The efficiency of Logic Optimization (LO) has become one of the key bottlenecks in chip design. To prompt efficient LO, previous studies propose using a key scoring function to predict and prune a large number of ineffective nodes of the LO heuristics. However, the existing scoring functions struggle to balance inference efficiency, interpretability, and generalization performance, which severely hinders their application to modern LO tools. To address this challenge, we propose a novel data-driven circuit symbolic learning framework, namely CMO, to learn lightweight, interpretable, and generalizable scoring functions. The major challenge of developing CMO is to discover symbolic functions that can well generalize to unseen circuits, i.e., the circuit symbolic generalization problem. Thus, the major technical contribution of CMO is the novel Graph Enhanced Symbolic Discovery framework, which distills dark knowledge from a well-designed Graph Neural Network (GNN) to enhance the generalization capability of the learned symbolic functions. To the best of our knowledge, CMO is *the first* graph-enhanced approach for discovering lightweight and interpretable symbolic functions that can well generalize to unseen circuits in LO. Experiments on three challenging circuit benchmarks show that the *interpretable* symbolic functions learned by CMO outperform previous state-of-the-art (SOTA) GPU-based and human-designed approaches in terms of *inference efficiency* and *generalization capability*. Moreover, we integrate CMO with the Mfs2 heuristic---one of the most time-consuming LO heuristics. The empirical results demonstrate that CMO significantly improves its efficiency while keeping comparable optimization performance when executed on a CPU-based machine, achieving up to 2.5× faster runtime.

Optimization modeling is one of the most crucial but technical parts of operations research (OR). To automate the modeling process, existing works have leveraged large language models (LLMs), prompting them to break down tasks into steps for generating variables, constraints, and objectives. However, due to the highly complex mathematical structures inherent in OR problems, standard fixed-step decomposition often fails to achieve high performance. To address this challenge, we introduce OptiTree, a novel tree search approach designed to enhance modeling capabilities for complex problems through adaptive problem decomposition into simpler subproblems. Specifically, we develop a modeling tree that organizes a wide range of OR problems based on their hierarchical problem taxonomy and complexity, with each node representing a problem category and containing relevant high-level modeling thoughts. Given a problem to model, we recurrently search the tree to identify a series of simpler subproblems and synthesize the global modeling thoughts by adaptively integrating the hierarchical thoughts. Experiments show that OptiTree significantly improves the modeling accuracy compared to the state-of-the-art, achieving over 10% improvements on the challenging benchmarks.

Cutting planes (cuts) are essential for solving mixed-integer linear programming (MILP) problems, as they tighten the feasible solution space and accelerate the solving process. Modern MILP solvers offer diverse cutting plane separators to generate cuts, enabling users to leverage their potential complementary strengths to tackle problems with different structures. Recent machine learning approaches learn to configure separators based on problem-specific features, selecting effective separators and deactivating ineffective ones to save unnecessary computing time. However, they ignore the dynamics of separator efficacy at different stages of cut generation and struggle to adapt the configurations for the evolving problems after multiple rounds of cut generation. To address this challenge, we propose a novel **dyn**amic **sep**arator configuration (**DynSep**) method that models separator configuration in different rounds as a reinforcement learning task, making decisions based on an incremental triplet graph updated by iteratively added cuts. Specifically, we tokenize the incremental subgraphs and utilize a decoder-only Transformer as our policy to autoregressively predict when to halt separation and which separators to activate at each round. Evaluated on synthetic and large-scale real-world MILP problems, DynSep speeds up average solving time by 64% on easy and medium datasets, and reduces primal-dual gap integral within the given time limit by 16% on hard datasets. Moreover, experiments demonstrate that DynSep well generalizes to MILP instances of significantly larger sizes than those seen during training.

As one of the most critical components in modern LP solvers, presolve in linear programming (LP) employs a rich set of presolvers to remove different types of redundancy in input problems by equivalent transformations. We found from extensive experiments that the presolve routine-that is, the method determining (P1) which presolvers to select, (P2) in what order to execute, and (P3) when to stop-significantly impacts the efficiency of solving LPs. However, designing high-quality presolve routines is highly challenging due to the enormous search space, and further optimizing the routines on different tasks for high performance demands extensive domain knowledge and manual tuning. To tackle this problem, we propose the first learning based framework-that is, reinforcement learning for presolve (RL4Presolve)-to learn high-quality presolve routines. An appealing feature is that we employ a novel adaptive action sequence that learns complex routines efficiently by generating combinations of presolvers automatically at each step. Extensive experiments demonstrate that RL4Presolve achieves significant improvement (up to roughly 90% ) in the efficiency of solving LPs. Furthermore, we extract routines from learned policies for simple and efficient deployment without GPU resources to Huawei's supply chain, where extensive manual tuning for each separate task was required previously due to the high economic value.

Sample efficiency remains a key challenge for the deployment of deep reinforcement learning (RL) in real-world scenarios. A common approach is to learn efficient representations through future prediction tasks, facilitating the agent to make farsighted decisions that benefit its long-term performance. Existing methods extract predictive features by predicting multi-step future state signals. However, they do not fully exploit the structural information inherent in sequential state signals, which can potentially improve the quality of long-term decision-making but is difficult to discern in the time domain. To tackle this problem, we introduce a new perspective that leverages the frequency domain of state sequences to extract the underlying patterns in time series data. We theoretically show that state sequences contain structural information closely tied to policy performance and signal regularity and analyze the fitness of the frequency domain for extracting these two types of structural information. Inspired by that, we propose a novel representation learning method, State Sequences Prediction via Fourier Transform (SPF), which extracts long-term features by predicting the Fourier transform of infinite-step future state sequences. The appealing features of our frequency prediction objective include: 1) simple to implement due to a recursive relationship; 2) providing an upper bound on the performance difference between the optimal policy and the latent policy in the representation space. Experiments on standard and goal-conditioned RL tasks demonstrate that the proposed method outperforms several state-of-the-art algorithms in terms of both sample efficiency and performance.

Machine learning (ML) has been shown to successfully accelerate solving NP-hard combinatorial optimization (CO) problems under the branch and bound framework. However, the high training and inference cost and limited interpretability of ML approaches severely limit their wide application to modern exact CO solvers. In contrast, human-designed policies---though widely integrated in modern CO solvers due to their compactness and reliability---can not capture data-driven patterns for higher performance. To combine the advantages of the two paradigms, we propose the first symbolic discovery framework---namely, deep symbolic discovery for exact combinatorial optimization solver (Symb4CO)---to learn high-performance symbolic policies on the branching task. Specifically, we show the potential existence of small symbolic policies empirically, employ a large neural network to search in the high-dimensional discrete space, and compile the learned symbolic policies directly for fast deployment. Experiments show that the Symb4CO learned purely CPU-based policies consistently achieve *comparable* performance to previous GPU-based state-of-the-art approaches. Furthermore, the appealing features of Symb4CO include its high training (*ten training instances*) and inference (*one CPU core*) efficiency and good interpretability (*one-line expressions*), making it simple and reliable for deployment. The results show encouraging potential for the *wide* deployment of ML to modern CO solvers.

Machine learning (ML) approaches have been successfully applied to accelerating exact combinatorial optimization (CO) solvers. However, many of them fail to explain what patterns they have learned that accelerate the CO algorithms due to the black-box nature of ML models like neural networks, and thus they prevent researchers from further understanding the tasks they are interested in. To tackle this problem, we propose the first graph-based algorithm discovery framework—namely, graph symbolic discovery for exact combinatorial optimization solver (GS4CO)—that learns interpretable branching policies directly from the general bipartite graph representation of CO problems. Specifically, we design a unified representation for symbolic policies with graph inputs, and then we employ a Transformer with multiple tree-structural encodings to generate symbolic trees end-to-end, which effectively reduces the cumulative error from iteratively distilling graph neural networks. Experiments show that GS4CO learned interpretable and lightweight policies outperform all the baselines on CPU machines, including both the human-designed and the learning-based. GS4CO shows an encouraging step towards general algorithm discovery on modern CO solvers.

Deep reinforcement learning (DRL) algorithms can suffer from modeling errors between the simulation and the real world. Many studies use adversarial learning to generate perturbation during training process to model the discrepancy and improve the robustness of DRL. However, most of these approaches use a fixed parameter to control the intensity of the adversarial perturbation, which can lead to a trade-off between average performance and robustness. In fact, finding the optimal parameter of the perturbation is challenging, as excessive perturbations may destabilize training and compromise agent performance, while insufficient perturbations may not impart enough information to enhance robustness. To keep the training stable while improving robustness, we propose a simple but effective method, namely, Adaptive Adversarial Perturbation (A2P), which can dynamically select appropriate adversarial perturbations for each sample. Specifically, we propose an adaptive adversarial coefficient framework to adjust the effect of the adversarial perturbation during training. By designing a metric for the current intensity of the perturbation, our method can calculate the suitable perturbation levels based on the current relative performance. The appealing feature of our method is that it is simple to deploy in real-world applications and does not require accessing the simulator in advance. The experiments in MuJoCo show that our method can improve the training stability and learn a robust policy when migrated to different test environments. The code is available at https://github.com/Lqm00/A2P-SAC.

Mixed-integer linear programming (MILP) is one of the most popular mathematical formulations with numerous applications. In practice, improving the performance of MILP solvers often requires a large amount of high-quality data, which can be challenging to collect. Researchers thus turn to generation techniques to generate additional MILP instances. However, existing approaches do not take into account specific block structures—which are closely related to the problem formulations—in the constraint coefficient matrices (CCMs) of MILPs. Consequently, they are prone to generate computationally trivial or infeasible instances due to the disruptions of block structures and thus problem formulations. To address this challenge, we propose a novel MILP generation framework, called Block Structure Decomposition (MILP-StuDio), to generate high-quality instances by preserving the block structures. Specifically, MILP-StuDio begins by identifying the blocks in CCMs and decomposing the instances into block units, which serve as the building blocks of MILP instances. We then design three operators to construct new instances by removing, substituting, and appending block units in the original instances, enabling us to generate instances with flexible sizes. An appealing feature of MILP-StuDio is its strong ability to preserve the feasibility and computational hardness of the generated instances. Experiments on the commonly-used benchmarks demonstrate that using instances generated by MILP-StuDio is able to significantly reduce over 10% of the solving time for learning-based solvers.

Solving large-scale sparse linear systems is essential in fields like mathematics, science, and engineering. Traditional numerical solvers, mainly based on the Krylov subspace iteration algorithm, suffer from the low-efficiency problem, which primarily arises from the less-than-ideal iteration. To tackle this problem, we propose a novel method, namely **Neur**al **K**rylov **It**era**t**ion (**NeurKItt**), for accelerating linear system solving. Specifically, NeurKItt employs a neural operator to predict the invariant subspace of the linear system and then leverages the predicted subspace to accelerate linear system solving. To enhance the subspace prediction accuracy, we utilize QR decomposition for the neural operator outputs and introduce a novel projection loss function for training. NeurKItt benefits the solving by using the predicted subspace to guide the iteration process, significantly reducing the number of iterations. We provide extensive experiments and comprehensive theoretical analyses to demonstrate the feasibility and efficiency of NeurKItt. In our main experiments, NeurKItt accelerates the solving of linear systems across various settings and datasets, achieving up to a 5.5× speedup in computation time and a 16.1× speedup in the number of iterations.

Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), which formulate many important real-world applications. Cut selection heavily depends on (P1) which cuts to prefer and (P2) how many cuts to select. Although modern MILP solvers tackle (P1)-(P2) by human-designed heuristics, machine learning carries the potential to learn more effective heuristics. However, many existing learning-based methods learn which cuts to prefer, neglecting the importance of learning how many cuts to select. Moreover, we observe that (P3) what order of selected cuts to prefer significantly impacts the efficiency of MILP solvers as well. To address these challenges, we propose a novel hierarchical sequence/set model (HEM) to learn cut selection policies. Specifically, HEM is a bi-level model: (1) a higher-level module that learns how many cuts to select, (2) and a lower-level module-that formulates the cut selection as a sequence/set to sequence learning problem-to learn policies selecting an ordered subset with the cardinality determined by the higher-level module. To the best of our knowledge, HEM is the first data-driven methodology that well tackles (P1)-(P3) simultaneously. Experiments demonstrate that HEM significantly improves the efficiency of solving MILPs on eleven challenging MILP benchmarks, including two Huawei's real problems.

While deep reinforcement learning (RL) has been demonstrated effective in solving complex control tasks, sample efficiency remains a key challenge due to the large amounts of data required for remarkable performance. Existing research explores the application of representation learning for data-efficient RL, e.g., learning predictive representations by predicting long-term future states. However, many existing methods do not fully exploit the structural information inherent in sequential state signals, which can potentially improve the quality of long-term decision-making but is difficult to discern in the time domain. To tackle this problem, we propose State Sequences Prediction via Fourier Transform (SPF), a novel method that exploits the frequency domain of state sequences to extract the underlying patterns in time series data for learning expressive representations efficiently. Specifically, we theoretically analyze the existence of structural information in state sequences, which is closely related to policy performance and signal regularity, and then propose to predict the Fourier transform of infinite-step future state sequences to extract such information. One of the appealing features of SPF is that it is simple to implement while not requiring storage of infinite-step future states as prediction targets. Experiments demonstrate that the proposed method outperforms several state-of-the-art algorithms in terms of both sample efficiency and performance.

Reinforcement learning algorithms have achieved remarkable success in acquiring behavioral skills directly from pixel inputs. However, their application in real-world scenarios presents challenges due to their sensitivity to visual distractions (e.g., changes in viewpoint and light). A key factor contributing to this challenge is that the learned representations often suffer from overfitting task-irrelevant information. By comparing several representation learning methods, we find that the key to alleviating overfitting in representation learning is to choose proper prediction targets. Motivated by our comparison, we propose a novel representation learning approach—namely, reward sequence prediction (RSP)—that uses reward sequences or their transforms (e.g., discrete time Fourier transform) as prediction targets. RSP can efficiently learn robust representations as reward sequences rarely contain task-irrelevant information while providing a large number of supervised signals to accelerate representation learning. An appealing feature is that RSP makes no assumption about the type of distractions and thus can improve performance even when multiple types of distractions exist. We evaluate our approach in Distracting Control Suite. Experiments show that our method achieves state-of-the-art sample efficiency and generalization ability in tasks with distractions.

Deep reinforcement learning algorithms can perform poorly in real-world tasks due to the discrepancy between source and target environments. This discrepancy is commonly viewed as the disturbance in transition dynamics. Many existing algorithms learn robust policies by modeling the disturbance and applying it to source environments during training, which usually requires prior knowledge about the disturbance and control of simulators. However, these algorithms can fail in scenarios where the disturbance from target environments is unknown or is intractable to model in simulators. To tackle this problem, we propose a novel model-free actor-critic algorithm---namely, state-conservative policy optimization (SCPO)---to learn robust policies without modeling the disturbance in advance. Specifically, SCPO reduces the disturbance in transition dynamics to that in state space and then approximates it by a simple gradient-based regularizer. The appealing features of SCPO include that it is simple to implement and does not require additional knowledge about the disturbance or specially designed simulators. Experiments in several robot control tasks demonstrate that SCPO learns robust policies against the disturbance in transition dynamics.

Many recent reinforcement learning (RL) methods learn stochastic policies with entropy regularization for exploration and robustness. However, in continuous action spaces, integrating entropy regularization with expressive policies is challenging and usually requires complex inference procedures. To tackle this problem, we propose a novel regularization method that is compatible with a broad range of expressive policy architectures. An appealing feature is that, the estimation of our regularization terms is simple and efficient even when the policy distributions are unknown. We show that our approach can effectively promote the exploration in continuous action spaces. Based on our regularization, we propose an off-policy actor-critic algorithm. Experiments demonstrate that the proposed algorithm outperforms state-of-the-art regularized RL methods in continuous control tasks.