MadSBM: Minimal-action Discrete Schrödinger Bridge

• The paper introduces a novel framework that formulates peptide generation as a minimal-action transport problem using discrete Schrödinger bridge matching.
• It combines continuous-time Markov chains, rate-based optimal control, and classifier guidance to ensure transitions remain in high-likelihood regions.
• The method leverages pre-trained protein language models to optimize peptide sequences for functional applications with biologically informed dynamics.

Minimal-action discrete Schrödinger Bridge Matching (MadSBM) is a generative modeling framework that addresses the challenge of synthesizing discrete objects—specifically peptide sequences—by formulating generation as a minimal-action transport problem between a simple prior and the data distribution, where all trajectories traverse high-likelihood regions of the discrete state space. The method combines concepts from continuous-time Markov chains, Schrödinger bridge theory, and rate-based optimal control, yielding probabilistic generation processes that circumvent many difficulties of conventional discrete diffusion and flow models (Goel et al., 29 Jan 2026).

1. Discrete State Space and Markov Process Construction

MadSBM operates on the space of $L$-length peptides, each $x = (x_1, ..., x_L)$ where $x_i \in \mathcal{V}$, the alphabet of $20$ amino acids plus a mask token “$\sqcup$”. The underlying structure is an edit graph: edges $(x \to y)$ exist for all sequences $x, y$ differing at exactly one position. The dynamics over this space are modeled as a time-inhomogeneous continuous-time Markov chain (CTMC) with generator $Q^0(t)$.

The reference transition rate for an edit $(x\to y)$ at time $t$ and position $x = (x_1, ..., x_L)$0 is given by

$x = (x_1, ..., x_L)$1

where $x = (x_1, ..., x_L)$2 are masked language modeling logits from a frozen, pre-trained protein LLM (ESM-2). This structure encodes domain knowledge: transitions are driven by biologically plausible substitutions reflecting high-likelihood amino acid predictions.

2. Discrete Schrödinger Bridge and Minimal-Action Principle

The generative task is formulated as a minimal Kullback-Leibler (KL) divergence transport problem between endpoint marginals: the fully masked prior $x = (x_1, ..., x_L)$3 (all positions masked) and the data marginal $x = (x_1, ..., x_L)$4 (empirical peptide distribution). Specifically, one seeks a controlled path law $x = (x_1, ..., x_L)$5 over trajectories under controlled rate $x = (x_1, ..., x_L)$6 that solves

$x = (x_1, ..., x_L)$7

where $x = (x_1, ..., x_L)$8 denotes the reference CTMC path law.

The path-space relative entropy expands (via discrete CTMC Girsanov theory) to a minimal-action functional:

$x = (x_1, ..., x_L)$9

This measures the entropic cost of deviating from the reference process, penalizing unnecessary or low-likelihood transitions.

3. Optimal Control Field and Schrödinger System

The solution to the minimal-action Schrödinger bridge has a Doob-$x_i \in \mathcal{V}$0 transform structure: there exist strictly positive backward potentials $x_i \in \mathcal{V}$1 such that the optimal log-tilt parameter is $x_i \in \mathcal{V}$2. The optimal control modifies the reference generator multiplicatively:

$x_i \in \mathcal{V}$3

The backward potential $x_i \in \mathcal{V}$4 solves the backward Kolmogorov (adjoint) equation:

$x_i \in \mathcal{V}$5

with $x_i \in \mathcal{V}$6 chosen so that final-time marginal aligns with the data.

4. Simulation-Free Training via Masking Interpolation

MadSBM eschews explicit simulation of CTMCs in training. Instead, it leverages the structure of peptide masking: an observed peptide $x_i \in \mathcal{V}$7 is partially masked to $x_i \in \mathcal{V}$8, where each site is masked independently with probability $x_i \in \mathcal{V}$9. The model parameterizes $20$0 and defines time-dependent logits as

$20$1

for each position $20$2 and token $20$3. The one-step transition is trained to reconstruct the true token $20$4 using a cross-entropy loss summed across masked sites:

$20$5

Minimization of $20$6 ensures $20$7 converges to the optimal log-tilt up to an additive constant.

5. Rate-Based Generation and Jump Sampling Procedure

Generation from the model runs backward in time ($20$8) in $20$9 discrete steps. At each step, for each sequence position, exit rates $\sqcup$0 are computed via the learned $\sqcup$1 field. A stochastic jump process—parameterized by hyperparameters $\sqcup$2—decides whether to update each token. If updated, nucleus sampling with temperature is used to draw new tokens. This framework ensures sampled trajectories remain close to high-likelihood or biologically plausible sequence neighborhoods throughout generation (Goel et al., 29 Jan 2026).

6. Discrete Classifier Guidance

MadSBM introduces discrete classifier guidance for the first time in a Schrödinger bridge context. This mechanism allows generation to be steered toward target properties (e.g., functional affinity) by resampling candidate jumps based on a property classifier $\sqcup$3. The guided rate is

$\sqcup$4

In practice, M candidate successors are generated per position, scored, and resampled proportionally to the classifier output, implementing guidance through candidate selection rather than full re-tilting.

7. Context, Relation to Broader Discrete Schrödinger Bridge Paradigms, and Applications

MadSBM builds conceptually on the discrete Schrödinger bridge approach, which seeks minimal-action or entropy-regularized transport in discrete path space, generalizing frameworks previously developed for molecular graphs (Kim et al., 2024), categorical variables (2502.01416), and adversarially-trained generative models (Gushchin et al., 2024). In contrast to existing discrete diffusion or flow-based methods—which either require simulating long chains through implausible intermediate states or rely on fixed corruption paths—MadSBM constructs transport processes explicitly biased to remain in high-likelihood sequence regions, drawing on pre-trained protein LLMs for biological prior structure.

MadSBM is explicitly developed for peptide sequence design, synthesizing sequences that interpolate between a fully masked prior and empirical data, and, via classifier guidance, can extend to functional optimization of therapeutic peptides. Empirical results and theoretical justification are provided in (Goel et al., 29 Jan 2026).

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Table: Key Formal Components of MadSBM

Component	Mathematical Description	Biological/Algorithmic Role
Sequence Space	$\sqcup$5 (amino-acid strings + mask)	Set of all peptides considered
Reference Rates $\sqcup$6	From $\sqcup$7 of ESM-2 logits at masked sites	Biologically plausible edit dynamics
Control Field $\sqcup$8	Time-dependent parametric tilt on transitions	Steers paths toward data distribution
Loss $\sqcup$9	Cross-entropy between predicted/true token at masked sites	Learns optimal local update rates
Guidance Mechanism	Resampling using $(x \to y)$0 as property classifier	Conditional generation for target function

MadSBM represents an integration of minimal-action discrete Schrödinger bridge formulation, domain-informed reference processes, efficient simulation-free training, and function-classifier guidance. This enables efficient, plausible peptide generation while traversing low-entropy regions of the discrete Markov path space (Goel et al., 29 Jan 2026).