arXiv subjects

Arnab Maiti

Publications and source records attributed to Arnab Maiti.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Uncoupled Learning Dynamics with $\tilde{O}\!\left(T^{-1/4}\right)$ Last-Iterate Convergence in Bilinear Saddle-Point Problems over Convex Sets under Bandit Feedback

In this paper, we study last-iterate convergence of learning algorithms in bilinear saddle-point problems, a preferable notion of convergence that captures the day-to-day behavior of learning dynamics. We focus on the challenging setting where players select actions from compact convex sets and receive only bandit feedback. Our main contribution is the design of an uncoupled learning algorithm that guarantees last-iterate convergence to the Nash equilibrium with high probability. We establish a convergence rate of $\tilde{O}(T^{-1/4})$ up to polynomial factors in problem parameters. Crucially, our proposed algorithm is computationally efficient, requiring only an efficient linear optimization oracle over the players' compact action sets. The algorithm is obtained by combining techniques from experimental design and the classic Follow-The-Regularized-Leader (FTRL) framework, with a carefully chosen regularizer function tailored to the geometry of the action set of each learner.

stat.ML↗

arXiv2026

Catalytic Crosstalk: Cooperative Enzyme Dynamics in Artificial Crowded Environments

In cellular environments, enzymes operate under densely crowded conditions that often hinder catalytic efficiency by limiting substrate diffusion and essential conformational dynamics. While reports suggest that crowding can often lead to inhibition of enzyme's catalytic activity, persistent efficiency of cellular biochemistry hints at underlying cooperative mechanisms among these molecules. Here, we experimentally demonstrate catalytic crosstalk between two enzymes - catalase and urease - in artificially crowded environments. Our results reveal that when co-localized in dense media, these enzymes mutually enhance each other's catalytic activity and dynamic behavior. This cooperative interaction leads to a net increase in reaction rates and mobility, suggesting an emergent many-body effect in enzyme assemblies. Modeling enzymes as dimeric active particles, we propose a minimal simulation framework that qualitatively captures the observed synergy. Our findings show that inter-enzyme cooperation can counteract the detrimental effects of crowding, offering insights into how enzymatic efficiency is sustained in complex biological milieu.

cond-mat.soft↗

arXiv2026

Revisiting the Bertrand Paradox via Equilibrium Analysis of No-regret Learners

We study the discrete Bertrand pricing game with a non-increasing demand function. The game has $n \ge 2$ players who simultaneously choose prices from the set $\{1/k, 2/k, \ldots, 1\}$, where $k\in\mathbb{N}$. The player who sets the lowest price captures the entire demand; if multiple players tie for the lowest price, they split the demand equally. We study the Bertrand paradox, where classical theory predicts low prices, yet real markets often sustain high prices. To understand this gap, we analyze a repeated-game model in which firms set prices using no-regret learners. Our goal is to characterize the equilibrium outcomes that can arise under different no-regret learning guarantees. We are particularly interested in questions such as whether no-external-regret learners can converge to undesirable high-price outcomes, and how stronger guarantees such as no-swap regret shape the emergence of competitive low-price behavior. We address these and related questions through a theoretical analysis, complemented by experiments that support the theory and reveal surprising phenomena for no-swap regret learners.

cs.GT↗

arXiv2026

On the Power of Adaptivity for $\varepsilon$-Best Arm Identification in Linear Bandits

We study the minimax sample complexity of $\varepsilon$-best arm identification in linear bandits. Given a compact action set $\mathcal{X}$ that spans $\mathbb{R}^d$ and an unknown reward vector $θ\in\mathbb{R}^d$, the goal is to output an arm $\widehat{x}\in\mathcal{X}$ such that $\langle \widehat{x},θ\rangle \ge \max_{x\in\mathcal{X}} \langle x,θ\rangle - \varepsilon$ with probability at least $1-δ$, using as few samples as possible. First, we present a non-adaptive fixed-design method with sample complexity $\mathcal{O}\!\left(\frac{d\log(1/δ)}{\varepsilon^2}+\frac{w(\mathcal{X})^2}{\varepsilon^2}\right)$, where $w(\mathcal{X})$ is a Gaussian width term dependent on $\mathcal{X}$, and we prove a matching lower bound $Ω\!\left(\frac{d\log(1/δ)}{\varepsilon^2}+\frac{w(\mathcal{X})^2}{\varepsilon^2}\right)$ for all non-adaptive fixed-design methods. We then turn to adaptive sampling. We raise an important structural question: beyond the canonical basis, are there structured action sets for which adaptivity yields only logarithmic-factor improvements over the optimal non-adaptive rate? We answer in the affirmative for several natural action sets, namely the hypercube, the $\ell_2$ ball, $m$-sets, and multi-task multi-armed bandits. Finally, we provide the first construction of an action set $\mathcal{X}$ for which adaptivity yields a polynomial-factor improvement over every non-adaptive algorithm. A key ingredient behind this separation is an $\ell_2$-norm estimation subroutine: we design an adaptive algorithm that uses $\mathcal{O}\!\left(\frac{d\log(1/δ)}{\varepsilon^2}\right)$ samples from the unit $\ell_2$ ball in $\mathbb{R}^d$ and outputs an estimate $\widehat r$ satisfying $|\widehat r-\|θ\|_2|\le \varepsilon$ with probability at least $1-δ$, where $θ$ is the unknown reward vector.

cs.LG↗

arXiv2026

Adversarial Learning in Games with Bandit Feedback: Logarithmic Pure-Strategy Maximin Regret

Learning to play zero-sum games is a fundamental problem in game theory and machine learning. While significant progress has been made in minimizing external regret in the self-play settings or with full-information feedback, real-world applications often force learners to play against unknown, arbitrary opponents and restrict learners to bandit feedback where only the payoff of the realized action is observable. In such challenging settings, it is well-known that $Ω(\sqrt{T})$ external regret is unavoidable (where T is the number of rounds). To overcome this barrier, we investigate adversarial learning in zero-sum games under bandit feedback, aiming to minimize the deficit against the maximin pure strategy -- a metric we term Pure-Strategy Maximin Regret. We analyze this problem under two bandit feedback models: uninformed (only the realized reward is revealed) and informed (both the reward and the opponent's action are revealed). For uninformed bandit learning of normal-form games, we show that the Tsallis-INF algorithm achieves $O(c \log T)$ instance-dependent regret with a game-dependent parameter $c$. Crucially, we prove an information-theoretic lower bound showing that the dependence on c is necessary. To overcome this hardness, we turn to the informed setting and introduce Maximin-UCB, which obtains another regret bound of the form $O(c' \log T)$ for a different game-dependent parameter $c'$ that could potentially be much smaller than $c$. Finally, we generalize both results to bilinear games over an arbitrary, large action set, proposing Tsallis-FTRL-SPM and Maximin-LinUCB for the uninformed and informed setting respectively and establishing similar game-dependent logarithmic regret bounds.

cs.LG↗

arXiv2025

Adapting to Stochastic and Adversarial Losses in Episodic MDPs with Aggregate Bandit Feedback

We study online learning in finite-horizon episodic Markov decision processes (MDPs) under the challenging aggregate bandit feedback model, where the learner observes only the cumulative loss incurred in each episode, rather than individual losses at each state-action pair. While prior work in this setting has focused exclusively on worst-case analysis, we initiate the study of best-of-both-worlds (BOBW) algorithms that achieve low regret in both stochastic and adversarial environments. We propose the first BOBW algorithms for episodic tabular MDPs with aggregate bandit feedback. In the case of known transitions, our algorithms achieve $O(\log T)$ regret in stochastic settings and ${O}(\sqrt{T})$ regret in adversarial ones. Importantly, we also establish matching lower bounds, showing the optimality of our algorithms in this setting. We further extend our approach to unknown-transition settings by incorporating confidence-based techniques. Our results rely on a combination of FTRL over occupancy measures, self-bounding techniques, and new loss estimators inspired by recent advances in online shortest path problems. Along the way, we also provide the first individual-gap-dependent lower bounds and demonstrate near-optimal BOBW algorithms for shortest path problems with bandit feedback.

cs.LG↗

arXiv2025

On the Universal Near Optimality of Hedge in Combinatorial Settings

In this paper, we study the classical Hedge algorithm in combinatorial settings. In each round, the learner selects a vector $\boldsymbol{x}_t$ from a set $X \subseteq \{0,1\}^d$, observes a full loss vector $\boldsymbol{y}_t \in \mathbb{R}^d$, and incurs a loss $\langle \boldsymbol{x}_t, \boldsymbol{y}_t \rangle \in [-1,1]$. This setting captures several important problems, including extensive-form games, resource allocation, $m$-sets, online multitask learning, and shortest-path problems on directed acyclic graphs (DAGs). It is well known that Hedge achieves a regret of $O\big(\sqrt{T \log |X|}\big)$ after $T$ rounds of interaction. In this paper, we ask whether Hedge is optimal across all combinatorial settings. To that end, we show that for any $X \subseteq \{0,1\}^d$, Hedge is near-optimal--specifically, up to a $\sqrt{\log d}$ factor--by establishing a lower bound of $Ω\big(\sqrt{T \log(|X|)/\log d}\big)$ that holds for any algorithm. We then identify a natural class of combinatorial sets--namely, $m$-sets with $\log d \leq m \leq \sqrt{d}$--for which this lower bound is tight, and for which Hedge is provably suboptimal by a factor of exactly $\sqrt{\log d}$. At the same time, we show that Hedge is optimal for online multitask learning, a generalization of the classical $K$-experts problem. Finally, we leverage the near-optimality of Hedge to establish the existence of a near-optimal regularizer for online shortest-path problems in DAGs--a setting that subsumes a broad range of combinatorial domains. Specifically, we show that the classical Online Mirror Descent (OMD) algorithm, when instantiated with the dilated entropy regularizer, is iterate-equivalent to Hedge, and therefore inherits its near-optimal regret guarantees for DAGs.

cs.LG↗

arXiv2025

Learning to Incentivize in Repeated Principal-Agent Problems with Adversarial Agent Arrivals

We initiate the study of a repeated principal-agent problem over a finite horizon $T$, where a principal sequentially interacts with $K\geq 2$ types of agents arriving in an adversarial order. At each round, the principal strategically chooses one of the $N$ arms to incentivize for an arriving agent of unknown type. The agent then chooses an arm based on its own utility and the provided incentive, and the principal receives a corresponding reward. The objective is to minimize regret against the best incentive in hindsight. Without prior knowledge of agent behavior, we show that the problem becomes intractable, leading to linear regret. We analyze two key settings where sublinear regret is achievable. In the first setting, the principal knows the arm each agent type would select greedily for any given incentive. Under this setting, we propose an algorithm that achieves a regret bound of $O(\min\{\sqrt{KT\log N},K\sqrt{T}\})$ and provide a matching lower bound up to a $\log K$ factor. In the second setting, an agent's response varies smoothly with the incentive and is governed by a Lipschitz constant $L\geq 1$. Under this setting, we show that there is an algorithm with a regret bound of $\tilde{O}((LN)^{1/3}T^{2/3})$ and establish a matching lower bound up to logarithmic factors. Finally, we extend our algorithmic results for both settings by allowing the principal to incentivize multiple arms simultaneously in each round.

cs.GT↗

arXiv2025

On the Limitations and Possibilities of Nash Regret Minimization in Zero-Sum Matrix Games under Noisy Feedback

This paper studies a variant of two-player zero-sum matrix games, where, at each timestep, the row player selects row $i$, the column player selects column $j$, and the row player receives a noisy reward with expected value $A_{i,j}$, along with noisy feedback on the input matrix $A$. The row player's goal is to maximize their total reward against an adversarial column player. Nash regret, defined as the difference between the player's total reward and the game's Nash equilibrium value scaled by the time horizon $T$, is often used to evaluate algorithmic performance in zero-sum games. We begin by studying the limitations of existing algorithms for minimizing Nash regret. We show that standard algorithm--including Hedge, FTRL, and OMD--as well as the strategy of playing the Nash equilibrium of the empirical matrix--all incur $Ω(\sqrt{T})$ Nash regret, even when the row player receives noisy feedback on the entire matrix $A$. Furthermore, we show that UCB for matrix games, a natural adaptation of the well-known bandit algorithm, also suffers $Ω(\sqrt{T})$ Nash regret under bandit feedback. Notably, these lower bounds hold even in the simplest case of $2 \times 2$ matrix games, where the instance-dependent matrix parameters are constant. We next ask whether instance-dependent $\text{polylog}(T)$ Nash regret is achievable against adversarial opponents. We answer this affirmatively. In the full-information setting, we present the first algorithm for general $n \times m$ matrix games that achieves instance-dependent $\text{polylog}(T)$ Nash regret. In the bandit feedback setting, we design an algorithm with similar guarantees for the special case of $2 \times 2$ game--the same regime in which existing algorithms provably suffer $Ω(\sqrt{T})$ regret despite the simplicity of the instance. Finally, we validate our theoretical results with empirical evidence.

cs.LG↗

arXiv2025

Efficient Near-Optimal Algorithm for Online Shortest Paths in Directed Acyclic Graphs with Bandit Feedback Against Adaptive Adversaries

In this paper, we study the online shortest path problem in directed acyclic graphs (DAGs) under bandit feedback against an adaptive adversary. Given a DAG $G = (V, E)$ with a source node $v_{\mathsf{s}}$ and a sink node $v_{\mathsf{t}}$, let $X \subseteq \{0,1\}^{|E|}$ denote the set of all paths from $v_{\mathsf{s}}$ to $v_{\mathsf{t}}$. At each round $t$, we select a path $\mathbf{x}_t \in X$ and receive bandit feedback on our loss $\langle \mathbf{x}_t, \mathbf{y}_t \rangle \in [-1,1]$, where $\mathbf{y}_t$ is an adversarially chosen loss vector. Our goal is to minimize regret with respect to the best path in hindsight over $T$ rounds. We propose the first computationally efficient algorithm to achieve a near-minimax optimal regret bound of $\tilde O(\sqrt{|E|T\log |X|})$ with high probability against any adaptive adversary, where $\tilde O(\cdot)$ hides logarithmic factors in the number of edges $|E|$. Our algorithm leverages a novel loss estimator and a centroid-based decomposition in a nontrivial manner to attain this regret bound. As an application, we show that our algorithm for DAGs provides state-of-the-art efficient algorithms for $m$-sets, extensive-form games, the Colonel Blotto game, shortest walks in directed graphs, hypercubes, and multi-task multi-armed bandits, achieving improved high-probability regret guarantees in all these settings.

cs.LG↗

arXiv2024

Query-Efficient Algorithm to Find all Nash Equilibria in a Two-Player Zero-Sum Matrix Game

We study the query complexity of finding the set of all Nash equilibria $\mathcal X_\star \times \mathcal Y_\star$ in two-player zero-sum matrix games. Fearnley and Savani (2016) showed that for any randomized algorithm, there exists an $n \times n$ input matrix where it needs to query $Ω(n^2)$ entries in expectation to compute a single Nash equilibrium. On the other hand, Bienstock et al. (1991) showed that there is a special class of matrices for which one can query $O(n)$ entries and compute its set of all Nash equilibria. However, these results do not fully characterize the query complexity of finding the set of all Nash equilibria in two-player zero-sum matrix games. In this work, we characterize the query complexity of finding the set of all Nash equilibria $\mathcal X_\star \times \mathcal Y_\star$ in terms of the number of rows $n$ of the input matrix $A \in \mathbb{R}^{n \times n}$, row support size $k_1 := |\bigcup_{x \in \mathcal X_\star} \text{supp}(x)|$, and column support size $k_2 := |\bigcup_{y \in \mathcal Y_\star} \text{supp}(y)|$. We design a simple yet non-trivial randomized algorithm that, with probability $1 - δ$, returns the set of all Nash equilibria $\mathcal X_\star \times \mathcal Y_\star$ by querying at most $O(nk^5 \cdot \text{polylog}(n / δ))$ entries of the input matrix $A \in \mathbb{R}^{n \times n}$, where $k := \max\{k_1, k_2\}$. This upper bound is tight up to a factor of $\text{poly}(k)$, as we show that for any randomized algorithm, there exists an $n \times n$ input matrix with $\min\{k_1, k_2\} = 1$, for which it needs to query $Ω(nk)$ entries in expectation in order to find the set of all Nash equilibria $\mathcal X_\star \times \mathcal Y_\star$.

cs.GT↗

arXiv2024

Propagation of Enzyme-driven Active Fluctuations in Crowded Milieu

We investigated the energy transfer from active enzymes to their surroundings in crowded environments by measuring the diffusion of passive microscopic tracers in active solutions of ficoll and glycerol. Despite observing lower rates of substrate turnover and relatively smaller enhancement of passive tracer diffusion in artificial crowded media compared to those in aqueous solutions, we found a significantly higher relative diffusion enhancement in crowded environments in the presence of enzymatic activity. Our experimental observations, coupled with supporting analytical estimations, underscored the critical role of the intervening media in facilitating mechanical energy distribution around active enzymes.

cond-mat.soft↗

arXiv2024

Parameterized Algorithms for Kidney Exchange

In kidney exchange programs, multiple patient-donor pairs each of whom are otherwise incompatible, exchange their donors to receive compatible kidneys. The Kidney Exchange problem is typically modelled as a directed graph where every vertex is either an altruistic donor or a pair of patient and donor; directed edges are added from a donor to its compatible patients. The computational task is to find if there exists a collection of disjoint cycles and paths starting from altruistic donor vertices of length at most l_c and l_p respectively that covers at least some specific number t of non-altruistic vertices (patients). We study parameterized algorithms for the kidney exchange problem in this paper. Specifically, we design FPT algorithms parameterized by each of the following parameters: (1) the number of patients who receive kidney, (2) treewidth of the input graph + max{l_p, l_c}, and (3) the number of vertex types in the input graph when l_p <= l_c. We also present interesting algorithmic and hardness results on the kernelization complexity of the problem. Finally, we present an approximation algorithm for an important special case of Kidney Exchange.

cs.GT↗

arXiv2024

Query Complexity of Tournament Solutions

A directed graph where there is exactly one edge between every pair of vertices is called a {\em tournament}. Finding the "best" set of vertices of a tournament is a well studied problem in social choice theory. A {\em tournament solution} takes a tournament as input and outputs a subset of vertices of the input tournament. However, in many applications, for example, choosing the best set of drugs from a given set of drugs, the edges of the tournament are given only implicitly and knowing the orientation of an edge is costly. In such scenarios, we would like to know the best set of vertices (according to some tournament solution) by "querying" as few edges as possible. We, in this paper, precisely study this problem for commonly used tournament solutions: given an oracle access to the edges of a tournament T, find $f(T)$ by querying as few edges as possible, for a tournament solution f. We first show that the set of Condorcet non-losers in a tournament can be found by querying $2n-\lfloor \log n \rfloor -2$ edges only and this is tight in the sense that every algorithm for finding the set of Condorcet non-losers needs to query at least $2n-\lfloor \log n \rfloor -2$ edges in the worst case, where $n$ is the number of vertices in the input tournament. We then move on to study other popular tournament solutions and show that any algorithm for finding the Copeland set, the Slater set, the Markov set, the bipartisan set, the uncovered set, the Banks set, and the top cycle must query $Ω(n^2)$ edges in the worst case. On the positive side, we are able to circumvent our strong query complexity lower bound results by proving that, if the size of the top cycle of the input tournament is at most $k$, then we can find all the tournament solutions mentioned above by querying $O(nk + \frac{n\log n}{\log(1-\frac{1}{k})})$ edges only.

cs.DS↗

arXiv2024

On Binary Networked Public Goods Game with Altruism

In the classical Binary Networked Public Goods (BNPG) game, a player can either invest in a public project or decide not to invest. Based on the decisions of all the players, each player receives a reward as per his/her utility function. However, classical models of BNPG game do not consider altruism which players often exhibit and can significantly affect equilibrium behavior. Yu et al. (2021) extended the classical BNPG game to capture the altruistic aspect of the players. We, in this paper, first study the problem of deciding the existence of a Pure Strategy Nash Equilibrium (PSNE) in a BNPG game with altruism. This problem is already known to be NP-Complete. We complement this hardness result by showing that the problem admits efficient algorithms when the input network is either a tree or a complete graph. We further study the Altruistic Network Modification problem, where the task is to compute if a target strategy profile can be made a PSNE by adding or deleting a few edges. This problem is also known to be NP-Complete. We strengthen this hardness result by exhibiting intractability results even for trees. A perhaps surprising finding of our work is that the above problem remains NP-Hard even for bounded degree graphs when the altruism network is undirected but becomes polynomial-time solvable when the altruism network is directed. We also show some results on computing an MSNE and some parameterized complexity results. In summary, our results show that it is much easier to predict how the players in a BNPG game will behave compared to how the players in a BNPG game can be made to behave in a desirable way.

cs.GT↗

arXiv2023

Near-Optimal Pure Exploration in Matrix Games: A Generalization of Stochastic Bandits & Dueling Bandits

We study the sample complexity of identifying the pure strategy Nash equilibrium (PSNE) in a two-player zero-sum matrix game with noise. Formally, we are given a stochastic model where any learner can sample an entry $(i,j)$ of the input matrix $A\in[-1,1]^{n\times m}$ and observe $A_{i,j}+η$ where $η$ is a zero-mean 1-sub-Gaussian noise. The aim of the learner is to identify the PSNE of $A$, whenever it exists, with high probability while taking as few samples as possible. Zhou et al. (2017) presents an instance-dependent sample complexity lower bound that depends only on the entries in the row and column in which the PSNE lies. We design a near-optimal algorithm whose sample complexity matches the lower bound, up to log factors. The problem of identifying the PSNE also generalizes the problem of pure exploration in stochastic multi-armed bandits and dueling bandits, and our result matches the optimal bounds, up to log factors, in both the settings.

cs.LG↗

arXiv2023

Activity Induced Diffusion Recovery in Crowded Colloidal Suspension

We show that the force generated by active enzyme molecules are strong enough to influence the dynamics of their surroundings under artificial crowded environments. We measured the behavior of polymer microparticles in a quasi-two-dimensional system under aqueous environment, at various area fraction values of particles. In the presence of enzymatic activity not only the diffusion of the suspended particles at shorter time-scale regime enhanced, the system also showed a transition from sub-diffusive to diffusive dynamics at longer time-scale limits. Similar observations were also recorded with enzyme functionalized microparticles. Brownian dynamics simulations have been performed to support the experimental observations.

cond-mat.soft↗

arXiv2023

Instance-dependent Sample Complexity Bounds for Zero-sum Matrix Games

We study the sample complexity of identifying an approximate equilibrium for two-player zero-sum $n\times 2$ matrix games. That is, in a sequence of repeated game plays, how many rounds must the two players play before reaching an approximate equilibrium (e.g., Nash)? We derive instance-dependent bounds that define an ordering over game matrices that captures the intuition that the dynamics of some games converge faster than others. Specifically, we consider a stochastic observation model such that when the two players choose actions $i$ and $j$, respectively, they both observe each other's played actions and a stochastic observation $X_{ij}$ such that $\mathbb E[ X_{ij}] = A_{ij}$. To our knowledge, our work is the first case of instance-dependent lower bounds on the number of rounds the players must play before reaching an approximate equilibrium in the sense that the number of rounds depends on the specific properties of the game matrix $A$ as well as the desired accuracy. We also prove a converse statement: there exist player strategies that achieve this lower bound.

cs.GT↗