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Title: Split LBI for Deep Learning: Structural Sparsity via Differential Inclusion Paths

Paper content:
Over-parameterization is ubiquitous nowadays in training neural networks to benefit both optimization in seeking global optima and generalization in reducing prediction error.
However, compressive networks are desired in many real world applications and direct training of small networks may be trapped in local optima.
In this paper, instead of pruning or distilling over-parameterized models to compressive ones, we propose a new approach based on \emph{differential inclusions of inverse scale spaces}, that generates a family of models from simple to complex ones by coupling gradient descent and mirror descent to explore model structural sparsity.
It has a simple discretization, called the Split Linearized Bregman Iteration (SplitLBI), whose global convergence analysis in deep learning is established that from any initializations, algorithmic iterations converge to a critical point of empirical risks.
Experimental evidence shows that\ SplitLBI may achieve state-of-the-art performance in large scale training on ImageNet-2012 dataset etc., while with \emph{early stopping} it unveils effective subnet architecture with comparable test accuracies to dense models after retraining instead of pruning well-trained ones.
The expressive power of deep neural networks comes from the millions of parameters, which are optimized by Stochastic Gradient Descent (SGD) (Bottou, 2010) and variants like Adam (Kingma & Ba, 2015) .
Remarkably, model over-parameterization helps both optimization and generalization.
For optimization, over-parameterization may simplify the landscape of empirical risks toward locating global optima efficiently by gradient descent method (Mei et al., 2018; Venturi et al., 2018; Allen-Zhu et al., 2018; Du et al., 2018) .
On the other hand, over-parameterization does not necessarily result in a bad generalization or overfitting (Zhang et al., 2017) , especially when some weight-size dependent complexities are controlled (Bartlett, 1997; Bartlett et al., 2017; Golowich et al., 2018; Neyshabur et al., 2019) .
However, compressive networks are desired in many real world applications, e.g. robotics, selfdriving cars, and augmented reality.
Despite that 1 regularization has been applied to deep learning to enforce the sparsity on weights toward compact, memory efficient networks, it sacrifices some prediction performance (Collins & Kohli, 2014) .
This is because that the weights learned in neural networks are highly correlated, and 1 regularization on such weights violates the incoherence or irrepresentable conditions needed for sparse model selection (Donoho & Huo, 2001; Tropp, 2004; Zhao & Yu, 2006) , leading to spurious selections with poor generalization.
On the other hand, 2 regularization is often utilized for correlated weights as some low-pass filtering, sometimes in the form of weight decay (Loshchilov & Hutter, 2019) or early stopping (Yao et al., 2007; Wei et al., 2017) .
Furthermore, group sparsity regularization (Yuan & Lin, 2006) has also been applied to neural networks, such as finding optimal number of neuron groups (Alvarez & Salzmann, 2016) and exerting good data locality with structured sparsity (Wen et al., 2016; Yoon & Hwang, 2017 ).
Yet, without the aid of over-parameterization, directly training a compressive model architecture may meet the obstacle of being trapped in local optima in contemporary experience.
Alternatively, researchers in practice typically start from training a big model using common task datasets like ImageNet, and then prune or distill such big models to small ones without sacrificing too much of the performance (Jaderberg et al., 2014; Han et al., 2015; Zhu et al., 2017; Zhou et al., 2017; Zhang et al., 2016; Li et al., 2017; Abbasi-Asl & Yu, 2017; Yang et al., 2018; Arora et al., 2018) .
In particular, a recent study (Frankle & Carbin, 2019) created the lottery ticket hypothesis based on empirical observations: "dense, randomly-initialized, feed-forward networks contain subnetworks (winning tickets) that -when trained in isolation -reach test accuracy comparable to the original network in a similar number of iterations".
How to effectively reduce an over-parameterized model thus becomes the key to compressive deep learning.
Yet, Liu et al. (2019) raised a question, is it necessary to fully train a dense, over-parameterized model before finding important structural sparsity?
In this paper, we provide a novel answer by exploiting a dynamic approach to deep learning with structural sparsity.
We are able to establish a family of neural networks, from simple to complex, by following regularization paths as solutions of differential inclusions of inverse scale spaces.
Our key idea is to design some dynamics that simultaneously exploit over-parameterized models and structural sparsity.
To achieve this goal, the original network parameters are lifted to a coupled pair, with one weight set W of parameters following the standard gradient descend to explore the over-parameterized model space, while the other set of parameters learning structure sparsity in an inverse scale space, i.e., structural sparsity set Γ.
The large-scale important parameters are learned at a fast speed while the small unimportant ones are learned at a slow speed.
The two sets of parameters are coupled in an 2 regularization.
The dynamics enjoys a simple discretization, i.e. the Split Linearized Bregman Iteration (SplitLBI), with provable global convergence guarantee shown in this paper.
Here, SplitLBI is a natural extension of SGD with structural sparsity exploration: SplitLBI reduces to the standard gradient descent method when the coupling regularization is weak, while it leads to a sparse mirror descent when the coupling is strong.
Critically, SplitLBI enjoys a nice property that important subnet architecture can be rapidly learned via the structural sparsity parameter Γ following the iterative regularization path, without fully training a dense network first.
Particularly, the support set of structural sparsity parameter Γ learned in the early stage of this inverse scale space discloses important sparse subnet architectures.
Such architectures can be fine-tuned or retrained to achieve comparable test accuracy as the dense, over-parameterized networks.
As a result, the structural sparsity parameter Γ may enable us to rapidly find "winning tickets" in early training epochs for the "lottery" of identifying successful subnetworks that bear comparable test accuracy to the dense ones.
This point is empirically validated in our experiments.
Historically, the Linearized Bregman Iteration (LBI) was firstly proposed in applied mathematics as iterative regularization paths for image reconstruction and compressed sensing (Osher et al., 2005; Yin et al., 2008) , later applied to logistic regression (Shi et al., 2013) .
The convergence analysis was given for convex problems (Yin et al., 2008; Cai et al., 2009) , yet remaining open for non-convex problems met in deep learning.
Osher et al. (2016) established statistical model selection consistency for high dimensional linear regression under the same irrepresentable condition as Lasso, later extended to generalized linear models (Huang & Yao, 2018) .
To relax such conditions, SplitLBI was proposed by Huang et al. (2016) to learn structural sparsity in linear models under weaker conditions than generalized Lasso, that was successfully applied in medical image analysis (Sun et al., 2017) and computer vision (Zhao et al., 2018) .
In this paper, it is the first time that SplitLBI is exploited to train highly non-convex neural networks with structural sparsity, together with a global convergence analysis based on the Kurdyka-Łojasiewicz framework Łojasiewicz (1963) .
In this paper, a parsimonious deep learning method is proposed based on differential inclusions of inverse scale spaces.
Implemented by a variable splitting scheme, such a dynamics system can exploit over-parameterized models and structural sparsity simultaneously.
Besides, its simple discretization, i.e., the SplitLBI, has a proven global convergence and hence can be employed to train deep networks.
We have experimentally shown that it can achieve the state-of-the-art performance on many datasets including ImageNet-2012, with better interpretability than SGD.
What's more, equipped with early stopping, such a structural sparsity can unveil the "winning tickets" -the architecture of sub-networks which after re-training can achieve comparable and even better accuracy than original dense networks.
First of all, we reformulate Eq. (6) into an equivalent form.
Without loss of generality, consider Ω = Ω 1 in the sequel.
Denote R(P ) := Ω(Γ), then Eq. (6) can be rewritten as,
where
Thus SplitLBI is equivalent to the following iterations,
Exploiting the equivalent reformulation (16a-16c), one can establish the global convergence of (W k , Γ k , g k ) based on the Kurdyka-Łojasiewicz framework.
In this section, the following extended version of Theorem 1 is actually proved.

assistant: text: SplitLBI is applied to deep learning to explore model structural sparsity, achieving state-of-the-art performance in ImageNet-2012 and unveiling effective subnet architecture.