A Stochastic Trust Region Method for Non-convex Minimization
We target the problem of finding a local minimum in non-convex finite-sum minimization. Towards this goal, we first prove that the trust region method with inexact gradient and Hessian estimation can achieve a convergence rate of order as long as those differential estimations are sufficiently accurate. Combining such result with a novel Hessian estimator, we propose a sample-efficient stochastic trust region (STR) algorithm which finds an -approximate local minimum within stochastic Hessian oracle queries. This improves the state-of-the-art result by a factor of . Finally, we also develop Hessian-free STR algorithms which achieve the lowest runtime complexity. Experiments verify theoretical conclusions and the efficiency of the proposed algorithms.
The authors propose a new analysis for trust region methods with approximate models. Using this result, they propose a number of methods to create stochastic trust region methods by constructing approximate quadratic models (based on a stochastic first and second order estimate) which satisfy the requirements for convergence. The effectiveness of the derived methods is evaluated empirically on two standard non-convex regression problems. This paper is overall an interesting contribution which proposes a number of competitive methods for achieving approximate local minima in a stochastic regime, with both hessian based and “hessian-free” methods. A couple of minor points: - In the experiments, it would be helpful to also include some measure of uncertainty (such as standard error bars) in the plots given the stochastic nature of the problem (although I do not expect high variance given the construction of the algorithm). - It would be helpful to indicate which results still hold in the online setting (not finite sum). Indeed, from the proof of theorem 3.1, MetaAlgorithm 1 does not seem to rely on the finite sum setting, which is mostly used for analyzing the variance-reduced estimators. This would be helpful as it would enable MetaAlgorithm 1 to be used with appropriate variance-reduced estimators in settings beyond the finite-sum problems.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
A Stochastic Trust Region Method for Non-convex Minimization
We target the problem of finding a local minimum in non-convex finite-sum minimization. Towards this goal, we first prove that the trust region method with inexact gradient and Hessian estimation can achieve a convergence rate of order as long as those differential estimations are sufficiently accurate. Combining such result with a novel Hessian estimator, we propose a sample-efficient stochastic trust region (STR) algorithm which finds an -approximate local minimum within stochastic Hessian oracle queries. This improves the state-of-the-art result by a factor of . Finally, we also develop Hessian-free STR algorithms which achieve the lowest runtime complexity. Experiments verify theoretical conclusions and the efficiency of the proposed algorithms.
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>



The authors propose a new analysis for trust region methods with approximate models. Using this result, they propose a number of methods to create stochastic trust region methods by constructing approximate quadratic models (based on a stochastic first and second order estimate) which satisfy the requirements for convergence. The effectiveness of the derived methods is evaluated empirically on two standard non-convex regression problems. This paper is overall an interesting contribution which proposes a number of competitive methods for achieving approximate local minima in a stochastic regime, with both hessian based and “hessian-free” methods. A couple of minor points: - In the experiments, it would be helpful to also include some measure of uncertainty (such as standard error bars) in the plots given the stochastic nature of the problem (although I do not expect high variance given the construction of the algorithm). - It would be helpful to indicate which results still hold in the online setting (not finite sum). Indeed, from the proof of theorem 3.1, MetaAlgorithm 1 does not seem to rely on the finite sum setting, which is mostly used for analyzing the variance-reduced estimators. This would be helpful as it would enable MetaAlgorithm 1 to be used with appropriate variance-reduced estimators in settings beyond the finite-sum problems.
assistant
The authors propose a new analysis for trust region methods with approximate models. Using this result, they propose a number of methods to create stochastic trust region methods by constructing approximate quadratic models (based on a stochastic first and second order estimate) which satisfy the requirements for convergence. The effectiveness of the derived methods is evaluated empirically on two standard non-convex regression problems. This paper is overall an interesting contribution which proposes a number of competitive methods for achieving approximate local minima in a stochastic regime, with both hessian based and “hessian-free” methods. A couple of minor points: - In the experiments, it would be helpful to also include some measure of uncertainty (such as standard error bars) in the plots given the stochastic nature of the problem (although I do not expect high variance given the construction of the algorithm). - It would be helpful to indicate which results still hold in the online setting (not finite sum). Indeed, from the proof of theorem 3.1, MetaAlgorithm 1 does not seem to rely on the finite sum setting, which is mostly used for analyzing the variance-reduced estimators. This would be helpful as it would enable MetaAlgorithm 1 to be used with appropriate variance-reduced estimators in settings beyond the finite-sum problems.