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Title: Lookahead: A Far-sighted Alternative of Magnitude-based Pruning

Paper content:
Magnitude-based pruning is one of the simplest methods for pruning neural networks.
Despite its simplicity, magnitude-based pruning and its variants demonstrated remarkable performances for pruning modern architectures.
Based on the observation that the magnitude-based pruning indeed minimizes the Frobenius distortion of a linear operator corresponding to a single layer, we develop a simple pruning method, coined lookahead pruning, by extending the single layer optimization to a multi-layer optimization.
Our experimental results demonstrate that the proposed method consistently outperforms the magnitude pruning on various networks including VGG and ResNet, particularly in the high-sparsity regime.
The "magnitude-equals-saliency" approach has been long underlooked as an overly simplistic baseline among all imaginable techniques to eliminate unnecessary weights from over-parametrized neural networks.
Since the early works of LeCun et al. (1989) ; Hassibi & Stork (1993) which provided more theoretically grounded alternative of magnitude-based pruning (MP) based on second derivatives of loss function, a wide range of methods including Bayesian / information-theoretic approaches (Neal, 1996; Louizos et al., 2017; Molchanov et al., 2017; Dai et al., 2018) , pregularization (Wen et al., 2016; Liu et al., 2017; Louizos et al., 2018) , sharing redundant channels (Zhang et al., 2018; Ding et al., 2019) , and reinforcement learning approaches (Lin et al., 2017; Bellec et al., 2018; He et al., 2018) has been proposed as more sophisticated alternatives.
On the other hand, the capabilities of MP heuristics are gaining attention once more.
Combined with minimalistic techniques including iterative pruning (Han et al., 2015) and dynamic reestablishment of connections (Zhu & Gupta, 2017) , a recent large-scale study by Gale et al. (2019) claims that MP can achieve a state-of-the-art trade-off of sparsity and accuracy on ResNet-50.
The unreasonable effectiveness of magnitude scores often extends beyond the strict domain of network pruning; a recent experiment by Frankle & Carbin (2019) suggests an existence of an automatic subnetwork discovery mechanism underlying the standard gradient-based optimization procedures of deep, overparametrized neural networks by showing that the MP algorithm finds an efficient trainable subnetwork.
These observations constitute a call to revisit the "magnitude-equals-saliency" approach for a better understanding of deep neural network itself.
As an attempt to better understand the nature of MP methods, we study a generalization of magnitude scores under a functional approximation framework; by viewing MP as a relaxed minimization of distortion in layerwise operators introduced by zeroing out parameters, we consider a multi-layer extension of the distortion minimization problem.
Minimization of the newly suggested distortion measure which 'looks ahead' the impact of pruning on neighboring layers gives birth to a novel pruning strategy, coined lookahead pruning (LAP).
In this paper, we focus on comparison of the proposed LAP scheme to its MP counterpart.
We empirically demonstrate that LAP consistently outperforms the MP under various setups including linear networks, fully-connected networks, and deep convolutional and residual networks.
In particular, the LAP consistently enables more than ×2 gain in the compression rate of the considered models, with increasing benefits under the high-sparsity regime.
Apart from its performance, the lookahead pruning method enjoys additional attractive properties: • Easy-to-use: Like magnitude-based pruning, the proposed LAP is a simple score-based approach agnostic to model and data, which can be implemented by computationally light elementary tensor operations.
Unlike most Hessian-based methods, LAP does not rely on an availability of training data except for the retraining phase.
It also has no hyper-parameter to tune, in contrast to other sophisticated training-based and optimization-based schemes.
• Versatility: As our method simply replaces the "magnitude-as-saliency" criterion with a lookahead alternative, it can be deployed jointly with algorithmic tweaks developed for magnitudebased pruning, such as iterative pruning and retraining (Han et al., 2015) or joint pruning and training with dynamic reconnections (Zhu & Gupta, 2017; Gale et al., 2019) .
The remainder of this manuscript is structured as follows: In Section 2, we introduce a functional approximation perspective toward MP and motivate LAP and its variants as a generalization of MP for multiple layer setups; in Section 3 we explore the capabilities of LAP and its variants with simple models, then move on to apply LAP to larger-scale models.
In this work, we interpret magnitude-based pruning as a solution to the minimization of the Frobenius distortion of a single layer operation incurred by pruning.
Based on this framework, we consider the minimization of the Frobenius distortion of multi-layer operation, and propose a novel lookahead pruning (LAP) scheme as a computationally efficient algorithm to solve the optimization.
Although LAP was motivated from linear networks, it extends to nonlinear networks which indeed minimizes the root mean square lookahead distortion assuming i.
τ fraction in all fully-connected layers, except for the last layer where we use (1 + q)/2 instead.
For FCN, we use (p, q) = (0, 0.5).
For Conv-6, VGGs ResNets, and WRN, we use (0.85, 0.8).
For ResNet-{18, 50}, we do not prune the first convolutional layer.
The range of sparsity for reported figures in all tables is decided as follows: we start from τ where test error rate starts falling below that of an unpruned model and report the results at τ, τ + 1, τ + 2, . . . for FCN and Conv-6, τ, τ + 2, τ + 4, . . . for VGGs, ResNet-50, and WRN, and τ, τ + 3, τ + 6, . . . for ResNet-18.
In this section, we show that the optimization in Eq. (3) is NP-hard by showing the reduction from the following binary quadratic programming which is NP-hard (Murty & Kabadi, 1987) :

assistant: text: We study a multi-layer generalization of the magnitude-based pruning.