arXiv subjects

Jan Hązła

Publications and source records attributed to Jan Hązła.

At least 19 recordsLinked to original sources

arXiv2026

Randomization Helps in Online Graph Exploration: Breaking the Deterministic Lower Bound on Cycles

In online graph exploration, introduced by Kalyanasundaram and Pruhs (1994), an agent must visit all vertices of an initially unknown weighted graph and return to its starting position, while the graph is revealed only locally at visited vertices. Although the problem has attracted considerable attention, previous work has focused exclusively on deterministic algorithms. Randomized strategies are often substantially harder to analyze because of a fundamental challenge inherent to exploration. In this work, we give the first positive result showing that randomization can improve competitive guarantees in online graph exploration. To this end, we focus on cycles, a simple graph class which nevertheless captures a key difficulty of online exploration. Our main contribution is $\textsc{RandHeavyTest}$, a randomized algorithm for online exploration of cycles whose competitive ratio we prove to be at most 1.315. This establishes a strict separation from the deterministic setting, where the optimal competitive ratio is $\thickapprox 1.366$, and thus gives the first provable advantage of randomization in online graph exploration. A key step towards this result is a new, simplified optimal deterministic algorithm, $\textsc{HeavyTest}$, whose formulation naturally suggests the randomized variant. We complement our upper bounds with lower bounds of 1.115 for arbitrary randomized algorithms and 1.207 for the natural class of so-called forward-greedy algorithms, which includes $\textsc{RandHeavyTest}$.

cs.DS↗

arXiv2026

Inevitability of Polarization in Geometric Opinion Exchange

Polarization and unexpected correlations between opinions on diverse topics (including in politics, culture and consumer choices) are an object of sustained attention. However, numerous theoretical models do not seem to convincingly explain these phenomena. This paper is motivated by a recent line of work, studying models where polarization can be explained in terms of biased assimilation and geometric interplay between opinions on various topics. The agent opinions are represented as unit vectors on a multidimensional sphere and updated according to geometric rules. In contrast to previous work, we focus on the classical opinion exchange setting, where the agents update their opinions in discrete time steps, with a pair of agents interacting randomly at every step. The opinions are updated according to an update rule belonging to a general class. Our findings are twofold. First, polarization appears to be ubiquitous in the class of models we study, requiring only relatively modest assumptions reflecting biased assimilation. Second, there is a qualitative difference between two-dimensional dynamics on the one hand, and three or more dimensions on the other. Accordingly, we prove almost sure polarization for a large class of update rules in two dimensions. Then, we prove polarization in three and more dimensions in more limited cases and try to shed light on central difficulties that are absent in two dimensions.

cs.SI↗

arXiv2026

Weight distribution bounds to relate minimum distance, list decoding, and symmetric channel performance

We study relationships between worst-case and random-noise properties of error correcting codes. More concretely, we consider connections between minimum distance, list decoding radius, and block error probability on noisy channels. A recent result of Pernice, Sprumont, and Wootters established the tight connection between list decoding radius and symmetric channel performance for linear codes. We extend this result to general codes. The proof proceeds by directly bounding the weight distribution rather than by sharp threshold techniques. We then turn to the relation between minimum distance and symmetric channel performance. The results we just described imply that a $q$-ary code of relative distance $δ$ has vanishing error probability on the symmetric channel up to the Johnson radius $J_q(δ)$. We improve upon this bound in the case of linear codes, for a range $δ$, for $q\ge 4$. In our proof we consider the \emph{erasure} properties of codes, and bound their weight distribution through inequalities introduced by Samorodnitsky. The latter part of the proof gives a more general technique that bounds the symmetric channel performance of a linear code with constant relative distance and good erasure channel performance.

cs.IT↗

arXiv2025

Geometric opinion exchange polarizes in every dimension

A recent line of work studies models of opinion exchange where agent opinions about $d$ topics are tracked simultaneously. The opinions are represented as vectors on the unit $(d-1)$-sphere, and the update rule is based on the overall correlation between the relevant vectors. The update rule reflects the assumption of biased assimilation, i.e., a pair of opinions is brought closer together if their correlation is positive and further apart if the correlation is negative. This model seems to induce the polarization of opinions into two antipodal groups. This is in contrast to many other known models which tend to achieve consensus. The polarization property has been recently proved for $d=2$, but the general case of $d \ge 3$ remained open. In this work, we settle the general case, using a more detailed understanding of the model dynamics and tools from the theory of random processes.

cs.SI↗

arXiv2025

Learning High-Degree Parities: The Crucial Role of the Initialization

Parities have become a standard benchmark for evaluating learning algorithms. Recent works show that regular neural networks trained by gradient descent can efficiently learn degree $k$ parities on uniform inputs for constant $k$, but fail to do so when $k$ and $d-k$ grow with $d$ (here $d$ is the ambient dimension). However, the case where $k=d-O_d(1)$ (almost-full parities), including the degree $d$ parity (the full parity), has remained unsettled. This paper shows that for gradient descent on regular neural networks, learnability depends on the initial weight distribution. On one hand, the discrete Rademacher initialization enables efficient learning of almost-full parities, while on the other hand, its Gaussian perturbation with large enough constant standard deviation $σ$ prevents it. The positive result for almost-full parities is shown to hold up to $σ=O(d^{-1})$, pointing to questions about a sharper threshold phenomenon. Unlike statistical query (SQ) learning, where a singleton function class like the full parity is trivially learnable, our negative result applies to a fixed function and relies on an initial gradient alignment measure of potential broader relevance to neural networks learning.

cs.LG↗

arXiv2024

Intransitive dice tournament is not quasirandom

We settle a version of the conjecture about intransitive dice posed by Conrey, Gabbard, Grant, Liu and Morrison in 2016 and Polymath in 2017. We consider generalized dice with $n$ faces and we say that a die $A$ beats $B$ if a random face of $A$ is more likely to show a higher number than a random face of $B$. We study random dice with faces drawn iid from the uniform distribution on $[0,1]$ and conditioned on the sum of the faces equal to $n/2$. Considering the "beats" relation for three such random dice, Polymath showed that each of eight possible tournaments between them is asymptotically equally likely. In particular, three dice form an intransitive cycle with probability converging to $1/4$. In this paper we prove that for four random dice not all tournaments are equally likely and the probability of a transitive tournament is strictly higher than $3/8$.

math.PR↗

arXiv2024

A Quantitative Version of More Capable Channel Comparison

This paper introduces a quantitative generalization of the ``more capable'' comparison of broadcast channels, which is termed ``more capable with advantage''. Some basic properties are demonstrated (including tensorization on product channels), and a characterisation is given for the cases of Binary Symmetric Channel (BSC) and Binary Erasure Channel (BEC). It is then applied to two problems. First, a list decoding bound on the BSC is given that applies to transitive codes that achieve capacity on the BEC. Second, new lower bounds on entropy rates of binary hidden Markov processes are derived.

cs.IT↗

arXiv2023

Combinatorics of subgroups of Beidleman near-vector spaces

Combinatorial aspects of R-subgroups of finite dimensional Beidleman near-vector spaces over nearfields are studied. A characterization of R-subgroups is used to obtain the smallest possible size of a generating set of a subgroup, which is much smaller than its dimension. Furthermore, a formula for the number of R-subgroups of an n-dimensional near-vector space is developed.

math.RA↗

arXiv2022

Optimal list decoding from noisy entropy inequality

A noisy entropy inequality for boolean functions by Samorodnitsky is applied to binary codes. It is shown that a binary code that achieves capacity on the binary erasure channel admits optimal list size for list decoding on some binary symmetric channels (in a regime where this optimal list size is exponentially large).

cs.IT↗

arXiv2022

An initial alignment between neural network and target is needed for gradient descent to learn

This paper introduces the notion of ``Initial Alignment'' (INAL) between a neural network at initialization and a target function. It is proved that if a network and a Boolean target function do not have a noticeable INAL, then noisy gradient descent on a fully connected network with normalized i.i.d. initialization will not learn in polynomial time. Thus a certain amount of knowledge about the target (measured by the INAL) is needed in the architecture design. This also provides an answer to an open problem posed in [AS20]. The results are based on deriving lower-bounds for descent algorithms on symmetric neural networks without explicit knowledge of the target function beyond its INAL.

cs.LG↗

arXiv2022

A Johnson--Lindenstrauss Framework for Randomly Initialized CNNs

How does the geometric representation of a dataset change after the application of each randomly initialized layer of a neural network? The celebrated Johnson--Lindenstrauss lemma answers this question for linear fully-connected neural networks (FNNs), stating that the geometry is essentially preserved. For FNNs with the ReLU activation, the angle between two inputs contracts according to a known mapping. The question for non-linear convolutional neural networks (CNNs) becomes much more intricate. To answer this question, we introduce a geometric framework. For linear CNNs, we show that the Johnson--Lindenstrauss lemma continues to hold, namely, that the angle between two inputs is preserved. For CNNs with ReLU activation, on the other hand, the behavior is richer: The angle between the outputs contracts, where the level of contraction depends on the nature of the inputs. In particular, after one layer, the geometry of natural images is essentially preserved, whereas for Gaussian correlated inputs, CNNs exhibit the same contracting behavior as FNNs with ReLU activation.

cs.LG↗

arXiv2021

Regularization by Misclassification in ReLU Neural Networks

We study the implicit bias of ReLU neural networks trained by a variant of SGD where at each step, the label is changed with probability $p$ to a random label (label smoothing being a close variant of this procedure). Our experiments demonstrate that label noise propels the network to a sparse solution in the following sense: for a typical input, a small fraction of neurons are active, and the firing pattern of the hidden layers is sparser. In fact, for some instances, an appropriate amount of label noise does not only sparsify the network but further reduces the test error. We then turn to the theoretical analysis of such sparsification mechanisms, focusing on the extremal case of $p=1$. We show that in this case, the network withers as anticipated from experiments, but surprisingly, in different ways that depend on the learning rate and the presence of bias, with either weights vanishing or neurons ceasing to fire.

cs.LG↗

arXiv2021

Almost-Reed--Muller Codes Achieve Constant Rates for Random Errors

This paper considers '$δ$-almost Reed-Muller codes', i.e., linear codes spanned by evaluations of all but a $δ$ fraction of monomials of degree at most $d$. It is shown that for any $δ> 0$ and any $\varepsilon>0$, there exists a family of $δ$-almost Reed-Muller codes of constant rate that correct $1/2-\varepsilon$ fraction of random errors with high probability. For exact Reed-Muller codes, the analogous result is not known and represents a weaker version of the longstanding conjecture that Reed-Muller codes achieve capacity for random errors (Abbe-Shpilka-Wigderson STOC '15). Our approach is based on the recent polarization result for Reed-Muller codes, combined with a combinatorial approach to establishing inequalities between the Reed-Muller code entropies.

cs.IT↗

arXiv2021

A Geometric Model of Opinion Polarization

We introduce a simple, geometric model of opinion polarization. It is a model of political persuasion, as well as marketing and advertising, utilizing social values. It focuses on the interplay between different topics and persuasion efforts. We demonstrate that societal opinion polarization often arises as an unintended byproduct of influencers attempting to promote a product or idea. We discuss a number of mechanisms for the emergence of polarization involving one or more influencers, sending messages strategically, heuristically, or randomly. We also examine some computational aspects of choosing the most effective means of influencing agents, and the effects of those strategic considerations on polarization.

cs.SI↗

arXiv2020

On arithmetic progressions in symmetric sets in finite field model

We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S\subseteq\mathbb{Z}_q^n$ containing $|S|=μ\cdot q^n$ elements must contain at least $δ(q,μ)\cdot q^n\cdot 2^n$ arithmetic progressions $x,x+d,\ldots,x+(q-1)\cdot d$ such that the difference $d$ is restricted to lie in $\{0,1\}^n$. Second, we show that for prime $p$ a symmetric set $S\subseteq\mathbb{F}^n_p$ with $|S|=μ\cdot p^n$ elements contains at least $μ^{C(p)}\cdot p^{2n}$ arithmetic progressions of length $p$. This establishes that the qualitative behavior of longer arithmetic progressions in symmetric sets is the same as for progressions of length three.

math.CO↗

arXiv2020

The Probability of Intransitivity in Dice and Close Elections

We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for $n$-sided dice with pairwise ordering induced by the probability, relative to $1/2$, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on $\{1,\ldots,n\}$ and conditioned on the average of faces equal to $(n+1)/2$ are intransitive with asymptotic probability $1/4$. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution conditioned on the average of faces equal to $0$, then three dice are transitive with high probability. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index $H\in(0,1)$. Second, we pose an analogous model in the context of Condorcet voting. We consider $n$ voters who rank $k$ alternatives independently and uniformly at random. The winner between each two alternatives is decided by a majority vote based on the preferences. We show that in this model, if all pairwise elections are close to tied, then the asymptotic probability of obtaining any tournament on the $k$ alternatives is equal to $2^{-k(k-1)/2}$, which markedly differs from known results in the model without conditioning. We also explore the Condorcet voting model where methods other than simple majority are used for pairwise elections. We investigate some natural definitions of "close to tied" for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions.

math.PR↗

arXiv2019

Bayesian Decision Making in Groups is Hard

We study the computations that Bayesian agents undertake when exchanging opinions over a network. The agents act repeatedly on their private information and take myopic actions that maximize their expected utility according to a fully rational posterior belief. We show that such computations are NP-hard for two natural utility functions: one with binary actions, and another where agents reveal their posterior beliefs. In fact, we show that distinguishing between posteriors that are concentrated on different states of the world is NP-hard. Therefore, even approximating the Bayesian posterior beliefs is hard. We also describe a natural search algorithm to compute agents' actions, which we call elimination of impossible signals, and show that if the network is transitive, the algorithm can be modified to run in polynomial time.

math.ST↗

arXiv2018

Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics

There is a common theme to some research questions in additive combinatorics and noise stability. Both study the following basic question: Let $\mathcal{P}$ be a probability distribution over a space $Ω^\ell$ with all $\ell$ marginals equal. Let $\underline{X}^{(1)}, \ldots, \underline{X}^{(\ell)}$ where $\underline{X}^{(j)} = (X_1^{(j)}, \ldots, X_n^{(j)})$ be random vectors such that for every coordinate $i \in [n]$ the tuples $(X_i^{(1)}, \ldots, X_i^{(\ell)})$ are i.i.d. according to $\mathcal{P}$. A central question that is addressed in both areas is: - Does there exist a function $c_{\mathcal{P}}()$ independent of $n$ such that for every $f: Ω^n \to [0, 1]$ with $\mathrm{E}[f(X^{(1)})] = μ> 0$: \begin{align*} \mathrm{E} \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] \ge c(μ) > 0 \, ? \end{align*} Instances of this question include the finite field model version of Roth's and Szemerédi's theorems as well as Borell's result about the optimality of noise stability of half-spaces. Our goal in this paper is to interpolate between the noise stability theory and the finite field additive combinatorics theory and address the question above in further generality than considered before. In particular, we settle the question for $\ell = 2$ and when $\ell > 2$ and $\mathcal{P}$ has bounded correlation $ρ(\mathcal{P}) < 1$. Under the same conditions we also characterize the _obstructions_ for similar lower bounds in the case of $\ell$ different functions. Part of the novelty in our proof is the combination of analytic arguments from the theories of influences and hyper-contraction with arguments from additive combinatorics.

cs.DM↗