Early stopping of iterative algorithms is a widely-used form of regularization in statistical learning, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including $L^2$-boost, LogitBoost and AdaBoost, among others), we connect the performance of a stopped iterate to the localized Rademacher/Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

Early stopping of iterative algorithms is a widely-used form of regularization in statistical learning, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including $L^2$-boost, LogitBoost and AdaBoost, among others), we connect the performance of a stopped iterate to the localized Rademacher/Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

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Early stopping of iterative algorithms is an algorithmic regularization method to avoid over-fitting in estimation and classification. In this paper, we show that early stopping can also be applied to obtain the minimax optimal testing in a general non-parametric setup. Specifically, a Wald-type test statistic is obtained based on an iterated estimate produced by functional gradient descent algorithms in a reproducing kernel Hilbert space. A notable contribution is to establish a ``sharp'' stopping rule: when the number of iterations achieves an optimal order, testing optimality is achievable; otherwise, testing optimality becomes impossible. As a by-product, a similar sharpness result is also derived for minimax optimal estimation under early stopping. All obtained results hold for various kernel classes, including Sobolev smoothness classes and Gaussian kernel classes.

Early stopping of iterative algorithms is a widely-used form of regularization in statistics, commonly used in conjunction with boosting and related gradienttype algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization.

Auto-scheduling for tensor programs is a process where a search algorithm automatically explores candidate schedules (program transformations) for a given program on a target hardware platform to improve its performance. However this can be a very time consuming process depending on the complexity of the tensor program and the capacity of the target device, with often many thousands of program variants being explored. To address this, in this paper we introduce the idea of transfer-tuning, a novel approach to identify and reuse auto-schedules between tensor programs. We demonstrate this concept using Deep Neural Networks (DNNs), taking sets of auto-schedules from pre-tuned DNNs and using them to reduce the inference time of a new DNN. We compare transfer-tuning against the state-of-the-art Ansor auto-scheduler, defining the maximum possible speedup for a given DNN model as what Ansor achieves using its recommended full tuning time. On a server-class CPU and across 11 widely used DNN models, we observe that transfer-tuning achieves up to $88.41\%$ ($49.13\%$ on average) of this maximum speedup, while Ansor requires $6.5\times$ more search time on average to match it. We also evaluate transfer-tuning on a constrained edge CPU and observe that the differences in search time are exacerbated, with Ansor requiring $10.8\times$ more time on average to match transfer-tuning's speedup, which further demonstrates its value. Our code is available at https://www.github.com/gicLAB/transfer-tuning

We develop new upper and lower bounds on the $\varepsilon$-entropy of a unit ball in a reproducing kernel Hilbert space induced by some Mercer kernel $K$. Our bounds are based on the behaviour of eigenvalues of a corresponding integral operator. In our approach we exploit an ellipsoidal structure of a unit ball in RKHS and a previous work on covering numbers of an ellipsoid in the euclidean space obtained by Dumer, Pinsker and Prelov. We present a number of applications of our main bound, such as its tightness for a practically important case of the Gaussian kernel. Further, we develop a series of lower bounds on the $\varepsilon$-entropy that can be established from a connection between covering numbers of a ball in RKHS and a quantization of a Gaussian Random Field that corresponds to the kernel $K$ by the Kosambi-Karhunen-Lo\`eve transform.

Several recent works have shown separation results between deep neural networks, and hypothesis classes with inferior approximation capacity such as shallow networks or kernel classes. On the other hand, the fact that deep networks can efficiently express a target function does not mean this target function can be learned efficiently by deep neural networks. In this work we study the intricate connection between learnability and approximation capacity. We show that learnability with deep networks of a target function depends on the ability of simpler classes to approximate the target. Specifically, we show that a necessary condition for a function to be learnable by gradient descent on deep neural networks is to be able to approximate the function, at least in a weak sense, with shallow neural networks. We also show that a class of functions can be learned by an efficient statistical query algorithm if and only if it can be approximated in a weak sense by some kernel class. We give several examples of functions which demonstrate depth separation, and conclude that they cannot be efficiently learned, even by a hypothesis class that can efficiently approximate them.

Early stopping of iterative algorithms is a widely-used form of regularization in statistical learning, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including $L 2$-boost, LogitBoost and AdaBoost, among others), we connect the performance of a stopped iterate to the localized Rademacher/Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

Early stopping of iterative algorithms is an algorithmic regularization method to avoid over-fitting in estimation and classification. In this paper, we show that early stopping can also be applied to obtain the minimax optimal testing in a general non-parametric setup. Specifically, a Wald-type test statistic is obtained based on an iterated estimate produced by functional gradient descent algorithms in a reproducing kernel Hilbert space. A notable contribution is to establish a sharp'' stopping rule: when the number of iterations achieves an optimal order, testing optimality is achievable; otherwise, testing optimality becomes impossible. As a by-product, a similar sharpness result is also derived for minimax optimal estimation under early stopping. All obtained results hold for various kernel classes, including Sobolev smoothness classes and Gaussian kernel classes.