We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by (tfrac{1}{sigma } ) times that of the uniform distribution; nature then samples an input from this distribution. Here, σ is a parameter that interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of an input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm’s current state; this appears to rule out the standard proof approaches in smoothed analysis.
This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems:
(1) | Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of σ-smooth distributions and the hypothesis class has VC dimension d. We bound the regret by (tilde{O}big (sqrt {T dln (1/sigma)} + dln (T/sigma) big) ) and provide a near-matching lower bound. Our result shows that under smoothed analysis, learnability against adaptive adversaries is characterized by the finiteness of the VC dimension. This is as opposed to the worst-case analysis, where online learnability is characterized by Littlestone dimension (which is infinite even in the extremely restricted case of one-dimensional threshold functions). Our results fully answer an open question of Rakhlin et al. [64]. | ||||
(2) | Online discrepancy minimization: We consider the setting of the online Komlós problem, where the input is generated from an adaptive sequence of σ-smooth and isotropic distributions on the ℓ2 unit ball. We bound the ℓ∞ norm of the discrepancy vector by (tilde{O}big (ln ^2big (frac{nT}{sigma }big) big) ). This is as opposed to the worst-case analysis, where the tight discrepancy bound is (Theta (sqrt {T/n}) ). We show such polylog(nT/σ) discrepancy guarantees are not achievable for non-isotropic σ-smooth distributions. | ||||
(3) | Dispersion in online optimization: We consider online optimization with piecewise Lipschitz functions where fun
我们证明了平滑分析模型中若干在线问题的新算法保证。在该模型中,对手在每个时间步选择一个输入分布,该输入分布的密度函数以 (tfrac{1}{sigma } ) 倍于均匀分布的密度函数为界。这里,σ 是一个介于最坏情况分析和平均情况分析两个极端之间的参数。最重要的是,我们的结果适用于自适应对手,它们可以根据算法的决策和之前时间步骤中输入的实现情况来选择输入分布。自适应对手可以将不同时间步骤的输入与算法的当前状态非难地联系起来;这似乎排除了平滑分析中的标准证明方法。本文提出了一种证明针对自适应对手的平滑算法保证的通用技术,实际上是将自适应对手的设置简化为更简单的遗忘对手(即事先承诺整个输入分布序列的对手)。我们运用这一技术证明了三个不同问题的强平滑保证:(1) 在线学习:我们用 (tilde{O}big (sqrt {T dln (1/sigma)} + dln (T/sigma) big) )来约束遗憾,并提供了一个接近匹配的下限。我们的结果表明,在平滑分析下,针对自适应对手的可学习性是以 VC 维度的有限性为特征的。这与最坏情况分析相反,在最坏情况分析下,在线可学性的特征是利特尔斯通维度(即使在一维阈值函数这种极其有限的情况下,利特尔斯通维度也是无限的)。我们的结果完全回答了 Rakhlin 等人[64]提出的一个开放问题。(2) 在线差异最小化:我们考虑的是在线 Komlós 问题,输入由 ℓ2 单位球上的σ 平滑各向同性分布的自适应序列生成。我们用 (tilde{O}big (ln ^2big (frac{nT}{sigma }big) big) 约束差异向量的 ℓ∞ norm。)这与最坏情况分析相反,在最坏情况分析中,严格的差异约束是 (Theta (sqrt {T/n}) )。我们证明,对于非各向异性的σ光滑分布,这种polylog(nT/σ)差异保证是无法实现的。(3) 在线优化中的离散性:我们考虑了具有片状 Lipschitz 函数的在线优化,其中具有 ℓ 不连续性的函数是由平滑自适应对手选择的,并且证明了所得到的序列是 (бig ({sigma }/{sqrt {Tell }}, tilde{O}бig (sqrt {Tell } бig)бig) 分散的。也就是说,每个半径为 ({sigma }/{sqrt {Tell }}) 的球都被这些函数所做的分割的 (tilde{O}big (sqrt {Tell } big) ) 分割。这一结果与 Balcan 等人[13]针对遗忘平滑对手的分散参数相匹配,达到对数因子。另一方面,最坏情况序列的离散度是(0, T)。
引用次数: 0
Fast Multivariate Multipoint Evaluation Over All Finite Fields
在所有有限域上快速进行多变量多点评估
Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field (mathbb {F} ) that outputs the evaluations of an m-variate polynomial of degree less than d in each variable at N points in time [ (d^m+N)^{1+o(1)}cdot {rm poly}(m,d,log |mathbb {F}|) ] |