Haim Kaplan

Edith Cohen, Haim Kaplan, Yishay Mansour et al.

The Adaptive Data Analysis (ADA) problem formalizes the challenge of preventing false discovery and overfitting when a dataset is repeatedly reused. Formally, our input is a dataset containing $n$ i.i.d. samples from an unknown distribution $\mathcal{P}$ over a domain $\mathcal{X}$, and our goal is to answer a sequence of $k$ adaptively chosen statistical queries with respect to $\mathcal{P}$. The main question is how many queries we can support (i.e., how large $k$ can be), primarily as a function of the number of samples $n$. This question has been intensively studied and is relatively well-understood for randomized mechanisms: there are computationally efficient mechanisms that support $k \approx n^2$ queries, and no computationally efficient mechanism can answer $k \gg n^2$ queries. In this paper, we address a fundamental question: is randomness necessary for ADA? Despite a decade of work on ADA, this question remains open. A folklore observation dating back to the initial works on ADA is that randomness is not necessary when the analyst is computationally bounded. Yet, the necessity of randomness against computationally unbounded analysts has remained elusive. Our main contribution resolves this gap in the information-theoretic Random Oracle model. Perhaps surprisingly, we show that randomness is strictly necessary to answer a non-trivial number of adaptive queries: when the analyst is unbounded, any deterministic mechanism can be forced to fail after just $k = \tilde{O} (n)$ queries.

Tamara Kaplan, Nimrod Kaplan, Haim Kaplan

We introduce a streaming model that uses both catalytic and regular memory. In this model, we show how to exactly compute the frequency moments using a logarithmic number of bits of regular memory and a polynomial number of bits of catalytic memory. More generally, we show how to compute arbitrary polynomials of the item frequencies exactly within the same space bounds. As an application, we obtain catalytic streaming algorithms that exactly compute the number of distinct elements in a stream, count the number of triangles (or any other small subgraph) in a graph whose edges arrive in a stream, and identify heavy hitters. Our algorithms for frequency moments perform a constant number of passes over the stream, and for polynomial evaluation, we require one more pass than the degree of the polynomial. By relating our catalytic streaming model to the catalytic communication model introduced in Pyne et al., we show that catalytic memory is not useful for any one-pass streaming algorithm. For lower bounds on multipass streaming algorithms, the impossibility results of Pyne et al. are not strong enough. However, using a different technique, we show that under certain natural restrictions, no catalytic streaming algorithm can compute the second frequency moment in fewer than three passes. This definition of the restricted class of two-pass algorithms then guides us in the design of a two-pass algorithm for computing the second moment exactly that circumvents these restrictions and breaks the three-pass barrier.

Haim Kaplan, Shay Sapir, Uri Stemmer

We analyze a balls-and-bins game against an adaptive adversary that sequentially deletes bins. Starting with $n$ balls distributed across $n$ bins, the adversary deletes a bin in each step, forcing the algorithm to redistribute its balls to surviving bins. We prove that after $n/2$ rounds, uniform random redistribution yields optimal $O(n)$ recourse and $O(\frac{\log n}{\log \log n})$ maximum load. Furthermore, we show that applying the ``power of two choices'' reduces the maximum load to $O(\log \log n)$ while maintaining linear recourse. We also consider a variation of this game where the balls from the deleted bin are partitioned evenly among $d \ll n$ random bins rather than being redistributed independently. We demonstrate that keeping the balls together ($d=1$), which gives small maximum load and recourse against an oblivious adversary, fails against an adaptive adversary. Nevertheless, we show that splitting the balls into just two groups ($d=2$) is sufficient to recover linear recourse and efficient load balancing in the adaptive setting.

Haim Kaplan, David Naori, Yaniv Sadeh

In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $Δ$ with at most $Δ+c$ colors, given updates to the edges of the graph. An important objective is to minimize the \emph{recourse}, which is the number of edges being recolored. We study this problem on forests, which is a natural yet nontrivial restriction of the problem. We consider the problem in both \emph{incremental} (edges are only inserted) and \emph{fully dynamic} (edges may be deleted) models. In the deterministic setting, we show that the natural greedy algorithm achieves $O(\frac{1}{c + \sqrtΔ})$ amortized recourse in the incremental model, and this is tight up to tie-breaking. In contrast, in a fully dynamic forest, greedy can be forced to have $Ω(\log_Δn)$ amortized recourse. To partially alleviate this limitation of greedy, we show an optimal non-greedy algorithm with $O(1)$ amortized recourse for \emph{rooted} fully dynamic forests and $c = Δ- 2$. In the randomized setting, we give a natural distribution-maintaining algorithm that achieves $Θ(\frac{1}Δ)$ expected amortized recourse in the incremental model and $Θ(\min \{ \fracΔ{c}, \log_Δ n \})$ expected recourse in the dynamic model. These randomized results are optimal for $c=0$.