arXiv subjects

Ofer Shayevitz

Publications and source records attributed to Ofer Shayevitz.

At least 19 recordsLinked to original sources

arXiv2026

Maximal correlation under cardinality constraints

In this paper, we define and analyze the quantized maximal correlation, an extension of the notion of maximal correlation restricted to functions taking values in sets of bounded cardinality. We derive an upper bound on the quantized maximal correlation by showing that the correlation between any quantized functions of $X$ and $Y$ is related to the MMSE distortion in quantization of a particular linear combination of random variables. Following this, we leverage rate-distortion techniques and anti-concentration inequalities to further bound this MMSE, which results in explicit bounds on the quantized maximal correlation. Unlike the quantized maximal correlation itself, which does not generally tensorize, our bounds on the mean squared error do tensorize, resulting in a dimension-free upper bound on the quantized maximal correlation for product distributions. Our results also lead to improved bounds on the isoperimetric constants of reversible Markov chains and product chains, strengthening classical results such as those by Alon and Milman.

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arXiv2024

Memory Complexity of Estimating Entropy and Mutual Information

We observe an infinite sequence of independent identically distributed random variables $X_1,X_2,\ldots$ drawn from an unknown distribution $p$ over $[n]$, and our goal is to estimate the entropy $H(p)=-\mathbb{E}[\log p(X)]$ within an $\varepsilon$-additive error. To that end, at each time point we are allowed to update a finite-state machine with $S$ states, using a possibly randomized but time-invariant rule, where each state of the machine is assigned an entropy estimate. Our goal is to characterize the minimax memory complexity $S^*$ of this problem, which is the minimal number of states for which the estimation task is feasible with probability at least $1-\delta$ asymptotically, uniformly in $p$. Specifically, we show that there exist universal constants $C_1$ and $C_2$ such that $ S^* \leq C_1\cdot\frac{n (\log n)^4}{\varepsilon^2\delta}$ for $\varepsilon$ not too small, and $S^* \geq C_2 \cdot \max \{n, \frac{\log n}{\varepsilon}\}$ for $\varepsilon$ not too large. The upper bound is proved using approximate counting to estimate the logarithm of $p$, and a finite memory bias estimation machine to estimate the expectation operation. The lower bound is proved via a reduction of entropy estimation to uniformity testing. We also apply these results to derive bounds on the memory complexity of mutual information estimation.

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arXiv2023

Statistical Inference with Limited Memory: A Survey

The problem of statistical inference in its various forms has been the subject of decades-long extensive research. Most of the effort has been focused on characterizing the behavior as a function of the number of available samples, with far less attention given to the effect of memory limitations on performance. Recently, this latter topic has drawn much interest in the engineering and computer science literature. In this survey paper, we attempt to review the state-of-the-art of statistical inference under memory constraints in several canonical problems, including hypothesis testing, parameter estimation, and distribution property testing/estimation. We discuss the main results in this developing field, and by identifying recurrent themes, we extract some fundamental building blocks for algorithmic construction, as well as useful techniques for lower bound derivations.

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arXiv2023

Batches Stabilize the Minimum Norm Risk in High Dimensional Overparameterized Linear Regression

Learning algorithms that divide the data into batches are prevalent in many machine-learning applications, typically offering useful trade-offs between computational efficiency and performance. In this paper, we examine the benefits of batch-partitioning through the lens of a minimum-norm overparametrized linear regression model with isotropic Gaussian features. We suggest a natural small-batch version of the minimum-norm estimator and derive bounds on its quadratic risk. We then characterize the optimal batch size and show it is inversely proportional to the noise level, as well as to the overparametrization ratio. In contrast to minimum-norm, our estimator admits a stable risk behavior that is monotonically increasing in the overparametrization ratio, eliminating both the blowup at the interpolation point and the double-descent phenomenon. We further show that shrinking the batch minimum-norm estimator by a factor equal to the Weiner coefficient further stabilizes it and results in lower quadratic risk in all settings. Interestingly, we observe that the implicit regularization offered by the batch partition is partially explained by feature overlap between the batches. Our bound is derived via a novel combination of techniques, in particular normal approximation in the Wasserstein metric of noisy projections over random subspaces.

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arXiv2023

Planted Bipartite Graph Detection

We consider the task of detecting a hidden bipartite subgraph in a given random graph. This is formulated as a hypothesis testing problem, under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi random graph over $n$ vertices with edge density $q$. Under the alternative, there exists a planted $k_{\mathsf{R}} \times k_{\mathsf{L}}$ bipartite subgraph with edge density $p>q$. We characterize the statistical and computational barriers for this problem. Specifically, we derive information-theoretic lower bounds, and design and analyze optimal algorithms matching those bounds, in both the dense regime, where $p,q = \Theta\left(1\right)$, and the sparse regime where $p,q = \Theta\left(n^{-\alpha}\right), \alpha \in \left(0,2\right]$. We also consider the problem of testing in polynomial-time. As is customary in similar structured high-dimensional problems, our model undergoes an "easy-hard-impossible" phase transition and computational constraints penalize the statistical performance. To provide an evidence for this statistical computational gap, we prove computational lower bounds based on the low-degree conjecture, and show that the class of low-degree polynomials algorithms fail in the conjecturally hard region.

cs.DS↗

arXiv2022

Deterministic Finite-Memory Bias Estimation

In this paper we consider the problem of estimating a Bernoulli parameter using finite memory. Let $X_1,X_2,\ldots$ be a sequence of independent identically distributed Bernoulli random variables with expectation $\theta$, where $\theta \in [0,1]$. Consider a finite-memory deterministic machine with $S$ states, that updates its state $M_n \in \{1,2,\ldots,S\}$ at each time according to the rule $M_n = f(M_{n-1},X_n)$, where $f$ is a deterministic time-invariant function. Assume that the machine outputs an estimate at each time point according to some fixed mapping from the state space to the unit interval. The quality of the estimation procedure is measured by the asymptotic risk, which is the long-term average of the instantaneous quadratic risk. The main contribution of this paper is an upper bound on the smallest worst-case asymptotic risk any such machine can attain. This bound coincides with a lower bound derived by Leighton and Rivest, to imply that $\Theta(1/S)$ is the minimax asymptotic risk for deterministic $S$-state machines. In particular, our result disproves a longstanding $\Theta(\log S/S)$ conjecture for this quantity, also posed by Leighton and Rivest.

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arXiv2022

On The Memory Complexity of Uniformity Testing

In this paper we consider the problem of uniformity testing with limited memory. We observe a sequence of independent identically distributed random variables drawn from a distribution $p$ over $[n]$, which is either uniform or is $\varepsilon$-far from uniform under the total variation distance, and our goal is to determine the correct hypothesis. At each time point we are allowed to update the state of a finite-memory machine with $S$ states, where each state of the machine is assigned one of the hypotheses, and we are interested in obtaining an asymptotic probability of error at most $0<\delta<1/2$ uniformly under both hypotheses. The main contribution of this paper is deriving upper and lower bounds on the number of states $S$ needed in order to achieve a constant error probability $\delta$, as a function of $n$ and $\varepsilon$, where our upper bound is $O(\frac{n\log n}{\varepsilon})$ and our lower bound is $\Omega (n+\frac{1}{\varepsilon})$. Prior works in the field have almost exclusively used collision counting for upper bounds, and the Paninski mixture for lower bounds. Somewhat surprisingly, in the limited memory with unlimited samples setup, the optimal solution does not involve counting collisions, and the Paninski prior is not hard. Thus, different proof techniques are needed in order to attain our bounds.

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arXiv2022

On the Number of Graphs with a Given Histogram

Let $G$ be a large (simple, unlabeled) dense graph on $n$ vertices. Suppose that we only know, or can estimate, the empirical distribution of the number of subgraphs $F$ that each vertex in $G$ participates in, for some fixed small graph $F$. How many other graphs would look essentially the same to us, i.e., would have a similar local structure? In this paper, we derive upper and lower bounds on the number of graphs whose empirical distribution lies close (in the Kolmogorov-Smirnov distance) to that of $G$. Our bounds are given as solutions to a maximum entropy problem on random graphs of a fixed size $k$ that does not depend on $n$, under $d$ global density constraints. The bounds are asymptotically close, with a gap that vanishes with $d$ at a rate that depends on the concentration function of the center of the Kolmogorov-Smirnov ball.

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arXiv2021

On lossy Compression of Directed Graphs

The method of types presented by Csiszar and Korner is a central tool used to develop and analyze the basic properties and constraints on sequences of data over finite alphabets. A central problem considered using these tools is that of data compression, and specifically lossy data compression. In this work we consider this very problem, however, instead of sequences of data we consider directed graphs. We show that given a more natural distortion measure, fitting the data structure of a directed graph, the method of types cannot be applied. The suggested distortion measure aims to preserves the local structure of a directed graph. We build on the recent work of Barvinok and extend the method of types to the two dimensional setting of directed graphs. We see that the extension is quite natural in many ways. Given this extension we provide a lower and upper bound on the rate-distortion problem of lossy compression given the suggested distortion measure.

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arXiv2021

Learning User Preferences in Non-Stationary Environments

Recommendation systems often use online collaborative filtering (CF) algorithms to identify items a given user likes over time, based on ratings that this user and a large number of other users have provided in the past. This problem has been studied extensively when users' preferences do not change over time (static case); an assumption that is often violated in practical settings. In this paper, we introduce a novel model for online non-stationary recommendation systems which allows for temporal uncertainties in the users' preferences. For this model, we propose a user-based CF algorithm, and provide a theoretical analysis of its achievable reward. Compared to related non-stationary multi-armed bandit literature, the main fundamental difficulty in our model lies in the fact that variations in the preferences of a certain user may affect the recommendations for other users severely. We also test our algorithm over real-world datasets, showing its effectiveness in real-world applications. One of the main surprising observations in our experiments is the fact our algorithm outperforms other static algorithms even when preferences do not change over time. This hints toward the general conclusion that in practice, dynamic algorithms, such as the one we propose, might be beneficial even in stationary environments.

cs.LG↗

arXiv2020

Minimum Guesswork with an Unreliable Oracle

We study a guessing game where Alice holds a discrete random variable $X$, and Bob tries to sequentially guess its value. Before the game begins, Bob can obtain side-information about $X$ by asking an oracle, Carole, any binary question of his choosing. Carole's answer is however unreliable, and is incorrect with probability $ε$. We show that Bob should always ask Carole whether the index of $X$ is odd or even with respect to a descending order of probabilities -- this question simultaneously minimizes all the guessing moments for any value of $ε$. In particular, this result settles a conjecture of Burin and Shayevitz. We further consider a more general setup where Bob can ask a multiple-choice $M$-ary question, and then observe Carole's answer through a noisy channel. When the channel is completely symmetric, i.e., when Carole decides whether to lie regardless of Bob's question and has no preference when she lies, a similar question about the ordered index of $X$ (modulo $M$) is optimal. Interestingly however, the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels. We provide supporting evidence for this difficulty, by showing that a core property required in our proofs becomes NP-hard to test in the general $M$-ary case. We establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution, which we prove to be NP-hard for $M\geq 3$.

cs.DM↗

arXiv2020

A Note on the Probability of Rectangles for Correlated Binary Strings

Consider two sequences of $n$ independent and identically distributed fair coin tosses, $X=(X_1,\ldots,X_n)$ and $Y=(Y_1,\ldots,Y_n)$, which are $ρ$-correlated for each $j$, i.e. $\mathbb{P}[X_j=Y_j] = {1+ρ\over 2}$. We study the question of how large (small) the probability $\mathbb{P}[X \in A, Y\in B]$ can be among all sets $A,B\subset\{0,1\}^n$ of a given cardinality. For sets $|A|,|B| = Θ(2^n)$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|A|,|B| = 2^{Θ(n)}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb{P}[X \in A, Y\in B]$ in the regime of $ρ\to 1$. We also prove a similar tight lower bound, i.e. show that for $ρ\to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb{P}[X \in A, Y\in B]$.

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arXiv2020

Simple Modulo can Significantly Outperform Deep Learning-based Deepcode

Deepcode (H.Kim et al.2018) is a recently suggested Deep Learning-based scheme for communication over the AWGN channel with noisy feedback, claimed to be superior to all previous schemes in the literature. Deepcode's use of nonlinear coding (via Deep Learning) has been inspired by known shortcomings (Y.-H. Kim et al 2007) of linear feedback schemes. In 2014, we presented a nonlinear feedback coding scheme based on a combination of the classical SK scheme and modulo-arithmetic, using a small number of elementary operations without any type of neural network. This Modulo-SK scheme has been omitted from the performance comparisons made in the Deepcode paper, due to its use of common randomness (dither), and in a later version since it was incorrectly interpreted as a variable-length coding scheme. However, the dither in Modulo-SK was used only for the standard purpose of tractable performance analysis, and is not required in practice. In this short note, we show that a fully-deterministic Modulo-SK (without dithering) can outperform Deepcode. For example, to attain an error probability of 10^(-4) at rate 1/3 Modulo-SK requires 3dB less feedback SNR than Deepcode. To attain an error probability of 10^(-6) with noiseless feedback, Deepcode requires 150 rounds of communication, whereas Modulo-SK requires only 15 rounds, even if the feedback is noisy (with 27dB SNR). We further address the numerical stability issues of the original SK scheme reported in the Deepcode paper, and explain how they can be avoided. We augment this report with an online-available, fully-functional Matlab simulation for both the classical and Modulo-SK schemes. Finally, note that Modulo-SK is by no means claimed to be the best possible solution; in particular, using deep learning in conjunction with modulo-arithmetic might lead to better designs, and remains a fascinating direction for future research.

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arXiv2020

Sharp Thresholds of the Information Cascade Fragility Under a Mismatched Model

We analyze a sequential decision making model in which decision makers (or, players) take their decisions based on their own private information as well as the actions of previous decision makers. Such decision making processes often lead to what is known as the \emph{information cascade} or \emph{herding} phenomenon. Specifically, a cascade develops when it seems rational for some players to abandon their own private information and imitate the actions of earlier players. The risk, however, is that if the initial decisions were wrong, then the whole cascade will be wrong. Nonetheless, information cascade are known to be fragile: there exists a sequence of \emph{revealing} probabilities $\{p_{\ell}\}_{\ell\geq1}$, such that if with probability $p_{\ell}$ player $\ell$ ignores the decisions of previous players, and rely on his private information only, then wrong cascades can be avoided. Previous related papers which study the fragility of information cascades always assume that the revealing probabilities are known to all players perfectly, which might be unrealistic in practice. Accordingly, in this paper we study a mismatch model where players believe that the revealing probabilities are $\{q_\ell\}_{\ell\in\mathbb{N}}$ when they truly are $\{p_\ell\}_{\ell\in\mathbb{N}}$, and study the effect of this mismatch on information cascades. We consider both adversarial and probabilistic sequential decision making models, and derive closed-form expressions for the optimal learning rates at which the error probability associated with a certain decision maker goes to zero. We prove several novel phase transitions in the behaviour of the asymptotic learning rate.

cs.LG↗

arXiv2020

Binary Hypothesis Testing with Deterministic Finite-Memory Decision Rules

In this paper we consider the problem of binary hypothesis testing with finite memory systems. Let $X_1,X_2,\ldots$ be a sequence of independent identically distributed Bernoulli random variables, with expectation $p$ under $\mathcal{H}_0$ and $q$ under $\mathcal{H}_1$. Consider a finite-memory deterministic machine with $S$ states that updates its state $M_n \in \{1,2,\ldots,S\}$ at each time according to the rule $M_n = f(M_{n-1},X_n)$, where $f$ is a deterministic time-invariant function. Assume that we let the process run for a very long time ($n\rightarrow \infty)$, and then make our decision according to some mapping from the state space to the hypothesis space. The main contribution of this paper is a lower bound on the Bayes error probability $P_e$ of any such machine. In particular, our findings show that the ratio between the maximal exponential decay rate of $P_e$ with $S$ for a deterministic machine and for a randomized one, can become unbounded, complementing a result by Hellman.

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arXiv2019

On the Interactive Capacity of Finite-State Protocols

The interactive capacity of a noisy channel is the highest possible rate at which arbitrary interactive protocols can be simulated reliably over the channel. Determining the interactive capacity is notoriously difficult, and the best known lower bounds are far below the associated Shannon capacity, which serves as a trivial (and also generally the best known) upper bound. This paper considers the more restricted setup of simulating finite-state protocols. It is shown that all two-state protocols, as well as rich families of arbitrary finite-state protocols, can be simulated at the Shannon capacity, establishing the interactive capacity for those families of protocols.

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arXiv2019

On the Non-Adaptive Zero-Error Capacity of the Discrete Memoryless Two-Way Channel

We study the problem of communicating over a discrete memoryless two-way channel using non-adaptive schemes, under a zero probability of error criterion. We derive single-letter inner and outer bounds for the zero-error capacity region, based on random coding, linear programming, linear codes, and the asymptotic spectrum of graphs. Among others, we provide a single-letter outer bound based on a combination of Shannon's vanishing-error capacity region and a two-way analogue of the linear programming bound for point-to-point channels, which in contrast to the one-way case, is generally better than both. Moreover, we establish an outer bound for the zero-error capacity region of a two-way channel via the asymptotic spectrum of graphs and show that this bound could be achieved for certain cases.

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arXiv2019

A Lower Bound on the Essential Interactive Capacity of Binary Memoryless Symmetric Channels

The essential interactive capacity of a discrete memoryless channel is defined in this paper as the maximal rate at which the transcript of any interactive protocol can be reliably simulated over the channel, using a deterministic coding scheme. In contrast to other interactive capacity definitions in the literature, this definition makes no assumptions on the order of speakers (which can be adaptive) and does not allow any use of private / public randomness; hence, the essential interactive capacity is a function of the channel model only. It is shown that the essential interactive capacity of any binary memoryless symmetric (BMS) channel is at least $0.0302$ its Shannon capacity. To that end, we present a simple coding scheme, based on extended-Hamming codes combined with error detection, that achieves the lower bound in the special case of the binary symmetric channel (BSC). We then adapt the scheme to the entire family of BMS channels, and show that it achieves the same lower bound using extremes of the Bhattacharyya parameter.

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