The Implicit Bias of Gradient Descent on Separable Data
We show that gradient descent on an unregularized logistic regression problem, for almost all separable datasets, converges to the same direction as the max-margin solution. The result generalizes also to other monotone decreasing loss functions with an infimum at infinity, and we also discuss a multi-class generalizations to the cross entropy loss. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization in more complex models and with other optimization methods.
 Paper focuses on characterising behaviour of the log loss minimisation on the linearly separable data. As we know, optimisation like this does not converge in a strict mathematical sense, as the norm of the model will grow to infinity. However, one can still hope for a convergence of normalised solution (or equivalently - convergence in term of separator angle, rather than parametrisation). This paper shows that indeed, log-loss (and some other similar losses), minimised with gradient descent, leads to convergence (in the above sense) to the max-margin solution. On one hand it is an interesting property of model we train in practice, and on the other - provides nice link between two separate learning theories. Pros: - easy to follow line of argument - very interesting result of mapping "solution" of unregularised logistic regression (under gradient descent optimisation) onto hard max margin one Cons: - it is not clear in the abstract, and beginning of the paper what "convergence" means, as in the strict sense logistic regression optimisation never converges on separable data. It would be beneficial for the clarity if authors define what they mean by convergence (normalised weight vector, angle, whichever path seems most natural) as early in the paper as possible.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
The Implicit Bias of Gradient Descent on Separable Data
We show that gradient descent on an unregularized logistic regression problem, for almost all separable datasets, converges to the same direction as the max-margin solution. The result generalizes also to other monotone decreasing loss functions with an infimum at infinity, and we also discuss a multi-class generalizations to the cross entropy loss. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization in more complex models and with other optimization methods.
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>



 Paper focuses on characterising behaviour of the log loss minimisation on the linearly separable data. As we know, optimisation like this does not converge in a strict mathematical sense, as the norm of the model will grow to infinity. However, one can still hope for a convergence of normalised solution (or equivalently - convergence in term of separator angle, rather than parametrisation). This paper shows that indeed, log-loss (and some other similar losses), minimised with gradient descent, leads to convergence (in the above sense) to the max-margin solution. On one hand it is an interesting property of model we train in practice, and on the other - provides nice link between two separate learning theories. Pros: - easy to follow line of argument - very interesting result of mapping "solution" of unregularised logistic regression (under gradient descent optimisation) onto hard max margin one Cons: - it is not clear in the abstract, and beginning of the paper what "convergence" means, as in the strict sense logistic regression optimisation never converges on separable data. It would be beneficial for the clarity if authors define what they mean by convergence (normalised weight vector, angle, whichever path seems most natural) as early in the paper as possible.
assistant
 Paper focuses on characterising behaviour of the log loss minimisation on the linearly separable data. As we know, optimisation like this does not converge in a strict mathematical sense, as the norm of the model will grow to infinity. However, one can still hope for a convergence of normalised solution (or equivalently - convergence in term of separator angle, rather than parametrisation). This paper shows that indeed, log-loss (and some other similar losses), minimised with gradient descent, leads to convergence (in the above sense) to the max-margin solution. On one hand it is an interesting property of model we train in practice, and on the other - provides nice link between two separate learning theories. Pros: - easy to follow line of argument - very interesting result of mapping "solution" of unregularised logistic regression (under gradient descent optimisation) onto hard max margin one Cons: - it is not clear in the abstract, and beginning of the paper what "convergence" means, as in the strict sense logistic regression optimisation never converges on separable data. It would be beneficial for the clarity if authors define what they mean by convergence (normalised weight vector, angle, whichever path seems most natural) as early in the paper as possible.