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

Vasileios Sevetlidis, George Pavlidis

TL;DR

This paper reframes SGD dynamics as a classical process-tensor and introduces a two-step A/B protocol to detect memory via back-flow of distinguishability, quantified by $\\Delta_{BF} = D_2 - D_1$ with $D \\in \\{TV, JS, Hell\\}$ on probe predictions. By implementing a causal break that resets optimizer buffers, the authors provide a falsifiable test: a positive back-flow implies observable memory that can be collapsed by breaking memory channels, thus falsifying the operational Markov condition at the observable level. They validate the approach across CIFAR-100 and Imagenette with multiple architectures, finding that momentum and data overlap amplify back-flow while resets reduce it, with sign flips indicating nontrivial order-sensitivity. The work yields a model-agnostic diagnostic for training memory, enabling principled comparisons of optimizers, curricula, and schedules and offering guidance for memory-aware curriculum design. Overall, the back-flow witness connects mechanism and measurement to quantify and potentially control non-Markovian effects in practical SGD training.

Abstract

This work proposes neural training as a \emph{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 \emph{back-flow of distinguishability}. In a controlled two-step protocol, we compare outcome distributions after one intervention versus two; the increase $Δ_{\mathrm{BF}} = D_2 - D_1>0$ (with $D\in\{\mathrm{TV}, \mathrm{JS}, \mathrm{H}\}$ 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 \emph{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 \emph{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.

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

TL;DR

This paper reframes SGD dynamics as a classical process-tensor and introduces a two-step A/B protocol to detect memory via back-flow of distinguishability, quantified by with on probe predictions. By implementing a causal break that resets optimizer buffers, the authors provide a falsifiable test: a positive back-flow implies observable memory that can be collapsed by breaking memory channels, thus falsifying the operational Markov condition at the observable level. They validate the approach across CIFAR-100 and Imagenette with multiple architectures, finding that momentum and data overlap amplify back-flow while resets reduce it, with sign flips indicating nontrivial order-sensitivity. The work yields a model-agnostic diagnostic for training memory, enabling principled comparisons of optimizers, curricula, and schedules and offering guidance for memory-aware curriculum design. Overall, the back-flow witness connects mechanism and measurement to quantify and potentially control non-Markovian effects in practical SGD training.

Abstract

This work proposes neural training as a \emph{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 \emph{back-flow of distinguishability}. In a controlled two-step protocol, we compare outcome distributions after one intervention versus two; the increase (with 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 \emph{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 \emph{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.
Paper Structure (24 sections, 3 theorems, 23 equations, 9 figures, 4 tables)

This paper contains 24 sections, 3 theorems, 23 equations, 9 figures, 4 tables.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. Methodology
  4. State spaces, instruments, and kernels
  5. Operational Markov condition at the observable
  6. A back-flow witness of operational memory
  7. Causal break and a sufficient condition for restoring OMC
  8. Perturbative scaling predictions
  9. Divergences and contractivity
  10. Inference at the observable level
  11. What a positive back-flow means (and what it does not)
  12. EXPERIMENTAL PROCEDURE
  13. Micro-step protocol ($A$ then $B$)
  14. Probe set and observable distances
  15. Repeats, seeds, precision targets
  16. ...and 9 more sections

Key Result

Proposition 3.1

If the two-time process is operationally Markov at the observable for $B$, then for any contractive divergence $D$, the back-flow of distinguishability satisfies $\Delta^{(I,B)} \leq 0$ for all $I$.

Figures (9)

  • Figure 1: Distribution of TV mean $\Delta_{\!\mathrm{BF}}$ across all setups. Histogram for no break and break conditions. The mass shifts toward attenuation under a causal break.
  • Figure 2: TV mean effect: no break vs. break. Each point is one {dataset, model, regime, base_stage}. Many points lie below the diagonal and $28\%$ cross quadrants (sign flips), indicating that optimizer carryover causally drives amplification.
  • Figure 3: Average TV effect by regime and optimizer condition. Points are means across datasets and models; ribbons show bootstrap 95% CIs. High-momentum/high-overlap regimes amplify without a break and attenuate with a break.
  • Figure 4: Non-commute curves. TV$(AB,BA)$ vs. micro-steps $k$ for CIFAR-100/ViT-B/16/resonant strong. Left: no break. Right: break.
  • Figure 5: Amplification vs. order sensitivity (left). Configurations with a steeper $\mathrm{TV}(AB,BA)$ slope (no break) exhibit larger $\Delta$. Amplification vs. momentum alignment (right). Each point corresponds to a configuration. Greater pre-$B$ alignment predicts larger $\Delta$. Least-squares fits are shown.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Definition 1
  • Proposition 3.1: Proposition 3.1
  • Proposition 3.2: No back-flow under OMC
  • Definition 2: Observable sufficiency for $B$
  • Theorem 3.1