# Process-Tensor Tomography of SGD: Measuring Non-Markovian Memory via Back-Flow of Distinguishability

arXiv: https://arxiv.org/abs/2601.16563 · HTML: https://mlsanity.com/paper/2601.16563/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 23 Jan 2026.

## Builds on (most cited first, 16 of 16)

- [Extracting Training Data from Large Language Models](https://mlsanity.com/paper/2012.07805/) (2020)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [The Secret Sharer: Evaluating and Testing Unintended Memorization in Neural Networks](https://mlsanity.com/paper/1802.08232/) (2018)
- [Estimating Training Data Influence by Tracing Gradient Descent](https://mlsanity.com/paper/2002.08484/) (2020)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Representer Point Selection for Explaining Deep Neural Networks](https://mlsanity.com/paper/1811.09720/) (2018)
- [Stochastic modified equations and adaptive stochastic gradient algorithms](https://mlsanity.com/paper/1511.06251/) (2015)
- [Why Random Reshuffling Beats Stochastic Gradient Descent](https://mlsanity.com/paper/1510.08560/) (2015)
- [Closing the convergence gap of SGD without replacement](https://mlsanity.com/paper/2002.10400/) (2020)
- [Stochastic Gradient Descent under Markovian Sampling Schemes](https://mlsanity.com/paper/2302.14428/) (2023)
- [Adapting to Mixing Time in Stochastic Optimization with Markovian Data](https://mlsanity.com/paper/2202.04428/) (2022)
- [Leave-one-out Distinguishability in Machine Learning](https://mlsanity.com/paper/2309.17310/) (2023)
- [Finite-Time Analysis of Stochastic Gradient Descent under Markov Randomness](https://mlsanity.com/paper/2003.10973/) (2020)

## Built on by (most cited first, 2 of 2)

- [Optimizer Memory Makes Shuffle Order a First-Order Source of Fine-Tuning Noise](https://mlsanity.com/paper/2606.29554/) (2026)
- [Stored in Optimizer State, Valued by Later Training: A Causal Account of Subliminal Trait Transfer](https://mlsanity.com/paper/2608.20442/) (2026)

## Abstract (arXiv)

This work proposes neural training as a process tensor: a multi-time map that takes a sequence of controllable instruments (batch choices, augmentations, optimizer micro-steps) and returns an observable of the trained model. Building on this operational lens, we introduce a simple, model-agnostic witness of training memory based on back-flow of distinguishability. In a controlled two-step protocol, we compare outcome distributions after one intervention versus two; the increase DeltaBF = D2 - D1>0 (with D∈TV, JS, H measured on softmax predictions over a fixed probe set) certifies non-Markovianity. We observe consistent positive back-flow with tight bootstrap confidence intervals, amplification under higher momentum, larger batch overlap, and more micro-steps, and collapse under a causal break (resetting optimizer state), directly attributing the effect to optimizer/data-state memory. The witness is robust across TV/JS/Hellinger, inexpensive to compute, and requires no architectural changes. We position this as a measurement contribution: a principled diagnostic and empirical evidence that practical SGD deviates from the Markov idealization. An exploratory case study illustrates how the micro-level signal can inform curriculum orderings. "Data order matters" turns into a testable operator with confidence bounds, our framework offers a common stage to compare optimizers, curricula, and schedules through their induced training memory.