اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNearly Tight Bounds for Robust Proper Learning of Halfspaces with a Margin
84
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study the problem of {em properly} learning large margin halfspaces in the agnostic PAC model. In more detail, we study the complexity of properly learning $d$-dimensional halfspaces on the unit ball within misclassification error $alpha cdot mathrm{OPT}_{gamma} + epsilon$, where $mathrm{OPT}_{gamma}$ is the optimal $gamma$-margin error rate and $alpha geq 1$ is the approximation ratio. We give learning algorithms and computational hardness results for this problem, for all values of the approximation ratio $alpha geq 1$, that are nearly-matching for a range of parameters. Specifically, for the natural setting that $alpha$ is any constant bigger than one, we provide an essentially tight complexity characterization. On the positive side, we give an $alpha = 1.01$-approximate proper learner that uses $O(1/(epsilon^2gamma^2))$ samples (which is optimal) and runs in time $mathrm{poly}(d/epsilon) cdot 2^{tilde{O}(1/gamma^2)}$. On the negative side, we show that {em any} constant factor approximate proper learner has runtime $mathrm{poly}(d/epsilon) cdot 2^{(1/gamma)^{2-o(1)}}$, assuming the Exponential Time Hypothesis.
قيم البحث
اقرأ أيضاً
We study the computational complexity of adversarially robust proper learning of halfspaces in the distribution-independent agnostic PAC model, with a focus on $L_p$ perturbations. We give a computationally efficient learning algorithm and a nearly m
atching computational hardness result for this problem. An interesting implication of our findings is that the $L_{infty}$ perturbations case is provably computationally harder than the case $2 leq p < infty$.
We study efficient PAC learning of homogeneous halfspaces in $mathbb{R}^d$ in the presence of malicious noise of Valiant~(1985). This is a challenging noise model and only until recently has near-optimal noise tolerance bound been established under t
he mild condition that the unlabeled data distribution is isotropic log-concave. However, it remains unsettled how to obtain the optimal sample complexity simultaneously. In this work, we present a new analysis for the algorithm of Awasthi~et~al.~(2017) and show that it essentially achieves the near-optimal sample complexity bound of $tilde{O}(d)$, improving the best known result of $tilde{O}(d^2)$. Our main ingredient is a novel incorporation of a matrix Chernoff-type inequality to bound the spectrum of an empirical covariance matrix for well-behaved distributions, in conjunction with a careful exploration of the localization schemes of Awasthi~et~al.~(2017). We further extend the algorithm and analysis to the more general and stronger nasty noise model of Bshouty~et~al.~(2002), showing that it is still possible to achieve near-optimal noise tolerance and sample complexity in polynomial time.
We study {em online} active learning of homogeneous halfspaces in $mathbb{R}^d$ with adversarial noise where the overall probability of a noisy label is constrained to be at most $ u$. Our main contribution is a Perceptron-like online active learning
algorithm that runs in polynomial time, and under the conditions that the marginal distribution is isotropic log-concave and $ u = Omega(epsilon)$, where $epsilon in (0, 1)$ is the target error rate, our algorithm PAC learns the underlying halfspace with near-optimal label complexity of $tilde{O}big(d cdot polylog(frac{1}{epsilon})big)$ and sample complexity of $tilde{O}big(frac{d}{epsilon} big)$. Prior to this work, existing online algorithms designed for tolerating the adversarial noise are subject to either label complexity polynomial in $frac{1}{epsilon}$, or suboptimal noise tolerance, or restrictive marginal distributions. With the additional prior knowledge that the underlying halfspace is $s$-sparse, we obtain attribute-efficient label complexity of $tilde{O}big( s cdot polylog(d, frac{1}{epsilon}) big)$ and sample complexity of $tilde{O}big(frac{s}{epsilon} cdot polylog(d) big)$. As an immediate corollary, we show that under the agnostic model where no assumption is made on the noise rate $ u$, our active learner achieves an error rate of $O(OPT) + epsilon$ with the same running time and label and sample complexity, where $OPT$ is the best possible error rate achievable by any homogeneous halfspace.
We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples $(mathbf{x}, y)$ from an unknown distribution on $mathbb{R}^d times { pm 1}$, whose marginal distr
ibution on $mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $mathrm{OPT}+epsilon$, where $mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. In the latter problem, given labeled examples $(mathbf{x}, y)$ from an unknown distribution on $mathbb{R}^d times mathbb{R}$, whose marginal distribution on $mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with square loss $mathrm{OPT}+epsilon$, where $mathrm{OPT}$ is the square loss of the best-fitting ReLU. We prove Statistical Query (SQ) lower bounds of $d^{mathrm{poly}(1/epsilon)}$ for both of these problems. Our SQ lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.
In the Best-$k$-Arm problem, we are given $n$ stochastic bandit arms, each associated with an unknown reward distribution. We are required to identify the $k$ arms with the largest means by taking as few samples as possible. In this paper, we make pr
ogress towards a complete characterization of the instance-wise sample complexity bounds for the Best-$k$-Arm problem. On the lower bound side, we obtain a novel complexity term to measure the sample complexity that every Best-$k$-Arm instance requires. This is derived by an interesting and nontrivial reduction from the Best-$1$-Arm problem. We also provide an elimination-based algorithm that matches the instance-wise lower bound within doubly-logarithmic factors. The sample complexity of our algorithm strictly dominates the state-of-the-art for Best-$k$-Arm (module constant factors).
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
