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Title: Online Hyperparameter Adaptation via Amortized Proximal Optimization

Paper content:
Effective performance of neural networks depends critically on effective tuning of optimization hyperparameters, especially learning rates (and schedules thereof).
We present Amortized Proximal Optimization (APO), which takes the perspective that each optimization step should approximately minimize a proximal objective (similar to the ones used to motivate natural gradient and trust region policy optimization).
Optimization hyperparameters are adapted to best minimize the proximal objective after one weight update.
We show that an idealized version of APO (where an oracle minimizes the proximal objective exactly) achieves global convergence to stationary point and locally second-order convergence to global optimum for neural networks.
APO incurs minimal computational overhead.
We experiment with using APO to adapt a variety of optimization hyperparameters online during training, including (possibly layer-specific) learning rates, damping coefficients, and gradient variance exponents.
For a variety of network architectures and optimization algorithms (including SGD, RMSprop, and K-FAC), we show that with minimal tuning, APO performs competitively with carefully tuned optimizers.
Tuning optimization hyperparameters can be crucial for effective performance of a deep learning system.
Most famously, carefully selected learning rate schedules have been instrumental in achieving state-of-the-art performance on challenging datasets such as ImageNet BID6 and WMT BID36 .
Even algorithms such as RMSprop BID34 and Adam (Kingma & Ba, 2015) , which are often interpreted in terms of coordinatewise adaptive learning rates, still have a global learning rate parameter which is important to tune.
A wide variety of learning rate schedules have been proposed BID24 BID14 BID2 .
Seemingly unrelated phenomena have been explained in terms of effective learning rate schedules BID35 .
Besides learning rates, other hyperparameters have been identified as important, such as the momentum decay factor BID31 , the batch size BID28 , and the damping coefficient in second-order methods BID20 BID19 .There
have been many attempts to adapt optimization hyperparameters to minimize the training error after a small number of updates BID24 BID1 BID2 . This
approach faces two fundamental obstacles: first, learning rates and batch sizes have been shown to affect generalization performance because stochastic updates have a regularizing effect BID5 BID18 BID27 BID35 . Second
, minimizing the short-horizon expected loss encourages taking very small steps to reduce fluctuations at the expense of long-term progress BID37 . While
these effects are specific to learning rates, they present fundamental obstacles to tuning any optimization hyperparameter, since basically any optimization hyperparameter somehow influences the size of the updates.In this paper, we take the perspective that the optimizer's job in each iteration is to approximately minimize a proximal objective which trades off the loss on the current batch with the average change in the predictions. Specifically
, we consider proximal objectives of the form J(φ) = h(f (g(θ, φ))) + λD(f (θ), f (g(θ, φ))), where f is a model with parameters θ, h is an approximation to the objective function, g is the base optimizer update with hyperparameters φ, and D is a distance metric. Indeed, approximately
solving such a proximal objective motivated the natural gradient algorithm BID0 , as well as proximal reinforcement learning algorithms BID26 . We introduce Amortized
Proximal Optimization (APO), an approach which adapts optimization hyperparameters to minimize the proximal objective in each iteration. We use APO to tune hyperparameters
of SGD, RMSprop, and K-FAC; the hyperparameters we consider include (possibly layer-specific) learning rates, damping coefficients, and the power applied to the gradient covariances.Notice that APO has a hyperparameter λ which controls the aggressiveness of the updates. We believe such a hyperparameter is
necessary until the aforementioned issues surrounding stochastic regularization and short-horizon bias are better understood. However, in practice we find that by
performing a simple grid search over λ, we can obtain automatically-tuned learning rate schedules that are competitive with manual learning rate decay schedules. Furthermore, APO can automatically adapt
several optimization hyperparameters with only a single hand-tuned hyperparameter.We provide theoretical justification for APO by proving strong convergence results for an oracle which solves the proximal objective exactly in each iteration. In particular, we show global linear convergence
and locally quadratic convergence under mild assumptions. These results motivate the proximal objective as
a useful target for meta-optimization.We evaluate APO on real-world tasks including image classification on MNIST, CIFAR-10, CIFAR-100, and SVHN. We show that adapting learning rates online via
APO yields faster training convergence than the best fixed learning rates for each task, and is competitive with manual learning rate decay schedules. Although we focus on fast optimization of the training
objective, we also find that the solutions found by APO generalize at least as well as those found by fixed hyperparameters or fixed schedules.
We introduced amortized proximal optimization (APO), a method for online adaptation of optimization hyperparameters, including global and per-layer learning rates, and damping parameters for approximate second-order methods.
We evaluated our approach on real-world neural network optimization tasks-training MLP and CNN models-and showed that it converges faster and generalizes better than optimal fixed learning rates.
Empirically, we showed that our method overcomes short horizon bias and performs well with sensible default values for the meta-optimization parameters.Guodong Zhang, Chaoqi Wang, Bowen Xu, and Roger Grosse.
Three mechanisms of weight decay regularization.
arXiv preprint arXiv:1810.12281, 2018.A PROOF OF THEOREM 1We first introduce the following lemma:Lemma 1.
Assume the manifold is smooth with C-bounded curvature, the gradient norm of loss function L is upper bounded by G. If the effective gradient at point Z k ∈ M is g k , then for any DISPLAYFORM0 Proof.
We construct the Z satisfying the above inequality.
Consider the following point in R d : DISPLAYFORM1 We show that Z is a point satisfying the inequality in the lemma.
Firstly, we notice that DISPLAYFORM2 This is because when we introduce the extra curveṽ DISPLAYFORM3 Here we use the fact thatv = 0 and v ≤ C. Therefore we have DISPLAYFORM4 Here the first equality is by introducing the extra Y , the first inequality is by triangle inequality, the second equality is by the definition of g k being ∇ Z L(Z k ) projecting onto a plane, the second inequality is due to the above bound of Y − Z , the last inequality is due to DISPLAYFORM5 , there is therefore DISPLAYFORM6 which completes the proof.

assistant: text: We introduce amortized proximal optimization (APO), a method to adapt a variety of optimization hyperparameters online during training, including learning rates, damping coefficients, and gradient variance exponents.