AN EXPONENTIAL LEARNING RATE SCHEDULE FOR BATCH NORMALIZED NETWORKS
Intriguing empirical evidence exists that deep learning can work well with exotic schedules for varying the learning rate. This paper suggests that the phenomenon may be due to Batch Normalization or BN(Ioffe & Szegedy, 2015), which is ubiq- uitous and provides benefits in optimization and generalization across all standard architectures. The following new results are shown about BN with weight decay and momentum (in other words, the typical use case which was not considered in earlier theoretical analyses of stand-alone BN (Ioffe & Szegedy, 2015; Santurkar et al., 2018; Arora et al., 2018) • Training can be done using SGD with momentum and an exponentially in- creasing learning rate schedule, i.e., learning rate increases by some (1 + ?) factor in every epoch for some ? > 0. (Precise statement in the paper.) To the best of our knowledge this is the first time such a rate schedule has been successfully used, let alone for highly successful architectures. As ex- pected, such training rapidly blows up network weights, but the net stays well-behaved due to normalization. • Mathematical explanation of the success of the above rate schedule: a rigor- ous proof that it is equivalent to the standard setting of BN + SGD + Standard Rate Tuning + Weight Decay + Momentum. This equivalence holds for other normalization layers as well, Group Normalization(Wu & He, 2018), Layer Normalization(Ba et al., 2016), Instance Norm(Ulyanov et al., 2016), etc. • A worked-out toy example illustrating the above linkage of hyper- parameters. Using either weight decay or BN alone reaches global minimum, but convergence fails when both are used.
%%% Update to the review %%% Thanks for your clarification and the revision - the paper looks good! With regards to your comment on accelerating hyper-parameter search, note that there are fairly subtle issues owing to the use of SGD - refer to a recent work of Ge et al *The Step Decay Schedule: A Near Optimal, Geometrically Decaying Learning Rate Procedure For Least Squares* (2019). %%% The paper makes an interesting observation connecting the use of weight decay + normalization to training the same network (without regularization) using an exponentially increasing learning rate schedule, under an assumption of scale invariance that is satisfied by normalization techniques including batch norm, layer norm and other variants. An example is also provided where the joint use of batch norm and weight decay can lead to non-convergence to a global minimum whereas the use of one (without the other) converges to a global minimum - serving to indicate various interdependencies between the hyper-parameters that one needs to be careful about. While the connection of scale invariant models to novel schemes of learning rates is interesting (and novel), the paper will benefit quite a bit in its contributions through attempting a convergence analysis towards a stationary point even for solving a routine smooth non-convex stochastic optimization problem. Owing to the equivalence described in the paper, this enables us to understand the behavior of the combination of batch norm (or some scale invariance property) + weight decay + momentum (+ step decay of the learning rate), which, to my knowledge isn’t present in the literature of non-convex optimization. The paper is reasonably written (the proof of the main claim is fairly easy to follow), but needs to be carefully read through because I see typos and ill-formed sentences that should be rectified - e.g. see point 3. in appendix A.1 - some facts about equation 4, missing citation in definition 1.2 amongst others. I went over the proof of the main result and this appears to be correct. Furthermore, I find the connections to other learning rates (such as the cosine learning rate) to be rather hard to understand/interpret, in the current shape of the paper.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
AN EXPONENTIAL LEARNING RATE SCHEDULE FOR BATCH NORMALIZED NETWORKS
Intriguing empirical evidence exists that deep learning can work well with exotic schedules for varying the learning rate. This paper suggests that the phenomenon may be due to Batch Normalization or BN(Ioffe & Szegedy, 2015), which is ubiq- uitous and provides benefits in optimization and generalization across all standard architectures. The following new results are shown about BN with weight decay and momentum (in other words, the typical use case which was not considered in earlier theoretical analyses of stand-alone BN (Ioffe & Szegedy, 2015; Santurkar et al., 2018; Arora et al., 2018) • Training can be done using SGD with momentum and an exponentially in- creasing learning rate schedule, i.e., learning rate increases by some (1 + ?) factor in every epoch for some ? > 0. (Precise statement in the paper.) To the best of our knowledge this is the first time such a rate schedule has been successfully used, let alone for highly successful architectures. As ex- pected, such training rapidly blows up network weights, but the net stays well-behaved due to normalization. • Mathematical explanation of the success of the above rate schedule: a rigor- ous proof that it is equivalent to the standard setting of BN + SGD + Standard Rate Tuning + Weight Decay + Momentum. This equivalence holds for other normalization layers as well, Group Normalization(Wu & He, 2018), Layer Normalization(Ba et al., 2016), Instance Norm(Ulyanov et al., 2016), etc. • A worked-out toy example illustrating the above linkage of hyper- parameters. Using either weight decay or BN alone reaches global minimum, but convergence fails when both are used.
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>



%%% Update to the review %%% Thanks for your clarification and the revision - the paper looks good! With regards to your comment on accelerating hyper-parameter search, note that there are fairly subtle issues owing to the use of SGD - refer to a recent work of Ge et al *The Step Decay Schedule: A Near Optimal, Geometrically Decaying Learning Rate Procedure For Least Squares* (2019). %%% The paper makes an interesting observation connecting the use of weight decay + normalization to training the same network (without regularization) using an exponentially increasing learning rate schedule, under an assumption of scale invariance that is satisfied by normalization techniques including batch norm, layer norm and other variants. An example is also provided where the joint use of batch norm and weight decay can lead to non-convergence to a global minimum whereas the use of one (without the other) converges to a global minimum - serving to indicate various interdependencies between the hyper-parameters that one needs to be careful about. While the connection of scale invariant models to novel schemes of learning rates is interesting (and novel), the paper will benefit quite a bit in its contributions through attempting a convergence analysis towards a stationary point even for solving a routine smooth non-convex stochastic optimization problem. Owing to the equivalence described in the paper, this enables us to understand the behavior of the combination of batch norm (or some scale invariance property) + weight decay + momentum (+ step decay of the learning rate), which, to my knowledge isn’t present in the literature of non-convex optimization. The paper is reasonably written (the proof of the main claim is fairly easy to follow), but needs to be carefully read through because I see typos and ill-formed sentences that should be rectified - e.g. see point 3. in appendix A.1 - some facts about equation 4, missing citation in definition 1.2 amongst others. I went over the proof of the main result and this appears to be correct. Furthermore, I find the connections to other learning rates (such as the cosine learning rate) to be rather hard to understand/interpret, in the current shape of the paper.
assistant
%%% Update to the review %%% Thanks for your clarification and the revision - the paper looks good! With regards to your comment on accelerating hyper-parameter search, note that there are fairly subtle issues owing to the use of SGD - refer to a recent work of Ge et al *The Step Decay Schedule: A Near Optimal, Geometrically Decaying Learning Rate Procedure For Least Squares* (2019). %%% The paper makes an interesting observation connecting the use of weight decay + normalization to training the same network (without regularization) using an exponentially increasing learning rate schedule, under an assumption of scale invariance that is satisfied by normalization techniques including batch norm, layer norm and other variants. An example is also provided where the joint use of batch norm and weight decay can lead to non-convergence to a global minimum whereas the use of one (without the other) converges to a global minimum - serving to indicate various interdependencies between the hyper-parameters that one needs to be careful about. While the connection of scale invariant models to novel schemes of learning rates is interesting (and novel), the paper will benefit quite a bit in its contributions through attempting a convergence analysis towards a stationary point even for solving a routine smooth non-convex stochastic optimization problem. Owing to the equivalence described in the paper, this enables us to understand the behavior of the combination of batch norm (or some scale invariance property) + weight decay + momentum (+ step decay of the learning rate), which, to my knowledge isn’t present in the literature of non-convex optimization. The paper is reasonably written (the proof of the main claim is fairly easy to follow), but needs to be carefully read through because I see typos and ill-formed sentences that should be rectified - e.g. see point 3. in appendix A.1 - some facts about equation 4, missing citation in definition 1.2 amongst others. I went over the proof of the main result and this appears to be correct. Furthermore, I find the connections to other learning rates (such as the cosine learning rate) to be rather hard to understand/interpret, in the current shape of the paper.