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Title: Deep Frank-Wolfe For Neural Network Optimization

Paper content:
Learning a deep neural network requires solving a challenging optimization problem: it is a high-dimensional, non-convex and non-smooth minimization problem with a large number of terms.
The current practice in neural network optimization is to rely on the stochastic gradient descent (SGD) algorithm or its adaptive variants.
However, SGD requires a hand-designed schedule for the learning rate.
In addition, its adaptive variants tend to produce solutions that generalize less well on unseen data than SGD with a hand-designed schedule.
We present an optimization method that offers empirically the best of both worlds: our algorithm yields good generalization performance while requiring only one hyper-parameter.
Our approach is based on a composite proximal framework, which exploits the compositional nature of deep neural networks and can leverage powerful convex optimization algorithms by design.
Specifically, we employ the Frank-Wolfe (FW) algorithm for SVM, which computes an optimal step-size in closed-form at each time-step.
We further show that the descent direction is given by a simple backward pass in the network, yielding the same computational cost per iteration as SGD.
We present experiments on the CIFAR and SNLI data sets, where we demonstrate the significant superiority of our method over Adam, Adagrad, as well as the recently proposed BPGrad and AMSGrad.
Furthermore, we compare our algorithm to SGD with a hand-designed learning rate schedule, and show that it provides similar generalization while often converging faster.
The code is publicly available at https://github.com/oval-group/dfw.
Since the introduction of back-propagation BID23 , stochastic gradient descent (SGD) has been the most commonly used optimization algorithm for deep neural networks.
While yielding remarkable performance on a variety of learning tasks, a downside of the SGD algorithm is that it requires a schedule for the decay of its learning rate.
In the convex setting, curvature properties of the objective function can be used to design schedules that are hyper-parameter free and guaranteed to converge to the optimal solution (Bubeck, 2015) .
However, there is no analogous result of practical interest for the non-convex optimization problem of a deep neural network.
An illustration of this issue is the diversity of learning rate schedules used to train deep convolutional networks with SGD: BID25 and He et al. (2016) adapt the learning rate according to the validation performance, while BID27 , BID3 and BID8 use pre-determined schedules, which are respectively piecewise constant, geometrically decaying, and cyclic with a cosine annealing.
While these protocols result in competitive or state-of-the-art results on their learning task, there does not seem to be a consistent methodology.
As a result, finding such a schedule for a new setting is a time-consuming and computationally expensive effort.To alleviate this issue, adaptive gradient methods have been developed BID36 BID4 BID21 , and borrowed from online convex optimization (Duchi et al., 2011) .
Typically, these methods only require the tuning of the initial learning rate, the other hyper-parameters being considered robust across applications.
However, it has been shown that such adaptive gradient methods obtain worse generalization than SGD BID32 .
This observation is corroborated by our experimental results.In order to bridge this performance gap between existing adaptive methods and SGD, we introduce a new optimization algorithm, called Deep Frank-Wolfe (DFW).
The DFW algorithm exploits the composite structure of deep neural networks to design an optimization algorithm that leverages efficient convex solvers.
In more detail, we consider a composite (nested) optimization problem, with the loss as the outer function and the function encoded by the neural network as the inner one.
At each iteration, we define a proximal problem with a first-order approximation of the neural network (linearized inner function), while keeping the loss function in its exact form (exact outer function).
When the loss is the hinge loss, each proximal problem created by our formulation is exactly a linear SVM.
This allows us to employ the powerful Frank-Wolfe (FW) algorithm as the workhorse of our procedure.There are two by-design advantages to our method compared to the SGD algorithm.
First, each iteration exploits more information about the learning objective, while preserving the same computational cost as SGD.
Second, an optimal step-size is computed in closed-form by using the FW algorithm in the dual (Frank & Wolfe, 1956 BID5 .
Consequently, we do not need a hand-designed schedule for the learning rate.
As a result, our algorithm is the first to provide competitive generalization error compared to SGD, all the while requiring a single hyper-parameter and often converging significantly faster.We present two additional improvements to customize the use of the DFW algorithm to deep neural networks.
First, we show how to smooth the loss function to avoid optimization difficulties arising from learning deep models with SVMs (Berrada et al., 2018) .
Second, we incorporate Nesterov momentum (Nesterov, 1983) to accelerate our algorithm.We demonstrate the efficacy of our method on image classification with the CIFAR data sets (Krizhevsky, 2009) using two architectures: wide residual networks BID35 and densely connected convolutional neural networks BID3 ; we also provide experiments on natural language inference with a Bi-LSTM on the SNLI corpus (Bowman et al., 2015) .
We show that the DFW algorithm often strongly outperforms previous methods based on adaptive learning rates.
Furthermore, it provides comparable or better accuracy to SGD with hand-designed learning rate schedules.In conclusion, our contributions can be summed up as follows:• We propose a proximal framework which preserves information from the loss function.•
For the first time for deep neural networks, we demonstrate how our formulation gives at each iteration (
i) an optimal step-size in closed form and
(ii) an update at the same computational cost as SGD.•
We design a novel smoothing scheme for the dual optimization of SVMs.•
To the best of our knowledge, the resulting DFW algorithm is the first to offer comparable or better generalization to SGD with a hand-designed schedule on the CIFAR data sets, all the while converging several times faster and requiring only a single hyperparameter.
Our empirical evidence indicates that the initial learning rate can be a crucial hyper-parameter for good generalization.
We have observed in our experiments that such a choice of high learning rate provides a consistent improvement for convolutional neural networks: accurate minimization of the training objective with large initial steps usually leads to good generalization.
Furthermore, as mentioned in the previous section, it is sometimes beneficial to even increase the batch-size in order to be able to train the model using large initial steps.In the case of recurrent neural networks, however, this effect is not as distinct.
Additional experiments on different recurrent architectures have showed variations in the impact of the learning rate and in the best-performing optimizer.
Further analysis would be required to understand the effects at play.
We have introduced DFW, an efficient algorithm to train deep neural networks.
DFW predominantly outperforms adaptive gradient methods, and obtains similar performance to SGD without requiring a hand-designed learning rate schedule.We emphasize the generality of our framework in Section 3, which enables the training of deep neural networks to benefit from any advance on optimization algorithms for linear SVMs.
This framework could also be applied to other loss functions that yield efficiently solvable proximal problems.
In particular, our algorithm already supports the use of structured prediction loss functions BID28 BID30 , which can be used, for instance, for image segmentation.We have mentioned the intricate relationship between optimization and generalization in deep learning.
This illustrates a major difficulty in the design of effective optimization algorithms for deep neural networks: the learning objective does not include all the regularization needed for good generalization.
We believe that in order to further advance optimization for deep neural networks, it is essential to alleviate this problem and expose a clear objective function to optimize.

assistant: text: We train neural networks by locally linearizing them and using a linear SVM solver (Frank-Wolfe) at each iteration.