arXiv subjects

Rahul Singh

Publications and source records attributed to Rahul Singh.

At least 19 recordsLinked to original sources

arXiv2026

$O(1/n)$ Convergence Rate for RVI $Q$-Learning

We establish finite-time convergence guarantee for synchronous relative value iteration (RVI) $Q$-learning for finite weakly communicating Markov decision processes (MDPs). For stochastic Krasnosel'ski\uı--Mann iterations driven by a non-expansive, linearly regular map on a finite-dimensional normed space, we establish last-iterate mean-squared residual bounds under martingale-difference noise whose conditional second moment grows at most linearly with the squared residual. Suitable polynomial stepsizes of order $n^{-a}$ yield an $O(n^{-a})$ rate for $0<a\leq1$. We verify linear regularity and the noise condition for the centered Bellman operator for finite weakly communicating MDPs. Applying the general result gives an $O(n^{-1})$ bound on the mean-squared Bellman span residual of RVI $Q$-learning iterates when a suitable stepsize sequence is used. The positive gap between optimal and suboptimal stationary deterministic policies then yields an $O(n^{-1})$ bound on the expected suboptimality of the greedy policy. Consequently, $O(ε^{-1})$ synchronous iterations suffice for an $ε$-optimal policy in expectation. These guarantees are instance-dependent, with constants that depend on the hitting-time parameters and the suboptimality gap of the second-best policy.

cs.LG↗

arXiv2026

p-DCIM: A 28nm Probabilistic SRAM DCIM based Ising Machine for Near-Memory Context Selection in LLM-RAG Pipeline

We present a fabricated SRAM DCIM Ising machine that performs set-level context selection for large language model (LLM) with retrieval-augmented generation (RAG). In RAG, the evidence-selection stage is increasingly delegated to repeated language-model reranking, which adds substantial energy to the inference pipeline. We reformulate evidence selection as a quadratic unconstrained binary optimization (QUBO) problem and solve it on a dual-mode digital compute-in-memory (DCIM) macro: the same multiply-and-accumulate datapath computes the embedding-similarity scores that populate the QUBO coefficients and evaluates the per-spin Hamiltonian differences that drive the anneal. The annealing temperature is set by a digital plane-count control along with the SRAM pseudo-read random bit flips, giving graded cooling entirely from on-chip digital configuration. The macro is taped-out in a 28 nm process [0.16 mm2, 0.41 mW] and is proposed to be integrated as part of the logic base-die in hybrid high-bandwidth memory (HBM) and high-bandwidth flash (HBF) stacks in a GPU package, where only the selected passage identifiers return to the host. Replacing the language-model selection pass with the near-memory Ising solver yields a macro normalized selection-energy estimate more than six orders of magnitude lower than the LLM-based baseline and reduces the evaluated total energy per query by 67%, while maintaining comparable token-level F1 on the evaluated examples.

cs.AR↗

arXiv2026

Stationary Bias and Extrapolation in Nonlinear Two-Timescale Stochastic Approximation

Constant-step stochastic approximation generally has a nonzero stationary mean error that persists under time averaging. This paper studies that error for nonlinear two-timescale recursions driven by an exogenous finite-state Markov chain. Under stated smoothness assumptions and conditions on the stationary distribution, we derive a first-order bias expansion whose error bound remains uniform as the slow step size becomes much smaller than the fast step size. Fast-manifold coordinates keep the associated covariance equation regular in this limit. For fast step $η$ and slow step $\varepsilon$, the expansion reveals a mixed contribution $\varepsilon^2/η$ alongside terms linear in each step size. This dependence matters for bias reduction: along power-law step-size paths, the bias exponents need not be integers, so Richardson--Romberg extrapolation requires weights matched to the path. An exactly solvable nonlinear Markov example verifies the coefficients. We verify localization for temporal-difference learning and compare finite-run extrapolation at equal update budgets. For finite runs, we bound the initialization error of tail averages on both timescales under an additional coupling assumption. In the special case of additive independent noise, signed third-moment cancellation yields a sharper remainder.

cs.LG↗

arXiv2026

Logarithmic Regret via Passive Change Detection in Piecewise-Stationary Self-Tuning Regulation

We study minimum-variance control of an unknown autoregressive system with exogenous inputs and coefficients that change at unknown times. Under bounded independent disturbances, fixed detection gaps, stability and feasibility conditions, and sufficient time between changes, we prove $O((C+1)\log((T+1)/δ))$ regret with probability at least $1-δ$, where $T$ is the horizon and $C$ the number of changes. Unlike switching bandits, where unselected arms can change unobserved, admissible plant changes provide information during exploitation: the correct feasible controller leaves only the disturbance in the output, whereas a detectable change raises output energy under the old controller. PIECE-CD explores initially and after alarms, then uses gated recursive least squares for control. Its energy test compares windowed output power with a threshold above the noise floor; the extension to unstable controller mismatches also monitors the reference controller's input proposal. We control false alarms across the horizon and prove logarithmic detection delay. Inputs are clipped to prescribed bounds. Logarithmic regret also holds under an explicit condition ensuring that clipping becomes inactive after a finite burn-in. Under the stated feasibility conditions, the extended detector covers destabilizing changes with detectable excess energy over a fixed window.

cs.LG↗

arXiv2026

$q$-Oper Structures on a Formal Punctured Disc

Let $G$ be a connected reductive complex algebraic group and let $q\in\mathbb{C}^{\times}$ be not a root of unity. We prove that every $(G,q)$-connection on a formal punctured disc admits a $(G,q)$-oper structure.

math.RT↗

arXiv2026

Finite-Time Concentration and Convergence Rates for Projected Two-Time-Scale Stochastic Approximation with Markov Noise

We study finite-time concentration and convergence rates for projected two-time-scale stochastic approximation driven by a controlled Markov chain. The averaged fast map is contractive, while the slow iterate is projected onto a compact convex polyhedron. The associated projected ordinary differential equation may have a discontinuous vector field at the boundary, preventing a direct application of standard analyses based on Lipschitz vector fields. Using the Skorokhod map, we establish explicit high-probability bounds for tracking the moving fast equilibrium and the projected slow dynamics. These bounds separate martingale fluctuations, Markov-noise residuals, and the bias due to time-scale separation. A Lipschitz Lyapunov function satisfying a uniform decrease condition over fixed time intervals yields almost-sure convergence, with explicit last-iterate rates when the decrease admits a power lower bound. Under uniform Lyapunov contraction, polynomial step sizes yield joint fast-tracking and slow Lyapunov-error exponents arbitrarily close to $1/3$. Under the additional assumption that the reduced slow update map is a Euclidean contraction, logarithmically separated step sizes improve the joint rate to $O(n^{-1/2}\log n)$ almost surely, including for boundary equilibria. The same rate holds under a distinct geometric condition involving a strictly attracting face of a box and a fast equilibrium that is constant on that face. An actor-critic application achieves an almost-sure value-gap rate of $O(n^{-1}\log n)$ relative to the optimum within the constrained policy class. Further applications include projected TD(0) and projected stochastic gradient descent. We also extend the analysis to projection of the fast recursion under Euclidean contractivity.

math.OC↗

arXiv2026

Bayesian Bradley-Terry Framework for Ranking Multi-Indicator Entities: Application to Indian States and Union Territories

Often, real-world entities are explained by multiple indicators, and ranking those is challenging, e.g., administrative units, sport teams, and machine learning models. Existing methodologies rely on single indicators or composite indices, leaving a gap for frameworks capable of handling multi-indicator scenarios. We present a ranking methodology motivated by the Bayesian Bradley-Terry (BT) model. We convert indicators to paired comparisons to utilize the BT framework. The prior covariance of the BT latent merit parameters is modeled as a function of a covariate to accommodate prior information. For parameter estimation, we use a Markov chain Monte Carlo (MCMC) algorithm that employs a hybrid Metropolis--Hastings scheme with a preconditioned Crank--Nicolson proposal and Gibbs sampling. We show that the proposed estimator converges. Furthermore, we propose a problem-specific stopping rule based on ranking stability. To demonstrate the methodology's practical utility, we apply it to rank the states and union territories (UTs) of India using multi-indicator data from the National Family Health Survey-5. By estimating and comparing rankings across various economic regimes, including low-income, mid-income, and outlier-excluded subsets, our application uncovers meaningful deviations between regional economic standing and overall developmental performance, illustrating the framework's broad value for resolving complex multidimensional ranking problems.

stat.ME↗

arXiv2026

Stable Policy Learning

In evidence-based policymaking, typically one experimental sample is observed, then a learned policy recommendation is implemented at scale. Policies learned from the experimental data can perform well in expected welfare, yet random sampling in the experiment can produce recommendations with poor welfare outcomes. In this paper, we ask: how should policy learning algorithms balance expected welfare against sampling risk? Our main contribution is to show that algorithmic stability plays a central role in characterizing and navigating the tradeoff. Intuitively, if a policy learning algorithm's recommendation remains stable when one experimental unit is replaced, then that algorithm has limited sampling risk. We propose a method for policy learning called policy-vote bagging, which learns treatment decisions on many subsamples then averages their votes into treatment probabilities. Relative to using one subsample, averaging across subsamples preserves expected welfare and improves expected utility for a risk-averse researcher. We derive sharp bounds linking estimation accuracy, subsample size, and welfare variation, including an exact guarantee under CARA utility.

econ.EM↗

arXiv2026

q-Opers and Quantum/Classical Duality Beyond Type A

Using the language of q-opers, we propose an algebro-geometric description of the duality between Bethe Ansatz equations for quantum XXZ spin chains and many-body trigonometric Ruijsenaars-Schneider/Macdonald systems for classical groups. Upon this duality, the energy level sets of the classical B/C/D-type Hamiltonians are found to be in bijection with the set of solutions of the XXZ Bethe Ansatz equations of type A, albeit with open boundary conditions. This dictionary generalizes previous results in which a GL(N) XXZ spin chain with twisted periodic boundary conditions was dual to an N-body Ruijsenaars-Schneider system. Our construction implements a $\mathbb{Z}/2\mathbb{Z}$ folding both at the level of the (GL(N),q)-oper data as well as at the level of the N-body trigonometric Ruijsenaars-Schneider model. The nonreduced BC-type systems appear in our analysis as well.

math.AG↗

arXiv2026

Scalable Statistical Inference in Stochastic Gradient Descent

Constructing confidence regions for stochastic gradient descent (SGD) ideally requires estimating the asymptotic covariance matrix, a severe computational bottleneck in high dimensions. Traditional cancellation-based batch means methods bypass this estimation but require inverting a sample batch covariance matrix. This introduces strict mathematical degeneracy when the parameter dimension exceeds the number of batches. To address this problem, we utilize equal batch size batch means method and propose a simultaneous, marginal-friendly framework. The proposed marginal statistics has a asymptotic Student's $t$-distribution, and eliminates the matrix inversion step, entirely circumventing high-dimensional degeneracy. To achieve valid simultaneous coverage, we present an algorithm utilizing wild bootstrap samples drawn from a statistic as a function of only the diagonals of the variance-covariance estimator, and to further incorporate the contribution of cross-dependencies, we introduce an efficient Quasi-Monte Carlo procedure utilizing a $t$-copula approximation. Additionally, we integrate a Lugsail variance estimator to aggressively correct finite-sample bias and under-coverage. The proposed methodology delivers interpretable, simultaneous hyper-rectangular confidence regions that are statistically robust, memory-efficient, and strictly scalable for high-dimensional inference. The theoretical results are supported by extensive numerical simulation analysis through various aspects of dimension, number of batches and error structure.

stat.CO↗

arXiv2026

SemanticXR: Low Power and Real-time Queryable Semantic Mapping with an Object-Level Device-Cloud Architecture

Semantic mapping is a core service that enables grounded interactions in emerging Extended Reality (XR) applications such as AI assistants. Deploying this capability on mobile XR devices requires a system that is open-vocabulary, real-time, and low-power. Existing approaches are compute-intensive and assume server-class resources. Cloud offloading offers a practical path, but no existing system splits semantic mapping between the device and the cloud, and current approaches do not address how to manage communication, execution, and memory footprint across the device-cloud boundary. We present SemanticXR, the first device-cloud system for real-time, open-vocabulary semantic mapping and querying under XR power, bandwidth, and memory constraints. Our key insight is to elevate semantically identifiable objects to first-class units of system design, governing how the system communicates, executes, and manages memory across the device and the server. Evaluation against a new, aggressive device-cloud baseline shows that object-level system organization improves server-side mapping latency by 2.2x at equivalent semantic quality. Object-level depth-mapping co-design maintains upstream bandwidth under 2.5 Mbps. On the device, an object-level sparse local map with incremental updates and update prioritization enables sub-100 ms query latency for up to 10,000 objects even under network drops, supports tens of thousands of objects within 500 MB memory footprint, and scales downstream bandwidth with map changes rather than total scene size. The system adds only about 2% to idle device power.

cs.DC↗

arXiv2026

Automatic Debiased Machine Learning for Dynamic Treatment Effects and General Nested Functionals

Many canonical models in causal inference and structural econometrics have recursive identification formulas. In causal inference, recursion arises when identification requires both pre- and post-treatment covariates. For example, short-term surrogate outcomes are measured after the treatment, and serve as necessary covariates when identifying long-term effects. Post-treatment covariates are also required for identification of dynamic difference-in-differences designs, time-varying treatment regimes, and mediation analysis. In structural econometrics, recursion arises through evolving state variables, for example in dynamic sample selection models and dynamic discrete choice models. In this paper, we propose an automatic and recursive method for inference, applicable to such formulas, allowing for flexible estimation by neural networks and random forests. As a technical contribution, we introduce recursive Riesz representers.

econ.EM↗

arXiv2026

Kernel Ridge Regression Inference

We provide uniform confidence bands for kernel ridge regression (KRR), a widely used nonparametric regression estimator for nonstandard data such as preferences, sequences, and graphs. Despite the prevalence of these data--e.g., student preferences in school matching mechanisms--the inferential theory of KRR is not fully known. We construct valid and sharp confidence sets that shrink at nearly the minimax rate, allowing nonstandard regressors. Our bootstrap procedure uses anti-symmetric multipliers for computational efficiency and for validity under mis-specification. We use the procedure to develop a test for match effects, i.e. whether students benefit more from the schools they rank highly.

math.ST↗

arXiv2026

Optimal Scheduling for Remote State Estimation over Hybrid Channels

We study optimal scheduling for remote state estimation over a network with two heterogeneous communication channels: a fast but unreliable channel and a slow but reliable channel. To capture temporal correlations in packet losses, we model the unreliable channel as a Gilbert-Elliott (GE) channel. The remote estimation setup consists of a source, a sensor, and a remote estimator. The source evolves as a discrete-time autoregressive (AR) process, and the sensor decides at each time whether to use the fast unreliable channel or the slow reliable channel. We formulate the scheduling problem faced by the sensor as a Markov decision process (MDP) with a continuous state-space and consider minimizing the infinite horizon average cost criterion, where the cost consists of the squared estimation error and the transmission energy consumed. We establish the existence of an optimal stationary policy. We then characterize the structure of an optimal policy, and show that it has a threshold structure with respect to the estimation error. An optimal policy chooses from amongst the two channels based on whether the error exceeds certain thresholds, where the threshold value depends upon the GE channel state. When the system parameters are unknown, we propose an actor-critic (AC) learning algorithm that exploits the threshold structure of an optimal policy. Numerical results demonstrate that the proposed AC algorithm learns the policy structure effectively and achieves performance close to that of the optimal policy computed using the relative value iteration (RVI).

math.OC↗

arXiv2026

QUBO-Optimized Evidence Selection for Retrieval-Augmented Question Answering with Unconventional Solvers

Retrieval-augmented question answering depends on selecting evidence passages that jointly support answer generation. However, many RAG pipelines rely on top-$k$ ranking, where passages are selected mainly by individual relevance scores, even though multi-hop questions often require complementary evidence satisfying multiple information requirements. Recent LLM-based selectors address this by treating retrieval as set selection, but using an LLM for this intermediate stage can be costly and difficult to scale. In this work, we formulate evidence selection as a Quadratic Unconstrained Binary Optimization (QUBO) problem. Given a question, candidate passages, and decomposed information requirements, our method constructs an energy function that balances relevance, requirement coverage, support strength, redundancy, complementarity, and compactness. Low-energy solutions correspond to compact evidence subsets that cover the needed requirements while avoiding unnecessary or repetitive context. The selected passages are then passed to a downstream language model for answer generation, separating combinatorial evidence selection from semantic answer generation. We evaluate the proposed QUBO selector on HotpotQA and compare it with LLM-based set selectors and non-LLM baselines including BM25, relevance top-$k$, maximal marginal relevance, hybrid lexical--semantic ranking, greedy coverage, and random selection. The QUBO selector achieves competitive exact-match and token-F1 performance relative to LLM-based selectors while providing a solver-compatible formulation for structured evidence selection. These results suggest that multi-hop evidence selection can be cast as discrete optimization, opening a path toward RAG pipelines where LLMs are reserved for semantic processing and answer generation, while context selection is handled by Ising/QUBO-compatible solvers.

cs.CL↗

arXiv2026

Graph Bispectrum for Nonlinear Mode Interactions

We introduce a graph bispectrum formulation for characterizing higher-order interactions in graph signals. While conventional graph spectral methods capture only second-order structure, many graph signals exhibit nonlinear interactions that are not reflected in covariance or graph power spectra. Motivated by classical higher-order spectral analysis, we define a graph bispectrum tensor based on third-order moments of graph Fourier coefficients and derive a compact graph bicoherence measure that summarizes nonlinear mode interactions in a low-dimensional and scale-invariant form. We establish key properties of the proposed quantities, including vanishing third-order moments for Gaussian graph signals and a dynamical interpretation in terms of nonlinear mode coupling. Experiments on synthetic random graph signals demonstrate that the proposed measures detect complementary nonlinear dependencies even when second-order statistics are similar. We further apply the method to EEG recordings from the CHB-MIT Scalp EEG Database and show that ictal activity exhibits substantially increased nonlinear graph spectral coupling compared to interictal periods. The proposed approach provides an interpretable and computationally efficient tool for higher-order interaction analysis for graph signals.

eess.SP↗

arXiv2026

Placebo Discontinuity Design

Standard regression discontinuity design (RDD) models rely on the continuity of expected potential outcomes at the cutoff. The standard continuity assumption can be violated by strategic manipulation of the running variable, which is realistic when the cutoff is widely known and when the treatment of interest is a social program or government benefit. In this work, we identify the treatment effect despite such a violation, by leveraging a placebo treatment and a placebo outcome. We introduce a local instrumental variable estimator. Our estimator decomposes into two terms: the standard RDD estimator of the target outcome's discontinuity, and a new adjustment term based on the placebo outcome's discontinuity. We show that our estimator is consistent, and we justify a robust bias-corrected inference procedure. Our method expands the applicability of RDD to settings with strategic behavior around the cutoff, which commonly arise in social science.

econ.EM↗

arXiv2026

Uniform inference for kernel instrumental variable regression

Instrumental variable regression is a foundational tool for causal analysis across the social and biomedical sciences. Recent advances use kernel methods to estimate nonparametric causal relationships, with general data types, while retaining a simple closed-form expression. Empirical researchers ultimately need reliable inference on causal estimates; however, uniform confidence sets for the method remain unavailable. To fill this gap, we develop valid and sharp confidence sets for kernel instrumental variable regression, allowing general nonlinearities and data types. Computationally, our bootstrap procedure requires only a single run of the kernel instrumental variable regression estimator. Theoretically, it relies on the same key assumptions. Overall, we provide a practical procedure for inference that substantially increases the value of kernel methods for causal analysis.

math.ST↗