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Title: Flipout: Efficient Pseudo-Independent Weight Perturbations on Mini-Batches

Paper content:
Stochastic neural net weights are used in a variety of contexts, including regularization, Bayesian neural nets, exploration in reinforcement learning, and evolution strategies.
Unfortunately, due to the large number of weights, all the examples in a mini-batch typically share the same weight perturbation, thereby limiting the variance reduction effect of large mini-batches.
We introduce flipout, an efficient method for decorrelating the gradients within a mini-batch by implicitly sampling pseudo-independent weight perturbations for each example.
Empirically, flipout achieves the ideal linear variance reduction for fully connected networks, convolutional networks, and RNNs.
We find significant speedups in training neural networks with multiplicative Gaussian perturbations.
We show that flipout is effective at regularizing LSTMs, and outperforms previous methods.
Flipout also enables us to vectorize evolution strategies: in our experiments, a single GPU with flipout can handle the same throughput as at least 40 CPU cores using existing methods, equivalent to a factor-of-4 cost reduction on Amazon Web Services.
Stochasticity is a key component of many modern neural net architectures and training algorithms.
The most widely used regularization methods are based on randomly perturbing a network's computations BID29 BID7 .
Bayesian neural nets can be trained with variational inference by perturbing the weights BID4 BID0 .
Weight noise was found to aid exploration in reinforcement learning BID20 BID2 .
Evolution strategies (ES) minimizes a black-box objective by evaluating many weight perturbations in parallel, with impressive performance on robotic control tasks BID25 .
Some methods perturb a network's activations BID29 BID7 , while others perturb its weights BID4 BID0 BID20 BID2 BID25 .
Stochastic weights are appealing in the context of regularization or exploration because they can be viewed as a form of posterior uncertainty about the parameters.
However, compared with stochastic activations, they have a serious drawback: because a network typically has many more weights than units, it is very expensive to compute and store separate weight perturbations for every example in a mini-batch.
Therefore, stochastic weight methods are typically done with a single sample per mini-batch.
In contrast, activations are easy to sample independently for different training examples within a mini-batch.
This allows the training algorithm to see orders of magnitude more perturbations in a given amount of time, and the variance of the stochastic gradients decays as 1/N , where N is the mini-batch size.
We believe this is the main reason stochastic activations are far more prevalent than stochastic weights for neural net regularization.
In other settings such as Bayesian neural nets and evolution strategies, one is forced to use weight perturbations and live with the resulting inefficiency.In order to achieve the ideal 1/N variance reduction, the gradients within a mini-batch need not be independent, but merely uncorrelated.
In this paper, we present flipout, an efficient method for decorrelating the gradients between different examples without biasing the gradient estimates.
Flipout applies to any perturbation distribution that factorizes by weight and is symmetric around 0-including DropConnect, multiplicative Gaussian perturbations, evolution strategies, and variational Bayesian neural nets-and to many architectures, including fully connected nets, convolutional nets, and RNNs.In Section 3, we show that flipout gives unbiased stochastic gradients, and discuss its efficient vectorized implementation which incurs only a factor-of-2 computational overhead compared with shared perturbations.
We then analyze the asymptotics of gradient variance with and without flipout, demonstrating strictly reduced variance.
In Section 4, we measure the variance reduction effects on a variety of architectures.
Empirically, flipout gives the ideal 1/N variance reduction in all architectures we have investigated, just as if the perturbations were done fully independently for each training example.
We demonstrate speedups in training time in a large batch regime.
We also use flipout to regularize the recurrent connections in LSTMs, and show that it outperforms methods based on dropout.
Finally, we use flipout to vectorize evolution strategies BID25 , allowing a single GPU to handle the same throughput as 40 CPU cores using existing approaches; this corresponds to a factor-of-4 cost reduction on Amazon Web Services.
We have introduced flipout, an efficient method for decorrelating the weight gradients between different examples in a mini-batch.
We showed that flipout is guaranteed to reduce the variance compared with shared perturbations.
Empirically, we demonstrated significant variance reduction in the large batch setting for a variety of network architectures, as well as significant speedups in training time.
We showed that flipout outperforms dropout-based methods for regularizing LSTMs.
Flipout also makes it practical to apply GPUs to evolution strategies, resulting in substantially increased throughput for a given computational cost.
We believe flipout will make weight perturbations practical in the large batch setting favored by modern accelerators such as Tensor Processing Units (Jouppi et al., 2017) .
DISPLAYFORM0 In this section, we provide the proof of Theorem 2 (Variance Decomposition Theorem).Proof
. We use
the notations from Section 3.2. Let x,
x denote two training examples from the mini-batch B, and ∆W, ∆W denote the weight perturbations they received. We begin
with the decomposition into data and estimation terms (Eqn. 6), which we repeat here for convenience: DISPLAYFORM1 The data term from Eqn. 13 can be
simplified: DISPLAYFORM2 We break the estimation term from Eqn. 13 into variance
and covariance terms: DISPLAYFORM3 We now separately analyze the cases of fully independent perturbations, shared perturbations, and flipout.Fully independent perturbations. If the perturbations
are fully independent, the second term in Eqn. 15 disappears. Hence
, combining Eqns
. 13, 14, and 15, we are
left with DISPLAYFORM4 which is just α/N .

assistant: text: We introduce flipout, an efficient method for decorrelating the gradients computed by stochastic neural net weights within a mini-batch by implicitly sampling pseudo-independent weight perturbations for each example.