VALG Tackles COLT 2026 Open Problems With Agentic ML Theory Proofs

A Hong Kong led team has released VALG, an agentic system aimed at a problem most AI research agents sidestep: informal machine learning theory, where the theorem statement, assumptions, and proof strategy co evolve. Posted to arXiv on August 13, 2026 as paper 2608.13060, the work evaluates VALG on nine subproblems drawn from five COLT 2026 open problem papers. The authors report 22 internally finalized theorem candidates across the runs, with two matching the full scope of their source briefs and seven yielding restricted methods, special cases, or conditional results.

Why ML theory is a distinct agent challenge

Unlike fixed target formal proving in Lean or Coq, ML theory papers specify learning phenomena through data models, oracle access, training protocols, and asymptotic regimes. When a proof attempt fails, researchers must decide whether the obstruction is local to a derivation, structural in the proof graph, or fundamental to the formulation itself. Silent reformulation can produce a technically true theorem that no longer answers the original open question.

VALG names three design pillars: multi level verification, adaptive formulation of learning theory problems, and graph structured proof development. Each source relative branch fixes a mathematical specification, represents proof dependencies as a typed graph, and routes failures hierarchically. Derivation failures trigger local repair; graph failures trigger architectural revision; only formulation level obstructions spawn an explicitly related variant or relaxation tied back to the originating brief.

COLT 2026 open problems as a testbed

The evaluation spans tensor decomposition and ALS/GD overparameterization, distribution independent deep learning versus linear dimension complexity, non adaptive one bit mean estimation, differential privacy PAC learning, and online optimization of piecewise Lipschitz functions. These are live unresolved questions from the COLT open problem track rather than synthetic benchmark lemmas.

Across nine runs, VALG produced candidates the authors classify as source scope matches, relaxations, conditional theorems, or blocked branches. Two runs finalized theorems that fully address their subproblems. The remaining seven document partial progress: restricted method results, special cases, or conditional statements that still clarify why the full problem resists a direct attack. For each candidate the paper reports setup, statement, and explicit progress relative to the originating brief.

Positioning against prior agent stacks

The related work section draws a careful line between general scientific agents, natural language proof councils, and formal prover loops. Systems like Danus coordinate parallel proof search with a shared fact graph but typically terminate when a supplied target is verified or refuted. VALG instead treats formulation revision as a first class outcome and records the mathematical relation between a relaxation and its source problem.

The authors also connect to AI4SLT style formalization of statistical learning theory in Lean, noting that formal verification assumes a fixed encoded target whereas VALG operates earlier when the learning setup is still under construction. Open source artifacts include skill definitions and case study folders on GitHub under DechenZhang's VALG ML Theory Agent repository.

Limits and what practitioners should take away

The paper is explicit that internal finalization is not peer review. Reviewers still must adjudicate correctness, novelty, and whether a conditional result genuinely advances a COLT open problem. Nonetheless, VALG offers a concrete workflow template for theory groups experimenting with agent assistance: keep branches source relative, separate proof architecture from local derivations, and never collapse a failed attempt into an silently weaker claim.

For the broader AI research community, the August 2026 release is a datapoint that agentic systems are moving from code generation and literature synthesis into mathematically structured domains where problem definition is part of the search space. Whether VALG becomes a daily tool for theory groups will depend on reproducibility across problem classes and human expert time saved versus time spent auditing agent drafted proofs.

Sources

Dechen Zhang et al., VALG agentic ML theory paper, arxiv.org/abs/2608.13060

COLT 2026 open problems track, learningtheory.org

GitHub, DechenZhang VALG ML Theory Agent repository, github.com