arXiv subjects

Gowtham R. Kurri

Publications and source records attributed to Gowtham R. Kurri.

At least 19 recordsLinked to original sources

arXiv2026

Converse Bounds for Sun-Jafar-type Weak Private Information Retrieval

Building on the well-established capacity-achieving schemes of Sun-Jafar (for replicated storage) and the closely related scheme of Banawan-Ulukus (for MDS-coded setting), a recent work by Anand et al. proposed new classes of weak private information retrieval (WPIR) schemes for the collusion-free (replication and MDS-coded) setting, as well as for the $T$-colluding scenario. In their work, Anand et al. characterized the expressions for the rate-privacy trade-offs for these classes of WPIR schemes, under the mutual information leakage and maximal leakage metrics. Explicit achievable trade-offs for the same were also presented, which were shown to be competitive or better than prior WPIR schemes. However, the class-wise optimality of the reported trade-offs was unknown. In this work, we show that the explicit rate-privacy trade-offs reported for the Sun-Jafar-type schemes by Anand et al. are class-wise optimal for the non-colluding and replicated setting. Furthermore, we prove the class-wise optimality for Banawan-Ulukus-type MDS-WPIR and Sun-Jafar-type $T$-colluding WPIR schemes, under threshold-constraints on the system parameters. When these threshold-constraints do not hold, we present counter-examples which show that even higher rates than those reported before can be achieved.

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arXiv2026

Extreme Points of the $(0,δ)$-LDP Polytope with Small Input Size and Arbitrary Output Sizes

The structure of locally differentially private (LDP) mechanisms can be understood through the geometry of the corresponding privacy polytope. While the extreme points of the $ (ε,0)$-LDP polytope are well characterized (Kairouz \emph{et al.}, 2014; Holohan \emph{et al.}, 2017; Pensia \emph{et al.}, 2017), comparatively little is known for the $(ε,δ)$-LDP polytope with $δ>0$. Recent work (Elangovan and Jog, 2024) has shown that even in the special case $ε=0$, the $ (0,δ) $-LDP privacy polytope exhibits fundamentally different behaviour. In this work, we provide complete characterizations of the extreme points for the low-input-alphabet regime $k=2$ and $k=3$ and with arbitrary output alphabet size $m $. We also identify new extreme mechanisms for larger input alphabet sizes $k$, of the star configuration type, as introduced by Elangovan and Jog (2024).

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arXiv2026

Secure Distributed Hypothesis Testing

In distributed hypothesis testing, a central server performs hypothesis testing based on information received from distributed sensors/clients. We study a secure variant of this problem in which the central server determines the hypothesis class of an underlying distribution without learning any additional information about the distribution itself. We prove that, in its standard form, this is impossible to achieve, even for simple and highly restricted cases. To bypass this impossibility, we augment the model with a shared secret key available to clients but hidden from the server. We show that a single-bit secret key enables perfectly secure testing for simple classes by reducing the test distributions to a symmetric, canonical instance. Finally, for arbitrary hypothesis classes over finite domains, we establish a reduction to standard hypothesis testing using Private Simultaneous Messages (PSM) protocols, achieving polynomial communication and key lengths.

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arXiv2026

From Submodularity to Matrix Determinants: Strengthening Han's, Szász's, and Fischer's Inequalities

Dembo, Cover, and Thomas (1991) developed an elegant information-theoretic framework for proving determinantal inequalities for positive definite matrices, which relies on the structural inequalities of differential entropy. Submodular functions, which subsume entropy, inherently satisfy these structural inequalities because they obey generalized forms of the fundamental properties of entropy -- a chain rule and the property that conditioning reduces the function's value (under an appropriate definition of conditioning). Applying subadditivity, Han's inequality (1978), and partition subadditivity (i.e., subadditivity over a partition) yields Hadamard's, Szász's, and Fischer's inequalities, respectively. Furthermore, this framework recovers Ky Fan's inequality (1955), a strengthening of Hadamard's inequality. This improvement fundamentally arises because conditional subadditivity yields a tighter upper bound on the joint entropy than the one obtained via unconditional subadditivity. In this paper, we establish conditional strengthenings of Han's inequality and partition subadditivity in the general setting of submodular functions. We derive equality conditions for these strengthened bounds and characterize when they strictly improve their unconditional counterparts. We specialize these results to differential entropy and apply them to establish strengthened versions of Szász's and Fischer's inequalities. The strengthening of Szász's inequality recovers Ky Fan's inequality as a special case, and is strictly stronger than the classical Szász's inequality for any non-diagonal positive definite matrix. We also derive an inequality concerning eigenvalues, which generalizes and strictly strengthens a corresponding eigenvalue inequality of Ky Fan. We provide numerical examples to explicitly illustrate the tightness of our proposed matrix determinantal bounds.

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arXiv2026

On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns

A private information retrieval protocol (PIR) scheme under an arbitrary collusion pattern $\mathcal{P}$ enables a client to retrieve one message from a library of $K$ equal-sized messages duplicated in $N$ servers, while keeping the index of the desired message private from any colluding set in $\mathcal{P}$. Although achieving high rates typically requires sufficiently large message sizes, smaller message sizes also desirable due to reduced implementation complexity and fewer constraints. By characterizing the capacity-achieving schemes, Tian, Sun, and Chen (2019) showed that the optimal message size for uniformly decomposable PIR schemes under no-collusion setting is $N-1$. However, comparable results are not yet available for more general collusion settings. In this work, we present a complete characterization of the properties of capacity-achieving decomposable PIR schemes under arbitrary collusion patterns. Building on this characterization, we derive a general lower bound on the optimal message size for capacity-achieving uniformly decomposable PIR schemes under an arbitrary collusion pattern $\mathcal{P}$, expressed in terms of the hitting number of a newly defined family of subsets of servers determined by the collusion pattern $\mathcal{P}$. Finally, we specialize the lower bound to several important classes of collusion patterns, including $T$-collusion, disjoint collections of colluding sets, cyclically $T$-contiguous collusion, and disjoint collections of cyclically contiguous colluding sets. For the last two collusion patterns, we present matching achievable schemes that attain the corresponding bounds, thereby providing a complete characterization of the optimal message size.

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arXiv2026

Generalized Information Inequalities via Submodularity, and Two Combinatorial Problems

It is well known that there is a strong connection between entropy inequalities and submodularity, since the entropy of a collection of random variables is a submodular function. Unifying frameworks for information inequalities arising from submodularity were developed by Madiman and Tetali (2010) and Sason (2022). Madiman and Tetali (2010) established strong and weak fractional inequalities that subsume classical results such as Han's inequality and Shearer's lemma. Sason (2022) introduced a convex-functional framework for generalizing Han's inequality, and derived unified inequalities for submodular and supermodular functions. In this work, we build on these frameworks and make three contributions. First, we establish convex-functional generalizations of the strong and weak Madiman and Tetali inequalities for submodular functions. Second, using a special case of the strong Madiman-Tetali inequality, we derive a new Loomis-Whitney-type projection inequality for finite point sets in $\mathbb{R}^d$, which improves upon the classical Loomis-Whitney bound by incorporating slice-level structural information. Finally, we study an extremal graph theory problem that recovers and extends the previously known results of Sason (2022) and Boucheron et al., employing Shearer's lemma in contrast to the use of Han's inequality in those works.

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arXiv2025

Generalized Dual Discriminator GANs

Dual discriminator generative adversarial networks (D2 GANs) were introduced to mitigate the problem of mode collapse in generative adversarial networks. In D2 GANs, two discriminators are employed alongside a generator: one discriminator rewards high scores for samples from the true data distribution, while the other favors samples from the generator. In this work, we first introduce dual discriminator $α$-GANs (D2 $α$-GANs), which combines the strengths of dual discriminators with the flexibility of a tunable loss function, $α$-loss. We further generalize this approach to arbitrary functions defined on positive reals, leading to a broader class of models we refer to as generalized dual discriminator generative adversarial networks. For each of these proposed models, we provide theoretical analysis and show that the associated min-max optimization reduces to the minimization of a linear combination of an $f$-divergence and a reverse $f$-divergence. This generalizes the known simplification for D2-GANs, where the objective reduces to a linear combination of the KL-divergence and the reverse KL-divergence. Finally, we perform experiments on 2D synthetic data and use multiple performance metrics to capture various advantages of our GANs.

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arXiv2025

Fractional Subadditivity of Submodular Functions: Equality Conditions and Their Applications

Submodular functions are known to satisfy various forms of fractional subadditivity. This work investigates the conditions for equality to hold exactly or approximately in the fractional subadditivity of submodular functions. We establish that a small gap in the inequality implies that the function is close to being modular, and that the gap is zero if and only if the function is modular. We then present natural implications of these results for special cases of submodular functions, such as entropy, relative entropy, and matroid rank. As a consequence, we characterize the necessary and sufficient conditions for equality to hold in Shearer's lemma, recovering a result of Ellis \emph{et al.} (2016) as a special case. We leverage our results to propose a new multivariate mutual information, which generalizes Watanabe's total correlation (1960), Han's dual total correlation (1978), and Csiszár and Narayan's shared information (2004), and analyze its properties. Among these properties, we extend Watanabe's characterization of total correlation as the maximum correlation over partitions to fractional partitions. When applied to matrix determinantal inequalities for positive definite matrices, our results recover the equality conditions of the classical determinantal inequalities of Hadamard, Szász, and Fischer as special cases.

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arXiv2025

Sun-Jafar-Type Schemes for Weak Private Information Retrieval

In information-theoretic private information retrieval (PIR), a client wants to retrieve one desired file out of $M$ files, stored across $N$ servers, while keeping the index of the desired file private from each $T$-sized subset of servers. A PIR protocol must ideally maximize the rate, which is the ratio of the file size to the total quantum of the download from the servers, while ensuring such privacy. In Weak-PIR (WPIR), the criterion of perfect information-theoretic privacy is relaxed. This enables higher rates to be achieved, while some information about the desired file index leaks to the servers. This leakage is captured by various known privacy metrics. By leveraging the well-established capacity-achieving schemes of Sun and Jafar under non-colluding ($T=1$) and colluding ($1<T\leq N$) scenarios, we present WPIR protocols for these scenarios. We also present a new WPIR scheme for the MDS scenario, by building upon the scheme by Banawan and Ulukus for this scenario. We present corresponding explicit rate-privacy trade-offs for these setups, under the mutual-information and the maximal leakage privacy metrics. In the collusion-free setup, our presented rate-privacy trade-off under maximal leakage matches that of the previous state of the art. With respect to the MDS scenario under the maximal leakage metric, we compare with the non-explicit trade-off in the literature, and show that our scheme performs better for some numerical examples. For the $T$-collusion setup (under both privacy metrics) and for the MDS setup under the mutual information metric, our rate-privacy trade-offs are the first in the literature, to the best of our knowledge.

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arXiv2024

Maximal Guesswork Leakage

We introduce the study of information leakage through \emph{guesswork}, the minimum expected number of guesses required to guess a random variable. In particular, we define \emph{maximal guesswork leakage} as the multiplicative decrease, upon observing $Y$, of the guesswork of a randomized function of $X$, maximized over all such randomized functions. We also study a pointwise form of the leakage which captures the leakage due to the release of a single realization of $Y$. We also study these two notions of leakage with oblivious (or memoryless) guessing. We obtain closed-form expressions for all these leakage measures, with the exception of one. Specifically, we are able to obtain closed-form expression for maximal guesswork leakage for the binary erasure source only; deriving expressions for arbitrary sources appears challenging. Some of the consequences of our results are -- a connection between guesswork and differential privacy and a new operational interpretation to maximal $α$-leakage in terms of guesswork.

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arXiv2024

Unifying Privacy Measures via Maximal $(α,β)$-Leakage (M$α$beL)

We introduce a family of information leakage measures called maximal $(α,β)$-leakage (M$α$beL), parameterized by real numbers $α$ and $β$ greater than or equal to 1. The measure is formalized via an operational definition involving an adversary guessing an unknown (randomized) function of the data given the released data. We obtain a simplified computable expression for the measure and show that it satisfies several basic properties such as monotonicity in $β$ for a fixed $α$, non-negativity, data processing inequalities, and additivity over independent releases. We highlight the relevance of this family by showing that it bridges several known leakage measures, including maximal $α$-leakage $(β=1)$, maximal leakage $(α=\infty,β=1)$, local differential privacy (LDP) $(α=\infty,β=\infty)$, and local Renyi differential privacy (LRDP) $(α=β)$, thereby giving an operational interpretation to local Renyi differential privacy. We also study a conditional version of M$α$beL on leveraging which we recover differential privacy and Renyi differential privacy. A new variant of LRDP, which we call maximal Renyi leakage, appears as a special case of M$α$beL for $α=\infty$ that smoothly tunes between maximal leakage ($β=1$) and LDP ($β=\infty$). Finally, we show that a vector form of the maximal Renyi leakage relaxes differential privacy under Gaussian and Laplacian mechanisms.

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arXiv2023

An Operational Approach to Information Leakage via Generalized Gain Functions

We introduce a \emph{gain function} viewpoint of information leakage by proposing \emph{maximal $g$-leakage}, a rich class of operationally meaningful leakage measures that subsumes recently introduced leakage measures -- {maximal leakage} and {maximal $α$-leakage}. In maximal $g$-leakage, the gain of an adversary in guessing an unknown random variable is measured using a {gain function} applied to the probability of correctly guessing. In particular, maximal $g$-leakage captures the multiplicative increase, upon observing $Y$, in the expected gain of an adversary in guessing a randomized function of $X$, maximized over all such randomized functions. We also consider the scenario where an adversary can make multiple attempts to guess the randomized function of interest. We show that maximal leakage is an upper bound on maximal $g$-leakage under multiple guesses, for any non-negative gain function $g$. We obtain a closed-form expression for maximal $g$-leakage under multiple guesses for a class of concave gain functions. We also study maximal $g$-leakage measure for a specific class of gain functions related to the $α$-loss. In particular, we first completely characterize the minimal expected $α$-loss under multiple guesses and analyze how the corresponding leakage measure is affected with the number of guesses. Finally, we study two variants of maximal $g$-leakage depending on the type of adversary and obtain closed-form expressions for them, which do not depend on the particular gain function considered as long as it satisfies some mild regularity conditions. We do this by developing a variational characterization for the Rényi divergence of order infinity which naturally generalizes the definition of pointwise maximal leakage to incorporate arbitrary gain functions.

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arXiv2023

Addressing GAN Training Instabilities via Tunable Classification Losses

Generative adversarial networks (GANs), modeled as a zero-sum game between a generator (G) and a discriminator (D), allow generating synthetic data with formal guarantees. Noting that D is a classifier, we begin by reformulating the GAN value function using class probability estimation (CPE) losses. We prove a two-way correspondence between CPE loss GANs and $f$-GANs which minimize $f$-divergences. We also show that all symmetric $f$-divergences are equivalent in convergence. In the finite sample and model capacity setting, we define and obtain bounds on estimation and generalization errors. We specialize these results to $α$-GANs, defined using $α$-loss, a tunable CPE loss family parametrized by $α\in(0,\infty]$. We next introduce a class of dual-objective GANs to address training instabilities of GANs by modeling each player's objective using $α$-loss to obtain $(α_D,α_G)$-GANs. We show that the resulting non-zero sum game simplifies to minimizing an $f$-divergence under appropriate conditions on $(α_D,α_G)$. Generalizing this dual-objective formulation using CPE losses, we define and obtain upper bounds on an appropriately defined estimation error. Finally, we highlight the value of tuning $(α_D,α_G)$ in alleviating training instabilities for the synthetic 2D Gaussian mixture ring as well as the large publicly available Celeb-A and LSUN Classroom image datasets.

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arXiv2023

$(α_D,α_G)$-GANs: Addressing GAN Training Instabilities via Dual Objectives

In an effort to address the training instabilities of GANs, we introduce a class of dual-objective GANs with different value functions (objectives) for the generator (G) and discriminator (D). In particular, we model each objective using $α$-loss, a tunable classification loss, to obtain $(α_D,α_G)$-GANs, parameterized by $(α_D,α_G)\in (0,\infty]^2$. For sufficiently large number of samples and capacities for G and D, we show that the resulting non-zero sum game simplifies to minimizing an $f$-divergence under appropriate conditions on $(α_D,α_G)$. In the finite sample and capacity setting, we define estimation error to quantify the gap in the generator's performance relative to the optimal setting with infinite samples and obtain upper bounds on this error, showing it to be order optimal under certain conditions. Finally, we highlight the value of tuning $(α_D,α_G)$ in alleviating training instabilities for the synthetic 2D Gaussian mixture ring and the Stacked MNIST datasets.

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arXiv2022

An Alphabet of Leakage Measures

We introduce a family of information leakage measures called maximal $α,β$-leakage, parameterized by real numbers $α$ and $β$. The measure is formalized via an operational definition involving an adversary guessing an unknown function of the data given the released data. We obtain a simple, computable expression for the measure and show that it satisfies several basic properties such as monotonicity in $β$ for a fixed $α$, non-negativity, data processing inequalities, and additivity over independent releases. Finally, we highlight the relevance of this family by showing that it bridges several known leakage measures, including maximal $α$-leakage $(β=1)$, maximal leakage $(α=\infty,β=1)$, local differential privacy $(α=\infty,β=\infty)$, and local Renyi differential privacy $(α=β)$.

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arXiv2022

Multiple Access Channel Simulation

We study the problem of simulating a two-user multiple-access channel (MAC) over a multiple access network of noiseless links. Two encoders observe independent and identically distributed (i.i.d.) copies of a source random variable each, while a decoder observes i.i.d. copies of a side-information random variable. There are rate-limited noiseless communication links between each encoder and the decoder, and there is independent pairwise shared randomness between all the three possible pairs of nodes. The decoder has to output approximately i.i.d. copies of another random variable jointly distributed with the two sources and the side information. We are interested in the rate tuples which permit this simulation. This setting can be thought of as a multi-terminal generalization of the point-to-point channel simulation problem studied by Bennett et al. (2002) and Cuff (2013). When the pairwise shared randomness between the encoders is absent, the setting reduces to a special case of MAC simulation using another MAC studied by Haddadpour et al.~(2013). We establish that the presence of encoder shared randomness can strictly improve the communication rate requirements. We first show that the inner bound derived from Haddadpour et al.~(2013) is tight when the sources at the encoders are conditionally independent given the side-information at the decoder. This result recovers the existing results on point-to-point channel simulation and function computation over such multi-terminal networks. We then explicitly compute the communication rate regions for an example both with and without the encoder shared randomness and demonstrate that its presence strictly reduces the communication rates. Inner and outer bounds for the general case are also obtained.

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arXiv2022

$α$-GAN: Convergence and Estimation Guarantees

We prove a two-way correspondence between the min-max optimization of general CPE loss function GANs and the minimization of associated $f$-divergences. We then focus on $α$-GAN, defined via the $α$-loss, which interpolates several GANs (Hellinger, vanilla, Total Variation) and corresponds to the minimization of the Arimoto divergence. We show that the Arimoto divergences induced by $α$-GAN equivalently converge, for all $α\in \mathbb{R}_{>0}\cup\{\infty\}$. However, under restricted learning models and finite samples, we provide estimation bounds which indicate diverse GAN behavior as a function of $α$. Finally, we present empirical results on a toy dataset that highlight the practical utility of tuning the $α$ hyperparameter.

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arXiv2022

A Variational Formula for Infinity-Rényi Divergence with Applications to Information Leakage

We present a variational characterization for the Rényi divergence of order infinity. Our characterization is related to guessing: the objective functional is a ratio of maximal expected values of a gain function applied to the probability of correctly guessing an unknown random variable. An important aspect of our variational characterization is that it remains agnostic to the particular gain function considered, as long as it satisfies some regularity conditions. Also, we define two variants of a tunable measure of information leakage, the maximal $α$-leakage, and obtain closed-form expressions for these information measures by leveraging our variational characterization.

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