arXiv subjects

Hoang Huy Nguyen

Publications and source records attributed to Hoang Huy Nguyen.

6 recordsLinked to original sources

arXiv2026

Almost Sure Convergence of Stochastic Approximation: An Interplay of Noise and Step Size

We study the almost sure convergence of the Stochastic Approximation algorithm to the fixed point $x^\star$ of a nonlinear operator under a negative drift condition and a general noise sequence with finite $p$-th moment for some $p > 1$. Classical almost sure convergence results of Stochastic Approximation are mostly analyzed for the square-integrable noise setting, and it is shown that any non-summable but square-summable step size sequence is sufficient to obtain almost sure convergence. However, such a limitation prevents wider algorithmic application. In particular, many applications in Machine Learning and Operations Research admit heavy-tailed noise with infinite variance, rendering such guarantees inapplicable. On the other hand, when a stronger condition on the noise is available, such guarantees on the step size would be too conservative, as practitioners would like to pick a larger step size for a more preferable convergence behavior. To this end, we show that any non-summable but $p$-th power summable step size sequence is sufficient to guarantee almost sure convergence, covering the gap in the literature. Our guarantees are obtained using a universal Lyapunov drift argument. For the regime $p \in (1, 2)$, we show that using the Lyapunov function $\norm{x-x^\star}^p$ and applying a Taylor-like bound suffice. For $p > 2$, such an approach is no longer applicable, and therefore, we introduce a novel iterate projection technique to control the nonlinear terms produced by high-moment bounds and multiplicative noise. We believe our proof techniques and their implications could be of independent interest and pave the way for finite-time analysis of Stochastic Approximation under a general noise condition.

math.OC↗

arXiv2025

Creativity or Brute Force? Using Brainteasers as a Window into the Problem-Solving Abilities of Large Language Models

Accuracy remains a standard metric for evaluating AI systems, but it offers limited insight into how models arrive at their solutions. In this work, we introduce a benchmark based on brainteasers written in long narrative form to probe more deeply into the types of reasoning strategies that models use. Brainteasers are well-suited for this goal because they can be solved with multiple approaches, such as a few-step solution that uses a creative insight or a longer solution that uses more brute force. We investigate large language models (LLMs) across multiple layers of reasoning, focusing not only on correctness but also on the quality and creativity of their solutions. We investigate many aspects of the reasoning process: (1) semantic parsing of the brainteasers into precise mathematical competition style formats; (2) generating solutions from these mathematical forms; (3) self-correcting solutions based on gold solutions; (4) producing step-by-step sketches of solutions; and (5) making use of hints. We find that LLMs are in many cases able to find creative, insightful solutions to brainteasers, suggesting that they capture some of the capacities needed to solve novel problems in creative ways. Nonetheless, there also remain situations where they rely on brute force despite the availability of more efficient, creative solutions, highlighting a potential direction for improvement in the reasoning abilities of LLMs.

cs.AI↗

arXiv2025

Stochastic Approximation for Nonlinear Discrete Stochastic Control: Finite-Sample Bounds

We consider a nonlinear discrete stochastic control system, and our goal is to design a feedback control policy in order to lead the system to a prespecified state. We adopt a stochastic approximation viewpoint of this problem. It is known that by solving the corresponding continuous-time deterministic system, and using the resulting feedback control policy, one ensures almost sure convergence to the prespecified state in the discrete system. In this paper, we adopt such a control mechanism and provide its finite-sample convergence bounds whenever a Lyapunov function is known for the continuous system. In particular, we consider four cases based on whether the Lyapunov function for the continuous system gives exponential or sub-exponential rates and based on whether it is smooth or not. We provide the finite-time bounds in all cases. Our proof relies on constructing a Lyapunov function for the discrete system based on the given Lyapunov function for the continuous system. We do this by appropriately smoothing the given function using the Moreau envelope. We present numerical experiments corresponding to the various cases, which validate the rates we establish.

math.OC↗

arXiv2025

Finite-Time Behavior of Erlang-C Model: Mixing Time, Mean Queue Length and Tail Bounds

Service systems like data centers and ride-hailing are popularly modeled as queueing systems in the literature. Such systems are primarily studied in the steady state due to their analytical tractability. However, almost all applications in real life do not operate in a steady state, so there is a clear discrepancy in translating theoretical queueing results to practical applications. To this end, we provide a finite-time convergence for Erlang-C systems (also known as $M/M/n$ queues), providing a stepping stone towards understanding the transient behavior of more general queueing systems. We obtain a bound on the Chi-square distance between the finite time queue length distribution and the stationary distribution for a finite number of servers. We then use these bounds to study the behavior in the many-server heavy-traffic asymptotic regimes. The Erlang-C model exhibits a phase transition at the so-called Halfin-Whitt regime. We show that our mixing rate matches the limiting behavior in the Super-Halfin-Whitt regime, and matches up to a constant factor in the Sub-Halfin-Whitt regime. To prove such a result, we employ the Lyapunov-Poincaré approach, where we first carefully design a Lyapunov function to obtain a negative drift outside a finite set. Within the finite set, we develop different strategies depending on the properties of the finite set to get a handle on the mixing behavior via a local Poincaré inequality. A key aspect of our methodological contribution is in obtaining tight guarantees in these two regions, which when combined give us tight mixing time bounds. We believe that this approach is of independent interest for studying mixing in reversible countable-state Markov chains more generally.

math.PR↗

arXiv2024

Stochastic Constrained Decentralized Optimization for Machine Learning with Fewer Data Oracles: a Gradient Sliding Approach

In modern decentralized applications, ensuring communication efficiency and privacy for the users are the key challenges. In order to train machine-learning models, the algorithm has to communicate to the data center and sample data for its gradient computation, thus exposing the data and increasing the communication cost. This gives rise to the need for a decentralized optimization algorithm that is communication-efficient and minimizes the number of gradient computations. To this end, we propose the primal-dual sliding with conditional gradient sliding framework, which is communication-efficient and achieves an $\varepsilon$-approximate solution with the optimal gradient complexity of $O(1/\sqrt{\varepsilon}+σ^2/{\varepsilon^2})$ and $O(\log(1/\varepsilon)+σ^2/\varepsilon)$ for the convex and strongly convex setting respectively and an LO (Linear Optimization) complexity of $O(1/\varepsilon^2)$ for both settings given a stochastic gradient oracle with variance $σ^2$. Compared with the prior work \cite{wai-fw-2017}, our framework relaxes the assumption of the optimal solution being a strict interior point of the feasible set and enjoys wider applicability for large-scale training using a stochastic gradient oracle. We also demonstrate the efficiency of our algorithms with various numerical experiments.

math.OC↗

arXiv2021

Projection-Based Reduced Order Model and Machine Learning Closure for Transient Simulations of High-Re Flows

The paper presents a Projection-Based Reduced-Order Model for simulations of high Reynolds turbulent flows. The PBROM are enhanced by incorporating various models of turbulent viscosity and residual closures to model the effects of interactions among the modes and energy dissipations. Remarkable improvements in prediction accuracies are achieved with a suitable turbulent viscosity model and a residual closure. The enhanced PBROM models are demonstrated for high-Re flows past a cylinder in two- and three- dimensions. These enhancements have shown capable of capturing complex flow features and removing unnecessary ones, while not affecting the efficiency of the overall model.

cs.CE↗