Akshat Valse · stochastic foundations of machine learning

Fig. 1

Object: overdamped Langevin
Potential: V = cos x + cos y
Integrator: Euler–Maruyama, h = 0.004
Checked: TV 1.0×10⁻³ vs Gibbs

Color: potential V

I’m an undergraduate researcher at Iowa State University working on reliable machine learning and scientific computing. I build Python simulations and GPU experiments to measure when methods fail, explain why, and test ways to improve them. My projects span LLM serving, statistical forecasting, and numerical algorithms.

I work on the stochastic foundations of machine learning: the theory of stochastic differential equations, applied to the algorithms that now generate and sample. A diffusion model is a reverse-time SDE, stochastic gradient descent and its Langevin variants are diffusions in a rough loss landscape, and a PAC-Bayes bound is the Donsker–Varadhan formula applied to a posterior over hypotheses. Three questions occupy me: what the discretization of a Langevin sampler actually costs, how to estimate probabilities too small for direct simulation to reach, and whether a PDE solver with a network inside it will hold together when you iterate it. Underneath all three is the same question, which is what can be proved about the objects practitioners already use: rates, stability, and error bounds in place of heuristics. A second set of projects, in computer science, carries the same habits into the systems that run large language models: whether a burst of agent traffic can tip a serving engine into a self-sustaining slow state, how to calibrate a forecast when its feedback arrives early but one-sided, and when restarting a stalled agent beats waiting. Everything moving on this page is one of the stochastic objects, running live and graded against a closed-form answer.

The band above is running in your browser from a fixed seed. It integrates

d X_{t} = - \nabla V (X_{t}) d t + 2 β^{- 1} d W_{t}

on the torus. Annealed Langevin sampling runs this same dynamics at each noise level, and it is the continuous-time limit of SGLD when

V

is the loss. Replace

\nabla V

by a learned score and you have a generative model. Because the score here is exact, the answer can be checked. Particles collect where

e^{- β V}

is large and hop between wells at the Kramers rate, and across 48 bins the empirical measure sits within a total variation of

1.0 \times 1 0^{- 3}

of the Gibbs law, with a discretization bias of

9 \times 1 0^{- 4}

on

E [cos X]

against the closed form

- I_{1} (β) / I_{0} (β)

.

Fig. 2

Object: VP-SDE, reverse time
Score: exact, not learned
Steps: 320 Euler–Maruyama
Graded on: mode coverage

A diffusion model with an answer key

Generative models are hard to evaluate because the target is unknown, so a sampler can only be judged on whether its output looks plausible. Make the target a Gaussian mixture and that changes. The mixture is closed under the variance-preserving forward kernel, so every marginal $p_{t}$ is again a mixture and the score $\nabla lo g p_{t}$ is available in closed form. Anderson's time reversal

d X = [- \frac{1}{2} β (t) X - β (t) \nabla lo g p_{t} (X)] d t + β (t) d \overset{ˉ}{W}

then runs with an oracle score, so whatever error remains belongs to the sampler and not to a network. The weights below are deliberately unequal, which is what makes mode collapse detectable at all. A sampler that drops the 0.08 component still emits individually plausible points, and only the proportions give it away.

**Fig. 2.** Ancestral sampling from a continuous-time diffusion model. The forward variance-preserving SDE noises a six-component Gaussian mixture into 𝒩(0, I); the reverse SDE carries 640 samples back over 320 Euler–Maruyama steps. Because a mixture stays a mixture under the VP kernel, the score used here is exact rather than learned, which leaves discretization as the only source of error and turns the mixture weights into something gradable. True weights (0.28, 0.22, 0.17, 0.14, 0.11, 0.08); the readout reports max_k |ŵ_k − w_k| at the end of each pass, against a Monte Carlo floor of 0.018 at n = 640. Color and contours encode −log p₀.

Over 16,384 samples the recovered weights match to $max_{k} ∣ \overset{w}{^}_{k} - w_{k} ∣ = 0.006$ against a Monte Carlo standard error of $0.0035$ , the component means to $0.010$ , and the within-component spread to $0.262$ against $σ = 0.26$ . This is what a toy target with a closed form buys you. The sampler either returns the weights or it does not, and no quantity of plausible-looking output makes up the difference.

research

Research

Langevin dynamics: sampling, discretization bias, transport coefficients

David Herzog · Mathematics · ongoing

The sampler underneath score-based generative models and stochastic-gradient Langevin dynamics, analyzed where its errors are computable: discretization bias against exact stationary laws, and estimator design for effective diffusivity and mobility.

Large-deviation asymptotics for mean-field interacting systems

Ruoyu Wu · Mathematics · ongoing

Rate functionals and the variational characterization of rare events for the empirical measure of a mean-field particle system. This is the construction that sits one level above the mean-field limit of wide-network training, and the machinery underneath PAC-Bayes.

Stability of neural approximations to evolution equations

Jue Yan · Mathematics · ongoing

Conservation in a cell-average neural network solver is architectural. Stability is not, and the training loss does not separate the networks that survive a long rollout from the ones that come apart. Adapting von Neumann analysis to say which, from the weights alone.

Metastable failures in prefix-cached LLM agent serving

Ali Jannesari · Computer Science · ongoing

An agent re-sends its whole growing conversation every turn, and the serving engine keeps that prefix in a GPU cache so the next turn stays cheap. Under enough concurrent sessions, evictions feed on themselves. Whether a short burst can tip a healthy server into a self-sustaining slow state, and what prevents it, studied by simulation, a queueing model, and real experiments on A100s.

Online calibration of forecasts with early, one-sided feedback

Hongyang Gao · Computer Science · ongoing

A forecast of a cumulative quantity is scored only when its horizon ends, but a miss becomes certain the moment the running sum crosses the bound. A calibration layer that credits those certain misses early and keeps accounts of what is still pending, with its coverage guarantee proved and checked.

Reliability of tool-using agents

Yang Li · Computer Science · ongoing

When an agent stalls, when is restarting it worth more than waiting, and how do you estimate the probability of a failure too rare for direct simulation to reach? Two pilots on real agents, with the direction still being chosen.

Full descriptions →

elsewhere

Elsewhere

Email: [email protected]
GitHub: AkshatValse
Code: langevin-samplers
CV: /cv

Preprints will be posted here as they become available.