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We establish optimal Statistical Query (SQ) lower bounds for robustly learning certain families of discrete high-dimensional distributions. In particular, we show that no efficient SQ algorithm with access to an $\epsilon$-corrupted binary product distribution can learn its mean within $\ell_2$-error $o(\epsilon \sqrt{\log(1/\epsilon)})$. Similarly, we show that no efficient SQ algorithm with access to an $\epsilon$-corrupted ferromagnetic high-temperature Ising model can learn the model to total variation distance $o(\epsilon \log(1/\epsilon))$. Our SQ lower bounds match the error guarantees of known algorithms for these problems, providing evidence that current upper bounds for these tasks are best possible. At the technical level, we develop a generic SQ lower bound for discrete high-dimensional distributions starting from low dimensional moment matching constructions that we believe will find other applications. Additionally, we introduce new ideas to analyze these moment-matching constructions for discrete univariate distributions.

This work focuses on decentralized stochastic optimization in the presence of Byzantine attacks. During the optimization process, an unknown number of malfunctioning or malicious nodes, which we term as Byzantine workers, disobey the algorithmic protocol and send wrong messages to their neighbors. Even though various Byzantine-resilient algorithms have been developed for distributed stochastic optimization, we show that there are still two major challenges during the designation of robust aggregation rules suitable for decentralized stochastic optimization: disagreement and non-doubly stochastic mixing matrix. This paper provides a comprehensive analysis disclosing the negative effects of these two issues, and gives guidelines of designing favorable Byzantine-resilient decentralized stochastic optimization algorithms. Following the guidelines, we propose an iterative filtering-based robust aggregation rule termed iterative outlier scissor (IOS), which has provable Byzantine-resilience. Numerical experiments demonstrate the effectiveness of IOS.

Information about action costs is critical for real-world AI planning applications. Rather than rely solely on declarative action models, recent approaches also use black-box external action cost estimators, often learned from data, that are applied during the planning phase. These, however, can be computationally expensive, and produce uncertain values. In this paper we suggest a generalization of deterministic planning with action costs that allows selecting between multiple estimators for action cost, to balance computation time against bounded estimation uncertainty. This enables a much richer -- and correspondingly more realistic -- problem representation. Importantly, it allows planners to bound plan accuracy, thereby increasing reliability, while reducing unnecessary computational burden, which is critical for scaling to large problems. We introduce a search algorithm, generalizing $A^*$, that solves such planning problems, and additional algorithmic extensions. In addition to theoretical guarantees, extensive experiments show considerable savings in runtime compared to alternatives.

In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods. Our unified theorem only requires to verify several representative conditions and is not tailored to any specific algorithm. As a direct application, we recover expected and almost sure convergence results of the stochastic gradient method (SGD) and random reshuffling (RR) under more general settings. Moreover, we establish new expected and almost sure convergence results for the stochastic proximal gradient method (prox-SGD) and stochastic model-based methods (SMM) for nonsmooth nonconvex optimization problems. These applications reveal that our unified theorem provides a plugin-type convergence analysis and strong convergence guarantees for a wide class of stochastic optimization methods.

Exponential generalization bounds with near-tight rates have recently been established for uniformly stable learning algorithms. The notion of uniform stability, however, is stringent in the sense that it is invariant to the data-generating distribution. Under the weaker and distribution dependent notions of stability such as hypothesis stability and $L_2$-stability, the literature suggests that only polynomial generalization bounds are possible in general cases. The present paper addresses this long standing tension between these two regimes of results and makes progress towards relaxing it inside a classic framework of confidence-boosting. To this end, we first establish an in-expectation first moment generalization error bound for potentially randomized learning algorithms with $L_2$-stability, based on which we then show that a properly designed subbagging process leads to near-tight exponential generalization bounds over the randomness of both data and algorithm. We further substantialize these generic results to stochastic gradient descent (SGD) to derive improved high-probability generalization bounds for convex or non-convex optimization problems with natural time decaying learning rates, which have not been possible to prove with the existing hypothesis stability or uniform stability based results.

Objectives: This paper develops two algorithms to achieve federated generalized linear mixed effect models (GLMM), and compares the developed model's outcomes with each other, as well as that from the standard R package (`lme4'). Methods: The log-likelihood function of GLMM is approximated by two numerical methods (Laplace approximation and Gaussian Hermite approximation), which supports federated decomposition of GLMM to bring computation to data. Results: Our developed method can handle GLMM to accommodate hierarchical data with multiple non-independent levels of observations in a federated setting. The experiment results demonstrate comparable (Laplace) and superior (Gaussian-Hermite) performances with simulated and real-world data. Conclusion: We developed and compared federated GLMMs with different approximations, which can support researchers in analyzing biomedical data to accommodate mixed effects and address non-independence due to hierarchical structures (i.e., institutes, region, country, etc.).

We study to what extent may stochastic gradient descent (SGD) be understood as a "conventional" learning rule that achieves generalization performance by obtaining a good fit to training data. We consider the fundamental stochastic convex optimization framework, where (one pass, without-replacement) SGD is classically known to minimize the population risk at rate $O(1/\sqrt n)$, and prove that, surprisingly, there exist problem instances where the SGD solution exhibits both empirical risk and generalization gap of $\Omega(1)$. Consequently, it turns out that SGD is not algorithmically stable in any sense, and its generalization ability cannot be explained by uniform convergence or any other currently known generalization bound technique for that matter (other than that of its classical analysis). We then continue to analyze the closely related with-replacement SGD, for which we show that an analogous phenomenon does not occur and prove that its population risk does in fact converge at the optimal rate. Finally, we interpret our main results in the context of without-replacement SGD for finite-sum convex optimization problems, and derive upper and lower bounds for the multi-epoch regime that significantly improve upon previously known results.

In machine learning, we traditionally evaluate the performance of a single model, averaged over a collection of test inputs. In this work, we propose a new approach: we measure the performance of a collection of models when evaluated on a $\textit{single input point}$. Specifically, we study a point's $\textit{profile}$: the relationship between models' average performance on the test distribution and their pointwise performance on this individual point. We find that profiles can yield new insights into the structure of both models and data -- in and out-of-distribution. For example, we empirically show that real data distributions consist of points with qualitatively different profiles. On one hand, there are "compatible" points with strong correlation between the pointwise and average performance. On the other hand, there are points with weak and even $\textit{negative}$ correlation: cases where improving overall model accuracy actually $\textit{hurts}$ performance on these inputs. We prove that these experimental observations are inconsistent with the predictions of several simplified models of learning proposed in prior work. As an application, we use profiles to construct a dataset we call CIFAR-10-NEG: a subset of CINIC-10 such that for standard models, accuracy on CIFAR-10-NEG is $\textit{negatively correlated}$ with accuracy on CIFAR-10 test. This illustrates, for the first time, an OOD dataset that completely inverts "accuracy-on-the-line" (Miller, Taori, Raghunathan, Sagawa, Koh, Shankar, Liang, Carmon, and Schmidt 2021)

Recent studies have shown that heavy tails can emerge in stochastic optimization and that the heaviness of the tails has links to the generalization error. While these studies have shed light on interesting aspects of the generalization behavior in modern settings, they relied on strong topological and statistical regularity assumptions, which are hard to verify in practice. Furthermore, it has been empirically illustrated that the relation between heavy tails and generalization might not always be monotonic in practice, contrary to the conclusions of existing theory. In this study, we establish novel links between the tail behavior and generalization properties of stochastic gradient descent (SGD), through the lens of algorithmic stability. We consider a quadratic optimization problem and use a heavy-tailed stochastic differential equation as a proxy for modeling the heavy-tailed behavior emerging in SGD. We then prove uniform stability bounds, which reveal the following outcomes: (i) Without making any exotic assumptions, we show that SGD will not be stable if the stability is measured with the squared-loss $x\mapsto x^2$, whereas it in turn becomes stable if the stability is instead measured with a surrogate loss $x\mapsto |x|^p$ with some $p<2$. (ii) Depending on the variance of the data, there exists a \emph{`threshold of heavy-tailedness'} such that the generalization error decreases as the tails become heavier, as long as the tails are lighter than this threshold. This suggests that the relation between heavy tails and generalization is not globally monotonic. (iii) We prove matching lower-bounds on uniform stability, implying that our bounds are tight in terms of the heaviness of the tails. We support our theory with synthetic and real neural network experiments.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.