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Abstract:We consider the problem of Neyman-Pearson classification which models unbalanced classification settings where error w.r.t. a distribution $\mu_1$ is to be minimized subject to low error w.r.t. a different distribution $\mu_0$. Given a fixed VC class $\mathcal{H}$ of classifiers to be minimized over, we provide a full characterization of possible distribution-free rates, i.e., minimax rates over the space of all pairs $(\mu_0, \mu_1)$. The rates involve a dichotomy between hard and easy classes $\mathcal{H}$ as characterized by a simple geometric condition, a three-points-separation condition, loosely related to VC dimension.

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Abstract:We consider the problem of sufficient dimension reduction (SDR) for multi-index models. The estimators of the central mean subspace in prior works either have slow (non-parametric) convergence rates, or rely on stringent distributional conditions (e.g., the covariate distribution $P_{\mathbf{X}}$ being elliptical symmetric). In this paper, we show that a fast parametric convergence rate of form $C_d \cdot n^{-1/2}$ is achievable via estimating the \emph{expected smoothed gradient outer product}, for a general class of distribution $P_{\mathbf{X}}$ admitting Gaussian or heavier distributions. When the link function is a polynomial with a degree of at most $r$ and $P_{\mathbf{X}}$ is the standard Gaussian, we show that the prefactor depends on the ambient dimension $d$ as $C_d \propto d^r$.

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Abstract:A critical barrier to learning an accurate decision rule for outlier detection is the scarcity of outlier data. As such, practitioners often turn to the use of similar but imperfect outlier data from which they might transfer information to the target outlier detection task. Despite the recent empirical success of transfer learning approaches in outlier detection, a fundamental understanding of when and how knowledge can be transferred from a source to a target outlier detection task remains elusive. In this work, we adopt the traditional framework of Neyman-Pearson classification -- which formalizes supervised outlier detection -- with the added assumption that one has access to some related but imperfect outlier data. Our main results are as follows: We first determine the information-theoretic limits of the problem under a measure of discrepancy that extends some existing notions from traditional balanced classification; interestingly, unlike in balanced classification, seemingly very dissimilar sources can provide much information about a target, thus resulting in fast transfer. We then show that, in principle, these information-theoretic limits are achievable by adaptive procedures, i.e., procedures with no a priori information on the discrepancy between source and target outlier distributions.

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Abstract:Many recent theoretical works on \emph{meta-learning} aim to achieve guarantees in leveraging similar representational structures from related tasks towards simplifying a target task. Importantly, the main aim in theory works on the subject is to understand the extent to which convergence rates -- in learning a common representation -- \emph{may scale with the number $N$ of tasks} (as well as the number of samples per task). First steps in this setting demonstrate this property when both the shared representation amongst tasks, and task-specific regression functions, are linear. This linear setting readily reveals the benefits of aggregating tasks, e.g., via averaging arguments. In practice, however, the representation is often highly nonlinear, introducing nontrivial biases in each task that cannot easily be averaged out as in the linear case. In the present work, we derive theoretical guarantees for meta-learning with nonlinear representations. In particular, assuming the shared nonlinearity maps to an infinite-dimensional RKHS, we show that additional biases can be mitigated with careful regularization that leverages the smoothness of task-specific regression functions,

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Abstract:We study nonparametric contextual bandits where Lipschitz mean reward functions may change over time. We first establish the minimax dynamic regret rate in this less understood setting in terms of number of changes $L$ and total-variation $V$, both capturing all changes in distribution over context space, and argue that state-of-the-art procedures are suboptimal in this setting. Next, we tend to the question of an adaptivity for this setting, i.e. achieving the minimax rate without knowledge of $L$ or $V$. Quite importantly, we posit that the bandit problem, viewed locally at a given context $X_t$, should not be affected by reward changes in other parts of context space $\cal X$. We therefore propose a notion of change, which we term experienced significant shifts, that better accounts for locality, and thus counts considerably less changes than $L$ and $V$. Furthermore, similar to recent work on non-stationary MAB (Suk & Kpotufe, 2022), experienced significant shifts only count the most significant changes in mean rewards, e.g., severe best-arm changes relevant to observed contexts. Our main result is to show that this more tolerant notion of change can in fact be adapted to.

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Abstract:We consider the problem of \emph{pruning} a classification tree, that is, selecting a suitable subtree that balances bias and variance, in common situations with inhomogeneous training data. Namely, assuming access to mostly data from a distribution $P_{X, Y}$, but little data from a desired distribution $Q_{X, Y}$ with different $X$-marginals, we present the first efficient procedure for optimal pruning in such situations, when cross-validation and other penalized variants are grossly inadequate. Optimality is derived with respect to a notion of \emph{average discrepancy} $P_{X} \to Q_{X}$ (averaged over $X$ space) which significantly relaxes a recent notion -- termed \emph{transfer-exponent} -- shown to tightly capture the limits of classification under such a distribution shift. Our relaxed notion can be viewed as a measure of \emph{relative dimension} between distributions, as it relates to existing notions of information such as the Minkowski and Renyi dimensions.

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Abstract:Theoretical studies on transfer learning or domain adaptation have so far focused on situations with a known hypothesis class or model; however in practice, some amount of model selection is usually involved, often appearing under the umbrella term of hyperparameter-tuning: for example, one may think of the problem of tuning for the right neural network architecture towards a target task, while leveraging data from a related source task. Now, in addition to the usual tradeoffs on approximation vs estimation errors involved in model selection, this problem brings in a new complexity term, namely, the transfer distance between source and target distributions, which is known to vary with the choice of hypothesis class. We present a first study of this problem, focusing on classification; in particular, the analysis reveals some remarkable phenomena: adaptive rates, i.e., those achievable with no distributional information, can be arbitrarily slower than oracle rates, i.e., when given knowledge on distances.

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Abstract:In bandits with distribution shifts, one aims to automatically detect an unknown number $L$ of changes in reward distribution, and restart exploration when necessary. While this problem remained open for many years, a recent breakthrough of Auer et al. (2018, 2019) provide the first adaptive procedure to guarantee an optimal (dynamic) regret $\sqrt{LT}$, for $T$ rounds, with no knowledge of $L$. However, not all distributional shifts are equally severe, e.g., suppose no best arm switches occur, then we cannot rule out that a regret $O(\sqrt{T})$ may remain possible; in other words, is it possible to achieve dynamic regret that optimally scales only with an unknown number of severe shifts? This unfortunately has remained elusive, despite various attempts (Auer et al., 2019, Foster et al., 2020). We resolve this problem in the case of two-armed bandits: we derive an adaptive procedure that guarantees a dynamic regret of order $\tilde{O}(\sqrt{\tilde{L} T})$, where $\tilde L \ll L$ captures an unknown number of severe best arm changes, i.e., with significant switches in rewards, and which last sufficiently long to actually require a restart. As a consequence, for any number $L$ of distributional shifts outside of these severe shifts, our procedure achieves regret just $\tilde{O}(\sqrt{T})\ll \tilde{O}(\sqrt{LT})$. Finally, we note that our notion of severe shift applies in both classical settings of stochastic switching bandits and of adversarial bandits.

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Abstract:We consider nonparametric classification with smooth regression functions, where it is well known that notions of margin in $E[Y|X]$ determine fast or slow rates in both active and passive learning. Here we elucidate a striking distinction between the two settings. Namely, we show that some seemingly benign nuances in notions of margin -- involving the uniqueness of the Bayes classifier, and which have no apparent effect on rates in passive learning -- determine whether or not any active learner can outperform passive learning rates. In particular, for Audibert-Tsybakov's margin condition (allowing general situations with non-unique Bayes classifiers), no active learner can gain over passive learning in commonly studied settings where the marginal on $X$ is near uniform. Our results thus negate the usual intuition from past literature that active rates should improve over passive rates in nonparametric settings.

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Abstract:Insecure Internet of things (IoT) devices pose significant threats to critical infrastructure and the Internet at large; detecting anomalous behavior from these devices remains of critical importance, but fast, efficient, accurate anomaly detection (also called "novelty detection") for these classes of devices remains elusive. One-Class Support Vector Machines (OCSVM) are one of the state-of-the-art approaches for novelty detection (or anomaly detection) in machine learning, due to their flexibility in fitting complex nonlinear boundaries between {normal} and {novel} data. IoT devices in smart homes and cities and connected building infrastructure present a compelling use case for novelty detection with OCSVM due to the variety of devices, traffic patterns, and types of anomalies that can manifest in such environments. Much previous research has thus applied OCSVM to novelty detection for IoT. Unfortunately, conventional OCSVMs introduce significant memory requirements and are computationally expensive at prediction time as the size of the train set grows, requiring space and time that scales with the number of training points. These memory and computational constraints can be prohibitive in practical, real-world deployments, where large training sets are typically needed to develop accurate models when fitting complex decision boundaries. In this work, we extend so-called Nystr\"om and (Gaussian) Sketching approaches to OCSVM, by combining these methods with clustering and Gaussian mixture models to achieve significant speedups in prediction time and space in various IoT settings, without sacrificing detection accuracy.

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