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Title: Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions

Paper content:
Hyperparameter optimization can be formulated as a bilevel optimization problem, where the optimal parameters on the training set depend on the hyperparameters.
We aim to adapt regularization hyperparameters for neural networks by fitting compact approximations to the best-response function, which maps hyperparameters to optimal weights and biases.
We show how to construct scalable best-response approximations for neural networks by modeling the best-response as a single network whose hidden units are gated conditionally on the regularizer.
We justify this approximation by showing the exact best-response for a shallow linear network with L2-regularized Jacobian can be represented by a similar gating mechanism.
We fit this model using a gradient-based hyperparameter optimization algorithm which alternates between approximating the best-response around the current hyperparameters and optimizing the hyperparameters using the approximate best-response function.
Unlike other gradient-based approaches, we do not require differentiating the training loss with respect to the hyperparameters, allowing us to tune discrete hyperparameters, data augmentation hyperparameters, and dropout probabilities.
Because the hyperparameters are adapted online, our approach discovers hyperparameter schedules that can outperform fixed hyperparameter values.
Empirically, our approach outperforms competing hyperparameter optimization methods on large-scale deep learning problems.
We call our networks, which update their own hyperparameters online during training, Self-Tuning Networks (STNs).
Regularization hyperparameters such as weight decay, data augmentation, and dropout (Srivastava et al., 2014) are crucial to the generalization of neural networks, but are difficult to tune.
Popular approaches to hyperparameter optimization include grid search, random search BID3 , and Bayesian optimization (Snoek et al., 2012) .
These approaches work well with low-dimensional hyperparameter spaces and ample computational resources; however, they pose hyperparameter optimization as a black-box optimization problem, ignoring structure which can be exploited for faster convergence, and require many training runs.We can formulate hyperparameter optimization as a bilevel optimization problem.
Let w denote parameters (e.g. weights and biases) and λ denote hyperparameters (e.g. dropout probability).
Let L T and L V be functions mapping parameters and hyperparameters to training and validation losses, respectively.
We aim to solve 1 : DISPLAYFORM0 Substituting the best-response function w * (λ) = arg min w L T (λ, w) gives a single-level problem: DISPLAYFORM1 If the best-response w * is known, the validation loss can be minimized directly by gradient descent using Equation 2, offering dramatic speed-ups over black-box methods.
However, as the solution to a high-dimensional optimization problem, it is difficult to compute w * even approximately.Following Lorraine & Duvenaud (2018) , we propose to approximate the best-response w * directly with a parametric functionŵ φ .
We jointly optimize φ and λ, first updating φ so thatŵ φ ≈ w * in a neighborhood around the current hyperparameters, then updating λ by usingŵ φ as a proxy for w * in Eq. 2: DISPLAYFORM2 Finding a scalable approximationŵ φ when w represents the weights of a neural network is a significant challenge, as even simple implementations entail significant memory overhead.
We show how to construct a compact approximation by modelling the best-response of each row in a layer's weight matrix/bias as a rank-one affine transformation of the hyperparameters.
We show that this can be interpreted as computing the activations of a base network in the usual fashion, plus a correction term dependent on the hyperparameters.
We justify this approximation by showing the exact best-response for a shallow linear network with L 2 -regularized Jacobian follows a similar structure.
We call our proposed networks Self-Tuning Networks (STNs) since they update their own hyperparameters online during training.STNs enjoy many advantages over other hyperparameter optimization methods.
First, they are easy to implement by replacing existing modules in deep learning libraries with "hyper" counterparts which accept an additional vector of hyperparameters as input 2 .
Second, because the hyperparameters are adapted online, we ensure that computational effort expended to fit φ around previous hyperparameters is not wasted.
In addition, this online adaption yields hyperparameter schedules which we find empirically to outperform fixed hyperparameter settings.
Finally, the STN training algorithm does not require differentiating the training loss with respect to the hyperparameters, unlike other gradient-based approaches (Maclaurin et al., 2015; Larsen et al., 1996) , allowing us to tune discrete hyperparameters, such as the number of holes to cut out of an image BID12 , data-augmentation hyperparameters, and discrete-noise dropout parameters.
Empirically, we evaluate the performance of STNs on large-scale deep-learning problems with the Penn Treebank (Marcus et al., 1993) and CIFAR-10 datasets (Krizhevsky & Hinton, 2009) , and find that they substantially outperform baseline methods.
We introduced Self-Tuning Networks (STNs), which efficiently approximate the best-response of parameters to hyperparameters by scaling and shifting their hidden units.
This allowed us to use gradient-based optimization to tune various regularization hyperparameters, including discrete hyperparameters.
We showed that STNs discover hyperparameter schedules that can outperform fixed hyperparameters.
We validated the approach on large-scale problems and showed that STNs achieve better generalization performance than competing approaches, in less time.
We believe STNs offer a compelling path towards large-scale, automated hyperparameter tuning for neural networks.

assistant: text: We use a hypernetwork to predict optimal weights given hyperparameters, and jointly train everything together.