Temporal-difference learning for nonlinear value function approximation in the lazy training regime
We discuss the approximation of the value function for infinite-horizon discounted Markov Reward Processes (MRP) with nonlinear functions trained with the Temporal-Difference (TD) learning algorithm. We consider this problem under a certain scaling of the approximating function, leading to a regime called lazy training. In this regime the parameters of the model vary only slightly during the learning process, a feature that has recently been observed in the training of neural networks, where the scaling we study arises naturally, implicit in the initialization of their parameters. Both in the under- and over-parametrized frameworks, we prove exponential convergence to local, respectively global minimizers of the above algorithm in the lazy training regime. We then give examples of such convergence results in the case of models that diverge if trained with non-lazy TD learning, and in the case of neural networks.
The paper considers TD learning with function approximation, and establishes convergence results for over- and under-parameterized models in the lazy regime, and illustrates the theory on simple numerical examples. Although the obtained results are interesting and the paper is well written, the contribution is quite incremental, in that it simply combines prior work on TD learning with linear function approximation with lazy training in order to show that models with a certain scaling can lead to convergence. This scaling makes the models essentially linear in the parameters (with a non-linear feature map given by initialization), so that it is not surprising that convergence can be reached, given the prior work on linear function approximation. I encourage the authors to further explain their motivation in studying such a setting. It is claimed that the over-parameterized regime is only useful for finite state spaces, which seems quite limiting, since one cannot obtain global convergence in the under-parameterized case. When considering neural networks at infinite width, would the results be applicable if one assumes that V* belongs to the RKHS of the corresponding neural tangent kernel?
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### Title + Abstract
Temporal-difference learning for nonlinear value function approximation in the lazy training regime
We discuss the approximation of the value function for infinite-horizon discounted Markov Reward Processes (MRP) with nonlinear functions trained with the Temporal-Difference (TD) learning algorithm. We consider this problem under a certain scaling of the approximating function, leading to a regime called lazy training. In this regime the parameters of the model vary only slightly during the learning process, a feature that has recently been observed in the training of neural networks, where the scaling we study arises naturally, implicit in the initialization of their parameters. Both in the under- and over-parametrized frameworks, we prove exponential convergence to local, respectively global minimizers of the above algorithm in the lazy training regime. We then give examples of such convergence results in the case of models that diverge if trained with non-lazy TD learning, and in the case of neural networks.
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 paper considers TD learning with function approximation, and establishes convergence results for over- and under-parameterized models in the lazy regime, and illustrates the theory on simple numerical examples. Although the obtained results are interesting and the paper is well written, the contribution is quite incremental, in that it simply combines prior work on TD learning with linear function approximation with lazy training in order to show that models with a certain scaling can lead to convergence. This scaling makes the models essentially linear in the parameters (with a non-linear feature map given by initialization), so that it is not surprising that convergence can be reached, given the prior work on linear function approximation. I encourage the authors to further explain their motivation in studying such a setting. It is claimed that the over-parameterized regime is only useful for finite state spaces, which seems quite limiting, since one cannot obtain global convergence in the under-parameterized case. When considering neural networks at infinite width, would the results be applicable if one assumes that V* belongs to the RKHS of the corresponding neural tangent kernel?
assistant
The paper considers TD learning with function approximation, and establishes convergence results for over- and under-parameterized models in the lazy regime, and illustrates the theory on simple numerical examples. Although the obtained results are interesting and the paper is well written, the contribution is quite incremental, in that it simply combines prior work on TD learning with linear function approximation with lazy training in order to show that models with a certain scaling can lead to convergence. This scaling makes the models essentially linear in the parameters (with a non-linear feature map given by initialization), so that it is not surprising that convergence can be reached, given the prior work on linear function approximation. I encourage the authors to further explain their motivation in studying such a setting. It is claimed that the over-parameterized regime is only useful for finite state spaces, which seems quite limiting, since one cannot obtain global convergence in the under-parameterized case. When considering neural networks at infinite width, would the results be applicable if one assumes that V* belongs to the RKHS of the corresponding neural tangent kernel?