arXiv subjects

Ali Shehper

Publications and source records attributed to Ali Shehper.

10 recordsLinked to original sources

arXiv2026

Hierarchical Reinforcement Learning for Sparse-Reward Search in Commutative Algebra

Applying machine learning techniques to solving long-standing mathematical conjectures can be particularly challenging due to their extreme reward sparsity. As an illustrative example, we consider Kalai's algebraic Hirsch conjecture and recast the construction of its counterexamples as a sparse-reward reinforcement learning problem on graphs. We propose a constrained options-based HRL framework with an equivariant graph neural network policy, which allows us to learn useful temporal abstractions for this task. We evaluate our approach over a wide range of degrees and demonstrate that it consistently outperforms classical RL algorithms as well as greedy search. By exploiting the hierarchical structure of the problem, we effectively provide a first-of-its-kind application of HRL to a problem in commutative algebra.

cs.LG↗

arXiv2026

The Two-Hump Problem: Bridging the Difficulty Gap in Mathematical Reinforcement Learning

Mathematical search problems present a unique challenge for Reinforcement Learning (RL) due to vast search spaces and sparse rewards. In previous works, the Andrews-Curtis (AC) conjecture was established as an illustrative example of such problems. In this work, we identify a critical structural barrier in the AC landscape: a "Two-Hump" distribution, where problem instances are either trivially solvable or effectively impossible, with a scarcity of intermediate "hard-but-solvable" instances required for effective learning. We tackle this challenge through two primary avenues: novel data generation techniques to populate the difficulty gap, and significant algorithmic enhancements including the introduction of supermoves and Transformer-based architectures. We demonstrate substantial performance improvements over previous baselines, and release new comprehensive benchmark datasets including AC-19 (125,192 AC-trivial presentations of varying difficulty with length at most 19) and AC-1M (1,136,154 hard AC-trivial presentations of length at most 30), the first large-scale, publicly available datasets of this kind.

cs.LG↗

arXiv2026

A Training-Time Diagnostic for Generalization via the Log-Alignment Ratio

We study the log-alignment ratio (LAR), a measure of parameter-activation alignment, introduced in parameterization theory. We reformulate it as the overlap between a weight spectrum $p$ of the normalized squared singular values of a matrix and an activation spectrum $q$ of the normalized squared projections of inputs onto its singular directions. We show that unembedding LAR tracks the transition between memorization and generalization in two different settings by capturing the spread of $p$ and $q$ during training. In grokking, LAR predicts the effective dimension of the learned function: $k \approx n^{2(1-\text{LAR})}$, where $n$ is the input dimension of the matrix. In 3B-parameter language model pre-training, its deviation from a non-overfitting baseline tracks the generalization gap, and its rate of decline increases as overfitting approaches. LAR is computable from quantities available during the forward pass with negligible computational overhead, and requires no held-out validation data.

cs.LG↗

arXiv2024

What makes math problems hard for reinforcement learning: a case study

Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.

cs.LG↗

arXiv2021

Nonabelian Twists of the $D_4$ Theory

We study theories of type $D_4$ in class-S, with nonabelian outer-automorphism twists around various cycles of the punctured Riemann surface $C$. We propose an extension of previous formul\ae\ for the superconformal index to cover this case and classify the SCFTs corresponding to fixtures (3-punctured spheres). We then go on to study families of SCFTs corresponding to once-punctured tori and 4-punctured spheres. These exhibit new behaviours, not seen in previous investigations. In particular, the generic theory with 4 punctures on the sphere from non-commuting $\mathbb{Z}_2$ twisted sectors has six distinct weakly-coupled descriptions.

hep-th↗

arXiv2021

Symmetries of 2d TQFTs and Equivariant Verlinde Formulae for General Groups

We study (generalized) discrete symmetries of 2d semisimple TQFTs. These are 2d TQFTs whose fusion rules can be diagonalized. We show that, in this special basis, the 0-form symmetries always act as permutations while 1-form symmetries act by phases. This leads to an explicit description of the gauging of these symmetries. One application of our results is a generalization of the equivariant Verlinde formula to the case of general Lie groups. The generalized formula leads to many predictions for the geometry of Hitchin moduli spaces, which we explicitly check in several cases with low genus and SO(3) gauge group.

hep-th↗

arXiv2020

Distinquishing 4d N=2 SCFTs

We construct a family of examples of pairs of 4d N=2 SCFTs whose graded Coulomb branch dimensions, Weyl-anomaly coefficients and flavour symmetry algebras and levels coincide, but which are nonetheless distinct SCFTs. The difference (detectable by the superconformal index) can occur at arbitrarily high order. We argue that it is, however, reflected in a difference in the global form of the flavour symmetry groups.

hep-th↗

arXiv2020

Deformations of surface defect moduli spaces

Given a 4d ${\mathcal N}=2$ supersymmetric theory with an ${\mathcal N}=(2,2)$ supersymmetric surface defect, a marginal perturbation of the bulk theory induces a complex structure deformation of the defect moduli space. We describe a concrete way of computing this deformation using the bulk-defect OPE.

hep-th↗

arXiv2015

Brane Webs and Random Processes

We study $(p,q)$ 5-brane webs dual to certain $N$ M5-brane configurations and show that the partition function of these brane webs gives rise to cylindric Schur process with period $N$. This generalizes the previously studied case of period $1$. We also show that open string amplitudes corresponding to these brane webs are captured by the generating function of cylindric plane partitions with profile determined by the boundary conditions imposed on the open string amplitudes.

hep-th↗

arXiv2015

Topological Field Theory Amplitudes for $A_{N-1}$ Fibration

We study the partition function ${\cal N}=1$ 5D $U(N)$ gauge theory with $g$ adjoint hypermultiplets and show that for massless adjoint hypermultiplets it is equal to the partition function of a two dimensional topological field on a genus $g$ Riemann surface. We describe the topological field theory by its amplitudes associated with cap, propagator and pair of pants. These basic amplitudes are open topological string amplitudes associated with certain Calabi-Yau threefolds in the presence of Lagrangian branes.

hep-th↗