arXiv subjects

Aprameyo Chakrabartty

Publications and source records attributed to Aprameyo Chakrabartty.

5 recordsLinked to original sources

arXiv2026

Nested Convex-Body Chasing for Online Optimization with Evolving Feasible Sets

We study online optimization with nested shrinking feasible regions in two settings: convex optimization with nested evolving feasible sets (CONES) and adversarial constrained online convex optimization (COCO). Our algorithms separate loss control from geometric movement: constrained minimizers and cumulative-loss tests preserve regret guarantees, while a deterministic resettable nested convex-body chaser limits movement. For CONES with a $G$-Lipschitz, $μ$-strongly convex objective on a diameter-$D$ domain, we chase intersections of the current feasible set with adaptive objective sublevel sets. Using the Euclidean chasing ratio $O(\sqrt{d\log(1+d)})$, we obtain nonpositive regret at every prefix and movement $O(\sqrt{d\log(1+d)\,GD\log(eT)/μ})$. The bound adapts to the increase in the constrained optimum value. In dimension two, with all other parameters fixed, every randomized algorithm with terminal expected regret $O(T^β)$, $β<1$, suffers $Ω(\sqrt{\log T})$ expected movement on some deterministic nested sequence, proving optimal horizon dependence. Under linear growth away from the constrained minimizer set, Steiner-point tracking yields movement independent of $T$. For general convex COCO, one-step-delayed chasing with regularized-leader resets gives regret $O(G_fD\sqrt{d\log(1+d)T})$ and cumulative constraint violation $O(G_gD\sqrt{d\log(1+d)T})$. For strongly convex losses, both are $O(d\log(1+d)\log(eT))$ when other parameters are fixed. These reductions replace the $O(d^{d/2})$ projection-path factor in prior analyses by the polynomial dimension dependence of Euclidean nested convex-body chasing.

cs.AI↗

arXiv2026

High-Probability PL-SGD with Markovian Noise: Optimal Mixing and Tail Dependence

We study first-order methods for smooth objectives satisfying the Polyak-Łojasiewicz (PL) condition when gradient samples are generated by an exogenous Markov chain. In the light-tailed setting, prior uniform-in-time high-probability bounds for ordinary Stochastic Gradient Descent (SGD) under a standard growth envelope scale as $\widetilde{O}(t_{mix}^2/k)$, leaving a gap with the $\widetilde{O}(t_{mix}/k)$ expectation bounds. We close this gap using a lag-blocking argument to establish a uniform high-probability guarantee with a leading stochastic term of $\widetilde{O}(t_{mix}/(k+K_0))$ under geometric mixing. We prove this linear dependence on the mixing time is optimal via a matching $Ω(σ^2 t_{mix}/k)$ lower bound on a quadratic objective driven by a persistent two-state chain. We then extend this framework to heavy-tailed Markovian gradients satisfying a stationary finite-$p$-moment condition, $p \in (1,2]$. We design an all-samples clipped block method that uses every Markov transition while mitigating Markovian bias. Under a transition budget $T$, this algorithm achieves a high-probability stochastic error of $\widetilde{O}(σ_p^2(t_{mix}/T)^{2(p-1)/p})$. We establish a matching lower bound by reducing PL optimization to heavy-tailed mean estimation for a sticky Markov chain. Ultimately, this work tightly characterizes the optimal polynomial dependence on mixing time for light-tailed PL-SGD, and the optimal heavy-tail exponent and effective-sample-size dependence in the robust regime.

cs.LG↗

arXiv2026

Projection-free Algorithms for Online Convex Optimization with Adversarial Constraints

We study a generalization of the Online Convex Optimization (OCO) framework with time-varying adversarial constraints. In this setting, at each round, the learner selects an action from a convex decision set $X$, after which both a convex cost function and a convex constraint function are revealed. The objective is to design a computationally efficient learning policy that simultaneously achieves low regret with respect to the cost functions and low cumulative constraint violation (CCV) over a horizon of length $T$. A major computational bottleneck in standard OCO algorithms is the projection operation onto the decision set $X$. However, for many structured decision sets, linear optimization can be performed efficiently. Motivated by this, we propose a projection-free online conditional gradient (OCG)-based algorithm that requires only a single call to a linear optimization oracle over $X$ per round. Our approach improves upon the state of the art for projection-free online learning with adversarial constraints, achieving $\tilde{O}(T^{\frac{3}{4}})$ bounds for both regret and CCV. Our algorithm is conceptually simple. It constructs a surrogate cost function as a nonnegative linear combination of the cost and constraint functions, and feeds these surrogate costs into a novel adaptive online conditional gradient subroutine introduced in this paper. We further extend our framework to the bandit setting, where we show that a new form of surrogate loss is necessary to properly handle bandit feedback - an issue overlooked in prior work. Finally, we develop an efficient Follow-the-Perturbed-Leader (FTPL)-based algorithm, particularly well-suited for online combinatorial optimization problems with discrete actions, which also achieves $O(T^{\frac{3}{4}})$ regret and CCV.

cs.LG↗

arXiv2025

TAPS : Frustratingly Simple Test Time Active Learning for VLMs

Test-Time Optimization enables models to adapt to new data during inference by updating parameters on-the-fly. Recent advances in Vision-Language Models (VLMs) have explored learning prompts at test time to improve performance in downstream tasks. In this work, we extend this idea by addressing a more general and practical challenge: Can we effectively utilize an oracle in a continuous data stream where only one sample is available at a time, requiring an immediate query decision while respecting latency and memory constraints? To tackle this, we propose a novel Test-Time Active Learning (TTAL) framework that adaptively queries uncertain samples and updates prompts dynamically. Unlike prior methods that assume batched data or multiple gradient updates, our approach operates in a real-time streaming scenario with a single test sample per step. We introduce a dynamically adjusted entropy threshold for active querying, a class-balanced replacement strategy for memory efficiency, and a class-aware distribution alignment technique to enhance adaptation. The design choices are justified using careful theoretical analysis. Extensive experiments across 10 cross-dataset transfer benchmarks and 4 domain generalization datasets demonstrate consistent improvements over state-of-the-art methods while maintaining reasonable latency and memory overhead. Our framework provides a practical and effective solution for real-world deployment in safety-critical applications such as autonomous systems and medical diagnostics.

cs.CV↗

arXiv2025

Relation-Aware Slicing in Cross-Domain Alignment

The Sliced Gromov-Wasserstein (SGW) distance, aiming to relieve the computational cost of solving a non-convex quadratic program that is the Gromov-Wasserstein distance, utilizes projecting directions sampled uniformly from unit hyperspheres. This slicing mechanism incurs unnecessary computational costs due to uninformative directions, which also affects the representative power of the distance. However, finding a more appropriate distribution over the projecting directions (slicing distribution) is often an optimization problem in itself that comes with its own computational cost. In addition, with more intricate distributions, the sampling itself may be expensive. As a remedy, we propose an optimization-free slicing distribution that provides fast sampling for the Monte Carlo approximation. We do so by introducing the Relation-Aware Projecting Direction (RAPD), effectively capturing the pairwise association of each of two pairs of random vectors, each following their ambient law. This enables us to derive the Relation-Aware Slicing Distribution (RASD), a location-scale law corresponding to sampled RAPDs. Finally, we introduce the RASGW distance and its variants, e.g., IWRASGW (Importance Weighted RASGW), which overcome the shortcomings experienced by SGW. We theoretically analyze its properties and substantiate its empirical prowess using extensive experiments on various alignment tasks.

stat.ML↗