Transferring Optimality Across Data Distributions via Homotopy Methods
Homotopy methods, also known as continuation methods, are a powerful mathematical tool to efficiently solve various problems in numerical analysis, including complex non-convex optimization problems where no or only little prior knowledge regarding the localization of the solutions is available. In this work, we propose a novel homotopy-based numerical method that can be used to transfer knowledge regarding the localization of an optimum across different task distributions in deep learning applications. We validate the proposed methodology with some empirical evaluations in the regression and classification scenarios, where it shows that superior numerical performance can be achieved in popular deep learning benchmarks, i.e. FashionMNIST, CIFAR-10, and draw connections with the widely used fine-tuning heuristic. In addition, we give more insights on the properties of a general homotopy method when used in combination with Stochastic Gradient Descent by conducting a theoretical analysis in a simplified setting.
Contribution This paper proposes algorithm for transferring knowledge from easy -to-solved to complex tasks or from already solve to new tasks. It relies on homotopy functions and sequentially solves a sequence of optimization problems where the task distribution is gradually deformed from a source task to the target task. Theoretical guarantees are provided and proven in a strongly convex setting. The main results from the theory show that the distance between the final solution and its optimal are less or equal to relative to the distance of the initial source solution to its optimum. So a near optimal solution for the source task will lead to near optimal solution for the target task. Regression and Classification experimentations show competitive results compared to random and warm-start initialization schemes. Clarity Overall, the paper is well written, well motivated and well structured. The technical content is also very clear and excellent. Minor point: Seems that there is a notation error in proposition G.1 and its proof (i instead of i+1). Novelty The novelty in this work seems to be the application of homotopy methods to the transfer learning settings. The mathematical guarantees are also new and may even offer new ways to interpret fine tuning methods that have been so successful in recent literature. However, given the non-convexity of DNNs, it seems like the analysis in the non-convex settings and its implications should be part of the main text. Experiments: Overall, the experiments are very insightful but limited since you only show the training loss and the validation performance is not evaluated at all. Other things that would could be beneficial in better assessing the quality of your method are: comparison to Curriculum learning methods, more in depth analysis of the impact of k, and gamma in both regression and classification settings, and solving toy convex optimization problems to bridge the gap between theory and application. Preliminary rating: * Accept *
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Transferring Optimality Across Data Distributions via Homotopy Methods
Homotopy methods, also known as continuation methods, are a powerful mathematical tool to efficiently solve various problems in numerical analysis, including complex non-convex optimization problems where no or only little prior knowledge regarding the localization of the solutions is available. In this work, we propose a novel homotopy-based numerical method that can be used to transfer knowledge regarding the localization of an optimum across different task distributions in deep learning applications. We validate the proposed methodology with some empirical evaluations in the regression and classification scenarios, where it shows that superior numerical performance can be achieved in popular deep learning benchmarks, i.e. FashionMNIST, CIFAR-10, and draw connections with the widely used fine-tuning heuristic. In addition, we give more insights on the properties of a general homotopy method when used in combination with Stochastic Gradient Descent by conducting a theoretical analysis in a simplified setting.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



Contribution This paper proposes algorithm for transferring knowledge from easy -to-solved to complex tasks or from already solve to new tasks. It relies on homotopy functions and sequentially solves a sequence of optimization problems where the task distribution is gradually deformed from a source task to the target task. Theoretical guarantees are provided and proven in a strongly convex setting. The main results from the theory show that the distance between the final solution and its optimal are less or equal to relative to the distance of the initial source solution to its optimum. So a near optimal solution for the source task will lead to near optimal solution for the target task. Regression and Classification experimentations show competitive results compared to random and warm-start initialization schemes. Clarity Overall, the paper is well written, well motivated and well structured. The technical content is also very clear and excellent. Minor point: Seems that there is a notation error in proposition G.1 and its proof (i instead of i+1). Novelty The novelty in this work seems to be the application of homotopy methods to the transfer learning settings. The mathematical guarantees are also new and may even offer new ways to interpret fine tuning methods that have been so successful in recent literature. However, given the non-convexity of DNNs, it seems like the analysis in the non-convex settings and its implications should be part of the main text. Experiments: Overall, the experiments are very insightful but limited since you only show the training loss and the validation performance is not evaluated at all. Other things that would could be beneficial in better assessing the quality of your method are: comparison to Curriculum learning methods, more in depth analysis of the impact of k, and gamma in both regression and classification settings, and solving toy convex optimization problems to bridge the gap between theory and application. Preliminary rating: * Accept *
assistant
Contribution This paper proposes algorithm for transferring knowledge from easy -to-solved to complex tasks or from already solve to new tasks. It relies on homotopy functions and sequentially solves a sequence of optimization problems where the task distribution is gradually deformed from a source task to the target task. Theoretical guarantees are provided and proven in a strongly convex setting. The main results from the theory show that the distance between the final solution and its optimal are less or equal to relative to the distance of the initial source solution to its optimum. So a near optimal solution for the source task will lead to near optimal solution for the target task. Regression and Classification experimentations show competitive results compared to random and warm-start initialization schemes. Clarity Overall, the paper is well written, well motivated and well structured. The technical content is also very clear and excellent. Minor point: Seems that there is a notation error in proposition G.1 and its proof (i instead of i+1). Novelty The novelty in this work seems to be the application of homotopy methods to the transfer learning settings. The mathematical guarantees are also new and may even offer new ways to interpret fine tuning methods that have been so successful in recent literature. However, given the non-convexity of DNNs, it seems like the analysis in the non-convex settings and its implications should be part of the main text. Experiments: Overall, the experiments are very insightful but limited since you only show the training loss and the validation performance is not evaluated at all. Other things that would could be beneficial in better assessing the quality of your method are: comparison to Curriculum learning methods, more in depth analysis of the impact of k, and gamma in both regression and classification settings, and solving toy convex optimization problems to bridge the gap between theory and application. Preliminary rating: * Accept *