اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the Estimation of Derivatives Using Plug-in KRR Estimators
69
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study the problem of estimating the derivatives of the regression function, which has a wide range of applications as a key nonparametric functional of unknown functions. Standard analysis may be tailored to specific derivative orders, and parameter tuning remains a daunting challenge particularly for high-order derivatives. In this article, we propose a simple plug-in kernel ridge regression (KRR) estimator in nonparametric regression with random design that is broadly applicable for multi-dimensional support and arbitrary mixed-partial derivatives. We provide a non-asymptotic analysis to study the behavior of the proposed estimator, leading to two error bounds for a general class of kernels under the strong $L_infty$ norm. In a concrete example specialized to kernels with polynomially decaying eigenvalues, the proposed estimator recovers the minimax optimal rate up to a logarithmic factor for estimating derivatives of functions in Holder class. Interestingly, the proposed estimator achieves the optimal rate of convergence with the same choice of tuning parameter for any order of derivatives. Hence, the proposed estimator enjoys a remarkable textit{plug-in property} for derivatives in that it automatically adapts to the order of derivatives to be estimated, enabling easy tuning in practice. Our simulation studies show favorable finite sample performance of the proposed method relative to several existing methods.
قيم البحث
اقرأ أيضاً
Machine learning models have traditionally been developed under the assumption that the training and test distributions match exactly. However, recent success in few-shot learning and related problems are encouraging signs that these models can be ad
apted to more realistic settings where train and test distributions differ. Unfortunately, there is severely limited theoretical support for these algorithms and little is known about the difficulty of these problems. In this work, we provide novel information-theoretic lower-bounds on minimax rates of convergence for algorithms that are trained on data from multiple sources and tested on novel data. Our bounds depend intuitively on the information shared between sources of data, and characterize the difficulty of learning in this setting for arbitrary algorithms. We demonstrate these bounds on a hierarchical Bayesian model of meta-learning, computing both upper and lower bounds on parameter estimation via maximum-a-posteriori inference.
Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerat
The maximum mean discrepancy (MMD) is a kernel-based distance between probability distributions useful in many applications (Gretton et al. 2012), bearing a simple estimator with pleasing computational and statistical properties. Being able to effici
ently estimate the variance of this estimator is very helpful to various problems in two-sample testing. Towards this end, Bounliphone et al. (2016) used the theory of U-statistics to derive estimators for the variance of an MMD estimator, and differences between two such estimators. Their estimator, however, drops lower-order terms, and is unnecessarily biased. We show in this note - extending and correcting work of Sutherland et al. (2017) - that we can find a truly unbiased estimator for the actual variance of both the squared MMD estimator and the difference of two correlated squared MMD estimators, at essentially no additional computational cost.
Conditional estimation given specific covariate values (i.e., local conditional estimation or functional estimation) is ubiquitously useful with applications in engineering, social and natural sciences. Existing data-driven non-parametric estimators
mostly focus on structured homogeneous data (e.g., weakly independent and stationary data), thus they are sensitive to adversarial noise and may perform poorly under a low sample size. To alleviate these issues, we propose a new distributionally robust estimator that generates non-parametric local estimates by minimizing the worst-case conditional expected loss over all adversarial distributions in a Wasserstein ambiguity set. We show that despite being generally intractable, the local estimator can be efficiently found via convex optimization under broadly applicable settings, and it is robust to the corruption and heterogeneity of the data. Experiments with synthetic and MNIST datasets show the competitive performance of this new class of estimators.
The fastICA method is a popular dimension reduction technique used to reveal patterns in data. Here we show both theoretically and in practice that the approximations used in fastICA can result in patterns not being successfully recognised. We demons
trate this problem using a two-dimensional example where a clear structure is immediately visible to the naked eye, but where the projection chosen by fastICA fails to reveal this structure. This implies that care is needed when applying fastICA. We discuss how the problem arises and how it is intrinsically connected to the approximations that form the basis of the computational efficiency of fastICA.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
