arXiv subjects

Piyush Srivastava

Publications and source records attributed to Piyush Srivastava.

At least 19 recordsLinked to original sources

arXiv2026

Entropy Objectives in Markov Decision Processes

We consider the problem of synthesizing control policies that enforce a concentration property on the state distributions of a stochastic system. We present a formalization of this problem in terms of synthesizing strategies for maintaining an entropy-based objective in Markov Decision Processes (MDPs). We first show that even relaxed versions of this problem are complexity-theoretically hard. We then present a sound and (conditionally) relatively complete method to verify and synthesize strategies for such entropy objectives. The main challenge is the non-linear nature of such objectives, and our approach addresses this by exploiting and combining ideas from convex duality and invariant synthesis. We also investigate the role of memory and randomization in ensuring entropy objectives. Finally, we implement our ideas to evaluate our approach empirically on a few illustrative benchmarks.

cs.AI↗

arXiv2026

Innovation: An Almost Characterization of Hallucination

Hallucination is a central limitation of large language models (LLMs), and substantial effort has been devoted to understanding and mitigating it. Towards this, Kalai and Vempala (STOC 2024) introduced a probabilistic framework formalizing calibration and hallucination, and showed that, with high probability, calibrated LLMs hallucinate roughly at the rate of the "missing mass", a measure of how incomplete the training data is relative to its source. This raises two fundamental questions: (i) what property of a calibrated LLM makes hallucinations unavoidable? and (ii) can hallucinations be avoided by giving up calibration? We answer these questions by introducing a simpler property we call innovation that measures the tendency of a model to produce outputs outside the training data. We show that innovation is implied by the condition for hallucination identified by Kalai and Vempala, and, further, that it is an almost characterization of hallucination: hallucination implies innovation, and conversely, innovation implies hallucination with high probability. We also provide lower bounds on the hallucination rate based on the "innovation rate", and by relating innovation rate back to missing mass, we obtain new hallucination rate lower bounds based on missing mass that extend the results of Kalai and Vempala.

cs.LG↗

arXiv2026

The Rate-Immediacy Barrier in Explicit Tree Code Constructions

Since the introduction of tree codes by Schulman (STOC 1993), explicit construction of asymptotically good tree codes has remained a notorious challenge. A work by Cohen, Haeupler and Schulman (STOC 2018), as well as the state-of-the-art construction by Ben Yaacov, Cohen, and Yankovitz (STOC 2022) have achieved codes with rate $Ω(1/\log\log n)$, exponentially improving upon the original rate $Ω(1/\log n)$ construction of Evans, Klugerman and Schulman from 1994. All of these constructions rely, at least in part, on increasingly sophisticated methods of combining (block) error-correcting codes. In this work, we identify a fundamental barrier to constructing tree codes using known techniques. We introduce a key property which we call immediacy, that, while not required by the original definition of tree codes, is shared by all known constructions and inherently arises in recursive combinations of error-correcting codes. Our main technical contribution is the proof of a rate-immediacy trade-off, which, in particular, implies that any tree code with constant distance and non-trivial immediacy must necessarily have vanishing rate. By applying our rate-immediacy trade-off to existing constructions, we establish that their known rate analyses are essentially optimal given their actual error-correction properties. More broadly, our work highlights the need for fundamentally new ideas -- beyond the recursive use of error-correcting codes -- to achieve substantial progress in explicitly constructing asymptotically good tree codes.

cs.IT↗

arXiv2026

Deterministically approximating the volume of a Kostka polytope

Polynomial-time deterministic approximation of volumes of polytopes, up to an approximation factor that grows at most sub-exponentially with the dimension, remains an open problem. Recent work on this question has focused on identifying interesting classes of polytopes for which such approximation algorithms can be obtained. In this paper, we focus on one such class of polytopes: the Kostka polytopes. The volumes of Kostka polytopes appear naturally in questions of random matrix theory, in the context of evaluating the probability density that a random Hermitian matrix with fixed spectrum $λ$ has a given diagonal $μ$ (the so-called randomized Schur-Horn problem): the corresponding Kostka polytope is denoted $\mathrm{GT}(λ, μ)$. We give a polynomial-time deterministic algorithm for approximating the volume of a ($Ω(n^2)$ dimensional) Kostka polytope $\mathrm{GT}(λ, μ)$ to within a multiplicative factor of $\exp(O(n\log n))$, when $λ$ is an integral partition with $n$ parts, with entries bounded above by a polynomial in $n$, and $μ$ is an integer vector lying in the interior of the permutohedron (i.e., convex hull of all permutations) of $λ$. The algorithm thus gives asymptotically correct estimates of the log-volume of Kostka polytopes corresponding to such $(λ, μ)$. Our approach is based on a partition function interpretation of a continuous analogue of Schur polynomials.

math.CO↗

arXiv2025

A direct proof of a unified law of robustness for Bregman divergence losses

In contemporary deep learning practice, models are often trained to near zero loss i.e. to nearly interpolate the training data. However, the number of parameters in the model is usually far more than the number of data points n, the theoretical minimum needed for interpolation: a phenomenon referred to as overparameterization. In an interesting piece of work, Bubeck and Sellke considered a natural notion of interpolation: the model is said to interpolate when the model's training loss goes below the loss of the conditional expectation of the response given the covariate. For this notion of interpolation and for a broad class of covariate distributions (specifically those satisfying a natural notion of concentration of measure), they showed that overparameterization is necessary for robust interpolation i.e. if the interpolating function is required to be Lipschitz. Their main proof technique applies to regression with square loss against a scalar response, but they remark that via a connection to Rademacher complexity and using tools such as the Ledoux-Talagrand contraction inequality, their result can be extended to more general losses, at least in the case of scalar response variables. In this work, we recast the original proof technique of Bubeck and Sellke in terms of a bias-variance type decomposition, and show that this view directly unlocks a generalization to Bregman divergence losses (even for vector-valued responses), without the use of tools such as Rademacher complexity or the Ledoux-Talagrand contraction principle. Bregman divergences are a natural class of losses since for these, the best estimator is the conditional expectation of the response given the covariate, and include other practical losses such as the cross entropy loss. Our work thus gives a more general understanding of the main proof technique of Bubeck and Sellke and demonstrates its broad utility.

cs.LG↗

arXiv2025

Approximating the Total Variation Distance between Gaussians

The total variation distance is a metric of central importance in statistics and probability theory. However, somewhat surprisingly, questions about computing it algorithmically appear not to have been systematically studied until very recently. In this paper, we contribute to this line of work by studying this question in the important special case of multivariate Gaussians. More formally, we consider the problem of approximating the total variation distance between two multivariate Gaussians to within an $ε$-relative error. Previous works achieved a fixed constant relative error approximation via closed-form formulas. In this work, we give algorithms that given any two $n$-dimensional Gaussians $D_1,D_2$, and any error bound $ε> 0$, approximate the total variation distance $D := d_{TV}(D_1,D_2)$ to $ε$-relative accuracy in $\text{poly}(n,\frac{1}ε,\log \frac{1}{D})$ operations. The main technical tool in our work is a reduction that helps us extend the recent progress on computing the TV-distance between discrete random variables to our continuous setting.

cs.DS↗

arXiv2024

Sampling from convex sets with a cold start using multiscale decompositions

Running a random walk in a convex body $K\subseteq\mathbb{R}^n$ is a standard approach to sample approximately uniformly from the body. The requirement is that from a suitable initial distribution, the distribution of the walk comes close to the uniform distribution $π_K$ on $K$ after a number of steps polynomial in $n$ and the aspect ratio $R/r$ (i.e., when $rB_2 \subseteq K \subseteq RB_{2}$). Proofs of rapid mixing of such walks often require the probability density $η_0$ of the initial distribution with respect to $π_K$ to be at most $\mathrm{poly}(n)$: this is called a "warm start". Achieving a warm start often requires non-trivial pre-processing before starting the random walk. This motivates proving rapid mixing from a "cold start", wherein $η_0$ can be as high as $\exp(\mathrm{poly}(n))$. Unlike warm starts, a cold start is usually trivial to achieve. However, a random walk need not mix rapidly from a cold start: an example being the well-known "ball walk". On the other hand, Lovász and Vempala proved that the "hit-and-run" random walk mixes rapidly from a cold start. For the related coordinate hit-and-run (CHR) walk, which has been found to be promising in computational experiments, rapid mixing from a warm start was proved only recently but the question of rapid mixing from a cold start remained open. We construct a family of random walks inspired by classical decompositions of subsets of $\mathbb{R}^n$ into countably many axis-aligned dyadic cubes. We show that even with a cold start, the mixing times of these walks are bounded by a polynomial in $n$ and the aspect ratio. Our main technical ingredient is an isoperimetric inequality for $K$ for a metric that magnifies distances between points close to the boundary of $K$. As a corollary, we show that the CHR walk also mixes rapidly both from a cold start and from a point not too close to the boundary of $K$.

cs.DS↗

arXiv2024

Nuclear correlation functions using first-principle calculations of lattice quantum chromodynamics

Exploring nuclear physics through the fundamental constituents of the strong force -- quarks and gluons -- is a formidable challenge. While numerical calculations using lattice quantum chromodynamics offer the most promising approach for this pursuit, practical implementation is arduous, especially due to the uncontrollable growth of quark-combinatorics, the so-called Wick-contraction problem of nuclei. We present here two novel methods providing a state-of-the-art solution to this problem. In the first, we exploit randomized algorithms inspired from computational number theory to detect and eliminate redundancies that arise in Wick contraction computations. Our second method explores facilities for automation of tensor computations -- in terms of efficient utilization of specialized hardware, algorithmic optimizations, as well as ease of programming and the potential for automatic code generation -- that are offered by new programming models inspired by applications in machine learning (e.g., TensorFlow). We demonstrate the efficacy of our methods by computing two-point correlation functions for Deuteron, Helium-3, Helium-4 and Lithium-7, achieving at least an order of magnitude improvement over existing algorithms with efficient implementation on GPU-accelerators. Additionally, we discover an intriguing characteristic shared by all the nuclei we study: specific spin-color combinations dominate the correlation functions, hinting at a potential connection to an as-yet-unidentified symmetry in nuclei. Moreover finding them beforehand can reduce the computing time further and substantially. Our results, with the efficiency that we achieved, suggest the possibility of extending the applicability of our methods for calculating properties of light nuclei, potentially up to A ~12 and beyond.

hep-lat↗

arXiv2022

On complex roots of the independence polynomial

It is known from the work of Shearer (1985) (and also Scott and Sokal (2005)) that the independence polynomial $Z_G(λ)$ of a graph $G$ of maximum degree at most $d+1$ does not vanish provided that $\vertλ\vert \leq \frac{d^d}{(d+1)^{d+1}}$. Significant extensions of this result have recently been given in the case $\Re λ\geq 0$ by Peters and Regts (2019) and Bencs and Csikvári (arxiv:1807.08963). In this paper, our motivation is to further extend these results and find zero free regions when $\Re λ\leq 0$. We begin by giving new geometric criteria for establishing zero-free regions as well as for carrying out semi-rigorous numerical explorations. We then provide two examples of the (rigorous) use of these criteria, by establishing two new zero-free regions in the left-half plane. We also improve upon the results of Bencs and Csikvári (arxiv:1807.08963) for the right half-plane using our framework. By a direct application of the interpolation method of Barvinok, combined with extensions due to Patel and Regts, these results also imply deterministic polynomial time approximation algorithms for the independence polynomial of bounded degree graphs in the new zero-free regions.

cs.DM↗

arXiv2022

Universal Lower Bound for Learning Causal DAGs with Atomic Interventions

A well-studied challenge that arises in the structure learning problem of causal directed acyclic graphs (DAG) is that using observational data, one can only learn the graph up to a "Markov equivalence class" (MEC). The remaining undirected edges have to be oriented using interventions, which can be very expensive to perform in applications. Thus, the problem of minimizing the number of interventions needed to fully orient the MEC has received a lot of recent attention, and is also the focus of this work. Our first result is a new universal lower bound on the number of single-node interventions that any algorithm (whether active or passive) would need to perform in order to orient a given MEC. Our second result shows that this bound is, in fact, within a factor of two of the size of the smallest set of single-node interventions that can orient the MEC. Our lower bound is provably better than previously known lower bounds. Further, using simulations on synthetic graphs and by giving examples of special graph families, we show that our bound is often significantly better. To prove our lower bound, we develop the notion of clique-block shared-parents (CBSP) orderings, which are topological orderings of DAGs without v-structures and satisfy certain special properties. We also use the techniques developed here to extend our results to the setting of multi-node interventions.

cs.LG↗

arXiv2022

On the mixing time of coordinate Hit-and-Run

We obtain a polynomial upper bound on the mixing time $T_{CHR}(ε)$ of the coordinate Hit-and-Run random walk on an $n-$dimensional convex body, where $T_{CHR}(ε)$ is the number of steps needed in order to reach within $ε$ of the uniform distribution with respect to the total variation distance, starting from a warm start (i.e., a distribution which has a density with respect to the uniform distribution on the convex body that is bounded above by a constant). Our upper bound is polynomial in $n, R$ and $\frac{1}ε$, where we assume that the convex body contains the unit $\Vert\cdot\Vert_\infty$-unit ball $B_\infty$ and is contained in its $R$-dilation $R\cdot B_\infty$. Whether coordinate Hit-and-Run has a polynomial mixing time has been an open question.

cs.DS↗

arXiv2022

Correlation decay and partition function zeros: Algorithms and phase transitions

We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite $Δ$-regular tree) implies strong spatial mixing on all graphs of maximum degree $Δ$ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing. This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard core lattice gas. It also allows us to prove new results on the location of Lee-Yang zeros of the anti-ferromagnetic Ising model. We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs. In particular, we show that results on strong spatial mixing in the anti-ferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function. This extension allows us to give the first deterministic FPTAS for counting the number of $q$-colorings of a graph of maximum degree $Δ$ provided only that $q\ge 2Δ$. This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. We also give an improved version of this result for triangle-free graphs.

cs.CC↗

arXiv2020

City-Scale Agent-Based Simulators for the Study of Non-Pharmaceutical Interventions in the Context of the COVID-19 Epidemic

We highlight the usefulness of city-scale agent-based simulators in studying various non-pharmaceutical interventions to manage an evolving pandemic. We ground our studies in the context of the COVID-19 pandemic and demonstrate the power of the simulator via several exploratory case studies in two metropolises, Bengaluru and Mumbai. Such tools become common-place in any city administration's tool kit in our march towards digital health.

q-bio.PE↗

arXiv2020

The PSPACE-hardness of understanding neural circuits

In neuroscience, an important aspect of understanding the function of a neural circuit is to determine which, if any, of the neurons in the circuit are vital for the biological behavior governed by the neural circuit. A similar problem is to determine whether a given small set of neurons may be enough for the behavior to be displayed, even if all other neurons in the circuit are deactivated. Such a subset of neurons forms what is called a degenerate circuit for the behavior being studied. Recent advances in experimental techniques have provided researchers with tools to activate and deactivate subsets of neurons with a very high resolution, even in living animals. The data collected from such experiments may be of the following form: when a given subset of neurons is deactivated, is the behavior under study observed? This setting leads to the algorithmic question of determining the minimal vital or degenerate sets of neurons when one is given as input a description of the neural circuit. The algorithmic problem entails both figuring out which subsets of neurons should be perturbed (activated/deactivated), and then using the data from those perturbations to determine the minimal vital or degenerate sets. Given the large number of possible perturbations, and the recurrent nature of neural circuits, the possibility of a combinatorial explosion in such an approach has been recognized in the biology and the neuroscience literature. In this paper, we prove that the problems of finding minimal or minimum-size degenerate sets, and of finding the set of vital neurons, of a neural circuit given as input, are in fact PSPACE-hard. More importantly, we prove our hardness results by showing that a simpler problem, that of simulating such neural circuits, is itself PSPACE-hard.

cs.CC↗

arXiv2020

COVID-19 Epidemic Study II: Phased Emergence From the Lockdown in Mumbai

The nation-wide lockdown starting 25 March 2020, aimed at suppressing the spread of the COVID-19 disease, was extended until 31 May 2020 in three subsequent orders by the Government of India. The extended lockdown has had significant social and economic consequences and `lockdown fatigue' has likely set in. Phased reopening began from 01 June 2020 onwards. Mumbai, one of the most crowded cities in the world, has witnessed both the largest number of cases and deaths among all the cities in India (41986 positive cases and 1368 deaths as of 02 June 2020). Many tough decisions are going to be made on re-opening in the next few days. In an earlier IISc-TIFR Report, we presented an agent-based city-scale simulator(ABCS) to model the progression and spread of the infection in large metropolises like Mumbai and Bengaluru. As discussed in IISc-TIFR Report 1, ABCS is a useful tool to model interactions of city residents at an individual level and to capture the impact of non-pharmaceutical interventions on the infection spread. In this report we focus on Mumbai. Using our simulator, we consider some plausible scenarios for phased emergence of Mumbai from the lockdown, 01 June 2020 onwards. These include phased and gradual opening of the industry, partial opening of public transportation (modelling of infection spread in suburban trains), impact of containment zones on controlling infections, and the role of compliance with respect to various intervention measures including use of masks, case isolation, home quarantine, etc. The main takeaway of our simulation results is that a phased opening of workplaces, say at a conservative attendance level of 20 to 33\%, is a good way to restart economic activity while ensuring that the city's medical care capacity remains adequate to handle the possible rise in the number of COVID-19 patients in June and July.

q-bio.PE↗

arXiv2019

Online codes for analog signals

This paper revisits a classical scenario in communication theory: a waveform sampled at regular intervals is to be encoded so as to minimize distortion in its reconstruction, despite noise. This transformation must be online (causal), to enable real-time signaling; and should use no more power than the original signal. The noise model we consider is an "atomic norm" convex relaxation of the standard (discrete alphabet) Hamming-weight-bounded model: namely, adversarial $\ell_1$-bounded. In the "block coding" (noncausal) setting, such encoding is possible due to the existence of large almost-Euclidean sections in $\ell_1$ spaces, a notion first studied in the work of Dvoretzky in 1961. Our main result is that an analogous result is achievable even causally. Equivalently, our work may be seen as a "lower triangular" version of $\ell_1$ Dvoretzky theorems. In terms of communication, the guarantees are expressed in terms of certain time-weighted norms: the time-weighted $\ell_2$ norm imposed on the decoder forces increasingly accurate reconstruction of the distant past signal, while the time-weighted $\ell_1$ norm on the noise ensures vanishing interference from distant past noise. Encoding is linear (hence easy to implement in analog hardware). Decoding is performed by an LP analogous to those used in compressed sensing.

cs.DM↗

arXiv2018

Fisher zeros and correlation decay in the Ising model

We study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the "interaction parameter"; these are known as Fisher zeros in light of their introduction by Fisher in 1965. While the zeros of the partition function as a polynomial in the "field" parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros for general graphs. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy, or the logarithm of the partition function) and decay of correlations with distance. We also discuss the consequences of our result for efficient deterministic approximation of the partition function. Our proof relies heavily on algorithmic techniques, notably Weitz's self-avoiding walk tree, and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics.

math-ph↗

arXiv2018

The Ising Partition Function: Zeros and Deterministic Approximation

We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters $β$ (the interaction) and $λ$ (the external field), except for the case $\vertλ\vert=1$ (the "zero-field" case). A randomized algorithm (FPRAS) for all graphs, and all $β,λ$, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the "decay of correlations" property. Rather, we exploit and extend machinery developed recently by Barvinok, and Patel and Regts, based on the location of the complex zeros of the partition function, which can be seen as an algorithmic realization of the classical Lee-Yang approach to phase transitions. Our approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this extension, we establish a tight version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

cs.DS↗