We consider the problem of generating samples via Flow Matching (FM) with an additional requirement that the generated samples must satisfy given constraints. We consider two scenarios, viz.: (a) when a differentiable distance function to the constraint set is given, and (b) when the constraint set is only available via queries to a membership oracle. For case (a), we propose a simple adaptation of the FM objective with an additional term that penalizes the distance between the constraint set and the generated samples. For case (b), we propose to employ randomization and learn a mean flow that is numerically shown to have a high likelihood of satisfying the constraints. This approach deviates significantly from existing works that require simple convex constraints, knowledge of a barrier function, or a reflection mechanism to constrain the probability flow. Furthermore, in the proposed setting we show that a two-stage approach, where both stages approximate the same original flow but with only the second stage probing the constraints via randomization, is more computationally efficient. Through several synthetic cases of constrained generation, we numerically show that the proposed approaches achieve significant gains in terms of constraint satisfaction while matching the target distributions. As a showcase for a practical oracle-based constraint, we show how our approach can be used for training an adversarial example generator, using queries to a hard-label black-box classifier. We conclude with several future research directions. Our code is available at https://github.com/ZhengyanHuan/FM-RE.

In imperfect-information games, agents must make decisions based on partial knowledge of the game state. The Belief Stochastic Game model addresses this challenge by delegating state estimation to the game model itself. This allows agents to operate on externally provided belief states, thereby reducing the need for game-specific inference logic. This paper investigates two approaches to represent beliefs in games with hidden piece identities: a constraint-based model using Constraint Satisfaction Problems and a probabilistic extension using Belief Propagation to estimate marginal probabilities. We evaluated the impact of both representations using general-purpose agents across two different games. Our findings indicate that constraint-based beliefs yield results comparable to those of probabilistic inference, with minimal differences in agent performance. This suggests that constraint-based belief states alone may suffice for effective decision-making in many settings.

This paper considers online convex optimization with hard constraints and analyzes achievable regret and cumulative hard constraint violation (violation for short).

Our framework employs traditional regret minimizers as black-box components.

We consider the problem of assigning items to platforms where each item has a utility associated with each of the platforms to which it can be assigned. Each platform has a soft constraint over the total number of items it serves, modeled via a convex cost function. Additionally, items are partitioned into groups, and each platform also incurs group-specific convex cost over the number of items from each group that can be assigned to the platform. These costs promote group fairness by penalizing imbalances, yielding a soft variation of fairness notions introduced in prior work, such as Restricted Dominance and Minority protection. Restricted Dominance enforces upper bounds on group representation, while Minority protection enforces lower bounds. Our approach replaces such hard constraints with cost-based penalties, allowing more flexible trade-offs. Our model also captures Nash Social Welfare kind of objective. The cost of an assignment is the sum of the values of all the cost functions across all the groups and platforms. The objective is to find an assignment that minimizes the cost while achieving a total utility that is at least a user-specified threshold. The main challenge lies in balancing the overall platform cost with group-specific costs, both governed by convex functions, while meeting the utility constraint. We present an efficient polynomial-time approximation algorithm, supported by theoretical guarantees and experimental evaluation. Our algorithm is based on techniques involving linear programming and network flows. We also provide an exact algorithm for a special case with uniform utilities and establish the hardness of the general problem when the groups can intersect arbitrarily.

Neural networks excel at pattern recognition but struggle with reliable logical reasoning, often violating basic logical principles during inference. We address this limitation by developing a categorical framework that systematically constructs neural architectures with provable logical guarantees. Our approach treats logical theories as algebraic structures called Lawvere theories, which we transform into neural networks using categorical algebra in the 2-category of parametric maps. Unlike existing methods that impose logical constraints during training, our categorical construction embeds logical principles directly into the network's architectural structure, making logical violations mathematically impossible. We demonstrate this framework by constructing differentiable neural architectures for propositional logic that preserve boolean reasoning while remaining trainable via gradient descent. Our main theoretical result establishes a bijective correspondence between finitary logical theories and neural architectures, proving that every logically constrained network arises uniquely from our construction. This extends Categorical Deep Learning beyond geometric symmetries to semantic constraints, enabling automatic derivation of verified architectures from logical specifications. The framework provides mathematical foundations for trustworthy AI systems, with applications to theorem proving, formal verification, and safety-critical reasoning tasks requiring verifiable logical behavior.

In many sequential decision making applications, the change of decision would bring an additional cost, such as the wear-and-tear cost associated with changing server status. To control the switching cost, we introduce the problem of online convex optimization with continuous switching constraint, where the goal is to achieve a small regret given a budget on the overall switching cost. We first investigate the hardness of the problem, and provide a lower bound of order ( p T) when the switching cost budget S = ( p T), and (min { T/S, T }) when S = O ( p T), where T is the time horizon. The essential idea is to carefully design an adaptive adversary, who can adjust the loss function according to the cumulative switching cost of the player incurred so far based on the orthogonal technique. We then develop a simple gradient-based algorithm which enjoys the minimax optimal regret bound. Finally, we show that, for strongly convex functions, the regret bound can be improved to O (log T) for S = (log T), and O (min { T/ exp( S)+ S,T }) for S = O (log T) .