Recent methods have demonstrated that Large Language Models (LLMs) can solve reasoning tasks better when they are encouraged to solve subtasks of the main task first. In this paper we devise a similar strategy that breaks down reasoning tasks into a problem decomposition phase and a problem solving phase and show that the strategy is able to outperform a single stage solution. Further, we hypothesize that the decomposition should be easier to distill into a smaller model compared to the problem solving because the latter requires large amounts of domain knowledge while the former only requires learning general problem solving strategies. We propose methods to distill these two capabilities and evaluate their impact on reasoning outcomes and inference cost. We find that we can distill the problem decomposition phase and at the same time achieve good generalization across tasks, datasets, and models. However, it is harder to distill the problem solving capability without losing performance and the resulting distilled model struggles with generalization. These results indicate that by using smaller, distilled problem decomposition models in combination with problem solving LLMs we can achieve reasoning with cost-efficient inference and local adaptation.
Hierarchical federated learning (HFL) shows great advantages over conventional two-layer federated learning (FL) in reducing network overhead and interaction latency while still retaining the data privacy of distributed FL clients. However, the communication and energy overhead still pose a bottleneck for HFL performance, especially as the number of clients raises dramatically. To tackle this issue, we propose a non-orthogonal multiple access (NOMA) enabled HFL system under semi-synchronous cloud model aggregation in this paper, aiming to minimize the total cost of time and energy at each HFL global round. Specifically, we first propose a novel fuzzy logic based client orchestration policy considering client heterogenerity in multiple aspects, including channel quality, data quantity and model staleness. Subsequently, given the fuzzy based client-edge association, a joint edge server scheduling and resource allocation problem is formulated. Utilizing problem decomposition, we firstly derive the closed-form solution for the edge server scheduling subproblem via the penalty dual decomposition (PDD) method. Next, a deep deterministic policy gradient (DDPG) based algorithm is proposed to tackle the resource allocation subproblem considering time-varying environments. Finally, extensive simulations demonstrate that the proposed scheme outperforms the considered benchmarks regarding HFL performance improvement and total cost reduction.
Large Language Models (LLMs) prompted to generate chain-of-thought (CoT) exhibit impressive reasoning capabilities. Recent attempts at prompt decomposition toward solving complex, multi-step reasoning problems depend on the ability of the LLM to simultaneously decompose and solve the problem. A significant disadvantage is that foundational LLMs are typically not available for fine-tuning, making adaptation computationally prohibitive. We believe (and demonstrate) that problem decomposition and solution generation are distinct capabilites, better addressed in separate modules, than by one monolithic LLM. We introduce DaSLaM, which uses a decomposition generator to decompose complex problems into subproblems that require fewer reasoning steps. These subproblems are answered by a solver. We use a relatively small (13B parameters) LM as the decomposition generator, which we train using policy gradient optimization to interact with a solver LM (regarded as black-box) and guide it through subproblems, thereby rendering our method solver-agnostic. Evaluation on multiple different reasoning datasets reveal that with our method, a 175 billion parameter LM (text-davinci-003) can produce competitive or even better performance, compared to its orders-of-magnitude larger successor, GPT-4. Additionally, we show that DaSLaM is not limited by the solver's capabilities as a function of scale; e.g., solver LMs with diverse sizes give significant performance improvement with our solver-agnostic decomposition technique. Exhaustive ablation studies evince the superiority of our modular finetuning technique over exorbitantly large decomposer LLMs, based on prompting alone.
Large Language Models (LLMs) demonstrate impressive ability in handling reasoning tasks. However, unlike humans who can instinctively adapt their problem-solving strategies to the complexity of task, most LLM-based methods adopt a one-size-fits-all approach. These methods employ consistent models, sample sizes, prompting methods and levels of problem decomposition, regardless of the problem complexity. The inflexibility of these methods can bring unnecessary computational overhead or sub-optimal performance. To address this limitation, we introduce an Adaptive-Solver (AS) framework tha dynamically adapts solving strategies to suit various problems, enabling the flexible allocation of test-time computational resources. The framework functions with two primary modules. The initial evaluation module assesses the reliability of the current solution using answer consistency. If the solution is deemed unreliable, the subsequent adaptation module comes into play. Within this module, various types of adaptation strategies are employed collaboratively. Through such dynamic and multi-faceted adaptations, our framework can help reduce computational consumption and improve performance. Experimental results from complex reasoning benchmarks reveal that our method can significantly reduce API costs (up to 85%) while maintaining original performance. Alternatively, it achieves up to 4.5% higher accuracy compared to the baselines at the same cost. The code and dataset are available at https://github.com/john1226966735/Adaptive-Solver.
While large language models (LLMs) have demonstrated impressive performance in question-answering tasks, their performance is limited when the questions require knowledge that is not included in the model's training data and can only be acquired through direct observation or interaction with the real world. Existing methods decompose reasoning tasks through the use of modules invoked sequentially, limiting their ability to answer deep reasoning tasks. We introduce a method, Recursion based extensible LLM (REBEL), which handles open-world, deep reasoning tasks by employing automated reasoning techniques like dynamic planning and forward-chaining strategies. REBEL allows LLMs to reason via recursive problem decomposition and utilization of external tools. The tools that REBEL uses are specified only by natural language description. We further demonstrate REBEL capabilities on a set of problems that require a deeply nested use of external tools in a compositional and conversational setting.
This manuscript presents novel techniques for identifying the switch states, phase identification, and estimation of equipment parameters in multi-phase low voltage electrical grids, which is a major challenge in long-standing German low voltage grids that lack observability and are heavily impacted by modelling errors. The proposed methods are tailored for systems with a limited number of spatially distributed measuring devices, which measure voltage magnitudes at specific nodes and some line current magnitudes. The overall approach employs a problem decomposition strategy to divide the problem into smaller subproblems, which are addressed independently. The techniques for identifying switch states and system phases are based on heuristics and a binary optimization problem using correlation analysis of the measured time series. The estimation of equipment parameters is achieved through a data-driven regression approach and by an optimization problem, and the identification of cable types is solved using a Mixed-Integer Quadratic Programming solver. To validate the presented methods, a realistic grid is used and the presented techniques are evaluated for their resilience to data quality and time resolution, discussing the limitations of the proposed methods.
It is known that the multiplication of an $N \times M$ matrix with an $M \times P$ matrix can be performed using fewer multiplications than what the naive $NMP$ approach suggests. The most famous instance of this is Strassen's algorithm for multiplying two $2\times 2$ matrices in 7 instead of 8 multiplications. This gives rise to the constraint satisfaction problem of fast matrix multiplication, where a set of $R < NMP$ multiplication terms must be chosen and combined such that they satisfy correctness constraints on the output matrix. Despite its highly combinatorial nature, this problem has not been exhaustively examined from that perspective, as evidenced for example by the recent deep reinforcement learning approach of AlphaTensor. In this work, we propose a simple yet novel Constraint Programming approach to find non-commutative algorithms for fast matrix multiplication or provide proof of infeasibility otherwise. We propose a set of symmetry-breaking constraints and valid inequalities that are particularly helpful in proving infeasibility. On the feasible side, we find that exploiting solver performance variability in conjunction with a sparsity-based problem decomposition enables finding solutions for larger (feasible) instances of fast matrix multiplication. Our experimental results using CP Optimizer demonstrate that we can find fast matrix multiplication algorithms for matrices up to $3\times 3$ in a short amount of time.
The offline pickup and delivery problem with time windows (PDPTW) is a classical combinatorial optimization problem in the transportation community, which has proven to be very challenging computationally. Due to the complexity of the problem, practical problem instances can be solved only via heuristics, which trade-off solution quality for computational tractability. Among the various heuristics, a common strategy is problem decomposition, that is, the reduction of a large-scale problem into a collection of smaller sub-problems, with spatial and temporal decompositions being two natural approaches. While spatial decomposition has been successful in certain settings, effective temporal decomposition has been challenging due to the difficulty of stitching together the sub-problem solutions across the decomposition boundaries. In this work, we introduce a novel temporal decomposition scheme for solving a class of PDPTWs that have narrow time windows, for which it is able to provide both fast and high-quality solutions. We utilize techniques that have been popularized recently in the context of online dial-a-ride problems along with the general idea of rolling horizon optimization. To the best of our knowledge, this is the first attempt to solve offline PDPTWs using such an approach. To show the performance and scalability of our framework, we use the optimization of paratransit services as a motivating example. We compare our results with an offline heuristic algorithm using Google OR-Tools. In smaller problem instances, the baseline approach is as competitive as our framework. However, in larger problem instances, our framework is more scalable and can provide good solutions to problem instances of varying degrees of difficulty, while the baseline algorithm often fails to find a feasible solution within comparable compute times.
In recent years several learning approaches to point goal navigation in previously unseen environments have been proposed. They vary in the representations of the environments, problem decomposition, and experimental evaluation. In this work, we compare the state-of-the-art Deep Reinforcement Learning based approaches with Partially Observable Markov Decision Process (POMDP) formulation of the point goal navigation problem. We adapt the (POMDP) sub-goal framework proposed by [1] and modify the component that estimates frontier properties by using partial semantic maps of indoor scenes built from images' semantic segmentation. In addition to the well-known completeness of the model-based approach, we demonstrate that it is robust and efficient in that it leverages informative, learned properties of the frontiers compared to an optimistic frontier-based planner. We also demonstrate its data efficiency compared to the end-to-end deep reinforcement learning approaches. We compare our results against an optimistic planner, ANS and DD-PPO on Matterport3D dataset using the Habitat Simulator. We show comparable, though slightly worse performance than the SOTA DD-PPO approach, yet with far fewer data.
Step-by-step reasoning approaches like chain of thought (CoT) have proved to be very effective in inducing reasoning capabilities in large language models. However, the success of the CoT approach is fundamentally tied to the model size, and billion parameter-scale models are often needed to get CoT to work. In this paper, we propose a knowledge distillation approach that leverages the step-by-step CoT reasoning capabilities of larger models and distills these abilities into smaller models. In this work, we propose an alternative reasoning scheme, Socratic CoT, that learns a decomposition of the original problem into a sequence of subproblems and uses it to guide the intermediate reasoning steps. We use Socratic CoT to train a combination of two small distilled models: a problem decomposer and a subproblem solver. In practice, given a new problem, the two distilled models work in sync to decompose and solve complex problems. On multiple reasoning datasets (GSM8K, StrategyQA, and SVAMP), our proposed distillation strategies boosts the performance of smaller models over 70% compared to the baselines. Finally, we investigate when Socratic CoT is an effective alternative to CoT, demonstrating cases where a much smaller model (GPT-2 large) can outperform a 10X larger model (GPT-3 6B). Our code is available here: https://github.com/kumar-shridhar/Distiiling-LM
Multi-objective orienteering problems (MO-OPs) are classical multi-objective routing problems and have received a lot of attention in the past decades. This study seeks to solve MO-OPs through a problem-decomposition framework, that is, a MO-OP is decomposed into a multi-objective knapsack problem (MOKP) and a travelling salesman problem (TSP). The MOKP and TSP are then solved by a multi-objective evolutionary algorithm (MOEA) and a deep reinforcement learning (DRL) method, respectively. While the MOEA module is for selecting cities, the DRL module is for planning a Hamiltonian path for these cities. An iterative use of these two modules drives the population towards the Pareto front of MO-OPs. The effectiveness of the proposed method is compared against NSGA-II and NSGA-III on various types of MO-OP instances. Experimental results show that our method exhibits the best performance on almost all the test instances, and has shown strong generalization ability.
Real-world optimization problems may have a different underlying structure. In black-box optimization, the dependencies between decision variables remain unknown. However, some techniques can discover such interactions accurately. In Large Scale Global Optimization (LSGO), problems are high-dimensional. It was shown effective to decompose LSGO problems into subproblems and optimize them separately. The effectiveness of such approaches may be highly dependent on the accuracy of problem decomposition. Many state-of-the-art decomposition strategies are derived from Differential Grouping (DG). However, if a given problem consists of non-additively separable subproblems, DG-based strategies may discover many non-existing interactions. On the other hand, monotonicity checking strategies proposed so far do not report non-existing interactions for any separable subproblems but may miss discovering many of the existing ones. Therefore, we propose Incremental Recursive Ranking Grouping (IRRG) that suffers from none of these flaws. IRRG consumes more fitness function evaluations than the recent DG-based propositions, e.g., Recursive DG 3 (RDG3). Nevertheless, the effectiveness of the considered Cooperative Co-evolution frameworks after embedding IRRG or RDG3 was similar for problems with additively separable subproblems that are suitable for RDG3. After replacing the additive separability with non-additive, embedding IRRG leads to results of significantly higher quality.
With massive penetrations of active grid-edge technologies, distributed computing and optimization paradigm has gained significant attention to solve distribution-level optimal power flow (OPF) problems. However, the application of generic distributed optimization techniques to OPF problems leads to a very large number of macro-iterations or communication rounds among the distributed computing agents delaying the decision-making process or resulting in suboptimal solutions. Moreover, the existing distribution-level OPF problems typically model inverter-interfaced distributed energy resources (DERs) as grid-following inverters; grid-supporting and grid-forming functionalities have not been explicitly considered. The added complexities introduced by different inverter models require further attention to developing an appropriate model for new types of inverter-based DERs and computationally-tractable OPF algorithms. In this paper, we expand the distribution-level OPF model to include a combination of the grid-forming, grid-supporting, grid-following inverter-based DERs and also present the application of a domain-specific problem decomposition and distributed algorithm for the topologically radial power distribution systems to efficiently solve distribution-level OPF problem.
The Job-shop Scheduling Problem (JSP) is a well-known and challenging combinatorial optimization problem in which tasks sharing a machine are to be arranged in a sequence such that encompassing jobs can be completed as early as possible. In this paper, we investigate problem decomposition into time windows whose operations can be successively scheduled and optimized by means of multi-shot Answer Set Programming (ASP) solving. From a computational perspective, decomposition aims to split highly complex scheduling tasks into better manageable subproblems with a balanced number of operations such that good-quality or even optimal partial solutions can be reliably found in a small fraction of runtime. We devise and investigate a variety of decomposition strategies in terms of the number and size of time windows as well as heuristics for choosing their operations. Moreover, we incorporate time window overlapping and compression techniques into the iterative scheduling process to counteract optimization limitations due to the restriction to window-wise partial schedules. Our experiments on different JSP benchmark sets show that successive optimization by multi-shot ASP solving leads to substantially better schedules within tight runtime limits than single-shot optimization on the full problem. In particular, we find that decomposing initial solutions obtained with proficient heuristic methods into time windows leads to improved solution quality.
Student diversity, like academic background, learning styles, career and life goals, ethnicity, age, social and emotional characteristics, course load and work schedule, offers unique opportunities in education, like learning new skills, peer mentoring and example setting. But student diversity can be challenging too as it adds variability in the way in which students learn and progress over time. A single teaching approach is likely to be ineffective and result in students not meeting their potential. Automated support could address limitations of traditional teaching by continuously assessing student learning and implementing needed interventions. This paper discusses a novel methodology based on data analytics and Machine Learning to measure and causally diagnose the progress and shortcomings of student learning, and then utilizes the insight gained on individuals to optimize learning. Diagnosis pertains to dynamic diagnostic formative assessment, which aims to uncover the causes of learning shortcomings. The methodology groups learning difficulties into four categories: recall from memory, concept adjustment, concept modification, and problem decomposition into sub-goals (sub-problems) and concept combination. Data models are predicting the occurrence of each of the four challenge types, as well as a student's learning trajectory. The models can be used to automatically create real-time, student-specific interventions (e.g., learning cues) to address less understood concepts. We envision that the system will enable new adaptive pedagogical approaches to unleash student learning potential through customization of the course material to the background, abilities, situation, and progress of each student; and leveraging diversity-related learning experiences.
Given their ability to effectively learn non-linear mappings and perform fast inference, deep neural networks (NNs) have been proposed as a viable alternative to traditional simulation-driven approaches for solving high-dimensional eigenvalue equations (HDEs), which are the foundation for many scientific applications. Unfortunately, for the learned models in these scientific applications to achieve generalization, a large, diverse, and preferably annotated dataset is typically needed and is computationally expensive to obtain. Furthermore, the learned models tend to be memory- and compute-intensive primarily due to the size of the output layer. While generalization, especially extrapolation, with scarce data has been attempted by imposing physical constraints in the form of physics loss, the problem of model scalability has remained. In this paper, we alleviate the compute bottleneck in the output layer by using physics knowledge to decompose the complex regression task of predicting the high-dimensional eigenvectors into multiple simpler sub-tasks, each of which are learned by a simple "expert" network. We call the resulting architecture of specialized experts Physics-Guided Mixture-of-Experts (PG-MoE). We demonstrate the efficacy of such physics-guided problem decomposition for the case of the Schr\"{o}dinger's Equation in Quantum Mechanics. Our proposed PG-MoE model predicts the ground-state solution, i.e., the eigenvector that corresponds to the smallest possible eigenvalue. The model is 150x smaller than the network trained to learn the complex task while being competitive in generalization. To improve the generalization of the PG-MoE, we also employ a physics-guided loss function based on variational energy, which by quantum mechanics principles is minimized iff the output is the ground-state solution.
We are entering our tenth year of the current Artificial Intelligence (AI) spring, and, as with previous AI hype cycles, the threat of an AI winter looms. AI winters occurred because of ineffective approaches towards navigating the technology valley of death. The 6-D framework provides an end-to-end framework to successfully navigate this challenge. The 6-D framework starts with problem decomposition to identify potential AI solutions, and ends with considerations for deployment of AI-enabled systems. Each component of the 6-D framework and a precision medicine use case is described in this paper.
Reinforcement learning (RL) is a central problem in artificial intelligence. This problem consists of defining artificial agents that can learn optimal behaviour by interacting with an environment -- where the optimal behaviour is defined with respect to a reward signal that the agent seeks to maximize. Reward machines (RMs) provide a structured, automata-based representation of a reward function that enables an RL agent to decompose an RL problem into structured subproblems that can be efficiently learned via off-policy learning. Here we show that RMs can be learned from experience, instead of being specified by the user, and that the resulting problem decomposition can be used to effectively solve partially observable RL problems. We pose the task of learning RMs as a discrete optimization problem where the objective is to find an RM that decomposes the problem into a set of subproblems such that the combination of their optimal memoryless policies is an optimal policy for the original problem. We show the effectiveness of this approach on three partially observable domains, where it significantly outperforms A3C, PPO, and ACER, and discuss its advantages, limitations, and broader potential.
Problem decomposition plays a vital role when applying cooperative coevolution (CC) to large scale global optimization problems. However, most learning-based decomposition algorithms either only apply to additively separable problems or face the issue of false separability detections. Directing against these limitations, this study proposes a novel decomposition algorithm called surrogate-assisted variable grouping (SVG). SVG first designs a general-separability-oriented detection criterion according to whether the optimum of a variable changes with other variables. This criterion is consistent with the separability definition and thus endows SVG with broad applicability and high accuracy. To reduce the fitness evaluation requirement, SVG seeks the optimum of a variable with the help of a surrogate model rather than the original expensive high-dimensional model. Moreover, it converts the variable grouping process into a dynamic-binary-tree search one, which facilitates reutilizing historical separability detection information and thus reducing detection times. To evaluate the performance of SVG, a suite of benchmark functions with up to 2000 dimensions, including additively and non-additively separable ones, were designed. Experimental results on these functions indicate that, compared with six state-of-the-art decomposition algorithms, SVG possesses broader applicability and competitive efficiency. Furthermore, it can significantly enhance the optimization performance of CC.
We consider a task of scheduling a crawler to retrieve content from several sites with ephemeral content. A user typically loses interest in ephemeral content, like news or posts at social network groups, after several days or hours. Thus, development of timely crawling policy for such ephemeral information sources is very important. We first formulate this problem as an optimal control problem with average reward. The reward can be measured in the number of clicks or relevant search requests. The problem in its initial formulation suffers from the curse of dimensionality and quickly becomes intractable even with moderate number of information sources. Fortunately, this problem admits a Whittle index, which leads to problem decomposition and to a very simple and efficient crawling policy. We derive the Whittle index and provide its theoretical justification.