The sliding window model generalizes the standard streaming model and often performs better in applications where recent data is more important or more accurate than data that arrived prior to a certain time. We study the problem of approximating symmetric norms (a norm on $\mathbb{R}^n$ that is invariant under sign-flips and coordinate-wise permutations) in the sliding window model, where only the $W$ most recent updates define the underlying frequency vector. Whereas standard norm estimation algorithms for sliding windows rely on the smooth histogram framework of Braverman and Ostrovsky (FOCS 2007), analyzing the smoothness of general symmetric norms seems to be a challenging obstacle. Instead, we observe that the symmetric norm streaming algorithm of Braverman et. al. (STOC 2017) can be reduced to identifying and approximating the frequency of heavy-hitters in a number of substreams. We introduce a heavy-hitter algorithm that gives a $(1+\epsilon)$-approximation to each of the reported frequencies in the sliding window model, thus obtaining the first algorithm for general symmetric norm estimation in the sliding window model. Our algorithm is a universal sketch that simultaneously approximates all symmetric norms in a parametrizable class and also improves upon the smooth histogram framework for estimating $L_p$ norms, for a range of large $p$. Finally, we consider the problem of overconstrained linear regression problem in the case that loss function that is an Orlicz norm, a symmetric norm that can be interpreted as a scale-invariant version of $M$-estimators. We give the first sublinear space algorithms that produce $(1+\epsilon)$-approximate solutions to the linear regression problem for loss functions that are Orlicz norms in both the streaming and sliding window models.
翻译:滑动窗口模式将标准流模式普遍化, 并且往往在应用中表现更好, 最近的数据比在某个时间之前到达的数据更重要或更准确。 我们研究滑动窗口模式中相似的对称规范( 符号翻转和协调一致式调整下的关于$mathbb{R ⁇ n$的规范) 的问题, 只有最近更新的美元可以定义隐藏频率矢量。 滑动窗口的标准标准标准标准标准值估算算法依赖于布拉弗曼和奥斯特罗夫斯基( FOCS 2007) 的平滑直方图框架框架。 分析一般对称规范的平滑动平滑动问题似乎是一个挑战性障碍。 相反, 我们观察到, 布拉弗曼等人等( STOC 2017) 的对正对正对调调调调调调调调调算法的算法, 在一个直流式的平流动方向模型中, 我们的对平流的平流法标准值值值的每平流值值函数解释, 也就是我们平流的平流的平流的平流法模型, 的对平流的平流法的平流法模型的平流法解释, 将所有平流法的平流法的平流法 的平流法 的平流法模型的平流法模型的平流的平流的平流的平流法, 使整个方向的平流法模型的平流法的平流法模型的平流法的平流法的平流法的平流法的平流的平流法的平流法 平流法 制的平的平的平的平流法的平流法, 使整个的平方的平方平方的平方的平方的平方的平方的平方的平方的平方的平方的平方的平方的平方的平方的平方的平方的平方的平方平方的平方的平方平方的平方体平方的平方的平方的平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方