Differentially Private Nonparametric Regression Under a Growth Condition

• URL: http://arxiv.org/abs/2111.12786v1
• Date: Wed, 24 Nov 2021 20:36:01 GMT
• Title: Differentially Private Nonparametric Regression Under a Growth Condition
• Authors: Noah Golowich
• Abstract summary: Given a real-valued hypothesis class $mathcalH$, we investigate under what conditions there is a differentially private algorithm which learns an optimal hypothesis. We show that under the relaxed condition $lim inf_eta downarrow 0 eta cdot rm sfat_eta(mathcalH)$ = 0$, $mathcalH$ is privately learnable.
• Score: 9.416757363901295
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Given a real-valued hypothesis class $\mathcal{H}$, we investigate under what conditions there is a differentially private algorithm which learns an optimal hypothesis from $\mathcal{H}$ given i.i.d. data. Inspired by recent results for the related setting of binary classification (Alon et al., 2019; Bun et al., 2020), where it was shown that online learnability of a binary class is necessary and sufficient for its private learnability, Jung et al. (2020) showed that in the setting of regression, online learnability of $\mathcal{H}$ is necessary for private learnability. Here online learnability of $\mathcal{H}$ is characterized by the finiteness of its $\eta$-sequential fat shattering dimension, ${\rm sfat}_\eta(\mathcal{H})$, for all $\eta > 0$. In terms of sufficient conditions for private learnability, Jung et al. (2020) showed that $\mathcal{H}$ is privately learnable if $\lim_{\eta \downarrow 0} {\rm sfat}_\eta(\mathcal{H})$ is finite, which is a fairly restrictive condition. We show that under the relaxed condition $\lim \inf_{\eta \downarrow 0} \eta \cdot {\rm sfat}_\eta(\mathcal{H}) = 0$, $\mathcal{H}$ is privately learnable, establishing the first nonparametric private learnability guarantee for classes $\mathcal{H}$ with ${\rm sfat}_\eta(\mathcal{H})$ diverging as $\eta \downarrow 0$. Our techniques involve a novel filtering procedure to output stable hypotheses for nonparametric function classes.

Related papers

• Revisiting Agnostic PAC Learning [30.67561230812141]
  PAC learning, dating back to Valiant'84 and Vapnik and Chervonenkis'64,'74, is a classic model for studying supervised learning. Empirical Risk Minimization (ERM) is a natural learning algorithm, where one simply outputs the hypothesis from $mathcalH$ making the fewest mistakes on the training data. We revisit PAC learning and first show that ERM is in fact sub-optimal if we treat the performance of the best hypothesis, denoted $tau:=Pr_mathcalD[hstar_math
  arXiv  Detail & Related papers  (2024-07-29T08:20:49Z)
• Scaling Up Differentially Private LASSO Regularized Logistic Regression via Faster Frank-Wolfe Iterations [51.14495595270775]
  We adapt the Frank-Wolfe algorithm for $L_1$ penalized linear regression to be aware of sparse inputs and to use them effectively. Our results demonstrate that this procedure can reduce runtime by a factor of up to $2,200times$, depending on the value of the privacy parameter $epsilon$ and the sparsity of the dataset.
  arXiv  Detail & Related papers  (2023-10-30T19:52:43Z)
• Estimation and Inference in Distributional Reinforcement Learning [28.253677740976197]
  We show that a dataset of size $widetilde Oleft(frac|mathcalS||mathcalA|epsilon2 (1-gamma)4right)$ suffices to ensure the Kolmogorov metric and total variation metric between $hatetapi$ and $etapi$ is below $epsilon$ with high probability. Our findings give rise to a unified approach to statistical inference of a wide class of statistical functionals of $etapi$.
  arXiv  Detail & Related papers  (2023-09-29T14:14:53Z)
• On the Optimality of Misspecified Kernel Ridge Regression [13.995944403996566]
  We show that KRR is minimax optimal for any $sin (0,1)$ when the $mathcalH$ is a Sobolev RKHS.
  arXiv  Detail & Related papers  (2023-05-12T04:12:12Z)
• On Optimal Learning Under Targeted Data Poisoning [48.907813854832206]
  In this work we aim to characterize the smallest achievable error $epsilon=epsilon(eta)$ by the learner in the presence of such an adversary. Remarkably, we show that the upper bound can be attained by a deterministic learner.
  arXiv  Detail & Related papers  (2022-10-06T06:49:48Z)
• Littlestone Classes are Privately Online Learnable [28.04975353867202]
  We consider the problem of online classification under a privacy constraint. In this setting a learner observes sequentially a stream of labelled examples $(x_t, y_t)$, for $1 leq t leq T$, and returns at each iteration a hypothesis $h_t$ which is used to predict the label of each new example $x_t$. The learner's performance is measured by her regret against a known hypothesis class $mathcalH$.
  arXiv  Detail & Related papers  (2021-06-25T09:08:33Z)
• Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry [69.24618367447101]
  Up to logarithmic factors the optimal excess population loss of any $(varepsilon,delta)$-differently private is $sqrtlog(d)/n + sqrtd/varepsilon n.$ We show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $sqrtlog(d)/n + (log(d)/varepsilon n)2/3.
  arXiv  Detail & Related papers  (2021-03-02T06:53:44Z)
• Optimal Regret Algorithm for Pseudo-1d Bandit Convex Optimization [51.23789922123412]
  We study online learning with bandit feedback (i.e. learner has access to only zeroth-order oracle) where cost/reward functions admit a "pseudo-1d" structure. We show a lower bound of $min(sqrtdT, T3/4)$ for the regret of any algorithm, where $T$ is the number of rounds. We propose a new algorithm sbcalg that combines randomized online gradient descent with a kernelized exponential weights method to exploit the pseudo-1d structure effectively.
  arXiv  Detail & Related papers  (2021-02-15T08:16:51Z)
• $Q$-learning with Logarithmic Regret [60.24952657636464]
  We prove that an optimistic $Q$-learning enjoys a $mathcalOleft(fracSAcdot mathrmpolyleft(Hright)Delta_minlogleft(SATright)right)$ cumulative regret bound, where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, $T$ is the total number of steps, and $Delta_min$ is the minimum sub-optimality gap.
  arXiv  Detail & Related papers  (2020-06-16T13:01:33Z)
• Agnostic Learning of a Single Neuron with Gradient Descent [92.7662890047311]
  We consider the problem of learning the best-fitting single neuron as measured by the expected square loss. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$.
  arXiv  Detail & Related papers  (2020-05-29T07:20:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.