such that achieving low regret in the game implies a high-probability upper bound on the excess likelihood of the true parameter of the GLM. This allows us to develop a scheme that we call online-to-confidence-set conversions, which effectively reduces the problem of proving the desired statistical claim to an algorithmic question. We study two varieties of this conversion scheme: 1) analytical conversions that only require proving the existence of algorithms with low regret and provide confidence sets centered at the maximum-likelihood estimator 2) algorithmic conversions that actively leverage the output of the online algorithm to construct confidence sets (and may be centered at other, adaptively constructed point estimators). The resulting methodology recovers all state-of-the-art confidence set constructions within a single framework, and also provides several new types of confidence sets that were previously unknown in the literature.

Wojciech Kotłowski, Marek Wydmuch, Erik Schultheis et al.

We consider sequential maximization of performance metrics that are general functions of a confusion matrix of a classifier (such as precision, F-measure, or G-mean). Such metrics are, in general, non-decomposable over individual instances, making their optimization very challenging. While they have been extensively studied under different frameworks in the batch setting, their analysis in the online learning regime is very limited, with only a few distinguished exceptions. In this paper, we introduce and analyze a general online algorithm that can be used in a straightforward way with a variety of complex performance metrics in binary, multi-class, and multi-label classification problems. The algorithm's update and prediction rules are appealingly simple and computationally efficient without the need to store any past data. We show the algorithm attains $\mathcal{O}(\frac{\ln n}{n})$ regret for concave and smooth metrics and verify the efficiency of the proposed algorithm in empirical studies.

Manfred K. Warmuth, Wojciech Kotłowski, Matt Jones et al.

It is well known that the class of rotation invariant algorithms are suboptimal even for learning sparse linear problems when the number of examples is below the "dimension" of the problem. This class includes any gradient descent trained neural net with a fully-connected input layer (initialized with a rotationally symmetric distribution). The simplest sparse problem is learning a single feature out of $d$ features. In that case the classification error or regression loss grows with $1-k/n$ where $k$ is the number of examples seen. These lower bounds become vacuous when the number of examples $k$ reaches the dimension $d$. We show that when noise is added to this sparse linear problem, rotation invariant algorithms are still suboptimal after seeing $d$ or more examples. We prove this via a lower bound for the Bayes optimal algorithm on a rotationally symmetrized problem. We then prove much lower upper bounds on the same problem for simple non-rotation invariant algorithms. Finally we analyze the gradient flow trajectories of many standard optimization algorithms in some simple cases and show how they veer toward or away from the sparse targets. We believe that our trajectory categorization will be useful in designing algorithms that can exploit sparse targets and our method for proving lower bounds will be crucial for analyzing other families of algorithms that admit different classes of invariances.

Ehsan Amid, Rohan Anil, Wojciech Kotłowski et al.

We present the surprising result that randomly initialized neural networks are good feature extractors in expectation. These random features correspond to finite-sample realizations of what we call Neural Network Prior Kernel (NNPK), which is inherently infinite-dimensional. We conduct ablations across multiple architectures of varying sizes as well as initializations and activation functions. Our analysis suggests that certain structures that manifest in a trained model are already present at initialization. Therefore, NNPK may provide further insight into why neural networks are so effective in learning such structures.

Tim van Erven, Sarah Sachs, Wouter M. Koolen et al.

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.

Manfred K. Warmuth, Wojciech Kotłowski, Ehsan Amid

It was conjectured that any neural network of any structure and arbitrary differentiable transfer functions at the nodes cannot learn the following problem sample efficiently when trained with gradient descent: The instances are the rows of a $d$-dimensional Hadamard matrix and the target is one of the features, i.e. very sparse. We essentially prove this conjecture: We show that after receiving a random training set of size $k < d$, the expected square loss is still $1-\frac{k}{(d-1)}$. The only requirement needed is that the input layer is fully connected and the initial weight vectors of the input nodes are chosen from a rotation invariant distribution. Surprisingly the same type of problem can be solved drastically more efficient by a simple 2-layer linear neural network in which the $d$ inputs are connected to the output node by chains of length 2 (Now the input layer has only one edge per input). When such a network is trained by gradient descent, then it has been shown that its expected square loss is $\frac{\log d}{k}$. Our lower bounds essentially show that a sparse input layer is needed to sample efficiently learn sparse targets with gradient descent when the number of examples is less than the number of input features.

Michał Kempka, Wojciech Kotłowski, Manfred K. Warmuth

We consider online learning with linear models, where the algorithm predicts on sequentially revealed instances (feature vectors), and is compared against the best linear function (comparator) in hindsight. Popular algorithms in this framework, such as Online Gradient Descent (OGD), have parameters (learning rates), which ideally should be tuned based on the scales of the features and the optimal comparator, but these quantities only become available at the end of the learning process. In this paper, we resolve the tuning problem by proposing online algorithms making predictions which are invariant under arbitrary rescaling of the features. The algorithms have no parameters to tune, do not require any prior knowledge on the scale of the instances or the comparator, and achieve regret bounds matching (up to a logarithmic factor) that of OGD with optimally tuned separate learning rates per dimension, while retaining comparable runtime performance.

Wojciech Kotłowski, Gergely Neu

We consider a partial-feedback variant of the well-studied online PCA problem where a learner attempts to predict a sequence of $d$-dimensional vectors in terms of a quadratic loss, while only having limited feedback about the environment's choices. We focus on a natural notion of bandit feedback where the learner only observes the loss associated with its own prediction. Based on the classical observation that this decision-making problem can be lifted to the space of density matrices, we propose an algorithm that is shown to achieve a regret of $O(d^{3/2}\sqrt{T})$ after $T$ rounds in the worst case. We also prove data-dependent bounds that improve on the basic result when the loss matrices of the environment have bounded rank or the loss of the best action is bounded. One version of our algorithm runs in $O(d)$ time per trial which massively improves over every previously known online PCA method. We complement these results by a lower bound of $Ω(d\sqrt{T})$.

Dirk van der Hoeven, Tim van Erven, Wojciech Kotłowski

A standard introduction to online learning might place Online Gradient Descent at its center and then proceed to develop generalizations and extensions like Online Mirror Descent and second-order methods. Here we explore the alternative approach of putting Exponential Weights (EW) first. We show that many standard methods and their regret bounds then follow as a special case by plugging in suitable surrogate losses and playing the EW posterior mean. For instance, we easily recover Online Gradient Descent by using EW with a Gaussian prior on linearized losses, and, more generally, all instances of Online Mirror Descent based on regular Bregman divergences also correspond to EW with a prior that depends on the mirror map. Furthermore, appropriate quadratic surrogate losses naturally give rise to Online Gradient Descent for strongly convex losses and to Online Newton Step. We further interpret several recent adaptive methods (iProd, Squint, and a variation of Coin Betting for experts) as a series of closely related reductions to exp-concave surrogate losses that are then handled by Exponential Weights. Finally, a benefit of our EW interpretation is that it opens up the possibility of sampling from the EW posterior distribution instead of playing the mean. As already observed by Bubeck and Eldan, this recovers the best-known rate in Online Bandit Linear Optimization.

Wojciech Kotłowski

We consider a variant of online convex optimization in which both the instances (input vectors) and the comparator (weight vector) are unconstrained. We exploit a natural scale invariance symmetry in our unconstrained setting: the predictions of the optimal comparator are invariant under any linear transformation of the instances. Our goal is to design online algorithms which also enjoy this property, i.e. are scale-invariant. We start with the case of coordinate-wise invariance, in which the individual coordinates (features) can be arbitrarily rescaled. We give an algorithm, which achieves essentially optimal regret bound in this setup, expressed by means of a coordinate-wise scale-invariant norm of the comparator. We then study general invariance with respect to arbitrary linear transformations. We first give a negative result, showing that no algorithm can achieve a meaningful bound in terms of scale-invariant norm of the comparator in the worst case. Next, we compliment this result with a positive one, providing an algorithm which "almost" achieves the desired bound, incurring only a logarithmic overhead in terms of the norm of the instances.

Wojciech Kotłowski, Wouter M. Koolen, Alan Malek

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 $Ω(T^{1/3})$ 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.

Wojciech Kotłowski, Manfred K. Warmuth

Most of machine learning deals with vector parameters. Ideally we would like to take higher order information into account and make use of matrix or even tensor parameters. However the resulting algorithms are usually inefficient. Here we address on-line learning with matrix parameters. It is often easy to obtain online algorithm with good generalization performance if you eigendecompose the current parameter matrix in each trial (at a cost of $O(n^3)$ per trial). Ideally we want to avoid the decompositions and spend $O(n^2)$ per trial, i.e. linear time in the size of the matrix data. There is a core trade-off between the running time and the generalization performance, here measured by the regret of the on-line algorithm (total gain of the best off-line predictor minus the total gain of the on-line algorithm). We focus on the key matrix problem of rank $k$ Principal Component Analysis in $\mathbb{R}^n$ where $k \ll n$. There are $O(n^3)$ algorithms that achieve the optimum regret but require eigendecompositions. We develop a simple algorithm that needs $O(kn^2)$ per trial whose regret is off by a small factor of $O(n^{1/4})$. The algorithm is based on the Follow the Perturbed Leader paradigm. It replaces full eigendecompositions at each trial by the problem finding $k$ principal components of the current covariance matrix that is perturbed by Gaussian noise.

Wojciech Kotłowski, Krzysztof Dembczyński

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 $F_β$-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 $\widehatθ$ 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 $\widehatθ$) 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.

Wojciech Kotłowski

In this paper, we theoretically justify an approach popular among participants of the Higgs Boson Machine Learning Challenge to optimize approximate median significance (AMS). The approach is based on the following two-stage procedure. First, a real-valued function is learned by minimizing a surrogate loss for binary classification, such as logistic loss, on the training sample. Then, a threshold is tuned on a separate validation sample, by direct optimization of AMS. We show that the regret of the resulting (thresholded) classifier measured with respect to the squared AMS, is upperbounded by the regret of the underlying real-valued function measured with respect to the logistic loss. Hence, we prove that minimizing logistic surrogate is a consistent method of optimizing AMS.