In the adversarially robust streaming model, a stream of elements is presented to an algorithm and is allowed to depend on the output of the algorithm at earlier times during the stream. In the classic insertion-only model of data streams, Ben-Eliezer et. al. (PODS 2020, best paper award) show how to convert a non-robust algorithm into a robust one with a roughly 1/%DIFDELCMD < \varepsilonfactor %%% overhead. This was subsequently improved to a 1/\sqrt{\varepsilon\mathcal{O}\left({\varepsilon^{4/p}}\cdot n^{1-2/p}}\right) \mathcal{O}\left({\varepsilon^{4/p}}\cdot n^{1-2/p}}\right)  the first optimal dependence on the approximation factor. As in previous work, we obtain a general transformation that applies to any non-robust streaming algorithm and depends on the so-called flip number. However, the key technical innovation is that we apply the transformation to what we call a difference estimator for the streaming problem, rather than an estimator for the streaming problem itself. We then develop the first difference estimators for a wide range of problems. Our difference estimator methodology is not only applicable to the adversarially robust model, but to other streaming models where temporal properties of the data play a central role.(Abstract shortened to meet arXiv limit.)\left({\varepsilon^2}\cdot n^{1-2/p}}\right) 
We study the classical problem of moment estimation of an underlying vector whose n coordinates are implicitly defined through a series of updates in a data stream. We show that if the updates to the vector arrive in the random-order insertion-only model %DIFDELCMD < \varepsilonfactor %%% , then there exist space efficient algorithms with improved dependencies on the approximation parameter \varepsilon. In particular, for any real p > 2, we first obtain an algorithm for F_p moment estimation using\mathcal{O}\left(\frac{1{\varepsilon^{4/p}}\cdot n^{1-2/p}}\right) bits of memory. Our techniques also give algorithms for F_p moment estimation with p>2 on arbitrary order insertion-only and turnstile streams, using\mathcal{O}\left(\frac{1{\varepsilon^{4/p}}\cdot n^{1-2/p}}\right) bits of space and two passes, which is the first optimal multi-pass F_p estimation algorithm up to \log n factors. Finally, we give an improved lower bound of \Omega\left(\frac{1{\varepsilon^2}\cdot n^{1-2/p}}\right) for one-pass insertion-only streams. Our results separate the complexity of this problem both between random and non-random orders, as well as one-pass and multi-pass streams.
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In the adversarially robust streaming model, a stream of elements is presented to an algorithm and is allowed to depend on the output of the algorithm at earlier times during the stream. In the classic
We study the classical problem of moment estimation of an underlying vector whose n coordinates are implicitly defined through a series of updates in a data stream. We show that if the updates to the vector arrive in the random-order
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of data streams, Ben-Eliezer et. al. (PODS 2020, best paper award) show how to convert a non-robust algorithm into a robust one with a roughly 1/
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overhead. This was subsequently improved to a 1/\sqrt{\varepsilon
, then there exist space efficient algorithms with improved dependencies on the approximation parameter \varepsilon. In particular, for any real p > 2, we first obtain an algorithm for F_p moment estimation using
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bits of memory. Our techniques also give algorithms for F_p moment estimation with p>2 on arbitrary order insertion-only and turnstile streams, using
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\frac{1
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bits of space and two passes, which is
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dependence on the approximation factor. As in previous work, we obtain a general transformation that applies to any non-robust streaming algorithm and depends on the so-called flip number. However, the key technical innovation is that we apply the transformation to what we call a difference estimator for the streaming problem, rather than an estimator for the streaming problem itself. We then develop the first difference estimators for a wide range of problems. Our difference estimator methodology is not only applicable to the adversarially robust model, but to other streaming models where temporal properties of the data play a central role.(Abstract shortened to meet arXiv limit.)
multi-pass F_p estimation algorithm up to \log n factors. Finally, we give an improved lower bound of \Omega
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\frac{1
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for one-pass insertion-only streams. Our results separate the complexity of this problem both between random and non-random orders, as well as one-pass and multi-pass streams.