We study data structure lower bounds for the Approximate Matrix Powering (AMP) problem. Given a substochastic, symmetric matrix 𝐌 ∈ ℝ^{n× n} and parameters k and α, the goal is to preprocess 𝐌 so as to answer entry queries (u,v)↦ 𝐌^{k}[u,v] up to additive error 1/n^{α}. We focus on AMP in the succinct and systematic regime, in which the data structure stores 𝐌 verbatim, uses an additional r bits of redundancy, and must answer queries by probing only a small number of entries of 𝐌. Our main conceptual contribution is a general framework for proving probe-redundancy trade-offs for systematic data structures. We introduce the query-with-sketch model and develop a min-entropy-based approach that lifts conditional min-entropy bounds in the absence of redundancy to probe lower bounds in the presence of redundancy. We then establish these min-entropy bounds using problem-specific analytic and algebraic tools, for the downstream applications to AMP and its variants. As a consequence, our results provide new unconditional evidence toward a conjecture of Pătraşcu and Roditty on the space required for constant-time set-disjointness queries [Patrascu and Roditty, 2010].

This work initiates the study of memory–query tradeoffs for graph problems, with a focus on correlation clustering. Correlation clustering asks for a partition of the vertices that minimizes disagreements: non‑edges inside clusters plus edges across clusters. Our first result is a tight query lower bound: to output a partition whose cost approximates the optimum up to an additive error of ε n², any algorithm requires Ω(n/ε²) adjacency-matrix queries. Under memory constraints, we show that even for the seemingly easier task of approximating the optimal clustering cost (without producing a partition), any algorithm in the random query model must make ≫ n/ε² adjacency-matrix queries. Finally, we prove the first general graph model query lower bound for correlation clustering, where algorithms are allowed adjacency-matrix, neighbor, and degree queries. The latter two bounds are not yet tight, leaving room for sharper results.

Motivated by recent work on the experts problem in the streaming model, we consider the experts problem in the sliding window model. The sliding window model is a well-studied model that captures applications such as traffic monitoring, epidemic tracking, and automated trading, where recent information is more valuable than older data. Formally, we have \(n\) experts, \(T\) days, the ability to query the predictions of \(q\) experts on each day, a limited amount of memory, and should achieve the (near-)optimal regret \(\sqrt{nW}\,\mathrm{polylog}(nT)\) regret over any window of the last \(W\) days. While it is impossible to achieve such regret with 1 query, we show that with 2 queries we can achieve such regret and with only \(\mathrm{polylog}(nT)\) bits of memory. Not only are our algorithms optimal for sliding windows, but we also show for every interval \(\mathcal{I}\) of days that we achieve \(\sqrt{n|\mathcal{I}|}\,\mathrm{polylog}(nT)\) regret with 2 queries and only \(\mathrm{polylog}(nT)\) bits of memory, providing an exponential improvement on the memory of previous interval regret algorithms. Building upon these techniques, we address the bandit problem in data streams, where \(q = 1\), achieving \(nT^{2/3}\,\mathrm{polylog}(T)\) regret with \(\mathrm{polylog}(nT)\) memory, which is the first sublinear regret in the streaming model in the bandit setting with polylogarithmic memory; this can be further improved to the optimal \(\mathcal{O}(\sqrt{nT})\) regret if the best expert’s losses are in a random order.

Motivated by recent advances in locally testable codes and quantum LDPCs based on robust testability of tensor product codes, we explore the local testability of tensor products of (an abstraction of) algebraic geometry codes. Such codes are parameterized by, in addition to standard parameters such as block length n and dimension k, their genus g. We show that the tensor product of two algebraic geometry codes is robustly locally testable provided n = Ω((k+g)²). Apart from Reed-Solomon codes, this seems to be the first explicit family of two-wise tensor codes of high dual distance that is robustly locally testable by the natural test that measures the expected distance of a random row/column from the underlying code.

We study the robust local testability of tensor products of two Algebraic-Geometry (AG) codes. In particular, we prove that constant rate AG codes are robust locally testable. This significantly generalizes the seminal result of Polishchuk-Spielman (1994), which proved robust local testability of Reed-Solomon codes. We establish an algebraic-geometric framework that enables us to geometrically interpret codewords in tensor products of AG codes. Thereby, we use tools from intersection theory of algebraic surfaces to prove a divisibility criterion for AG codes, that generalizes the bivariate divisibility result of Polishchuk-Spielman.Over the years, robust local testability of tensor products has played a key role in the development of classical locally testable codes (LTCs) as well as quantum Low Density Parity Check (qLDPC) codes and quantum Locally Testable Codes (qLTCs). To the best of our knowledge, after Reed-Solomon codes, our result provides the first explicit family of robustly locally testable codes with constant rate and linear dual-distance. Moreover, our result, when combined with Golowich-Guruswami (2024), yields new explicit families of good quantum CSS codes of length N which are locally testable with locality $O(\sqrt N )$ and constant soundness.

A recent line of work has shown a surprising connection between multicalibration, a multi- group fairness notion, and omniprediction, a learning paradigm that provides simultaneous loss minimization guarantees for a large family of loss functions [20, 19, 21, 18]. Prior work studies omniprediction in the batch setting. We initiate the study of omniprediction in the online adversarial setting. Although there exist algorithms for obtaining notions of multicalibration in the online adversarial setting [23], unlike batch algorithms, they work only for small finite classes of benchmark functions F, because they require enumerating every function f ∈ F at every round. In contrast, omniprediction is most interesting for learning theoretic hypothesis classes F, which are generally continuously (or at least exponentially) large.

We introduce a new notion of information complexity for multi-pass streaming problems and use it to resolve several important questions in data streams.

In this work, we show, for the well-studied problem of learning parity under noise, where a learner tries to learn x = (x₁,…,x_n) ∈ {0,1}ⁿ from a stream of random linear equations over 𝔽₂ that are correct with probability 1/2+ε and flipped with probability 1/2-ε (0 < ε < 1/2), that any learning algorithm requires either a memory of size Ω(n²/ε) or an exponential number of samples. In fact, we study memory-sample lower bounds for a large class of learning problems, as characterized by [Garg et al., 2018], when the samples are noisy. A matrix M: A × X → {-1,1} corresponds to the following learning problem with error parameter ε: an unknown element x ∈ X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a₁, b₁), (a₂, b₂) …, where for every i, a_i ∈ A is chosen uniformly at random and b_i = M(a_i,x) with probability 1/2+ε and b_i = -M(a_i,x) with probability 1/2-ε (0 < ε < 1/2). Assume that k,𝓁, r are such that any submatrix of M of at least 2^{-k} ⋅ |A| rows and at least 2^{-𝓁} ⋅ |X| columns, has a bias of at most 2^{-r}. We show that any learning algorithm for the learning problem corresponding to M, with error parameter ε, requires either a memory of size at least Ω((k⋅𝓁)/ε), or at least 2^{Ω(r)} samples. The result holds even if the learner has an exponentially small success probability (of 2^{-Ω(r)}). In particular, this shows that for a large class of learning problems, same as those in [Garg et al., 2018], any learning algorithm requires either a memory of size at least Ω(((log|X|)⋅(log|A|))/ε) or an exponential number of noisy samples. Our proof is based on adapting the arguments in [Ran Raz, 2017; Garg et al., 2018] to the noisy case.

In the coin problem we are asked to distinguish, with probability at least 2/3, between$n\ i.i.d$. coins which are heads with probability$\frac{1}{2}+\beta$from ones which are heads with probability$\frac{1}{2}-\beta$. We are interested in the space complexity of the coin problem, corresponding to the width of a read-once branching program solving the problem. The coin problem becomes more difficult as$\beta$becomes smaller. Statistically, it can be solved whenever$\beta= \Omega(n^{-1/2})$, using counting. It has been previously shown that for$\beta=O(n^{-1/2})$, counting is essentially optimal (equivalently, width$poly (n)$is necessary [Braverman-Garg-Woodruff FOCS'20]). On the other hand, the coin problem only requires$O(\log n)$width for$\beta > n^{-c}$for any constant$c > \log_{2}(\sqrt{5}-1)\approx 0.306$(following low-width simulation of AND-OR tree of [Valiant Journal of Algorithms'84]). In this paper, we close the gap between the bounds, showing a tight threshold between the values of$\beta=n^{-c}$where$O(\log n)$width suffices and the regime where$poly (n)$width is needed, with a transition at$c=1/3$. This gives a complete characterization (up to constant factors) of the memory complexity of solving the coin problem, for all values of bias$\beta$. We introduce new techniques in both bounds. For the upper bound, we give a construction based on recursive majority that does not require a memory stack of size$\log n$bits. For the lower bound, we introduce new combinatorial techniques for analyzing progression of the success probabilities in read-once branching programs.

In this work, we establish lower-bounds against memory bounded algorithms for distinguishing between natural pairs of related distributions from samples that arrive in a streaming setting. In our first result, we show that any algorithm that distinguishes between uniform distribution on $\{0,1\}^n$ and uniform distribution on an $n/2$-dimensional linear subspace of $\{0,1\}^n$ with non-negligible advantage needs $2^{Ω(n)}$ samples or $Ω(n^2)$ memory. Our second result applies to distinguishing outputs of Goldreich's local pseudorandom generator from the uniform distribution on the output domain. Specifically, Goldreich's pseudorandom generator $G$ fixes a predicate $P:\{0,1\}^k \rightarrow \{0,1\}$ and a collection of subsets $S_1, S_2, \ldots, S_m \subseteq [n]$ of size $k$. For any seed $x \in \{0,1\}^n$, it outputs $P(x_{S_1}), P(x_{S_2}), \ldots, P(x_{S_m})$ where $x_{S_i}$ is the projection of $x$ to the coordinates in $S_i$. We prove that whenever $P$ is $t$-resilient (all non-zero Fourier coefficients of $(-1)^P$ are of degree $t$ or higher), then no algorithm, with $

Consider the problem of computing the majority of a stream of n i.i.d. uniformly random bits. This problem, known as the coin problem, is central to a number of counting problems in different data stream models. We show that any streaming algorithm for solving this problem with large constant advantage must use Ω(log n) bits of space. We extend our lower bound to proving tight lower bounds for solving multiple, randomly interleaved copies of the coin problem, as well as for solving the OR of multiple copies of a variant of the coin problem. Our proofs involve new measures of information complexity that are well-suited for data streams. We use these lower bounds to obtain a number of new results for data streams. In each case there is an underlying d dimensional vector x with additive updates to its coordinates given in a stream of length m. The input streams arising from our coin lower bound have nice distributional properties, and consequently for many problems for which we only had lower bounds in general turnstile streams, we now obtain the same lower bounds in more natural models, such as the bounded deletion model, in which ||x||2never drops by a constant fraction of what it was earlier, or in the random order model, in which the updates are ordered randomly. In particular, in the bounded deletion model, we obtain nearly tight lower bounds for approximating ||x||∞up to additive error [1/(√k)]||x||2, approximating ||x||2up to a multiplicative ( 1+ε) factor (resolving a question of Jayaram and Woodruff in PODS 2018), and solving the Point Query and ℓ2-Heavy Hitters Problems. In the random order model, we also obtain new lower bounds for the Point Query and ℓ2-Heavy Hitters Problems. We also give new algorithms complementing our lower bounds and illustrating the tightness of the models we consider, including an algorithm for approximating ||x||∞up to additive error [1/(√k)]||x||2 in turnstile streams (resolving a question of Cormode in a 2006 IITK Workshop), and an algorithm for finding ℓ2-heavy hitters in randomly ordered insertion streams (which for random order streams, resolves a question of Nelson in a 2018 Warwick Workshop).

Nisan [ Combinatorica, 12 (1992), pp. 449--461] constructed a pseudorandom generator for length $n$, width $n$ read-once branching programs (ROBPs) with error $\varepsilon$ and seed length $O(\log^2{n} + \log{n} \cdot \log(1/\varepsilon))$. A major goal in complexity theory is to reduce the seed length, hopefully, to the optimal $O(\log{n}+\log(1/\varepsilon))$, or to construct improved hitting sets, as these would yield stronger derandomization of ${BPL}$ and ${RL}$, respectively. In contrast to a successful line of work in restricted settings, no progress has been made for general, unrestricted, ROBPs. Indeed, Nisan's construction is the best pseudorandom generator and, prior to this work, also the best hitting set for unrestricted ROBPs. In this work, we make the first improvement for the general case by constructing a hitting set with seed length $\widetilde{O}(\log^2{n}+\log(1/\varepsilon))$. That is, we decouple $\varepsilon$ and $n$, and obtain near-optimal dependence on the former. The regime of parameters in which our construction strictly improves upon prior works, namely, $\log(1/\varepsilon) \gg \log{n}$, is also motivated by the work of Saks and Zhou [ J. Comput. System Sci., 58 (1999), pp. 376--403], who use pseudorandom generators with error $\varepsilon$, for length $n$, width $w$ ROBPs, such that $w,1/\varepsilon = 2^{(\log{n})^{2}}$ in their proof for ${BPL} \subseteq \mathbf{L}^{3/2}$. In fact, we introduce and construct a new type of primitive we call pseudorandom pseudo-distributions. Informally, this is a generalization of pseudorandom generators in which one may assign negative and unbounded weights to paths, as opposed to working with probability distributions. We show that such a primitive yields hitting sets and, for derandomization purposes, can be used to derandomize two-sided error algorithms.

A line of recent works showed that for a large class of learning problems, any learning algorithm requires either super-linear memory size or a super-polynomial number of samples [Raz, 2016; Kol et al., 2017; Raz, 2017; Moshkovitz and Moshkovitz, 2018; Beame et al., 2018; Garg et al., 2018]. For example, any algorithm for learning parities of size n requires either a memory of size Omega(n^{2}) or an exponential number of samples [Raz, 2016]. All these works modeled the learner as a one-pass branching program, allowing only one pass over the stream of samples. In this work, we prove the first memory-samples lower bounds (with a super-linear lower bound on the memory size and super-polynomial lower bound on the number of samples) when the learner is allowed two passes over the stream of samples. For example, we prove that any two-pass algorithm for learning parities of size n requires either a memory of size Omega(n^{1.5}) or at least 2^{Omega(sqrt{n})} samples. More generally, a matrix M: A x X - > {-1,1} corresponds to the following learning problem: An unknown element x in X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a_1, b_1), (a_2, b_2) ..., where for every i, a_i in A is chosen uniformly at random and b_i = M(a_i,x). Assume that k,l, r are such that any submatrix of M of at least 2^{-k} * |A| rows and at least 2^{-l} * |X| columns, has a bias of at most 2^{-r}. We show that any two-pass learning algorithm for the learning problem corresponding to M requires either a memory of size at least Omega (k * min{k,sqrt{l}}), or at least 2^{Omega(min{k,sqrt{l},r})} samples.

We consider the following game between two players Alice and Bob, which we call the mirror game. Alice and Bob take turns saying numbers belonging to the set {1, 2, ...,N}. A player loses if they repeat a number that has already been said. Otherwise, after N turns, when all the numbers have been spoken, both players win. When N is even, Bob, who goes second, has a very simple (and memoryless) strategy to avoid losing: whenever Alice says x, respond with N+1-x. The question is: does Alice have a similarly simple strategy to win that avoids remembering all the numbers said by Bob? The answer is no. We prove a linear lower bound on the space complexity of any deterministic winning strategy of Alice. Interestingly, this follows as a consequence of the Eventown-Oddtown theorem from extremal combinatorics. We additionally demonstrate a randomized strategy for Alice that wins with high probability that requires only O~(sqrt N) space (provided that Alice has access to a random matching on K_N). We also investigate lower bounds for a generalized mirror game where Alice and Bob alternate saying 1 number and b numbers each turn (respectively). When 1+b is a prime, our linear lower bounds continue to hold, but when 1+b is composite, we show that the existence of a o(N) space strategy for Bob (when N != 0 mod (1+b)) implies the existence of exponential-sized matching vector families over Z^N_{1+b}.

Nisan (Combinatorica’92) constructed a pseudorandom generator for length n, width n read-once branching programs (ROBPs) with error ε and seed length O(log2n + logn · log(1/ε)). A major goal in complexity theory is to reduce the seed length, hopefully, to the optimal O(logn+log(1/ε)), or to construct improved hitting sets, as these would yield stronger derandomization of BPL and RL, respectively. In contrast to a successful line of work in restricted settings, no progress has been made for general, unrestricted, ROBPs. Indeed, Nisan’s construction is the best pseudorandom generator and, prior to this work, also the best hitting set for unrestricted ROBPs.

A matrix M: A × X → {−1,1} corresponds to the following learning problem: An unknown element x ∈ X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a1, b1), (a2, b2) …, where for every i, ai ∈ A is chosen uniformly at random and bi = M(ai,x).

While it is known that using network coding can significantly improve the throughput of directed networks, it is a notorious open problem whether coding yields any advantage over the multicommodity flow (MCF) rate in undirected networks. It was conjectured that the answer is no. In this paper we show that even a small advantage over MCF can be amplified to yield a near-maximum possible gap. We prove that any undirected network with k source-sink pairs that exhibits a (1+epsilon) gap between its MCF rate and its network coding rate can be used to construct a family of graphs G' whose gap is log(|G'|)^c for some constant c < 1. The resulting gap is close to the best currently known upper bound, log(|G'|), which follows from the connection between MCF and sparsest cuts. Our construction relies on a gap-amplifying graph tensor product that, given two graphs G1,G2 with small gaps, creates another graph G with a gap that is equal to the product of the previous two, at the cost of increasing the size of the graph. We iterate this process to obtain a gap of log(|G'|)^c from any initial gap.