The paper received positive feedback. After reading the rebuttal and discussing the paper, the general consensus is that the paper should be accepted. The area chair agrees with this assessement and follows the reviewer's recommendation. Several suggestions were made to improve the paper (see in particular R1's review), which will be good to take into account for the final version.

This paper proposes a formal langauge to describe the search space of architecture search problem. This langauge is a domain specific language embedded in python. Users can write modular, composable, and reusable search space by using this langauge. Originality: The contribution is new. This is the first work that tries to provide a formal langauge for the space definition.

Summary and Contributions: This work considers the problem of synthesizing programs from input/outputs, but where some of the components of the program might have continuous parameters, and where the entire program is differentiable with respect to these parameters. Neurosymbolic programs are a special case of this set up (symbolic programs which can call out to neural modules if needed). This is an especially challenging combinatorial search problem, because not only do we have to consider an infinitely large, discrete space of program structures, but we also have to consider an inner-loop optimization over continuous parameters. The approach they take is to perform an explicit symbolic graph search over the discrete space of partial programs. As a heuristic function for this graph search, they train neural networks to approximate the behavior of incomplete portions of the program syntax tree.

Weaknesses: Although I believe the math derivation of the novel minimax objective function is correct, I have two major concerns. My first concern is whether this minimax objective function provides some novel insight on network dynamics which cannot be captured by traditional framework that network dynamics is minimizing an "energy" function. My concern is resulted from that the minimax objective (Eq. It seems to me that the only difference between the minimax and minimized objective function is that the network state converges to the saddle point in the former case, while in later case the network state converges to a stable fixed point. I really hope the authors explain this and correct me if I understood something wrong.

The reviewers were originally divergent in their opinions of this paper, but came to some agreements in discussion. It was agreed that the paper provides an interesting contribution for neuroscience by extending the previous work of Seung et al. (1997) to more biologically realistic networks, but the actual theoretical insights beyond that original paper are not large. In the end, an "accept" decision was reached, but it was agreed that the authors should better clarify the strong links to the Seung paper and be more cautious in their claims of "detailed" or "tight" balance in cortical networks.

Relation to Prior Work: In addition to the papers mentioned above, the relation to the literature of monotone VI should be discussed in details. The convex-concave min-max falls into the category of monotone VIs and there are many works in the literature addressing that. For example, the following papers should be discussed: Ronald E Bruck Jr. Dual extrapolation and its applications to solving variational inequalities and related problems. Solving variational inequalities with monotone operators on domains given by linear minimization oracles.

Authors propose a novel algorithm that achieves linear convergence rate and improved dependence on the condition numbers, compared to prior work.

Reasoning language models (RLMs), also known as Large Reasoning Models (LRMs), such as OpenAI's o1 and o3, DeepSeek-V3, and Alibaba's QwQ, have redefined AI's problem-solving capabilities by extending LLMs with advanced reasoning mechanisms. Yet, their high costs, proprietary nature, and complex architectures - uniquely combining Reinforcement Learning (RL), search heuristics, and LLMs - present accessibility and scalability challenges. To address these, we propose a comprehensive blueprint that organizes RLM components into a modular framework, based on a survey and analysis of all RLM works. This blueprint incorporates diverse reasoning structures (chains, trees, graphs, and nested forms), reasoning strategies (e.g., Monte Carlo Tree Search, Beam Search), RL concepts (policy, value models and others), supervision schemes (Outcome-Based and Process-Based Supervision), and other related concepts (e.g., Test-Time Compute, Retrieval-Augmented Generation, agent tools). We also provide detailed mathematical formulations and algorithmic specifications to simplify RLM implementation. By showing how schemes like LLaMA-Berry, QwQ, Journey Learning, and Graph of Thoughts fit as special cases, we demonstrate the blueprint's versatility and unifying potential. To illustrate its utility, we introduce x1, a modular implementation for rapid RLM prototyping and experimentation. Using x1 and a literature review, we provide key insights, such as multi-phase training for policy and value models, and the importance of familiar training distributions. Finally, we discuss scalable RLM cloud deployments and we outline how RLMs can integrate with a broader LLM ecosystem. Our work demystifies RLM construction, democratizes advanced reasoning capabilities, and fosters innovation, aiming to mitigate the gap between "rich AI" and "poor AI" by lowering barriers to RLM design and experimentation.

A critical requirement for deep learning models is ensuring their robustness against adversarial attacks. These attacks commonly introduce noticeable perturbations, compromising the visual fidelity of adversarial examples. Another key challenge is that while white-box algorithms can generate effective adversarial perturbations, they require access to the model gradients, limiting their practicality in many real-world scenarios. Existing attack mechanisms struggle to achieve similar efficacy without access to these gradients. In this paper, we introduce GreedyPixel, a novel pixel-wise greedy algorithm designed to generate high-quality adversarial examples using only query-based feedback from the target model. GreedyPixel improves computational efficiency in what is typically a brute-force process by perturbing individual pixels in sequence, guided by a pixel-wise priority map. This priority map is constructed by ranking gradients obtained from a surrogate model, providing a structured path for perturbation. Our results demonstrate that GreedyPixel achieves attack success rates comparable to white-box methods without the need for gradient information, and surpasses existing algorithms in black-box settings, offering higher success rates, reduced computational time, and imperceptible perturbations. These findings underscore the advantages of GreedyPixel in terms of attack efficacy, time efficiency, and visual quality.

Similarity search, the task of finding similar vectors, has become a fundamental operation in machine learning, with applications in recommendation engines, semantic search systems, and more [1-3]. As datasets grow to billions of entries, the challenge of performing efficient searches on high-dimensional vectors becomes increasingly complex [4]. This is further compounded by the well-known curse of dimensionality [5], which affects the performance and accuracy of search algorithms as the number of dimensions increases. Approximate Nearest Neighbor (ANN) algorithms, such as Inverted File Index (IVF) [6] and Hierarchical Navigable Small World (HNSW) [7], have been developed to address scalability and performance issues. IVF segments the search space into smaller areas, called Voronoi cells [8], while HNSW constructs a navigable graph structure for efficient search space traversal. Despite their advancements, these methods often struggle to support complex, multi-dimensional filtering efficiently. This is crucial in practical scenarios where additional criteria beyond vector similarity are required to refine search results [6]. Examples of such scenarios include e-commerce product search and semantic search with filtering and recommendation systems.