Multi Armed Bandit (MAB) algorithms are a cornerstone of reinforcement learning and have been studied both theoretically and numerically. One of the most commonly used implementation uses a softmax mapping to prescribe the optimal policy and served as the foundation for downstream algorithms, including REINFORCE. Distinct from vanilla approaches, we consider here the L2 regularized softmax policy gradient where a quadratic term is subtracted from the mean reward. Previous studies exploiting convexity failed to identify a suitable theoretical framework to analyze its convergence when the regularization parameter vanishes. We prove here theoretical convergence results and confirm empirically that this regime makes the L2 regularization numerically advantageous on standard benchmarks.

Reinforcement Learning with Verifiable Rewards (RLVR) has achieved substantial gains in single-attempt accuracy (Pass@1) on reasoning tasks, yet often suffers from reduced multi-sample coverage (Pass@K), indicating diversity collapse. We identify a structural cause for this degradation: common RLVR objectives, such as GRPO, are indifferent to how probability mass is distributed among correct solutions. Combined with stochastic training dynamics, this indifference induces a self-reinforcing collapse, in which probability mass concentrates on a narrow subset of correct outputs while alternative valid solutions are suppressed. We formalize this collapse mechanism and further characterize the optimal policy structure under two complementary criteria: robustness and entropy-regularized optimality, which identify the Uniform-Correct Policy as uniquely optimal. Motivated by this analysis, we propose Uniform-Correct Policy Optimization (UCPO), a modification to GRPO that adds a conditional uniformity penalty on the policy's distribution over correct solutions. The penalty redistributes gradient signal toward underrepresented correct responses, encouraging uniform allocation of probability mass within the correct set. Across three models (1.5B-7B parameters) and five mathematical reasoning benchmarks, UCPO improves Pass@K and diversity while maintaining competitive Pass@1, achieving up to +10\% absolute improvement on AIME24 at Pass@64 and up to 45\% higher equation-level diversity within the correct set. The code is available at https://github.com/AnamikaLochab/UCPO.

Probabilistic Circuits (PCs) are a promising avenue for probabilistic modeling. They combine advantages of probabilistic graphical models (PGMs) with those of neural networks (NNs). Crucially, however, they are tractable probabilistic models, supporting efficient and exact computation of many probabilistic inference queries, such as marginals and MAP. Further, since PCs are structured computation graphs, they can take advantage of deep-learning-style parameter updates, which greatly improves their scalability. However, this innovation also makes PCs prone to overfitting, which has been observed in many standard benchmarks. Despite the existence of abundant regularization techniques for both PGMs and NNs, they are not effective enough when applied to PCs.

Left: The reward function is a mixture of Gaussian, and the offline data distribution is unbalanced with most samples located in low-reward states.

Scientific modeling applications often require estimating a distribution of parameters consistent with a dataset of observations--an inference task also known as source distribution estimation. This problem can be ill-posed, however, since many different source distributions might produce the same distribution of data-consistent simulations. To make a principled choice among many equally valid sources, we propose an approach which targets the maximum entropy distribution, i.e., prioritizes retaining as much uncertainty as possible.