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Samory Kpotufe

Publications and source records attributed to Samory Kpotufe.

35 records · Page 2Linked to original sources
arXiv2020

Self-Tuning Bandits over Unknown Covariate-Shifts

Bandits with covariates, a.k.a. contextual bandits, address situations where optimal actions (or arms) at a given time $t$, depend on a context $x_t$, e.g., a new patient's medical history, a consumer's past purchases. While it is understood that the distribution of contexts might change over time, e.g., due to seasonalities, or deployment to new environments, the bulk of studies concern the most adversarial such changes, resulting in regret bounds that are often worst-case in nature. Covariate-shift on the other hand has been considered in classification as a middle-ground formalism that can capture mild to relatively severe changes in distributions. We consider nonparametric bandits under such middle-ground scenarios, and derive new regret bounds that tightly capture a continuum of changes in context distribution. Furthermore, we show that these rates can be adaptively attained without knowledge of the time of shift nor the amount of shift.

stat.ML↗
arXiv2020

Feature Extraction for Novelty Detection in Network Traffic

Data representation plays a critical role in the performance of novelty detection (or ``anomaly detection'') methods in machine learning. The data representation of network traffic often determines the effectiveness of these models as much as the model itself. The wide range of novel events that network operators need to detect (e.g., attacks, malware, new applications, changes in traffic demands) introduces the possibility for a broad range of possible models and data representations. In each scenario, practitioners must spend significant effort extracting and engineering features that are most predictive for that situation or application. While anomaly detection is well-studied in computer networking, much existing work develops specific models that presume a particular representation -- often IPFIX/NetFlow. Yet, other representations may result in higher model accuracy, and the rise of programmable networks now makes it more practical to explore a broader range of representations. To facilitate such exploration, we develop a systematic framework, open-source toolkit, and public Python library that makes it both possible and easy to extract and generate features from network traffic and perform and end-to-end evaluation of these representations across most prevalent modern novelty detection models. We first develop and publicly release an open-source tool, an accompanying Python library (NetML), and end-to-end pipeline for novelty detection in network traffic. Second, we apply this tool to five different novelty detection problems in networking, across a range of scenarios from attack detection to novel device detection. Our findings general insights and guidelines concerning which features appear to be more appropriate for particular situations.

cs.NI↗
arXiv2020

A No-Free-Lunch Theorem for MultiTask Learning

Multitask learning and related areas such as multi-source domain adaptation address modern settings where datasets from $N$ related distributions $\{P_t\}$ are to be combined towards improving performance on any single such distribution ${\cal D}$. A perplexing fact remains in the evolving theory on the subject: while we would hope for performance bounds that account for the contribution from multiple tasks, the vast majority of analyses result in bounds that improve at best in the number $n$ of samples per task, but most often do not improve in $N$. As such, it might seem at first that the distributional settings or aggregation procedures considered in such analyses might be somehow unfavorable; however, as we show, the picture happens to be more nuanced, with interestingly hard regimes that might appear otherwise favorable. In particular, we consider a seemingly favorable classification scenario where all tasks $P_t$ share a common optimal classifier $h^*,$ and which can be shown to admit a broad range of regimes with improved oracle rates in terms of $N$ and $n$. Some of our main results are as follows: $\bullet$ We show that, even though such regimes admit minimax rates accounting for both $n$ and $N$, no adaptive algorithm exists; that is, without access to distributional information, no algorithm can guarantee rates that improve with large $N$ for $n$ fixed. $\bullet$ With a bit of additional information, namely, a ranking of tasks $\{P_t\}$ according to their distance to a target ${\cal D}$, a simple rank-based procedure can achieve near optimal aggregations of tasks' datasets, despite a search space exponential in $N$. Interestingly, the optimal aggregation might exclude certain tasks, even though they all share the same $h^*$.

cs.LG↗
arXiv2020

On the Value of Target Data in Transfer Learning

We aim to understand the value of additional labeled or unlabeled target data in transfer learning, for any given amount of source data; this is motivated by practical questions around minimizing sampling costs, whereby, target data is usually harder or costlier to acquire than source data, but can yield better accuracy. To this aim, we establish the first minimax-rates in terms of both source and target sample sizes, and show that performance limits are captured by new notions of discrepancy between source and target, which we refer to as transfer exponents.

cs.LG↗
arXiv2019

Gaussian Sketching yields a J-L Lemma in RKHS

The main contribution of the paper is to show that Gaussian sketching of a kernel-Gram matrix $\boldsymbol K$ yields an operator whose counterpart in an RKHS $\mathcal H$, is a \emph{random projection} operator---in the spirit of Johnson-Lindenstrauss (J-L) lemma. To be precise, given a random matrix $Z$ with i.i.d. Gaussian entries, we show that a sketch $Z\boldsymbol{K}$ corresponds to a particular random operator in (infinite-dimensional) Hilbert space $\mathcal H$ that maps functions $f \in \mathcal H$ to a low-dimensional space $\mathbb R^d$, while preserving a weighted RKHS inner-product of the form $\langle f, g \rangle_{\Sigma} \doteq \langle f, \Sigma^3 g \rangle_{\mathcal H}$, where $\Sigma$ is the \emph{covariance} operator induced by the data distribution. In particular, under similar assumptions as in kernel PCA (KPCA), or kernel $k$-means (K-$k$-means), well-separated subsets of feature-space $\{K(\cdot, x): x \in \cal X\}$ remain well-separated after such operation, which suggests similar benefits as in KPCA and/or K-$k$-means, albeit at the much cheaper cost of a random projection. In particular, our convergence rates suggest that, given a large dataset $\{X_i\}_{i=1}^N$ of size $N$, we can build the Gram matrix $\boldsymbol K$ on a much smaller subsample of size $n\ll N$, so that the sketch $Z\boldsymbol K$ is very cheap to obtain and subsequently apply as a projection operator on the original data $\{X_i\}_{i=1}^N$. We verify these insights empirically on synthetic data, and on real-world clustering applications.

stat.ML↗
arXiv2018

Quickshift++: Provably Good Initializations for Sample-Based Mean Shift

We provide initial seedings to the Quick Shift clustering algorithm, which approximate the locally high-density regions of the data. Such seedings act as more stable and expressive cluster-cores than the singleton modes found by Quick Shift. We establish statistical consistency guarantees for this modification. We then show strong clustering performance on real datasets as well as promising applications to image segmentation.

cs.LG↗
arXiv2018

Marginal Singularity, and the Benefits of Labels in Covariate-Shift

We present new minimax results that concisely capture the relative benefits of source and target labeled data, under covariate-shift. Namely, we show that the benefits of target labels are controlled by a transfer-exponent $\gamma$ that encodes how singular Q is locally w.r.t. P, and interestingly allows situations where transfer did not seem possible under previous insights. In fact, our new minimax analysis - in terms of $\gamma$ - reveals a continuum of regimes ranging from situations where target labels have little benefit, to regimes where target labels dramatically improve classification. We then show that a recently proposed semi-supervised procedure can be extended to adapt to unknown $\gamma$, and therefore requests labels only when beneficial, while achieving minimax transfer rates.

stat.ML↗
arXiv2017

Achieving the time of $1$-NN, but the accuracy of $k$-NN

We propose a simple approach which, given distributed computing resources, can nearly achieve the accuracy of $k$-NN prediction, while matching (or improving) the faster prediction time of $1$-NN. The approach consists of aggregating denoised $1$-NN predictors over a small number of distributed subsamples. We show, both theoretically and experimentally, that small subsample sizes suffice to attain similar performance as $k$-NN, without sacrificing the computational efficiency of $1$-NN.

math.ST↗
arXiv2017

An Adaptive Strategy for Active Learning with Smooth Decision Boundary

We present the first adaptive strategy for active learning in the setting of classification with smooth decision boundary. The problem of adaptivity (to unknown distributional parameters) has remained opened since the seminal work of Castro and Nowak (2007), which first established (active learning) rates for this setting. While some recent advances on this problem establish adaptive rates in the case of univariate data, adaptivity in the more practical setting of multivariate data has so far remained elusive. Combining insights from various recent works, we show that, for the multivariate case, a careful reduction to univariate-adaptive strategies yield near-optimal rates without prior knowledge of distributional parameters.

stat.ML↗
arXiv2017

Adaptivity to Noise Parameters in Nonparametric Active Learning

This work addresses various open questions in the theory of active learning for nonparametric classification. Our contributions are both statistical and algorithmic: -We establish new minimax-rates for active learning under common \textit{noise conditions}. These rates display interesting transitions -- due to the interaction between noise \textit{smoothness and margin} -- not present in the passive setting. Some such transitions were previously conjectured, but remained unconfirmed. -We present a generic algorithmic strategy for adaptivity to unknown noise smoothness and margin; our strategy achieves optimal rates in many general situations; furthermore, unlike in previous work, we avoid the need for \textit{adaptive confidence sets}, resulting in strictly milder distributional requirements.

stat.ML↗
arXiv2016

Modal-set estimation with an application to clustering

We present a first procedure that can estimate -- with statistical consistency guarantees -- any local-maxima of a density, under benign distributional conditions. The procedure estimates all such local maxima, or $\textit{modal-sets}$, of any bounded shape or dimension, including usual point-modes. In practice, modal-sets can arise as dense low-dimensional structures in noisy data, and more generally serve to better model the rich variety of locally-high-density structures in data. The procedure is then shown to be competitive on clustering applications, and moreover is quite stable to a wide range of settings of its tuning parameter.

stat.ML↗
arXiv2014

Consistent procedures for cluster tree estimation and pruning

For a density $f$ on ${\mathbb R}^d$, a {\it high-density cluster} is any connected component of $\{x: f(x) \geq \lambda\}$, for some $\lambda > 0$. The set of all high-density clusters forms a hierarchy called the {\it cluster tree} of $f$. We present two procedures for estimating the cluster tree given samples from $f$. The first is a robust variant of the single linkage algorithm for hierarchical clustering. The second is based on the $k$-nearest neighbor graph of the samples. We give finite-sample convergence rates for these algorithms which also imply consistency, and we derive lower bounds on the sample complexity of cluster tree estimation. Finally, we study a tree pruning procedure that guarantees, under milder conditions than usual, to remove clusters that are spurious while recovering those that are salient.

stat.ML↗
arXiv2013

Consistency of Causal Inference under the Additive Noise Model

We analyze a family of methods for statistical causal inference from sample under the so-called Additive Noise Model. While most work on the subject has concentrated on establishing the soundness of the Additive Noise Model, the statistical consistency of the resulting inference methods has received little attention. We derive general conditions under which the given family of inference methods consistently infers the causal direction in a nonparametric setting.

cs.LG↗
arXiv2012

Which Spatial Partition Trees are Adaptive to Intrinsic Dimension?

Recent theory work has found that a special type of spatial partition tree - called a random projection tree - is adaptive to the intrinsic dimension of the data from which it is built. Here we examine this same question, with a combination of theory and experiments, for a broader class of trees that includes k-d trees, dyadic trees, and PCA trees. Our motivation is to get a feel for (i) the kind of intrinsic low dimensional structure that can be empirically verified, (ii) the extent to which a spatial partition can exploit such structure, and (iii) the implications for standard statistical tasks such as regression, vector quantization, and nearest neighbor search.

stat.ML↗
arXiv2011

k-NN Regression Adapts to Local Intrinsic Dimension

Many nonparametric regressors were recently shown to converge at rates that depend only on the intrinsic dimension of data. These regressors thus escape the curse of dimension when high-dimensional data has low intrinsic dimension (e.g. a manifold). We show that k-NN regression is also adaptive to intrinsic dimension. In particular our rates are local to a query x and depend only on the way masses of balls centered at x vary with radius. Furthermore, we show a simple way to choose k = k(x) locally at any x so as to nearly achieve the minimax rate at x in terms of the unknown intrinsic dimension in the vicinity of x. We also establish that the minimax rate does not depend on a particular choice of metric space or distribution, but rather that this minimax rate holds for any metric space and doubling measure.

stat.ML↗
arXiv2011

Pruning nearest neighbor cluster trees

Nearest neighbor (k-NN) graphs are widely used in machine learning and data mining applications, and our aim is to better understand what they reveal about the cluster structure of the unknown underlying distribution of points. Moreover, is it possible to identify spurious structures that might arise due to sampling variability? Our first contribution is a statistical analysis that reveals how certain subgraphs of a k-NN graph form a consistent estimator of the cluster tree of the underlying distribution of points. Our second and perhaps most important contribution is the following finite sample guarantee. We carefully work out the tradeoff between aggressive and conservative pruning and are able to guarantee the removal of all spurious cluster structures at all levels of the tree while at the same time guaranteeing the recovery of salient clusters. This is the first such finite sample result in the context of clustering.

stat.ML↗