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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 $n = \Omega((k+g)^2)$. Apart from Reed-Solomon codes, this seems to be the first explicit family of codes of super-constant dual distance that is robustly locally testable., Comment: 12 pages

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 the coin problem, one sees a stream of $n$ i.i.d. uniform bits and one would like to compute the majority with constant advantage. We show that any constant pass algorithm must use $\Omega(\log n)$ bits of memory, significantly extending an earlier $\Omega(\log n)$ bit lower bound for single-pass algorithms of Braverman-Garg-Woodruff (FOCS, 2020). This also gives the first $\Omega(\log n)$ bit lower bound for the problem of approximating a counter up to a constant factor in worst-case turnstile streams for more than one pass. In the needle problem, one either sees a stream of $n$ i.i.d. uniform samples from a domain $[t]$, or there is a randomly chosen needle $\alpha \in[t]$ for which each item independently is chosen to equal $\alpha$ with probability $p$, and is otherwise uniformly random in $[t]$. The problem of distinguishing these two cases is central to understanding the space complexity of the frequency moment estimation problem in random order streams. We show tight multi-pass space bounds for this problem for every $p < 1/\sqrt{n \log^3 n}$, resolving an open question of Lovett and Zhang (FOCS, 2023); even for $1$-pass our bounds are new. To show optimality, we improve both lower and upper bounds from existing results. Our information complexity framework significantly extends the toolkit for proving multi-pass streaming lower bounds, and we give a wide number of additional streaming applications of our lower bound techniques, including multi-pass lower bounds for $\ell_p$-norm estimation, $\ell_p$-point query and heavy hitters, and compressed sensing problems., Comment: To appear in STOC 2024

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. 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, unlike batch algorithms, they work only for small finite classes of benchmark functions $F$, because they require enumerating every function $f \in F$ at every round. In contrast, omniprediction is most interesting for learning theoretic hypothesis classes $F$, which are generally continuously large. We develop a new online multicalibration algorithm that is well defined for infinite benchmark classes $F$, and is oracle efficient (i.e. for any class $F$, the algorithm has the form of an efficient reduction to a no-regret learning algorithm for $F$). The result is the first efficient online omnipredictor -- an oracle efficient prediction algorithm that can be used to simultaneously obtain no regret guarantees to all Lipschitz convex loss functions. For the class $F$ of linear functions, we show how to make our algorithm efficient in the worst case. Also, we show upper and lower bounds on the extent to which our rates can be improved: our oracle efficient algorithm actually promises a stronger guarantee called swap-omniprediction, and we prove a lower bound showing that obtaining $O(\sqrt{T})$ bounds for swap-omniprediction is impossible in the online setting. On the other hand, we give a (non-oracle efficient) algorithm which can obtain the optimal $O(\sqrt{T})$ omniprediction bounds without going through multicalibration, giving an information theoretic separation between these two solution concepts.

In this work, we show, for the well-studied problem of learning parity under noise, where a learner tries to learn $x=(x_1,\ldots,x_n) \in \{0,1\}^n$ from a stream of random linear equations over $\mathrm{F}_2$ that are correct with probability $\frac{1}{2}+\varepsilon$ and flipped with probability $\frac{1}{2}-\varepsilon$, that any learning algorithm requires either a memory of size $\Omega(n^2/\varepsilon)$ or an exponential number of samples. In fact, we study memory-sample lower bounds for a large class of learning problems, as characterized by [GRT'18], when the samples are noisy. A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the following learning problem with error parameter $\varepsilon$: an unknown element $x \in X$ is chosen uniformly at random. A learner tries to learn $x$ from a stream of samples, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen uniformly at random and $b_i = M(a_i,x)$ with probability $1/2+\varepsilon$ and $b_i = -M(a_i,x)$ with probability $1/2-\varepsilon$ ($0<\varepsilon< \frac{1}{2}$). Assume that $k,\ell, r$ are such that any submatrix of $M$ of at least $2^{-k} \cdot |A|$ rows and at least $2^{-\ell} \cdot |X|$ columns, has a bias of at most $2^{-r}$. We show that any learning algorithm for the learning problem corresponding to $M$, with error, requires either a memory of size at least $\Omega\left(\frac{k \cdot \ell}{\varepsilon} \right)$, or at least $2^{\Omega(r)}$ samples. In particular, this shows that for a large class of learning problems, same as those in [GRT'18], any learning algorithm requires either a memory of size at least $\Omega\left(\frac{(\log |X|) \cdot (\log |A|)}{\varepsilon}\right)$ or an exponential number of noisy samples. Our proof is based on adapting the arguments in [Raz'17,GRT'18] to the noisy case., Comment: 19 pages. To appear in RANDOM 2021. arXiv admin note: substantial text overlap with arXiv:1708.02639

We investigate the problem of designing optimal classifiers in the strategic classification setting, where the classification is part of a game in which players can modify their features to attain a favorable classification outcome (while incurring some cost). Previously, the problem has been considered from a learning-theoretic perspective and from the algorithmic fairness perspective. Our main contributions include 1. Showing that if the objective is to maximize the efficiency of the classification process (defined as the accuracy of the outcome minus the sunk cost of the qualified players manipulating their features to gain a better outcome), then using randomized classifiers (that is, ones where the probability of a given feature vector to be accepted by the classifier is strictly between 0 and 1) is necessary. 2. Showing that in many natural cases, the imposed optimal solution (in terms of efficiency) has the structure where players never change their feature vectors (the randomized classifier is structured in a way, such that the gain in the probability of being classified as a 1 does not justify the expense of changing one's features). 3. Observing that the randomized classification is not a stable best-response from the classifier's viewpoint, and that the classifier doesn't benefit from randomized classifiers without creating instability in the system. 4. Showing that in some cases, a noisier signal leads to better equilibria outcomes -- improving both accuracy and fairness when more than one subpopulation with different feature adjustment costs are involved. This is interesting from a policy perspective, since it is hard to force institutions to stick to a particular randomized classification strategy (especially in a context of a market with multiple classifiers), but it is possible to alter the information environment to make the feature signals inherently noisier., Comment: 22 pages. Appeared in FORC, 2020

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 $2^{\Omega(n)}$ samples or $\Omega(n^2)$ 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 $

As algorithmic prediction systems have become widespread, fears that these systems may inadvertently discriminate against members of underrepresented populations have grown. With the goal of understanding fundamental principles that underpin the growing number of approaches to mitigating algorithmic discrimination, we investigate the role of information in fair prediction. A common strategy for decision-making uses a predictor to assign individuals a risk score; then, individuals are selected or rejected on the basis of this score. In this work, we study a formal framework for measuring the information content of predictors. Central to this framework is the notion of a refinement, first studied by Degroot and Fienberg. Intuitively, a refinement of a predictor $z$ increases the overall informativeness of the predictions without losing the information already contained in $z$. We show that increasing information content through refinements improves the downstream selection rules across a wide range of fairness measures (e.g. true positive rates, false positive rates, selection rates). In turn, refinements provide a simple but effective tool for reducing disparity in treatment and impact without sacrificing the utility of the predictions. Our results suggest that in many applications, the perceived "cost of fairness" results from an information disparity across populations, and thus, may be avoided with improved information., Comment: Appeared at EC 2019

We consider a simple streaming game between two players Alice and Bob, which we call the mirror game. In this game, Alice and Bob take turns saying numbers belonging to the set $\{1, 2, \dots,2N\}$. A player loses if they repeat a number that has already been said. Bob, who goes second, has a very simple (and memoryless) strategy to avoid losing: whenever Alice says $x$, respond with $2N+1-x$. 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 $\tilde{O}(\sqrt N)$ space (provided that Alice has access to a random matching on $K_{2N}$). 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 implies the existence of exponential-sized matching vector families over $\mathbb{Z}^N_{1+b}$., Comment: 13 pages

A matrix $M: A \times X \rightarrow \{-1,1\}$ 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, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen uniformly at random and $b_i = M(a_i,x)$. Assume that $k,\ell, r$ are such that any submatrix of $M$ of at least $2^{-k} \cdot |A|$ rows and at least $2^{-\ell} \cdot |X|$ columns, has a bias of at most $2^{-r}$. We show that any learning algorithm for the learning problem corresponding to $M$ requires either a memory of size at least $\Omega\left(k \cdot \ell \right)$, or at least $2^{\Omega(r)}$ samples. The result holds even if the learner has an exponentially small success probability (of $2^{-\Omega(r)}$). In particular, this shows that for a large class of learning problems, any learning algorithm requires either a memory of size at least $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ or an exponential number of samples, achieving a tight $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ lower bound on the size of the memory, rather than a bound of $\Omega\left(\min\left\{(\log |X|)^2,(\log |A|)^2\right\}\right)$ obtained in previous works [R17,MM17b]. Moreover, our result implies all previous memory-samples lower bounds, as well as a number of new applications. Our proof builds on [R17] that gave a general technique for proving memory-samples lower bounds.

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 by Li and Li (2004) 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 $(1+\varepsilon)$ 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 $G_1,G_2$ 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., Comment: 20 pages

We give a new class of security definitions for authentication in the quantum setting. These definitions capture and strengthen existing definitions of security against quantum adversaries for both classical message authentication codes (MACs) and well as full quantum state authentication schemes. The main feature of our definitions is that they precisely characterize the effective behavior of any adversary when the authentication protocol accepts, including correlations with the key. Our definitions readily yield a host of desirable properties and interesting consequences; for example, our security definition for full quantum state authentication implies that the entire secret key can be re-used if the authentication protocol succeeds. Next, we present several protocols satisfying our security definitions. We show that the classical Wegman-Carter authentication scheme with 3-universal hashing is secure against superposition attacks, as well as adversaries with quantum side information. We then present conceptually simple constructions of full quantum state authentication. Finally, we prove a lifting theorem which shows that, as long as a protocol can securely authenticate the maximally entangled state, it can securely authenticate any state, even those that are entangled with the adversary. Thus, this shows that protocols satisfying a fairly weak form of authentication security automatically satisfy a stronger notion of security (in particular, the definition of Dupuis, et al (2012))., Comment: 50 pages, QCrypt 2016 - 6th International Conference on Quantum Cryptography, added a new lifting theorem that shows equivalence between a weak form of authentication security and a stronger notion that considers side information