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Title: Scheduling the Learning Rate Via Hypergradients: New Insights and a New Algorithm

Paper content:
We study the problem of fitting task-specific learning rate schedules from the perspective of hyperparameter optimization.
This allows us to explicitly search for schedules that achieve good generalization.
We describe the structure of the gradient of a validation error w.r.t.
the learning rates, the hypergradient, and based on this we introduce a novel online algorithm.
Our method adaptively interpolates between two recently proposed techniques (Franceschi et al., 2017; Baydin et al.,2018), featuring increased stability and faster convergence.
We show empirically that the proposed technique compares favorably with baselines and related methodsin terms of final test accuracy.
Learning rate (LR) adaptation for first-order optimization methods is one of the most widely studied aspects in optimization for learning methods -in particular neural networks -with early work dating back to the origins of connectionism (Jacobs, 1988; Vogl et al., 1988) .
More recent work focused on developing complex schedules that depend on a small number of hyperparameters (Loshchilov & Hutter, 2017; Orabona & Pál, 2016) .
Other papers in this area have focused on the optimization of the (regularized) training loss (Schaul et al., 2013; Baydin et al., 2018; Wu et al., 2018) .
While quick optimization is desirable, the true goal of supervised learning is to minimize the generalization error, which is commonly estimated by holding out part of the available data for validation.
Hyperparameter optimization (HPO), a related but distinct branch of the literature, specifically focuses on this aspect, with less emphasis on the goal of rapid convergence on a single task.
Research in this direction is vast (see Hutter et al. (2019) for an overview) and includes model-based (Snoek et al., 2012; Hutter et al., 2015) , model-free (Bergstra & Bengio, 2012; Hansen, 2016) , and gradientbased (Domke, 2012; Maclaurin et al., 2015) approaches.
Additionally, works in the area of learning to optimize (Andrychowicz et al., 2016; Wichrowska et al., 2017) have focused on the problem of tuning parameterized optimizers on whole classes of learning problems but require prior expensive optimization and are not designed to speed up training on a single specific task.
The goal of this paper is to automatically compute in an online fashion a learning rate schedule for stochastic optimization methods (such as SGD) only on the basis of the given learning task, aiming at producing models with associated small validation error.
We study the problem of finding a LR schedule under the framework of gradient-based hyperparameter optimization (Franceschi et al., 2017) : we consider as an optimal schedule η * = (η * 0 , . . . , η * T −1 ) ∈ R T + a solution to the following constrained optimization problem min{f T (η) = E(w T (η)) : η ∈ R T + } s.t. w 0 =w, w t+1 (η) = Φ t (w t (η), η t )
for t = {0, . . . , T − 1} = [T ] , where E : R d → R + is an objective function, Φ t :
is a (possibly stochastic) weight update dynamics,w ∈ R d represents the initial model weights (parameters) and finally w t are the weights after t iterations.
We can think of E as either the training or the validation loss of the model, while the dynamics Φ describe the update rule (such as SGD, SGD-Momentum, Adam etc.).
For example in the case of SGD, Φ t (w t , η t ) = w t − η t ∇L t (w t ), with L t (w t ) the (possibly regularized) training loss on the t-th minibatch.
The horizon T should be large enough so that the training error can be effectively minimized, in order to avoid underfitting.
Note that a too large value of T does not necessarily harm since η k = 0 for k >T is still a feasible solution, implementing early stopping in this setting.
Finding a good learning rate schedule is an old but crucially important issue in machine learning.
This paper makes a step forward, proposing an automatic method to obtain performing LR schedules that uses an adaptive moving average over increasingly long hypergradient approximations.
MARTHE interpolates between HD and RTHO taking the best of the two worlds.
The implementation of our algorithm is fairly simple within modern automatic differentiation and deep learning environments, adding only a moderate computational overhead over the underlying optimizer complexity.
In this work, we studied the case of optimizing the learning rate schedules for image classification tasks; we note, however, that MARTHE is a general technique for finding online hyperparameter schedules (albeit it scales linearly with the number of hyperparameters), possibly implementing a competitive alternative in other application scenarios, such as tuning regularization parameters (Luketina et al., 2016) .
We plan to further validate the method both in other learning domains for adapting the LR and also to automatically tune other crucial hyperparameters.
We believe that another interesting future research direction could be to learn the adaptive rules for µ and β in a meta learning fashion.

assistant: text: MARTHE: a new method to fit task-specific learning rate schedules from the perspective of hyperparameter optimization