Learning Multiple Initial Solutions to Optimization Problems

Thank you for taking the time to review our paper and for your feedback. We appreciate your recognition of our clear writing, motivation, and coherent algorithm.

In the following, we clarify misconceptions regarding our claims, address concerns about benchmark tasks, solver comparisons, and implementation details, and underscore the rigorous empirical validation of our approach, highlighting its suitability for publication within the broader ML community.

Performance Guarantee Clarification. You correctly point out that our claim in Sect. 4.2 focuses on the initial solution quality according to the selection function Λ. Indeed, we emphasize this explicitly in lines 210-215: The guarantee pertains to the initial solutions themselves and does not directly guarantee improved, optimized outcomes. In contrast, MISO is guaranteed to lead to an equal or better final solution in the multiple optimizers setting, as stated in lines 216-219 and empirically supported by results in lines 1071-1099.

Misinterpretation of Main Claims. We would like to clarify that we have not claimed that the primary issue is local optima. Instead, we explicitly state that our setting is optimization under tight runtime constraints (lines 11, 47, 128, 255), for which local optimization solvers' performance is sensitive to initial solutions (line 229). We clearly distinguish using a proxy-oracle, allowing an extended runtime or multiple initializations to obtain (near-)optimal solutions. Notably, our empirical results (Table 1) show that MISO frequently outperforms the proxy-oracle baseline, validating this claim.

Missing Details. We acknowledge that some details were concise in the main text but stress that comprehensive implementation descriptions, hyperparameters, and computational cost analyses are thoroughly provided in the appendix (lines 607-680), directly addressing this concern.

Benchmark Tasks. Our selected tasks are widely recognized and commonly used benchmarks in optimization and learning-based research. Notably, the nuPlan autonomous driving task is an 80-dim problem featuring realistic and complex urban environments with nonlinear dynamics, making it significantly challenging and relevant. Although some optimizers, e.g., iLQR, internally employ linearization during optimization, the underlying tasks themselves are still inherently nonlinear.

Comparison to Other Solvers. Your suggestion regarding comparisons to methods like convex QP or multi-starting solvers is well taken. However, some selected optimizers (DDP and iLQR) inherently include internal QP iterations (Tassa et al., 2014; Amos et al., 2018). Thus, our chosen baselines already implicitly encompass aspects of these suggested comparisons.

Contribution. We respectfully disagree regarding the perceived limitation of our key contributions. While prior work primarily focused on predicting single initial solutions, MISO represents the first systematic exploration explicitly designed to predict multiple ones. This novel aspect significantly addresses limitations inherent in single-initial-solution approaches. Our contributions are rigorously supported by extensive empirical evaluations demonstrating substantial performance improvements across multiple optimizers and tasks (Tables 1, 2).

Suitability. We appreciate your perspective that the paper is suited for a leading venue in control applications. However, our contribution stands to benefit the broader ML community – especially under "optimization methods" and "application-driven ML" tracks, as it explores new ways to predict and use multiple initial solutions in general optimization problems. Furthermore, the relevance is reinforced by several recent works that similarly bridge ML with optimization tools, including those published at ICML 23–24 (e.g., Ferber et al. 2023, Li et al. 2024, and Huang et al. 2024).

Non-learning-based Approaches. We have indeed compared MISO to several non-learning-based methods, such as warm-start, perturbed warm-start, and strong oracle-proxy baselines (Tables 1, 2), demonstrating the significant improvements that MISO achieves over non-learning-based approaches.

QD. While QD solutions are generally generated iteratively through evolutionary processes, MISO directly predicts multiple high-quality, diverse initial guesses in one shot. Hence, MISO does not depend on iterative, population-based exploration that underlies QD approaches.

We have made every effort to address your concerns comprehensively and clearly in our revision. If you feel your concerns have been resolved, we would greatly appreciate it if you could reflect this positively in your evaluation. Should any remaining questions or points need further clarification, please let us know—we are more than happy to provide additional information.

Again, Thank you for your constructive feedback and valuable suggestions, which have significantly contributed to strengthening our manuscript.

We thank you for your constructive feedback and for recognizing the key contributions of our work. We are pleased that you found the idea of learning multiple diverse initial solutions both valuable and well-motivated and that you appreciated the design of our methodology and the clarity of our presentation.

Below, we respond to your specific concerns and clarify key points that we hope further strengthen our manuscript:

Detailed Analysis of the WTA Loss. We appreciate your suggestion. Winner-Takes-All training (Guzman et al., 2012; Lee et al., 2016) promotes the learning of multiple specialized hypotheses by updating only the best-performing prediction head per training instance, effectively partitioning the output space into local conditional means resembling a centroidal Voronoi tessellation (Rupprecht et al., 2017). This selective feedback encourages hypothesis diversity, mitigates mode averaging, and preserves multi-modal distributions, particularly when combined with auxiliary scoring functions that estimate regional probability masses (Letzelter et al., 2024; Rupprecht et al., 2017). We will include a dedicated section in the appendix in the revised manuscript.

Scope and Generalizability of Experiments. We agree that demonstrating generalization beyond optimal control problems is valuable. While our current work focuses specifically on optimal control to ensure thorough experimentation, we are actively exploring broader optimization scenarios as part of future work (lines 430-436). We emphasize that our proposed MISO framework is task-agnostic by design, relying only on the structure of the underlying optimization rather than problem-specific details. Consequently, we anticipate strong potential for generalization to diverse optimization domains.

Theoretical Guarantees. Without guarantees on the generator and selection function and considering the runtime constraints, it is difficult to make solid theoretical claims; conversely, with such guarantees, the problem would be significantly simplified and perhaps trivial. Most methods in this research field are heuristic-based, aiming to improve performance rather than provide formal guarantees. Nevertheless, our approach allows for the inclusion of traditional initialization methods, such as warm-start heuristics, as part of the set of initializations (lines 208-219, which we demonstrate empirically in 1071-1099). This means our method does not risk degrading performance compared to established practices, effectively providing a safety net (lines 377-361). Our contribution demonstrates that learning multiple diverse initializations can empirically enhance optimizer performance across various tasks and optimizers.

Analysis of Optimization Efficiency. You raised a valid point regarding the analysis of optimization efficiency in terms of inference time and the number of iterations to convergence. While examining optimizer convergence could offer insights, metrics such as average additional iterations or convergence time are not particularly meaningful in our context. For instance, an optimizer might converge more quickly within the allocated budget but reach a suboptimal local minimum. In contrast, another optimizer might use the entire budget to find a better solution. Faster convergence does not necessarily equate to better performance. Additionally, there are no practical advantages to converging before the budget limit, as the optimizer would wait the remaining time until that solution is required. Furthermore, sample-based optimizers like MPPI operate with a fixed sample budget (line 660), eliminating the concept of iterations. Our main emphasis is on the quality of solutions within the given computational constraints, which is directly reflected in the cost metrics we report. We demonstrate that our method scales efficiently and consistently with the number of predicted initial solutions, K (lines 967-971), and we find that all outputs remain effective even as K increases (lines 896-903).

We have made every effort to address your concerns comprehensively and clearly in our revision. If you feel your concerns have been resolved, we would greatly appreciate it if you could reflect this positively in your evaluation. Should any remaining questions or points need further clarification, please let us know—we are more than happy to provide additional information.

Again, Thank you for your constructive feedback and valuable suggestions, which have significantly contributed to strengthening our manuscript.

Thank you for your constructive feedback and insightful comments. We appreciate your recognition of our extensive experiments, clarity in presenting methods and results, comprehensive baseline comparisons, and our unique contributions relative to existing literature.

In the following, we clarify concerns regarding task selection and dimensionality, emphasize the relevance of strict runtime constraints and the use of an oracle-proxy, address ambiguities and supplementary material layout, justify our choice of Transformer architectures, and discuss theoretical guarantees.

Tasks, Dimensionality, and Generalizability. Our study specifically targets sequential optimization tasks under strict runtime constraints commonly encountered in optimal control domains such as robotics and autonomous driving. The selected tasks, with their respective dimensions—Cart-Pole: 10, Reacher: 20, and Driving: 80—are standard benchmarks widely used for evaluating optimization methods. Notably, the nuPlan autonomous driving task features realistic and complex urban environments with nonlinear dynamics. Nonetheless, we acknowledge the value of investigating even higher-dimensional tasks (lines 434-439) and intend to explore such scenarios in future work. Additionally, generalizability is explicitly highlighted in lines 238-243. As stated in lines 281–285, training and testing instances differ in parameters such as initial states, targets, and reference trajectories, ensuring that improvements represent generalization rather than memorization. Moreover, MISO naturally extends to variations like obstacle locations and friction coefficients.

Runtime Constraints and Oracle Data. Our manuscript strongly emphasizes tight runtime constraints as central to our problem setting (lines 11, 47, 128, 255). Given these, directly using the (slow) oracle-proxy at test time is infeasible, underscoring the importance of our fast, learning-based initialization. We recognize obtaining data as a common limitation in BC/IL methods. Our method reduces reliance on an actual oracle by using an oracle-proxy—not necessarily globally optimal, but sufficiently effective (lines 173–177). Due to space limitations, we summarize the key evidence supporting our runtime-aware design choices: in lines 806-839, we compare inference times of ensemble and multi-output architectures, demonstrating the scalability advantages of our approach. We evaluate sampling-based methods in lines 806-839, illustrating their unsuitability under strict runtime constraints.

Supplementary, Layout, and Clarifications. We appreciate your thorough reading of the supplementary materials and suggestions for balancing the content. Given the strict page constraints, we welcome recommendations on specific parts that would be more impactful in the main text. We remain willing to revise accordingly. Additionally, thank you for pointing out several ambiguities; we will revise the manuscript to clarify the following:

• The "500K instances" refer to time steps, not trajectories.
• Indeed, the cost, c, matches the objective J (Eq. 1) and is distinct from the constraint C (Eq. 6).
• The uncertainty values reported represent the Standard Error (mentioned in Fig. 4)
• The oracle-proxy baseline (lines 296, 343, 738) corresponds to our expert policy.

Transformer Architecture. The core novelty of our approach is independent of a specific architecture and centers around the training objectives and the selection function. Transformers were chosen primarily due to their capability to effectively model dependencies within sequences. We conducted experiments (not included in the paper) with MLPs, which performed less robustly than Transformers. Further discussion is provided in lines 1062-1069.

Theoretical Bounds. We agree that such theoretical guarantees can yield valuable insights (Sambharya & Stellato, 2024; Sambharya et al., 2024). These approaches leverage PAC-Bayes methods or carefully designed sample-complexity arguments to provide guarantees—but typically for more specialized problem classes (e.g., convex or contractive operators). However, MISO targets a broader, nonconvex setting with runtime constraints where deriving general bounds typically requires overly restrictive assumptions. We, therefore, believe that providing fully general regret or sample-complexity bounds for MISO without imposing strong structural constraints remains an open research direction.

We have made every effort to address your concerns comprehensively and clearly in our revision. If you feel your concerns have been resolved, we would greatly appreciate it if you could reflect this positively in your evaluation. Should any remaining questions or points need further clarification, please let us know—we are more than happy to provide additional information.

Again, Thank you for your constructive feedback and valuable suggestions, which have significantly contributed to strengthening our manuscript.

We thank you for your thoughtful and constructive feedback. We greatly appreciate your positive comments regarding the extensive experimentation, soundness of the method, and comprehensive evaluation across multiple optimizers and benchmark tasks. We are particularly pleased you found the ablation studies insightful and agree that the method addresses an important problem in the optimization community.

Below we provide responses to your specific concerns and clarify key points that we hope further strengthen our manuscript:

Sizes of Optimization Problems. Below, we outline the size and constraints of each optimization task in our experiments, as detailed in appendix lines 630-681:

• Constraints: Bounded control inputs (e.g., acceleration, steering angle/rate), Limited rail or joint ranges, Physical/kinematic constraints (forces, torques, vehicle dynamics).
• Dimensionality of the optimization variable: Cart-Pole: 10, Reacher: 20, Driving: 80.

Generalizability to Other Optimization Problems. We agree that demonstrating generalization beyond optimal control problems is valuable. While our current work focuses specifically on optimal control to ensure thorough experimentation, we are actively exploring broader optimization scenarios as part of future work (lines 430-436). We emphasize that our proposed MISO framework is task-agnostic by design, relying only on the structure of the underlying optimization rather than problem-specific details. Consequently, we anticipate strong potential for generalization to diverse optimization domains.

Runtime Performance. We appreciate this insightful comment and believe the paper could benefit from clearer explanations regarding runtime performance. To clarify:

• If your concern pertains to solving a single optimization instance, you are correct that most instances typically converge rapidly (e.g., within 5 ms for the driving task). However, convergence time depends significantly on several factors, notably abrupt environmental changes between optimization steps and the quality of initialization (lines 139-144). We explicitly illustrate this in Figure 5 (left), demonstrating scenarios where certain initializations significantly impact convergence time (lines 363-374).
• If your comment pertains to comparing the runtime of different architectures (i.e., MISO versus alternative approaches), we include two detailed runtime comparisons in the appendix:
  • Appendix A.6 (lines 806-839) contrasts the inference speed between our multi-output architecture and an ensemble-based approach, demonstrating the superior scalability of MISO.
  • Appendix A.14 (lines 1061-1069) compares our approach against sampling-based (diffusion) methods, highlighting why sampling methods are infeasible for these real-time applications.

Relevant Related Work. Thank you for suggesting this additional reference. Unlike MISO, GENCO assumes access to gradients with respect to design variables, which may not be available in settings involving black-box simulators or non-differentiable objectives. We will incorporate it in the related work section.

Thank you once again for taking the time to carefully review our paper and for your insightful feedback. We have made every effort to address your concerns and have revised the manuscript accordingly. Should you have any remaining questions or require further clarification, we would be more than happy to provide additional details.

If you feel your concerns have been resolved, we would greatly appreciate it if you could reflect this positively in your evaluation.