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+    {
+        "type": "text",
+        "text": "SWITCHING LINEAR DYNAMICS FOR VARIATIONAL BAYES FILTERING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "System identification of complex and nonlinear systems is a central problem for model predictive control and model-based reinforcement learning. Despite their complexity, such systems can often be approximated well by a set of linear dynamical systems if broken into appropriate subsequences. This mechanism not only helps us find good approximations of dynamics, but also gives us deeper insight into the underlying system. Leveraging Bayesian inference and Variational Autoencoders, we show how to learn a richer and more meaningful state space, e.g. encoding joint constraints and collisions with walls in a maze, from partial and high-dimensional observations. This representation translates into a gain of accuracy of the learned dynamics which we showcase on various simulated tasks. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Learning dynamics from raw data (also known as system identification) is a key component of model predictive control and model-based reinforcement learning. Problematically, environments of interest often give rise to very complex and highly nonlinear dynamics which are seemingly difficult to approximate. However, switching linear dynamical systems (SLDS) approaches claim that those environments can often be broken down into simpler units made up of areas of equal and linear dynamics (Ackerson & Fu, 1970; Chang & Athans, 1978). Not only are those approaches capable of good predictive performance, which often is the sole goal of learning a system’s dynamics, they also encode valuable information into so called switching variables which determine the dynamics of the next transition. For example, when looking at the movement of an arm, one is intuitively aware of certain restrictions of possible movements, e.g. constraints to the movement due to joint constraints or obstacles. The knowledge is present without the need to simulate; it’s explicit. Exactly this kind of information will be encoded when successfully learning switching dynamics. Our goal in this work will therefore entail the search for richer representations in the form of latent state space models which encode knowledge about the underlying system dynamics. In turn, we expect this to improve the accuracy of our simulation as well. Such a representation alone could then be used in a reinforcement learning approach that possibly only takes advantage of the learned latent features but not necessarily its learned dynamics. ",
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+    {
+        "type": "text",
+        "text": "To learn richer representations, we identify one common problem with prevalent recurrent Variational Autoencoder models (Karl et al., 2017a; Krishnan et al., 2015; Chung et al., 2015; Fraccaro et al., 2016): the non-probabilistic treatment of the transition dynamics often modeled by a powerful nonlinear function approximator. From the history of the Autoencoder to the Variational Autoencoder, we know that in order to detect features in an unsupervised manner, probabilistic treatment of the latent space is paramount. As our starting point, we will build on previously proposed approaches by Krishnan et al. (2017) and Karl et al. (2017a). The latter already made use of locally linear dynamics, but only in a deterministic fashion. We extend their approaches by a stochastic switching LDS model and show that such treatment is vital for learning richer representations and simulation accuracy. ",
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We consider discretized time-series data consisting of continuous observations $x _ { t } \\in \\mathcal { X } \\subset \\mathbb { R } ^ { n _ { x } }$ and control inputs $u _ { t } \\in \\mathcal { U } \\subset \\mathbb { R } ^ { n _ { u } }$ that we would like to model by corresponding latent states $z _ { t } \\in \\mathcal { Z } \\subset \\mathbb { R } ^ { n _ { z } }$ . We’ll denote sequences of variables by $x _ { 1 : T } = ( x _ { 1 } , x _ { 2 } , . . . , x _ { T } )$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/da18197c9717341f6caf03dbb955c01e82140562065970885fef03304c9f77aa.jpg",
+        "image_caption": [
+            "Figure 1: (a) $s _ { t }$ denote discrete switch variables, $z _ { t }$ are continuous latent variables, $x _ { t }$ continuous observed variables, $u _ { t }$ are (optional) continuous control inputs. (b) By introducing a special latent variable $w$ used for initial state inference, we want to make explicit that the first step is treated differently from the rest of the sequence. "
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+    {
+        "type": "text",
+        "text": "2.1 SWITCHING LINEAR DYNAMICAL SYSTEMS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Switching Linear Dynamical System models (SLDS) enable us to model nonlinear time series data by splitting it into sequences of linear dynamical models. At each time $t = 1 , 2 , . . . , T$ , a discrete switch variable $s _ { t } \\in { 1 , . . . , M }$ chooses of a set LDSs a system which is to be used to transform our continuous latent state $z _ { t }$ to the next time step (Barber, 2012). ",
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+        "type": "equation",
+        "img_path": "images/528c362d6cd7f4191ab6bb40399b4ce081249e1a532df6581c5afc611e067a34.jpg",
+        "text": "$$\n\\begin{array} { r l r l } & { z _ { t } = A ( s _ { t } ) z _ { t - 1 } + B ( s _ { t } ) u _ { t - 1 } + \\epsilon ( s _ { t } ) \\quad } & & { \\epsilon ( s _ { t } ) \\sim { \\mathcal N } ( 0 , Q ( s _ { t } ) ) } \\\\ & { x _ { t } = H ( s _ { t } ) z _ { t } + \\eta ( s _ { t } ) \\quad } & & { \\eta ( s _ { t } ) \\sim { \\mathcal N } ( 0 , R ( s _ { t } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "Here $A \\in \\mathbb { R } ^ { n _ { z } \\times n _ { z } }$ is the state matrix, $B \\in \\mathbb { R } ^ { n _ { z } \\times n _ { u } }$ control matrix, $\\epsilon$ the transition noise with covariance matrix $Q$ and $\\eta$ the emission/sensor noise with covariance matrix $R$ . Finally, the observation matrix $H \\in \\mathbb { R } ^ { n _ { x } \\times n _ { z } }$ defines a linear mapping from latent to observation space which we will replace by a nonlinear transformation parameterized by a neural net. These equations imply the following joint distribution: ",
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+        "type": "equation",
+        "img_path": "images/1f4d07b243b40a6c4de193d05428698c2f7a671afc77f1609982111863707b7a.jpg",
+        "text": "$$\np \\left( { x _ { 1 : T } , z _ { 1 : T } , s _ { 1 : T } } \\mid { u _ { 1 : T } } \\right) = \\prod _ { t = 1 } ^ { T } p \\left( { x _ { t } } \\mid { z _ { t } } \\right) p \\left( { z _ { t } } \\mid { z _ { t - 1 } , u _ { t - 1 } , s _ { t } } \\right) p \\left( { s _ { t } } \\mid { z _ { t - 1 } , u _ { t - 1 } , s _ { t - 1 } } \\right)\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "with $p ( z _ { 1 } \\mid z _ { 0 } , u _ { 0 } , s _ { 1 } ) = p ( z _ { 1 } )$ being the initial state distribution. The corresponding graphical model is shown in figure 1a. ",
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+    {
+        "type": "text",
+        "text": "2.2 STOCHASTIC GRADIENT VARIATIONAL BAYES ",
+        "text_level": 1,
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+    {
+        "type": "equation",
+        "img_path": "images/6910e38a813a46afc897c97aa82c998dfe1d244bb5129735a0af5fbd3226bb32.jpg",
+        "text": "$$\np ( x ) = \\int p ( x , z ) \\mathrm { d } z = \\int p ( x \\mid z ) p ( z ) \\mathrm { d } z\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Given the simple graphical model in equation (3), Kingma & Welling (2014) and Rezende et al. (2014) introduced the Variational Autoencoder (VAE) which overcomes the intractability of posterior inference of $q ( z \\mid x )$ by maximizing the evidence lower bound (ELBO) of the model log-likelihood. ",
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+        "img_path": "images/f54fdc0841f282a006b20737fadbe7cc3d26f28c1642d92b94b7e5ac9bc9e9bf.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { E L B O } } ( x ; \\theta , \\phi ) = \\mathbb { E } _ { q _ { \\phi } ( z | x ) } [ \\ln p _ { \\theta } ( x \\mid z ) ] - D _ { \\mathrm { K L } } ( q _ { \\phi } ( z \\mid x ) \\mid | p ( z ) ) \\le \\log p ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Their main innovation was to approximate the intractable posterior distribution by a recognition network $q _ { \\phi } ( z | x )$ from which they can sample via the reparameterization trick to allow for stochastic backpropagation through both the recognition and generative model at once. Assuming that the latent state is normally distributed, a simple transformation allows us to obtain a Monte Carlo gradient estimate of $\\mathbb { E } _ { q _ { \\phi } ( z | x ) } \\left[ \\ln p _ { \\theta } ( x | z ) \\right]$ w.r.t. to $\\phi$ . Given that $z \\sim \\mathcal { N } ( \\mu , \\sigma ^ { 2 } )$ , we can generate samples by drawing from an auxiliary variable $\\epsilon \\sim \\mathcal { N } ( 0 , 1 )$ and applying the deterministic and differentiable transformation $z = \\mu + \\sigma \\epsilon$ . ",
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+        "text": "2.3 THE CONCRETE DISTRIBUTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "One simple and efficient way to obtain samples $d$ from a $k$ -dimensional categorical distribution with class probabilities $\\alpha$ is the Gumbel-Max trick: ",
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+        "img_path": "images/5dd53b6378aee5fa5c426ed728c1a53b13e7dceafdeb1ee91857c70298cf9da4.jpg",
+        "text": "$$\nd = \\mathrm { o n e \\_ h o t } \\left( \\mathrm { a r g m a x } \\big [ g _ { i } + \\log \\alpha _ { i } \\big ] \\right) , \\quad \\mathrm { w i t h } \\ g _ { 1 } , \\dots , g _ { k } \\sim \\mathrm { G u m b e l } ( 0 , 1 )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "However, since the derivative of the argmax is 0 everywhere except at the boundary of state changes, where it is undefined, we can’t learn a parameterization by backpropagation. The Gumbel-Softmax trick approximates the argmax by a softmax which gives us a probability vector (Maddison et al., 2017; Jang et al., 2017). We can then draw samples via ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/28ad63d5b6f2143a0df104a7542c8fa8f00da71750c30c40fc096f5c8cd23040.jpg",
+        "text": "$$\nd _ { k } = \\frac { \\exp ( ( \\log \\alpha _ { k } + g _ { k } ) / \\lambda ) } { \\sum _ { i = 1 } ^ { n } \\exp ( ( \\log \\alpha _ { i } + g _ { i } ) / \\lambda ) } , \\quad \\mathrm { w i t h ~ } g _ { 1 } , \\dots , g _ { k } \\sim \\mathrm { G u m b e l } ( 0 , 1 )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This softmax computation approaches the discrete argmax as temperature $\\lambda  0$ , for $\\lambda \\to \\infty$ it approaches a uniform distribution. ",
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+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our model can be viewed as a Deep Kalman Filter (Krishnan et al., 2015) with structured inference (Krishnan et al., 2017). In our case, structured inference entails another stochastic variable model with parameter sharing inspired by Karl et al. (2017b) and Karl et al. (2017a) which pointed out the importance of backpropagating the reconstruction error through the transition. We are different to a number of stochastic sequential models like Bayer & Osendorfer (2014); Chung et al. (2015); Shabanian et al. (2017); Goyal et al. (2017) by directly transitioning the stochastic latent variable over time instead of having an RNN augmented by stochastic inputs. Fraccaro et al. (2016) has a transition over both a deterministic and a stochastic latent state sequence, wanting to combine the best of both worlds. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Previous models (Watter et al., 2015; Karl et al., 2017a; Fraccaro et al., 2017) have already combined locally linear models with recurrent Variational Autoencoders, however they provide a weaker structural incentive for learning latent variables determining the transition function. Van Steenkiste et al. (2018) approach a similar multi bouncing ball problem (see section 5.1) by first distributing the representation of different balls into their own entities without supervision and then structurally hardwiring a transition function with interactions based on an attention mechanism. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recurrent switching linear dynamical systems (Linderman et al., 2016) uses message passing for approximate inference, but has restricted itself to low-dimensional observations and a multi-stage training process. Johnson et al. (2016) propose a similar model to ours but combine message passing for discrete switching variables with a neural network encoder for observations learned by stochastic backpropagation. Tackling the problem of propagating state uncertainty over time, various combinations of neural networks for inference and Gaussian processes for transition dynamics have been proposed (Eleftheriadis et al., 2017; Doerr et al., 2018). However, these models have not been demonstrated to work with high-dimensional observation spaces like images. One feature a switching LDS model may learn are interactions which have recently been approached by employing Graph Neural Networks (Battaglia et al., 2016; Kipf et al., 2018). These methods are similar in that they predict edges which encode interactions between components of the state space (nodes). ",
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+    {
+        "type": "text",
+        "text": "4 PROPOSED APPROACH",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our goal is to fit a series of continuous state $z _ { 1 : T }$ and switching variables $s _ { 2 : T }$ to a given sequence of observations $x _ { 1 : T }$ . We assume a nonlinear mapping between observations and latent space which we generally approximate by neural networks, apart from the transition which is modeled by a locally linear function. Our generative model is shown in figure 1b an our inference model in figure 2a. ",
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+    {
+        "type": "text",
+        "text": "4.1 GENERATIVE MODEL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our generative model for a single $x _ { t }$ is described by ",
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+    {
+        "type": "equation",
+        "img_path": "images/39b2c232fd3684b909531787ceabe9f2e9654f04763aebb2a7933af7ba4b69a5.jpg",
+        "text": "$$\np ( x _ { t } ) = \\int _ { s \\leq t } \\int _ { z \\leq t } p ( x _ { t } \\mid z _ { t } ) p ( z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } ) p ( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } ) p ( z _ { t - 1 } , s _ { t - 1 } ) d s _ { t } d t .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which is close to the one of the original SLDS model (see figure 1a). Latent states $z _ { t }$ are continuous and represent the state of the system while states $s _ { t }$ are the switching variables determining the transition. We approximate the discrete switching variables by a continuous relaxation, namely the Concrete distribution.1 Differently to the original model, we do not condition the likelihood of the current observation $p _ { \\theta } ( x _ { t } \\mid z _ { t } )$ directly on the switching variables. This limits the influence of the switching variables to choosing a proper transition dynamic for the continuous latent space. The likelihood model is parameterized by a neural network with either a Gaussian or a Bernoulli distribution as output depending on the data. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "There is both a transition on the continuous states $z _ { t }$ and discrete latent states $s _ { t }$ . For the continuous state transition $p ( z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } )$ we follow (1) and maintain a set of $M$ base matrices $\\{ \\left( A ^ { ( i ) } , B ^ { ( i ) } , Q ^ { ( i ) } \\right) | \\forall i . 0 < i < M \\}$ as our linear dynamical systems to choose from. For the transition on discrete latent states $p ( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } )$ , we usually require the learning of a Markov transition matrix. However, since we approximate our discrete switching variables by a continuous relaxation, we can parameterize this transition by a neural network. Therefore, our entire generative model can be learned end-to-end by (stochastic) backpropagation. Finally, the resulting dynamics matrices are computed through a linear combination of the base matrices: ",
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+        "img_path": "images/3c7f3eaeae5f8244cfcbf5debf5cf0eb86062a50669c73a83710683b3493afc2.jpg",
+        "text": "$$\nA _ { t } ( s _ { t } ) = \\sum _ { i = 1 } ^ { M } s _ { t } ^ { ( i ) } A ^ { ( i ) } , \\qquad B ( s _ { t } ) = \\sum _ { i = 1 } ^ { M } s _ { t } ^ { ( i ) } B ^ { ( i ) } , \\qquad Q ( s _ { t } ) = \\sum _ { i = 1 } ^ { M } s _ { t } ^ { ( i ) } Q ^ { ( i ) }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Both transition models – the continuous state transition $p _ { \\theta } ( z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } )$ and concrete switching variables transition $p _ { \\theta } \\big ( s _ { t } \\ | \\ s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } \\big ) -$ are shared with the inference model which is key for good performance. ",
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+        "img_path": "images/b384a07004aa2e777e67a1ac5764f7e0e8df55f7e7c19f8f91330e22612189c3.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\quad p _ { \\theta } ( z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } ) = \\mathcal { N } \\big ( \\mu , \\sigma ^ { 2 } \\big ) \\qquad \\mathrm { w h e r e } \\ [ \\mu , \\sigma ^ { 2 } ] = f _ { \\theta } ( z _ { t - 1 } , s _ { t } , u _ { t - 1 } ) } \\\\ & { \\quad p _ { \\theta } ( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } ) = \\mathrm { C o n c r e t e } ( \\alpha , \\lambda _ { \\mathrm { p r i o r } } ) \\qquad \\mathrm { w h e r e } \\ \\alpha = g _ { \\theta } ( z _ { t - 1 } , s _ { t - 1 } , u _ { t - 1 } ) } \\end{array}\n$$",
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+    {
+        "type": "text",
+        "text": "4.2 INFERENCE ",
+        "text_level": 1,
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+        "text": "4.2.1 STRUCTURED INFERENCE OF CONTINUOUS LATENT STATE ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We split our inference model $q _ { \\phi } ( z _ { t } \\mid z _ { t - 1 } , s _ { t } , x _ { \\geq t } , u _ { \\geq t - 1 } )$ into two parts: 1) transition model $q _ { \\mathrm { t r a n s } } \\bar { ( } z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } )$ and 2) inverse measurement model $q _ { \\mathrm { m e a s } } ( z _ { t } \\mid x _ { \\geq t } , u _ { \\geq t } )$ as previously proposed in Karl et al. (2017b). This split allows us to reuse our generative transition model in place of $q _ { \\mathrm { t r a n s } } ( z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } )$ . This sharing of variables is essential for good performance as it forces the reconstruction error to be backpropagated throughwe only share the computation of the transition mean $\\mu _ { \\mathrm { t r a n s } }$ nsition model. For prbut not the variance $\\sigma _ { \\mathrm { t r a n s } } ^ { 2 }$ l reasons,between inference and generative model. Both parts, and $q _ { \\mathrm { t r a n s } }$ , will give us independent predictions about the new state $z _ { t }$ which will be combined in a manner akin to a Bayesian update in a Kalman Filter. ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/5972ef4bcbc8193183f6b4f578222127f21ffd67d9e218d368c62c741110945e.jpg",
+        "text": "$$\n\\begin{array} { r } { l \\phi \\left( z _ { t } \\mid z _ { t - 1 } , s _ { t } , x _ { \\geq t } , u _ { \\geq t - 1 } \\right) \\propto q _ { \\operatorname* { m e a s } } \\left( z _ { t } \\mid x _ { \\geq t } , u _ { \\geq t } \\right) \\times q _ { \\mathrm { t r a n s } } \\left( z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } \\right) = \\mathcal { N } \\left( \\mu _ { q } , \\sigma _ { q } ^ { 2 } \\right) } \\\\ { q _ { \\operatorname* { m e a s } } \\left( z _ { t } \\mid x _ { \\geq t } , u _ { \\geq t } \\right) = \\mathcal { N } \\left( \\mu _ { \\operatorname* { m e a s } } , \\sigma _ { \\operatorname* { m e a s } } ^ { 2 } \\right) \\mathrm { ~ w h e r e ~ } \\left[ \\mu _ { \\operatorname* { m e a s } } , \\sigma _ { \\operatorname* { m e a s } } ^ { 2 } \\right] = h _ { \\phi } \\left( x _ { \\geq t } , u _ { \\geq t } \\right) \\quad \\mathrm { ( } 1 \\mathrm { ~ t ~ } } \\\\ { q _ { \\operatorname { t r a n s } } \\left( z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } \\right) = \\mathcal { N } \\left( \\mu _ { \\operatorname { t r a n s } } , \\sigma _ { \\operatorname* { t r a n s } } ^ { 2 } \\right) \\mathrm { ~ w h e r e ~ } \\left[ \\mu _ { \\operatorname { t r a n s } } , \\sigma _ { \\operatorname { t r a n s } } ^ { 2 } \\right] = f _ { \\theta } \\left( z _ { t - 1 } , s _ { t } , u _ { t - 1 } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The densities of $q _ { \\mathrm { m e a s } }$ and $q _ { \\mathrm { t r a n s } }$ are multiplied resulting in another Gaussian density: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/6a5bebbe71e753d29f75507dcfb97598ac8457e07119b655a1fc2ec9aa78bec7.jpg",
+        "text": "$$\n\\mu _ { q } = \\frac { \\mu _ { \\mathrm { t r a n s } } \\sigma _ { \\mathrm { m e a s } } ^ { 2 } + \\mu _ { \\mathrm { m e a s } } \\sigma _ { \\mathrm { t r a n s } } ^ { 2 } } { \\sigma _ { \\mathrm { m e a s } } ^ { 2 } + \\sigma _ { \\mathrm { t r a n s } } ^ { 2 } } , \\qquad \\sigma _ { q } ^ { 2 } = \\frac { \\sigma _ { \\mathrm { m e a s } } ^ { 2 } \\sigma _ { \\mathrm { t r a n s } } ^ { 2 } } { \\sigma _ { \\mathrm { m e a s } } ^ { 2 } + \\sigma _ { \\mathrm { t r a n s } } ^ { 2 } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This update scheme is highlighted in figure 2b. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We found empirically that conditioning the inverse measurement model $q _ { \\mathrm { m e a s } } ( z _ { t } \\mid x _ { \\geq t } , u _ { \\geq t } )$ solely on the current observation $x _ { t }$ instead of the entire remaining trajectory to lead to better results. We hypothesize that the recurrent model needlessly introduces very high-dimensional and complicated dynamics which are harder to approximate with our locally linear transition model. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the initial state $z _ { 1 }$ we do not have a conditional prior from the transition model as in the rest of the sequence. Other methods (Krishnan et al., 2015) have used a standard normal prior, however this is not a good fit. We therefore decided that instead of predicting $z _ { 1 }$ directly to predict an auxiliary variable $w$ that is then mapped deterministically to a starting state $z _ { 1 }$ . A standard Gaussian prior is then applied to $w$ . Alternatively, we could specify a more complex or learned prior for the initial state like the VampPrior (Tomczak & Welling, 2017). Empirically, this has lead to worse results. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c3ff26bcd7abd47ab476a05840ccb91530cce3f1a9a94bb09a0ed145e4d31533.jpg",
+        "text": "$$\n\\begin{array} { r l } { q _ { \\phi } ( w \\mid x _ { 1 : T } , u _ { 1 : T } ) = { \\mathcal { N } } { \\big ( } w ; \\mu _ { w } , \\sigma _ { w } ^ { 2 } { \\big ) } } & { { } { \\mathrm { w h e r e } } \\quad [ \\mu _ { w } , \\sigma _ { w } ^ { 2 } ] = i _ { \\phi } ( x _ { 1 : T } , u _ { 1 : T } ) } \\\\ { z _ { 1 } = f _ { \\phi } ( w ) } & { { } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "While we could condition on the entire sequence, we restrict it to just the first couple of observations. ",
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+    {
+        "type": "image",
+        "img_path": "images/77ec647f77f9f545bed126f45f7c914263e5c545ff3e46593cee575823abc31e.jpg",
+        "image_caption": [
+            "Figure 2: (a) Depicts the inference model. $b _ { t }$ is the hidden state of the backward RNN of $q _ { \\phi } \\mathbf { \\bar { ( } } s _ { t } \\mid x _ { \\geq t } , u _ { \\geq t } )$ . Initial inference of $w$ may be conditioned on the entire sequence of observations, or just a subsequence. We’ve omitted the arrows for sake of clarity for the rest of the graph. (b) Shows schematically how we combine the transition with the inverse measurement model in the inference network. Transitions (in blue) are (partially) shared with the generative model. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.2.2 INFERENCE OF SWITCHING VARIABLES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following Maddison et al. (2017) and Jang et al. (2017), we can reparameterize a discrete latent variable with the Gumbel-softmax trick. Again, we split our inference network $q _ { \\phi } ( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , x _ { \\geq t } , u _ { \\geq t - 1 } )$ in an identical fashion into two components: 1) Transition model $q _ { \\mathrm { t r a n s } } ( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } )$ and 2) inverse measurement model $q _ { \\mathrm { m e a s } } ( s _ { t } \\mid x _ { \\geq t } , u _ { \\geq t } )$ . The transition model is again shared with the generative model and is implemented via a neural network as we potentially require quick changes to chosen dynamics. The inverse measurement model is parametrized by a backward LSTM. However, for the case of concrete variables, we cannot do the same Gauss multiplication as in the previous case. Therefore, we let each network predict the logits of a Concrete distribution and our inverse measurement model $q _ { \\phi } ( s _ { t } \\mid x _ { \\geq t } , u _ { \\geq t } )$ produces an additional vector $\\gamma$ , which determines the value of a gate deciding how the two predictions are to be weighted: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd33379a8a3ccb0d71767781f8b1b69d701339b49a747a027414d6636311d73c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { q _ { \\phi } \\big ( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , x _ { \\geq t } , u _ { \\geq t - 1 } \\big ) = \\mathrm { C o n c r e t e } \\big ( \\alpha , \\lambda _ { \\mathrm { p o s t e r i o r } } \\big ) \\quad \\mathrm { w i t h } \\quad \\alpha = \\gamma \\alpha _ { \\mathrm { t r a n s } } + ( 1 - \\gamma ) \\alpha _ { \\mathrm { m e a s } } } \\\\ & { q _ { \\operatorname* { m e a s } } \\big ( s _ { t } \\mid x _ { \\geq t } , u _ { \\geq t } \\big ) = \\mathrm { C o n c r e t e } \\big ( \\alpha _ { \\mathrm { m e a s } } , \\lambda _ { \\mathrm { p o s t e r i o r } } \\big ) \\quad \\mathrm { w h e r e } \\quad \\big [ \\alpha _ { \\mathrm { m e a s } } , \\gamma \\big ] = k _ { \\phi } \\big ( x _ { \\geq t } , u _ { \\geq t } \\big ) \\qquad ( 1 3 ) } \\\\ & { q _ { \\mathrm { t r a n s } } \\big ( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } \\big ) = \\mathrm { C o n c r e t e } \\big ( \\alpha _ { \\mathrm { t r a n s } } , \\lambda _ { \\mathrm { p r i o r } } \\big ) \\quad \\mathrm { w h e r e } \\quad \\alpha = g _ { \\theta } \\big ( z _ { t - 1 } , s _ { t - 1 } , u _ { t - 1 } \\big ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The temperatures $\\lambda _ { \\mathrm { p o s t e r i o r } }$ and $\\lambda _ { \\mathrm { p r i o r } }$ are set as a hyperparameter and can be set differently for the prior and approximate posterior. The gating mechanism gives the model the option to balance between prior and approximate posterior. If the prior is good enough to explain the next observation, $\\gamma$ will be pushed to 1 which ignores the measurement and minimizes the KL between prior and posterior by only propagating the prior. If the prior is not sufficient, information from the inverse measurement model can flow by decreasing $\\gamma$ and incurring a KL penalty. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since the concrete distribution is a relaxation of the categorical, our sample will not be a one-hot vector, but a vector whose elements sum up to 1. We face two options here: we could take a categorical sample by choosing the linear system corresponding to the highest value in the sample (hard forward pass) and only use the relaxation for our backward pass. This, however, means that we will follow a biased gradient. Alternatively, we can use the relaxed version for our forward pass and aggregate the linear systems based on their corresponding weighting (see (8)). Here, we lose the discrete switching of linear systems, but maintain a valid lower bound. We note that the hard forward pass has led to worse results and focus on the soft forward pass for this paper. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Lastly, we could go further away from the theory and instead treat the switching variables also as normally distributed. If this worked better than the approach with Concrete variables, it would highlight still existing optimization problems of discrete random variables. As such, it will act as an ablation study for our model. The mixing coefficients for linear systems would then be determined by a linear combination of these latent variables: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/84b23d969b74ee1ba010f73a54d0d9807ce3b7af2c09f5c098cfe27326bdbb68.jpg",
+        "text": "$$\n\\alpha = \\operatorname { s o f t m a x } ( W s _ { t } + b ) \\in \\mathbb { R } ^ { M }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Our inference scheme for normally distributed switching variables is then identical to the one described in the previous section. We compare both approaches throughout our experimental section. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our objective function is the commonly used evidence lower bound for our hierarchical model. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/97d616905939eabefebeb531dd43f658e7064817adace691f180a9f38a78ca70.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathcal { L } _ { \\theta , \\phi } \\big ( x _ { 1 : T } \\ \\big | \\ u _ { 1 : T } \\big ) \\geq } & { \\mathbb { E } _ { q _ { \\phi } ( z _ { 1 : T } , s _ { 1 : T } \\mid x _ { 1 : T } ) } \\big [ \\log p _ { \\theta } \\big ( x _ { 1 : T } \\ \\big | \\ z _ { 1 : T } , s _ { 1 : T } , u _ { 1 : T } \\big ) \\big ] } \\\\ & { - D _ { \\mathrm { K L } } \\big ( q _ { \\phi } \\big ( z _ { 1 : T } , s _ { 1 : T } \\ \\big | \\ x _ { 1 : T } , u _ { 1 : T } \\big ) \\ \\big | \\ \\big | \\ p \\big ( z _ { 1 : T } , s _ { 1 : T } \\ \\big | \\ u _ { 1 : T } \\big ) \\big ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We choose to factorize over time, so the loss for a single observation $x _ { t }$ becomes: ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c8f4317b5930396d07e7e1332bc12fe650f322675d18a21a2ec4e5b9af14aa46.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\theta , \\phi } ( x _ { t } \\mid u _ { 1 : T } ) = \\mathbb { E } _ { q _ { \\phi } \\left( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , x _ { \\geq t } , u _ { \\geq t - 1 } \\right) } \\left[ \\mathbb { E } _ { q _ { \\phi } \\left( z _ { t } \\mid s _ { t } , z _ { t - 1 } , x _ { \\geq t } , u _ { \\geq t - 1 } \\right) } \\left[ \\log p _ { \\theta } ( x _ { t } \\mid z _ { t } ) \\right] \\right] \\qquad ( 1 6 ) } \\\\ & { \\qquad - \\mathbb { E } _ { s _ { t - 1 } } \\left[ \\mathbb { E } _ { z _ { t - 1 } } \\left[ D _ { \\mathrm { K L } } \\left( q _ { \\phi } \\left( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , x _ { \\geq t } , u _ { \\geq t - 1 } \\right) \\mid \\mid p _ { \\theta } \\left( s _ { t } \\mid s _ { t - 1 } , z _ { t - 1 } , u _ { t - 1 } \\right) \\right) \\right] \\right] } \\\\ & { \\qquad - \\mathbb { E } _ { z _ { t - 1 } } \\left[ \\mathbb { E } _ { s _ { t } } \\left[ D _ { \\mathrm { K L } } \\left( q _ { \\phi } \\left( z _ { t } \\mid z _ { t - 1 } , s _ { t } , x _ { \\geq t } , u _ { \\geq t - 1 } \\right) \\mid \\mid p _ { \\theta } \\left( z _ { t } \\mid z _ { t - 1 } , s _ { t } , u _ { t - 1 } \\right) \\right) \\right] \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "The full derivation can be found in appendix A. We learn the parameters of our model by backpropagation through time and we (generally) approximate the expectations with one sample by using the reparametrization trick. The exception is the KL between two Concrete random variables in which case we take 10 samples for the approximation. For the KL on the switching variables, we further introduce a scaling factor $\\beta < 1$ (as first suggested in Higgins et al. (2016), although they suggested increasing the KL term) to down weigh its importance. More details on the training procedure can be found in appendix B.2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In this section, we evaluate our approach on a diverse set of physics and robotics simulations based on partially observable system states or high-dimensional images as observations. We show that our model outperforms previous models and that our switching variables learn meaningful representations. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Models we compare to are Deep Variational Bayes Filter (DVBF) (Karl et al., 2017a), DVBF Fusion (Karl et al., 2017b) (called fusion as they do the same Gauss multiplication in the inference network) which is closest to our model but doesn’t have a stochastic treatment of the transition, the Kalman VAE (KVAE) (Fraccaro et al., 2017) and a LSTM (Hochreiter & Schmidhuber, 1997). ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e0194ebf04f7331684f7436c765cd1d05be1f6ea391f318e881011b153ff9194.jpg",
+        "image_caption": [
+            "(a) Multi agent maze envi- (b) Variable encoding free (c) Variable encoding walls (d) System activation for ronment. space for agent 2. for agent 1. deterministic transition. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 3: Figures (b) and (c) depict an agent’s position colored by the average value of a single latent variable $s$ marginalized over all control inputs $u$ and velocities. Figure (d) highlights a representative activation for a single transition system for the deterministic treatment of the transition dynamics. It doesn’t generalize to the entire maze and stays fairly active in proximity to the wall. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "5.1 MULTIPLE BOUNCING BALLS IN A MAZE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our first experiment is a custom 3-agent maze environment simulated with Box2D. Each agent is fully described by its $x$ and $y$ coordinates and its current velocity and has the capability to accelerate in either direction. We learn in a partially observable setting and limit the observations to the agents’ positions, therefore $x \\in \\mathbb { R } ^ { 6 }$ while the true state space is in $\\mathbb { R } ^ { 1 2 }$ and $u \\in \\mathbb { R } ^ { 6 }$ . First, we train a linear regression model on the latent space $z$ to see if we have recovered a linear encoding of the unobserved velocities. We achieve an R2 score of 0.92 averaged over all agents and velocity directions. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our focus shifts now to our switching variables which we expect to encode interactions with walls. We provide a visual confirmation of that in figure 3 where we see switching variables encoding all space where there is no interaction in the next time step, and variables which encode walls, distinguishing between vertical and horizontal ones. In figure 3d one can see show that if the choice of locally linear transition is treated deterministically, we don’t learn global features of the same kind. To confirm our visual inspection, we train a simple decision tree based on latent space $s$ in order to predict interaction with a wall. Here, we achieve an F1 score of 0.46. It is difficult to say what a good value should look like as collisions with low velocity are virtually indistinguishable from no collision. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We compare our prediction quality to several other methods in table 1 where we outperform all of our chosen baselines. Also, modeling switching variables by a Normal distribution outperforms the Concrete distribution in all of our experiments. Aside from known practical issues with training a discrete variable via backpropagation, we explore one reason why that may be in section 5.4, which is the greater susceptibility to the scale of temporal discretization. We provide plots of predicted trajectories in appendix D. Transitioning multiple agents with a single transition matrix comes with scalability issues with regards to switching dynamics which we explore further in appendix C. ",
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+        "page_idx": 6
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+    {
+        "type": "table",
+        "img_path": "images/53338daf175f862068e9bdea02c507acea46542efc7ab2c8545c0962dce55727.jpg",
+        "table_caption": [
+            "Table 1: Mean squared error (MSE) on predicting future observations. Static refers to constantly predicting the first observation of the sequence. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">REACHER</td><td colspan=\"3\">3-BALL MAZE</td></tr><tr><td>PREDICTION STEPS</td><td>1</td><td>5</td><td>10</td><td>1</td><td>5</td><td>10</td></tr><tr><td>STATIC</td><td>5.80E-02</td><td>5.36E-01</td><td>1.25E+00</td><td>1.40E-02</td><td>5.74E-01</td><td>2.65E+00</td></tr><tr><td>LSTM</td><td>3.07E-01</td><td>7.76E-01</td><td>1.22E+00</td><td>7.20E-02</td><td>1.58E-01</td><td>2.60E-01</td></tr><tr><td>DVBF</td><td>1.10E-01</td><td>3.08E-01</td><td>6.07E-01</td><td>6.20E-02</td><td>1.36E-01</td><td>1.82E-01</td></tr><tr><td>DVBFFUSION</td><td>4.90E-03</td><td>2.97E-02</td><td>8.25E-02</td><td>4.33E-03</td><td>2.03E-02</td><td>4.88E-02</td></tr><tr><td>OURS (CONCRETE)</td><td>1.06E-02</td><td>5.73E-02</td><td>1.56E-01</td><td>2.28E-03</td><td>1.22E-02</td><td>3.40E-02</td></tr><tr><td>OURS (NORMAL)</td><td>3.39E-03</td><td>1.85E-02</td><td>4.97E-02</td><td>1.30E-03</td><td>5.52E-03</td><td>1.38E-02</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 REACHER ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We then evaluate our model on the Roboschool reacher environment. To make things more interesting, we learn only on partial observations, removing time derivative information (velocities), leaving us with just the positions or angles of various joints as observations. Table 1 shows a comparison of various methods on predicting the next couple of time steps. One critical point is the possible collision2 between lower and upper joint which is one we’d like our model to capture. We again learn a linear classifier based on latent space $s$ to see if this is successfully encoded and reach an F1 score of 0.46. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 BALL IN A BOX ON IMAGE DATA ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we evaluate our method on high-dimensional image observations using the single bouncing ball environment used by Fraccaro et al. (2017). They simulated 5000 sequences of 20 time steps each of a ball moving in a two-dimensional box, where each video frame is a $3 2 \\times 3 2$ binary image. There are no forces applied to the ball, except for the fully elastic collisions with the walls. Initial position and velocity are randomly sampled. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e336e1bf737c0b1609bca209428a9329da2814fbc7a23379990da437d282bb0e.jpg",
+        "image_caption": [
+            "Figure 4: First row: data, second row: filtered reconstructions, third row: predictions. The first 4 steps are used to find a stable starting state, predictions start with step 5. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In figure 5a we compare our model to both the smoothed and generative version of the KVAE. The smoothed version receives the final state of the trajectory after the $n$ predicted steps which is fed into the smoothing capability of the KVAE. One can see that our model learns a better transition model, even outperforming the smoothed KVAE for longer sequences. For short sequences, KVAE performs better which highlights the value of it disentangling the latent space into separate object and dynamics representation. A sample trajectory is plotted in figure 4. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.4 SUSCEPTIBILITY TO THE SCALE OF TEMPORAL DISCRETIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we’d like to explore how the choice of $\\Delta t$ when discretizing a system influences our results. In particular, we’d expect our model with discrete (concrete) switching latent variables to be more susceptible to it than when modeled by a continuous distribution. This is because in the latter case the switching variables can scale the various matrices more freely, while in the former scaling up one system necessitates scaling down another. For empirical comparison, we go back to our custom maze environment (this time with only one agent as this is not pertinent to our question at hand) and learn the dynamics on various discretization scales. Then we compare the absolute error’s growth for both approaches in figure 5b which supports our hypothesis. While the discrete approximation even outperforms for small $\\Delta t$ , there is a point where it rapidly becomes worse and gets overtaken by the continuous approximation. This suggests that $\\Delta t$ was simply chosen to be too large in both the reacher and the ball in a box with image observations experiment. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0e452e2e6f69830aa478e0687ad1ffd2d7d2264958af6ea33ad244daeec98297.jpg",
+        "image_caption": [
+            "Figure 5: (a) Our dynamics model is outperforming even the smoothed KVAE for longer trajectories. (b) Modeling switching variables as Concrete random variables scales less favorably. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We want to emphasize some subtle differences to previously proposed architectures that make an empirical difference, in particular for the case when $s _ { t }$ is chosen to be continuous. In Watter et al. (2015) and Karl et al. (2017a), the latent space is already used to draw transition matrices, however they do not extract features such as walls or joint constraints. There are a few key differences from our approach. First, our latent switching variables $s _ { t }$ are only involved in predicting the current observation $x _ { t }$ through the transition selection process. The likelihood model therefore doesn’t need to learn to ignore some input dimensions which are only helpful for reconstructing future observations but not the current one. There is also a clearer restriction on how $s _ { t }$ and $z _ { t }$ may interact: $s _ { t }$ may now only influence $z _ { t }$ by determining the dynamics, while previously $z _ { t }$ influenced both the choice of transition function as well as acted inside the transition. These two opposing roles lead to conflicting gradients as to what should be improved. Furthermore, the learning signal for $s _ { t }$ is rather weak so that scaling down the KL-regularization was necessary to detect good features. Lastly, a (locally) linear transition may not be a good fit for variables determining dynamics as such variables may change very abruptly. ",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have shown that our construction of using switching variables encourages learning a richer and more interpretable latent space. In turn, the richer representation led to an improvement of simulation accuracy in various tasks. In the future, we’d like to look at other ways to approximate the discrete switching variables and exploit this approach for model-based control on real hardware systems. Furthermore, addressing the open problem of disentangling latent spaces is essential to fitting simple dynamics and would lead to significant improvements of this approach. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+        "text": "Manuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. In Advances in neural information processing systems, pp. 2746–2754, 2015. ",
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+    },
+    {
+        "type": "text",
+        "text": "A LOWER BOUND DERIVATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            446,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For brevity we omit conditioning on control inputs $u _ { 1 : T }$ . ",
+        "bbox": [
+            173,
+            152,
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+            169
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46fee95633a76fc398df7e16f83ffbaa41d93b491342486aac982859fcea1f8b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\log p ( x _ { T } ) = \\log \\int _ { z _ { 1 : T } } \\int _ { s _ { 1 : T } } q _ { \\phi } ( s _ { 1 : T } , z _ { 1 : T } \\mid x _ { 1 : T } ) \\frac { p _ { \\theta } ( x _ { 1 : T } \\mid z _ { 1 : T } ) p _ { \\theta } ( z _ { 1 : T } , s _ { 1 : T } ) } { q _ { \\phi } ( s _ { 1 : T } , z _ { 1 : T } \\mid x _ { 1 : T } ) } } \\\\ { \\displaystyle \\geq \\int _ { z _ { 1 : T } } \\int _ { s _ { 1 : T } } q _ { \\phi } ( s _ { 1 : T } , z _ { 1 : T } \\mid x _ { 1 : T } ) \\log \\frac { p _ { \\theta } ( x _ { 1 : T } \\mid z _ { 1 : T } ) p _ { \\theta } ( z _ { 1 : T } , s _ { 1 : T } ) } { q _ { \\phi } ( s _ { 1 : T } , z _ { 1 : T } \\mid x _ { 1 : T } ) } } \\\\ { \\displaystyle = \\sum _ { t = 1 } ^ { T } \\mathbb { E } _ { s _ { t } } [ \\mathbb { E } _ { z _ { t } } [ p ( x _ { t } \\mid z _ { t } , s _ { t } ) ] ] - D _ { \\mathrm { K L } } ( q ( z _ { 1 : T } , s _ { 1 : T } \\mid x _ { 1 : T } ) \\mid | p ( z _ { 1 : T } , s _ { 1 : T } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 FACTORIZATION OF THE KL DIVERGENCE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The dependencies on data $x _ { T }$ and $u _ { T }$ as well as parameters $\\phi$ and $\\theta$ are omitted in the following for convenience. ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/8e49b95be7fb2cebf829bed24f90f756358ac691cd732b07d90dcbda17acfaf1.jpg",
+        "text": "$$\nD _ { \\mathrm { K L } } ( q ( z _ { 1 } , s _ { 2 } , \\ldots , s _ { T } , z _ { T } ) \\parallel p ( z _ { 1 } , s _ { 2 } , \\ldots , s _ { T } , z _ { T } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2236e69753a1a54cf78444cd6f8796c3817be059108bec6c5e0d5e2dbcb328e2.jpg",
+        "text": "$$\n\\begin{array} { r l } { { } } & { { = \\displaystyle \\int _ { z _ { 1 } } \\int _ { s _ { 2 } } \\cdots \\int _ { s _ { T } } \\int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \\mid z _ { 1 } ) \\ldots q ( s _ { T } \\mid z _ { T - 1 } , s _ { T - 1 } ) q ( z _ { T } \\mid z _ { T - 1 } , s _ { T } ) } } \\\\ { { } } & { { \\phantom { = \\displaystyle \\int _ { z _ { 1 } } \\int _ { s _ { 2 } } \\cdots \\int _ { s _ { T } } \\int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \\mid z _ { 1 } ) \\ldots q ( s _ { T } \\mid z _ { T - 1 } , s _ { T - 1 } ) q ( z _ { T } \\mid z _ { T - 1 } , s _ { T } ) } } } \\\\ { { } } & { { \\phantom { = \\displaystyle \\int _ { z _ { 1 } } \\int _ { s _ { 2 } } \\cdots d \\int _ { z _ { T - 1 } } \\int _ { z _ { T - 1 } } s _ { T - 1 } \\int _ { z _ { T } } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(Factorization of the prior) ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/daca0f6b48330efe8627e783f0c5eb98c1992084b39e02f588797bc232d16939.jpg",
+        "text": "$$\n\\begin{array} { r l } { { } } & { { = { \\displaystyle \\int _ { z _ { 1 } } \\int _ { s _ { 2 } } \\cdots \\int _ { s _ { T } } \\int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \\mid z _ { 1 } ) \\ldots q ( s _ { T } \\mid z _ { T - 1 } , s _ { T - 1 } ) q ( z _ { T } \\mid z _ { T - 1 } , s _ { T } ) } } } \\\\ { { } } & { { { \\log \\frac { q ( z _ { 1 } ) q ( s _ { 2 } \\mid z _ { 1 } ) \\ldots q ( s _ { T } \\mid z _ { T - 1 } , s _ { T - 1 } ) q ( z _ { T } \\mid z _ { T - 1 } , s _ { T } ) } { p ( z _ { 1 } ) p ( s _ { 2 } \\mid z _ { 1 } ) \\ldots p ( s _ { T } \\mid z _ { T - 1 } , s _ { T - 1 } ) p ( z _ { T } \\mid z _ { T - 1 } , s _ { T } ) } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            592,
+            781,
+            662
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(Expanding the logarithm by the product rule) ",
+        "bbox": [
+            215,
+            662,
+            519,
+            678
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c04eb8d928288d23f552fc06e2faf837e5282a8c0d87d4f2d42d24f6e592a93d.jpg",
+        "text": "$$\n\\begin{array} { l } { { = \\displaystyle \\int _ { z _ { 1 } } q ( z _ { 1 } ) \\log \\frac { q ( z _ { 1 } ) } { p ( z _ { 1 } ) } + \\displaystyle \\int _ { z _ { 1 } } \\int _ { s _ { 1 } } q ( z _ { 1 } ) q ( s _ { 1 } \\mid z _ { 1 } ) \\log \\frac { q ( s _ { 1 } \\mid z _ { 1 } ) } { p ( s _ { 1 } \\mid z _ { 1 } ) } } } \\\\ { { + \\displaystyle \\sum _ { t = 2 } ^ { T } \\int _ { z _ { 1 } } \\int _ { s _ { 2 } } \\cdots \\int _ { s _ { T } } \\int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \\mid z _ { 1 } ) \\ldots q ( z _ { T } \\mid z _ { T - 1 } , s _ { T } ) \\log \\frac { q ( z _ { t } \\mid z _ { t - 1 } , s _ { t } ) } { p ( z _ { t } \\mid z _ { t - 1 } , s _ { t } ) } } } \\\\ { { + \\displaystyle \\sum _ { t = 3 } ^ { T } \\int _ { z _ { 1 } } \\int _ { s _ { 2 } } \\cdots \\int _ { s _ { T } } \\int _ { z _ { T } } q ( z _ { 1 } ) q ( s _ { 2 } \\mid z _ { 1 } ) \\ldots q ( z _ { T } \\mid z _ { T - 1 } , s _ { T } ) \\log \\frac { q ( s _ { t } \\mid z _ { t - 1 } , s _ { t - 1 } ) } { p ( s _ { t } \\mid z _ { t - 1 } , s _ { t - 1 } ) } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            681,
+            782,
+            805
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(Ignoring constants) ",
+        "bbox": [
+            214,
+            806,
+            352,
+            820
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/421c8a72b3727f9413d5ab4807cb8c40703c635642c6c338a772fbe15118719b.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle = D _ { \\mathrm { K L } } ( q ( z _ { 1 } ) \\mid | \\ p ( z _ { 1 } ) ) + \\mathbb { E } _ { z _ { 1 } \\sim q ( z _ { 1 } ) } [ D _ { \\mathrm { K L } } ( q ( s _ { 2 } \\mid z _ { 1 } ) \\mid | \\ p ( s _ { 2 } \\mid z _ { 1 } ) ) ] } } \\\\ { { \\displaystyle ~ + \\sum _ { t = 2 } ^ { T - 1 } \\mathbb { E } _ { s _ { t } , z _ { t - 1 } } [ D _ { \\mathrm { K L } } ( q ( z _ { t } \\mid z _ { t - 1 } , s _ { t } ) \\mid | \\ p ( z _ { t } \\mid z _ { t - 1 } , s _ { t } ) ) ] } } \\\\ { { \\displaystyle ~ + \\sum _ { t = 3 } ^ { T - 1 } \\mathbb { E } _ { s _ { t - 1 } , z _ { t - 1 } } [ D _ { \\mathrm { K L } } ( q ( s _ { t } \\mid z _ { t - 1 } , s _ { t - 1 } ) \\mid | \\ p ( s _ { t } \\mid z _ { t - 1 } , s _ { t - 1 } ) ) ] } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            821,
+            658,
+            929
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/3a9a6244447cadd7e5ef221bf24b639a74797971ffc37c258bdc8cb435728d36.jpg",
+        "table_caption": [
+            "Table 2: Dimensionality of environments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dimensionality of（</td><td>Observation Space</td><td>Control Input Space</td><td>Ground Truth State Space</td></tr><tr><td>Reacher</td><td>7</td><td>2</td><td>9</td></tr><tr><td>Hopper</td><td>8</td><td>3</td><td>15</td></tr><tr><td>Multi Agent Maze</td><td>4</td><td>6</td><td>12</td></tr><tr><td>Image Ball in BoX</td><td>32 × 32</td><td>0</td><td>4</td></tr></table>",
+        "bbox": [
+            186,
+            133,
+            808,
+            212
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B DETAILS OF THE EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            242,
+            545,
+            258
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1 ENVIRONMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            273,
+            331,
+            289
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1.1 ROBOSCHOOL REACHER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            300,
+            401,
+            314
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To generate data, we follow a Uniform distribution $\\mathcal { U } \\sim [ - 1 , 1 ]$ as the exploration policy. Before we record data, we take 20 warm-up steps in the environment to randomize our starting state. We take the data as is without any other preprocessing. ",
+        "bbox": [
+            174,
+            324,
+            825,
+            367
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1.2 MULTI AGENT MAZE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            382,
+            380,
+            397
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Observations are normalized to be in $[ - 1 , 1 ]$ . Both position and velocity is randomized for the starting state. We again follow a Uniform distribution $\\mathcal { U } \\sim [ - 1 , 1 ]$ as the exploration policy. ",
+        "bbox": [
+            174,
+            406,
+            821,
+            435
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.2 TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            453,
+            289,
+            468
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Overall, training the Concrete distribution has given us the biggest challenge as it was very susceptible to various hyperparameters. We made use of the fact that we can use a different temperature for the prior and approximate posterior (Maddison et al., 2017) and we do independent hyperparameter search over both. For us, the best values were 0.75 for the posterior and 2 for the prior. Additionally, we employ an exponential annealing scheme for the temperature hyperparameter of the Concrete distribution. This leads to a more uniform combination of base matrices early in training which has two desirable effects. First, all matrices are scaled to a similar magnitude, making initialization less critical. Second, the model initially tries to fit a globally linear model, leading to a good starting state for optimization. We also tried increasing the number of samples taken (up to 100) to approximate the KL between the Concrete distributions, however we have not observed an improvement of performance. We therefore restrict ourselves to 10 samples for all experiments. ",
+        "bbox": [
+            174,
+            479,
+            825,
+            632
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In all experiments, we train everything end-to-end with the ADAM optimizer.(Kingma & Ba, 2015) We start with learning rate of $5 \\mathrm { e } { - 4 }$ and use an exponential decay schedule with rate 0.97 every 2000 iterations. ",
+        "bbox": [
+            176,
+            640,
+            825,
+            681
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.3 NETWORK ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            699,
+            403,
+            713
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For most networks, we use MLPs implemented as residual nets (He et al., 2016) with ReLU activations. ",
+        "bbox": [
+            176,
+            724,
+            825,
+            753
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Networks used for the reacher and maze experiments. ",
+        "bbox": [
+            174,
+            760,
+            526,
+            775
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• $q _ { \\mathrm { m e a s } } ( z _ { t } \\mid \\cdot )$ : MLP consisting of two residual blocks with 256 neurons each. We only condition on the current observation $x _ { t }$ although we could condition on the entire sequence. This decision was taken based on empirical results.   \n） $q _ { \\mathrm { t r a n s } } ( z _ { t } \\mid \\cdot )$ : In the case of Concrete random variables, we just combine the base matrices and apply the transition dynamics to $z _ { t - 1 }$ . For the Normal case, the combination of matrices is preceded by a linear combination with softmax activation. (see equation 14) $q _ { \\mathrm { m e a s } } ( s _ { t } \\mid \\cdot )$ : is implemented by a backward LSTM with 256 hidden units. We reuse the preprocessing of $q _ { m e a s } ( z _ { t } \\mid x _ { t } )$ and take the last hidden layer of that network as the input to the LSTM.   \n• $q _ { \\mathrm { t r a n s } } ( s _ { t } \\mid \\cdot )$ : MLP consisting of one residual block with 256 neurons.   \n• $q _ { \\mathrm { i n i t i a l } } ( w \\mid \\cdot )$ : MLP consisting of two residual block with 256 neurons optionally followed by a backward LSTM. We only condition on the first 3 or 4 observations for our experiments.   \n• $q _ { \\mathrm { i n i t i a l } } ( s _ { 2 } )$ : The first switching variable in the sequence has no predecessor. We therefore require a replacement for $q _ { t r a n s } ( s _ { t } \\mid \\cdot )$ in the first time step, which we achieve by independently parameterizing another MLP.   \n• $p ( x _ { t } \\mid z _ { t } )$ : MLP consisting of two residual block with 256 neurons.   \n• $p ( \\boldsymbol { z } _ { t } \\mid \\cdot )$ : Shared parameters with $q _ { t r a n s } ( z _ { t } \\mid \\cdot )$ .   \n• $p ( s _ { t } \\mid \\cdot )$ : Shared parameters with $q _ { t r a n s } ( s _ { t } \\mid \\cdot )$ . ",
+        "bbox": [
+            215,
+            786,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            215,
+            102,
+            826,
+            256
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use the same architecture for the image ball in a box experiment, however we increase number of neurons of $q _ { \\mathrm { m e a s } } ( z _ { t } \\mid \\cdot )$ to 1024. ",
+        "bbox": [
+            173,
+            265,
+            825,
+            295
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.4 HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            310,
+            361,
+            325
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/1f1746708b21bc9d06889b6359b16d5a494473a00c6120e7380856e09b783e41.jpg",
+        "table_caption": [
+            "Table 3: Overview of hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Multi Agent Maze</td><td>Reacher</td><td>Image Ball in Box</td></tr><tr><td># episodes</td><td>50000</td><td>20000</td><td>5000</td></tr><tr><td>episode length</td><td>20</td><td>30</td><td>20</td></tr><tr><td>batch size</td><td>256</td><td>128</td><td>256</td></tr><tr><td>dimension of z</td><td>32</td><td>16</td><td>8</td></tr><tr><td>dimension of s</td><td>16</td><td>8</td><td>8</td></tr><tr><td>posterior temperature</td><td>0.75</td><td>0.75</td><td>0.67</td></tr><tr><td>prior temperature</td><td>2</td><td>2</td><td>2</td></tr><tr><td>temperature annealing steps</td><td>100</td><td>100</td><td>100</td></tr><tr><td>temperature annealing rate</td><td>0.97</td><td>0.97</td><td>0.98</td></tr><tr><td>β (KL-scaling of switching variables)</td><td>0.1</td><td>0.1</td><td>0.1</td></tr></table>",
+        "bbox": [
+            189,
+            372,
+            808,
+            536
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C ON SCALING ISSUES OF SWITCHING LINEAR DYNAMICAL SYSTEMS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            571,
+            769,
+            587
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Let’s consider a simple representation of a ball in a rectangular box where its state is represented by its position and velocity. Given a small enough $\\Delta t$ , we can approximate the dynamics decently by just 3 systems: no interaction with the wall, interaction with a vertical or horizontal wall (ignoring the corner case of interacting with two walls at the same time). Now consider the growth of required base systems if we increase the number of balls in the box (even if these balls cannot interact with each other). We would require a system for all combinations of a single ball’s possible states: $3 ^ { 2 }$ This will grow exponentially with the number of balls in the environment. ",
+        "bbox": [
+            173,
+            602,
+            825,
+            700
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "One way to alleviate this problem that requires only a linear growth in base systems is to independently turn individual systems on and off and let the resulting system the sum of all activated systems. A base system may then represent solely the transition for a single ball being in specific state, while the complete system is then a combination of $N$ such systems where $N$ is the number of balls. Practically, this can be achieved by replacing the softmax by a sigmoid activation function or by replacing the categorical variable $s$ of dimension $M$ by $M$ Bernoulli variables indicating whether a single system is active or not. We do this for our multiple agents in a maze environment. ",
+        "bbox": [
+            174,
+            707,
+            825,
+            804
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Theoretically, a preferred approach would be to disentangle multiple systems (like balls, joints) and apply transitions only to their respective states. This, however, would require a proper and unsupervised separation of (mostly) independent components. We defer this to future work. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            853
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D FURTHER RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            370,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.1 3-AGENT MAZE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            331,
+            148
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/a6cbeb7cfa725ccd9d3588936bebbdaf170a4b93f96ae105d0b31e890791f724.jpg",
+        "image_caption": [
+            "Figure 6: Comparison of actual and predicted 20 step trajectories. The diamond marker denotes the starting position of a trajectory. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            356,
+            164,
+            640,
+            376
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.2 IMAGE BALL IN A BOX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            445,
+            379,
+            460
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/591c3c42a554e02486eb4fda365270cf8b98b397e40d966b285967c2ff546d0a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            478,
+            818,
+            801
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 7: First row: data, second row: reconstructions, third row: predictions. The first 4 steps are used to find a stable starting state, predictions start with step 5. ",
+        "bbox": [
+            173,
+            814,
+            823,
+            843
+        ],
+        "page_idx": 13
+    }
+]

parse/train/SJi9WOeRb/SJi9WOeRb.md

@@ -0,0 +1,598 @@
+# GRADIENT ESTIMATORS FOR IMPLICIT MODELS
+
+Yingzhen Li & Richard E. Turner University of Cambridge Cambridge, CB2 1PZ, UK {yl494,ret26}@cam.ac.uk
+
+# ABSTRACT
+
+Implicit models, which allow for the generation of samples but not for point-wise evaluation of probabilities, are omnipresent in real-world problems tackled by machine learning and a hot topic of current research. Some examples include data simulators that are widely used in engineering and scientific research, generative adversarial networks (GANs) for image synthesis, and hot-off-the-press approximate inference techniques relying on implicit distributions. The majority of existing approaches to learning implicit models rely on approximating the intractable distribution or optimisation objective for gradient-based optimisation, which is liable to produce inaccurate updates and thus poor models. This paper alleviates the need for such approximations by proposing the Stein gradient estimator, which directly estimates the score function of the implicitly defined distribution. The efficacy of the proposed estimator is empirically demonstrated by examples that include gradient-free MCMC, meta-learning for approximate inference and entropy regularised GANs that provide improved sample diversity.
+
+# 1 INTRODUCTION
+
+Modelling is fundamental to the success of technological innovations for artificial intelligence. A powerful model learns a useful representation of the observations for a specified prediction task, and generalises to unknown instances that follow similar generative mechanics. A well established area of machine learning research focuses on developing prescribed probabilistic models (Diggle & Gratton, 1984), where learning is based on evaluating the probability of observations under the model. Implicit probabilistic models, on the other hand, are defined by a stochastic procedure that allows for direct generation of samples, but not for the evaluation of model probabilities. These are omnipresent in scientific and engineering research involving data analysis, for instance ecology, climate science and geography, where simulators are used to fit real-world observations to produce forecasting results. Within the machine learning community there is a recent interest in a specific type of implicit models, generative adversarial networks (GANs) (Goodfellow et al., 2014), which has been shown to be one of the most successful approaches to image and text generation (Radford et al., 2016; Yu et al., 2017; Arjovsky et al., 2017; Berthelot et al., 2017). Very recently, implicit distributions have also been considered as approximate posterior distributions for Bayesian inference, e.g. see Liu & Feng (2016); Wang & Liu (2016); Li & Liu (2016); Karaletsos (2016); Mescheder et al. (2017); Huszar (2017); Li et al. (2017); Tran et al. (2017). These examples demonstrate the su-´ perior flexibility of implicit models, which provide highly expressive means of modelling complex data structures.
+
+Whilst prescribed probabilistic models can be learned by standard (approximate) maximum likelihood or Bayesian inference, implicit probabilistic models require substantially more severe approximations due to the intractability of the model distribution. Many existing approaches first approximate the model distribution or optimisation objective function and then use those approximations to learn the associated parameters. However, for any finite number of data points there exists an infinite number of functions, with arbitrarily diverse gradients, that can approximate perfectly the objective function at the training datapoints, and optimising such approximations can lead to unstable training and poor results. Recent research on GANs, where the issue is highly prevalent, suggest that restricting the representational power of the discriminator is effective in stabilising training (e.g. see Arjovsky et al., 2017; Kodali et al., 2017). However, such restrictions often introduce undesirable biases, responsible for problems such as mode collapse in the context of GANs, and the underestimation of uncertainty in variational inference methods (Turner & Sahani, 2011).
+
+![](images/11baac80c9c0e12b40b681e1c4d420fc091d6a6fa494f594a8131dc46053395c.jpg)  
+Figure 1: A comparison between the two approximation schemes. Since in practice the optimiser only visits finite number of locations in the parameter space, it can lead to over-fitting if the neural network based functional approximator is not carefully regularised, and therefore the curvature information of the approximated loss can be very different from that of the original loss (shown in (a)). On the other hand, the gradient approximation scheme (b) can be more accurate since it only involves estimating the sensitivity of the loss function to the parameters in a local region.
+
+In this paper we explore approximating the derivative of the log density, known as the score function, as an alternative method for training implicit models. An accurate approximation of the score function then allows the application of many well-studied algorithms, such as maximum likelihood, maximum entropy estimation, variational inference and gradient-based MCMC, to implicit models. Concretely, our contributions include:
+
+• the Stein gradient estimator, a novel generalisation of the score matching gradient estimator (Hyvarinen, 2005), that includes both parametric and non-parametric forms; ¨ • a comparison of the proposed estimator with the score matching and the KDE plug-in estimators on performing gradient-free MCMC, meta-learning of approximate posterior samplers for Bayesian neural networks, and entropy based regularisation of GANs.
+
+# 2 LEARNING IMPLICIT PROBABILISTIC MODELS
+
+Given a dataset $\mathcal { D }$ containing i.i.d. samples we would like to learn a probabilistic model $p ( { \pmb x } )$ for the underlying data distribution $p _ { \mathcal { D } } ( \pmb { x } )$ . In the case of implicit models, $p ( { \pmb x } )$ is defined by a generative process. For example, to generate images, one might define a generative model $p ( { \pmb x } )$ that consists of sampling randomly a latent variable $z \sim p _ { 0 } ( z )$ and then defining $\mathbf { \boldsymbol { x } } = \mathbf { \boldsymbol { f } } _ { \boldsymbol { \theta } } ( \boldsymbol { z } )$ . Here $f$ is a function parametrised by $\pmb { \theta }$ , usually a deep neural network or a simulator. We assume $f$ to be differentiable w.r.t. $\pmb { \theta }$ . An extension to this scenario is presented by conditional implicit models, where the addition of a supervision signal $\textbf {  { y } }$ , such as an image label, allows us to define a conditional distribution $p ( { \pmb x } | { \pmb y } )$ implicitly by the transformation $\pmb { x } = f _ { \pmb { \theta } } ( \pmb { z } , \pmb { y } )$ . A related methodology, wild variational inference (Liu & Feng, 2016; Li $\&$ Liu, 2016) assumes a tractable joint density $p ( { \pmb x } , { \pmb z } )$ , but uses implicit proposal distributions to approximate an intractable exact posterior $p ( \boldsymbol { z } | \boldsymbol { x } )$ . Here the approximate posterior $q ( \pmb { z } | \pmb { x } )$ can likewise be represented by a deep neural network, but also by a truncated Markov chain, such as that given by Langevin dynamics with learnable step-size.
+
+Whilst providing extreme flexibility and expressive power, the intractability of density evaluation also brings serious optimisation issues for implicit models. This is because many learning algorithms, e.g. maximum likelihood estimation (MLE), rely on minimising a distance/divergence/discrepancy measure $\mathrm { D } [ p | | p _ { \mathcal { D } } ]$ , which often requires evaluating the model density (c.f. Ranganath et al., 2016; Liu & Feng, 2016). Thus good approximations to the optimisation procedure are the key to learning implicit models that can describe complex data structure. In the context of GANs, the Jensen-Shannon divergence is approximated by a variational lower-bound represented by a discriminator (Barber & Agakov, 2003; Goodfellow et al., 2014). Related work for wild variational inference (Li & Liu, 2016; Mescheder et al., 2017; Huszar, 2017; Tran et al., ´ 2017) uses a GAN-based technique to construct a density ratio estimator for $q / p _ { 0 }$ (Sugiyama et al., 2009; 2012; Uehara et al., 2016; Mohamed & Lakshminarayanan, 2016) and then approximates the KL-divergence term in the variational lower-bound:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { V I } } ( q ) = \mathbb { E } _ { q } \left[ \log p ( \pmb { x } | \pmb { z } ) \right] - \mathrm { K L } [ q _ { \phi } ( \pmb { z } | \pmb { x } ) | | p _ { 0 } ( \pmb { z } ) ] . } \end{array}
+$$
+
+In addition, Li & Liu (2016) and Mescheder et al. (2017) exploit the additive structure of the KLdivergence and suggest discriminating between $q$ and an auxiliary distribution that is close to $q$ , making the density ratio estimation more accurate. Nevertheless all these algorithms involve a minimax optimisation, and the current practice of gradient-based optimisation is notoriously unstable.
+
+The stabilisation of GAN training is itself a recent trend of related research (e.g. see Salimans et al., 2016; Arjovsky et al., 2017). However, as the gradient-based optimisation only interacts with gradients, there is no need to use a discriminator if an accurate approximation to the intractable gradients could be obtained. As an example, consider a variational inference task with the approximate posterior defined as $z \sim q _ { \phi } ( z | x ) \stackrel { \cdot } { \Leftrightarrow } \epsilon \sim \pi ( \epsilon ) , z = f _ { \phi } ( \epsilon , x )$ . Notice that the variational lower-bound can be rewritten as
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { V I } } ( q ) = \mathbb { E } _ { q } \left[ \log p ( \pmb { x } , z ) \right] + \mathbb { H } [ q _ { \phi } ( z | \pmb { x } ) ] , } \end{array}
+$$
+
+the gradient of the variational parameters $\phi$ can be computed by a sum of the path gradient of the first term (i.e. $\mathbb { E } _ { \boldsymbol { \pi } } \left[ \nabla _ { f } \log \bar { p } ( \boldsymbol { x } , f ( \boldsymbol { \epsilon } , \boldsymbol { x } ) ) ^ { \mathrm { T } } \nabla _ { \phi } f ( \boldsymbol { \epsilon } , \boldsymbol { x } ) \right] \bar { ) }$ and the gradient of the entropy term $\nabla _ { \phi } \mathbb { H } [ q ( \pmb { z } | \pmb { x } ) ]$ . Expanding the latter, we have
+
+$$
+\begin{array} { r l } & { \nabla _ { \phi } \mathbb { H } [ q _ { \phi } ( z | x ) ] = - \nabla _ { \phi } \mathbb { E } _ { \pi ( \epsilon ) } [ \log { q _ { \phi } ( f _ { \phi } ( \epsilon , x ) ) } ] } \\ & { \quad \quad \quad \quad \quad = - \mathbb { E } _ { \pi ( \epsilon ) } [ \nabla _ { \phi } \log { q _ { \phi } ( f _ { \phi } ( \epsilon , x ) ) } ] } \\ & { \quad \quad \quad \quad = - \mathbb { E } _ { \pi ( \epsilon ) } [ \nabla _ { \phi } \log { q _ { \phi } ( z | x ) } | _ { z = f _ { \phi } ( \epsilon , x ) } + \nabla _ { f } \log { q _ { \phi } ( f _ { \phi } ( \epsilon , x ) | x ) } \nabla _ { \phi } f _ { \phi } ( \epsilon , x ) ] } \\ & { \quad \quad \quad = - \mathbb { E } _ { q _ { \phi } ( z | x ) } [ \nabla _ { \phi } \log { q _ { \phi } ( z | x ) } ] - \mathbb { E } _ { \pi ( \epsilon ) } [ \nabla _ { f } \log { q _ { \phi } ( f _ { \phi } ( \epsilon , x ) | x ) } \nabla _ { \phi } f _ { \phi } ( \epsilon , x ) ] , } \end{array}
+$$
+
+in which the first term in the last line is zero (Roeder et al., 2017). As we typically assume the tractability of $\nabla _ { \phi } f$ , an accurate approximation to $\nabla _ { z } \log { q ( z | x ) }$ would remove the requirement of discriminators, speed-up the learning and obtain potentially a better model. Many gradient approximation techniques exist (Stone, 1985; Fan & Gijbels, 1996; Zhou & Wolfe, 2000; De Brabanter et al., 2013), and in particular, in the next section we will review kernel-based methods such as kernel density estimation (Singh, 1977) and score matching (Hyvarinen, 2005) in more detail, and ¨ motivate the main contribution of the paper.
+
+# 3 GRADIENT APPROXIMATION WITH THE STEIN GRADIENT ESTIMATOR
+
+We propose the Stein gradient estimator as a novel generalisation of the score matching gradient estimator. Before presenting it we first set-up the notation. Column vectors and matrices are boldfaced. The random variable under consideration is $\mathbf { \boldsymbol { x } } \in \mathcal { X }$ with $\mathcal { X } = \mathbb { R } ^ { d \times 1 }$ if not specifically mentioned. To avoid misleading notation we use the distribution $q ( { \pmb x } )$ to derive the gradient approximations for general cases. As Monte Carlo methods are heavily used for implicit models, in the rest of the paper we mainly consider approximating the gradient $\pmb { g } ( \pmb { x } ^ { k } ) : = \nabla _ { \pmb { x } ^ { k } } \mathrm { \tilde { l o g } } q ( \pmb { x } ^ { k } )$ for $\begin{array} { r } { \pmb { x } ^ { k } \sim q ( \pmb { x } ) , k = \hat { 1 } , . . . , K } \end{array}$ . We use $\boldsymbol { x } _ { j } ^ { i }$ to denote the $j$ th element of the ith sample $\mathbf { x } ^ { i }$ . We also denote the matrix form of the collected gradients as $\mathbf { G } : = \left( \nabla _ { \pmb { x } ^ { 1 } } \log q ( \pmb { x } ^ { 1 } ) , \cdots , \nabla _ { \pmb { x } ^ { K } } \log q ( \pmb { x } ^ { K } ) \right) ^ { \mathrm { T } } \in \mathbb { R } ^ { K \times d } .$ , and its approximation $\hat { \mathbf { G } } : = \left( \hat { g } ( \pmb { x } ^ { 1 } ) , \cdots , \hat { g } ( \pmb { x } ^ { K } ) \right) ^ { \mathrm { T } }$ with $\hat { g } ( \pmb { x } ^ { k } ) = \nabla _ { \pmb { x } ^ { k } } \log \hat { q } ( \pmb { x } ^ { k } )$ for some ${ \hat { q } } ( { \pmb x } )$ .
+
+# 3.1 STEIN GRADIENT ESTIMATOR: INVERTING STEIN’S IDENTITY
+
+We start from introducing Stein’s identity that was first developed for Gaussian random variables (Stein, 1972; 1981) then extended to general cases (Gorham $\&$ Mackey, 2015; Liu et al., 2016). Let $\pmb { h } : \mathbb { R } ^ { d \times 1 }  \mathbb { R } ^ { d ^ { \prime } \times 1 }$ be a differentiable multivariate test function which maps $_ { \textbf { \em x } }$ to a column vector $\pmb { h } ( \pmb { x } ) = [ h _ { 1 } ( \pmb { x } ) , h _ { 2 } ( \pmb { x } ) , . . . , h _ { d ^ { \prime } } ( \pmb { x } ) ] ^ { \mathrm { T } }$ . We further assume the boundary condition for $^ { h }$ :
+
+$$
+q ( \pmb { x } ) \pmb { h } ( \pmb { x } ) | _ { \partial \mathcal { X } } = \mathbf { 0 } , \mathrm { ~ o r ~ } \operatorname* { l i m } _ { \pmb { x }  \infty } q ( \pmb { x } ) \pmb { h } ( \pmb { x } ) = 0 \mathrm { ~ i f ~ } \mathcal { X } = \mathbb { R } ^ { d } .
+$$
+
+This condition holds for almost any test function if $q$ has sufficiently fast-decaying tails (e.g. Gaussian tails). Now we introduce Stein’s identity (Stein, 1981; Gorham & Mackey, 2015; Liu et al., 2016)
+
+$$
+\mathbb { E } _ { q } [ { \pmb h } ( { \pmb x } ) \nabla _ { \pmb x } \log q ( { \pmb x } ) ^ { \mathrm { T } } + \nabla _ { \pmb x } { \pmb h } ( { \pmb x } ) ] = { \bf 0 } ,
+$$
+
+in which the gradient matrix term $\nabla _ { \pmb { x } } \pmb { h } ( \pmb { x } ) = \left( \nabla _ { \pmb { x } } h _ { 1 } ( \pmb { x } ) , \cdots , \nabla _ { \pmb { x } } h _ { d ^ { \prime } } ( \pmb { x } ) \right) ^ { \mathrm { T } } \in \mathbb { R } ^ { d ^ { \prime } \times d }$ . This identity can be proved using integration by parts: for the ith row of the matrix $\pmb { h } ( \pmb { x } ) \nabla _ { \pmb { x } } \log q ( \pmb { x } ) ^ { \mathnormal { \mathrm { T } } }$ , we have
+
+$$
+\begin{array} { r l r } {  { \mathbb { E } _ { q } [ h _ { i } ( \pmb { x } ) \nabla _ { \pmb { x } } \log q ( \pmb { x } ) ^ { \mathrm { T } } ] = \int h _ { i } ( \pmb { x } ) \nabla _ { \pmb { x } } q ( \pmb { x } ) ^ { \mathrm { T } } d \pmb { x } } } \\ & { } & { = q ( \pmb { x } ) h _ { i } ( \pmb { x } ) | _ { \partial \mathscr { X } } - \int q ( \pmb { x } ) \nabla _ { \pmb { x } } h _ { i } ( \pmb { x } ) ^ { \mathrm { T } } d \pmb { x } } \\ & { } & { = - \mathbb { E } _ { q } [ \nabla _ { \pmb { x } } h _ { i } ( \pmb { x } ) ^ { \mathrm { T } } ] . } \end{array}
+$$
+
+Observing that the gradient term $\nabla _ { \pmb { x } } \log { q ( \pmb { x } ) }$ of interest appears in Stein’s identity (5), we propose the Stein gradient estimator by inverting Stein’s identity. As the expectation in (5) is intractable, we further approximate the above with Monte Carlo (MC):
+
+$$
+\frac { 1 } { K } \sum _ { k = 1 } ^ { K } - h ( \boldsymbol { x } ^ { k } ) \nabla _ { \boldsymbol { x } ^ { k } } \log q ( \boldsymbol { x } ^ { k } ) ^ { \mathrm { T } } + \mathrm { e r r } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \nabla _ { \boldsymbol { x } ^ { k } } h ( \boldsymbol { x } ^ { k } ) , \quad \boldsymbol { x } ^ { k } \sim q ( \boldsymbol { x } ^ { k } ) ,
+$$
+
+with err $\in \mathbb { R } ^ { d ^ { \prime } \times d }$ the random error due to MC approximation, which has mean 0 and vanishes as $K  + \infty$ . Now by temporarily denoting $\mathbf { H } ~ = ~ \bigl ( h ( x ^ { 1 } ) , \cdot \cdot \cdot , h ( x ^ { K } ) \bigr ) \in \mathbb { R } ^ { d ^ { \prime } \times K } , \quad \overline { { \nabla _ { x } h } } =$ $\begin{array} { r } { \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \nabla _ { \pmb { x } ^ { k } } h ( \pmb { x } ^ { k } ) \in \mathbb { R } ^ { d ^ { \prime } \times d } } \end{array}$ , equation (7) can be rewritten as $\begin{array} { r } { - \frac { 1 } { K } \mathbf { H } \mathbf G + \mathrm { e r r } = \overline { { \nabla _ { \mathbf { x } } h } } } \end{array}$ . Thus we consider a ridge regression method (i.e. adding an $\ell _ { 2 }$ regulariser) to estimate $\mathbf { G }$ :
+
+$$
+\hat { \mathbf { G } } _ { V } ^ { \mathrm { S t e i n } } : = \underset { \hat { \mathbf { G } } \in \mathbb { R } ^ { K \times d } } { \arg \operatorname* { m i n } } | | \overline { { \nabla _ { \mathbf { x } } h } } + \frac { 1 } { K } \mathbf { H } \hat { \mathbf { G } } | | _ { F } ^ { 2 } + \frac { \eta } { K ^ { 2 } } | | \hat { \mathbf { G } } | | _ { F } ^ { 2 } ,
+$$
+
+with $| | \cdot | | _ { F }$ the Frobenius norm of a matrix and $\eta \geq 0$ . Simple calculation shows that
+
+$$
+\hat { \mathbf { G } } _ { V } ^ { \mathrm { S t e i n } } = - ( \mathbf { K } + \eta \mathbf { I } ) ^ { - 1 } \langle \nabla , \mathbf { K } \rangle ,
+$$
+
+where $\mathbf { K } : = \mathbf { H } ^ { \mathrm { T } } \mathbf { H }$ , $\mathbf { K } _ { i j } = { \mathcal { K } } ( { \pmb x } ^ { i } , { \pmb x } ^ { j } ) : = h ( { \pmb x } ^ { i } ) ^ { \mathrm { T } } h ( { \pmb x } ^ { j } )$ , $\langle \nabla , { \bf K } \rangle : = K { \bf H } ^ { \mathrm { T } } \overline { { \nabla _ { \boldsymbol { x } } h } }$ , $\langle \nabla , { \bf K } \rangle _ { i j } =$ $\begin{array} { r l } {  { \sum _ { k = 1 } ^ { K } \nabla _ { x _ { j } ^ { k } } \mathcal { K } ( \pmb { x } ^ { i } , \pmb { x } ^ { k } ) } } & { { } } \end{array}$ . One can show that the RBF kernel satisfies Stein’s identity (Liu et al., 2016). In this case $\pmb { h } ( \pmb { x } ) = \mathcal { K } ( \pmb { x } , \cdot ) , d ^ { \prime } = + \infty$ and by the reproducing kernel property (Berlinet & ThomasAgnan, 2011), $\begin{array} { r } { h ( { \boldsymbol x } ) ^ { \mathrm { T } } h ( { \boldsymbol x } ^ { \prime } ) = \langle \mathcal { K } ( { \boldsymbol x } , \cdot ) , \mathcal { K } ( { \boldsymbol x } ^ { \prime } , \cdot ) \rangle _ { \mathcal { H } } = \mathcal { K } ( { \boldsymbol x } , { \boldsymbol x } ^ { \prime } ) . } \end{array}$ .
+
+# 3.2 STEIN GRADIENT ESTIMATOR MINIMISES THE KERNELISED STEIN DISCREPANCY
+
+In this section we derive the Stein gradient estimator again, but from a divergence/discrepancy minimisation perspective. Stein’s method also provides a tool for checking if two distributions $q ( { \pmb x } )$ and ${ \hat { q } } ( { \pmb x } )$ are identical. If the test function set $\mathcal { H }$ is sufficiently rich, then one can define a Stein discrepancy measure by
+
+$$
+\begin{array} { r } { S ( \boldsymbol { q } , \hat { \boldsymbol { q } } ) : = \displaystyle \operatorname* { s u p } _ { \boldsymbol { h } \in \mathcal { H } } \mathbb { E } _ { \boldsymbol { q } } \left[ \nabla _ { \boldsymbol { x } } \log \hat { q } ( \boldsymbol { x } ) ^ { \mathrm { T } } \boldsymbol { h } ( \boldsymbol { x } ) + \langle \nabla , \boldsymbol { h } \rangle \right] , } \end{array}
+$$
+
+see Gorham & Mackey (2015) for an example derivation. When $\mathcal { H }$ is defined as a unit ball in an RKHS induced by a kernel $\kappa ( { \pmb x } , \cdot )$ , Liu et al. (2016) and Chwialkowski et al. (2016) showed that the supremum in (10) can be analytically obtained as (with ${ \boldsymbol { \kappa } } _ { { \boldsymbol { x } } { \boldsymbol { x } } ^ { \prime } }$ shorthand for $\kappa ( { \pmb x } , { \pmb x } ^ { \prime } ) )$ :
+
+$$
+\mathcal { S } ^ { 2 } ( \boldsymbol { q } , \hat { \boldsymbol { q } } ) = \mathbb { E } _ { \boldsymbol { x } , \boldsymbol { x } ^ { \prime } \sim \boldsymbol { q } } \left[ ( \hat { g } ( \boldsymbol { x } ) - g ( \boldsymbol { x } ) ) ^ { \mathrm { T } } \mathcal { K } _ { \boldsymbol { x } \boldsymbol { x } ^ { \prime } } ( \hat { g } ( \boldsymbol { x } ^ { \prime } ) - \boldsymbol { g } ( \boldsymbol { x } ^ { \prime } ) ) \right] ,
+$$
+
+which is also named the kernelised Stein discrepancy (KSD). Chwialkowski et al. (2016) showed that for $C _ { 0 }$ -universal kernels satisfying the boundary condition, KSD is indeed a discrepancy measure: $S ^ { 2 } ( q , \hat { q } ) = 0 \Leftrightarrow q = \hat { q }$ . Gorham & Mackey (2017) further characterised the power of KSD on detecting non-convergence cases. Furthermore, if the kernel is twice differentiable, then using the same technique as to derive (16) one can compute KSD by
+
+$$
+\begin{array} { r } { \mathcal { S } ^ { 2 } ( \boldsymbol { q } , \boldsymbol { \hat { q } } ) = \mathbb { E } _ { \boldsymbol { x } , \boldsymbol { x } ^ { \prime } \sim \boldsymbol { q } } \left[ \hat { g } ( \boldsymbol { x } ) ^ { \top } \boldsymbol { K } _ { \boldsymbol { x } \boldsymbol { x } ^ { \prime } } \hat { g } ( \boldsymbol { x } ^ { \prime } ) + \hat { g } ( \boldsymbol { x } ) ^ { \top } \nabla _ { \boldsymbol { x } ^ { \prime } } \boldsymbol { K } _ { \boldsymbol { x } \boldsymbol { x } ^ { \prime } } + \nabla _ { \boldsymbol { x } } \boldsymbol { K } _ { \boldsymbol { x } \boldsymbol { x } ^ { \prime } } ^ { \top } \hat { g } ( \boldsymbol { x } ^ { \prime } ) + \mathrm { T r } ( \nabla _ { \boldsymbol { x } , \boldsymbol { x } ^ { \prime } } \boldsymbol { K } _ { \boldsymbol { x } \boldsymbol { x } ^ { \prime } } ) \right] . } \end{array}
+$$
+
+In practice KSD is estimated with samples $\{ \pmb { x } ^ { k } \} _ { k = 1 } ^ { K } \sim q$ , and simple derivations show that the Vstatistic of KSD can be reformulated as $\begin{array} { r } { S _ { V } ^ { 2 } ( q , \hat { q } ) = \frac { 1 } { K ^ { 2 } } \mathrm { T r } ( \hat { \mathbf { G } } ^ { \mathrm { T } } \mathbf { K } \hat { \mathbf { G } } + 2 \hat { \mathbf { G } } ^ { \mathrm { T } } \langle \nabla , \mathbf { K } \rangle ) + C } \end{array}$ . Thus the $l _ { 2 }$ error in (8) is equivalent to the $\mathrm { V } .$ -statistic of KSD if $\mathbf { \dot { h } } ( \mathbf { x } ) = \mathcal { K } ( \mathbf { x } , \cdot )$ , and we have the following:
+
+Theorem 1. $\hat { \mathbf { G } } _ { V } ^ { \mathrm { S t e i n } }$ is the solution of the following KSD V-statistic minimisation problem
+
+$$
+\hat { \mathbf { G } } _ { V } ^ { \mathrm { S t e i n } } = \underset { \hat { \mathbf { G } } \in \mathbb { R } ^ { K \times d } } { \arg \operatorname* { m i n } } S _ { V } ^ { 2 } ( q , \hat { q } ) + \frac { \eta } { K ^ { 2 } } | | \hat { \mathbf { G } } | | _ { F } ^ { 2 } .
+$$
+
+One can also minimise the U-statistic of KSD to obtain gradient approximations, and a full derivation of which, including the optimal solution, can be found in the appendix. In experiments we use $\mathrm { V } .$ - statistic solutions and leave comparisons between these methods to future work.
+
+# 3.3 COMPARISONS TO EXISTING KERNEL-BASED GRADIENT ESTIMATORS
+
+There exist other gradient estimators that do not require explicit evaluations of $\nabla _ { \pmb { x } } \log { q ( \pmb { x } ) }$ , e.g. the denoising auto-encoder (DAE) (Vincent et al., 2008; Vincent, 2011; Alain & Bengio, 2014) which, with infinitesimal noise, also provides an estimate of $\nabla _ { \pmb { x } } \log { q ( \pmb { x } ) }$ at convergence. However, applying such gradient estimators result in a double-loop optimisation procedure since the gradient approximation is repeatedly required for fitting implicit distributions, which can be significantly slower than the proposed approach. Therefore we focus on “quick and dirty” approximations and only include comparisons to kernel-based gradient estimators in the following.
+
+# 3.3.1 KDE GRADIENT ESTIMATOR: PLUG-IN ESTIMATOR WITH DENSITY ESTIMATION
+
+A naive approach for gradient approximation would first estimate the intractable density $\hat { q } ( { \pmb x } ) \approx$ $q ( { \pmb x } )$ (up to a constant), then approximate the exact gradient by $\nabla _ { \pmb { x } } \log \hat { q } ( \pmb { x } ) \approx \nabla _ { \pmb { x } } \log q ( \pmb { \dot { x } } )$ . Specifically, Singh (1977) considered kernel density estimation (KDE) $\begin{array} { r } { \hat { q } ( \pmb { x } ) = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } { K ( \pmb { x } , \pmb { x } ^ { k } ) \times { C } } . } \end{array}$ ., then differentiated through the KDE estimate to obtain the gradient estimator:
+
+$$
+\hat { \mathbf { G } } _ { i j } ^ { \mathrm { K D E } } = \sum _ { k = 1 } ^ { K } \nabla _ { x _ { j } ^ { i } } K ( { \pmb x } ^ { i } , { \pmb x } ^ { k } ) / \sum _ { k = 1 } ^ { K } K ( { \pmb x } ^ { i } , { \pmb x } ^ { k } ) .
+$$
+
+Interestingly for translation invariant kernels $\begin{array} { r } { K ( \pmb { x } , \pmb { x } ^ { \prime } ) = K ( \pmb { x } - \pmb { x } ^ { \prime } ) } \end{array}$ the $K D E$ gradient estimator (14) can be rewritten as $\hat { \mathbf { G } } ^ { \mathrm { K D E } } = - \mathrm { d i a g } \left( \mathbf { K 1 } \right) ^ { - 1 } \langle \nabla , \mathbf { K } \rangle$ . Inspecting and comparing it with the Stein gradient estimator (9), one might notice that the Stein method uses the full kernel matrix as the pre-conditioner, while the KDE method computes an averaged “kernel similarity” for the denominator. We conjecture that this difference is key to the superior performance of the Stein gradient estimator when compared to the KDE gradient estimator (see later experiments). The KDE method only collects the similarity information between $\scriptstyle { \boldsymbol { x } } ^ { k }$ and other samples $\bar { \boldsymbol { x } } ^ { j }$ to form an estimate of $\nabla _ { \pmb { x } ^ { k } } \log \mathbf { \dot { q } } ( \pmb { x } ^ { k } )$ , whereas for the Stein gradient estimator, the kernel similarity between $\mathbf { \Delta } _ { \mathbf { \boldsymbol { x } } ^ { i } }$ and $\mathbf { \boldsymbol { x } } ^ { j }$ for all $i , j \neq k$ are also incorporated. Thus it is reasonable to conjecture that the Stein method can be more sample efficient, which also implies higher accuracy when the same number of samples are collected.
+
+# 3.3.2 SCORE MATCHING GRADIENT ESTIMATOR: MINIMISING MSE
+
+The KDE gradient estimator performs indirect approximation of the gradient via density estimation, which can be inaccurate. An alternative approach directly approximates the gradient $\nabla _ { \pmb { x } } \log { q ( \pmb { x } ) }$ by minimising the expected $\ell _ { 2 }$ error w.r.t. the approximation $\hat { \pmb g } ( { \pmb x } ) = \left( \hat { g } _ { 1 } ( { \pmb x } ) , \cdots , \hat { g } _ { d } ( { \pmb x } ) \right) ^ { \top }$ :
+
+$$
+\mathcal { F } ( \pmb { \hat { g } } ) : = \mathbb { E } _ { q } \left[ | | \pmb { \hat { g } } ( \pmb { x } ) - \nabla _ { \pmb { x } } \log q ( \pmb { x } ) | | _ { 2 } ^ { 2 } \right] .
+$$
+
+It has been shown in Hyvarinen (2005) that this objective can be reformulated as ¨
+
+$$
+\mathcal { F } ( \hat { \pmb g } ) = \mathbb { E } _ { q } \left[ | | \hat { \pmb g } ( { \pmb x } ) | | _ { 2 } ^ { 2 } + 2 \langle \nabla , \hat { \pmb g } ( { \pmb x } ) \rangle \right] + C , \quad \langle \nabla , \hat { \pmb g } ( { \pmb x } ) \rangle = \sum _ { j = 1 } ^ { d } \nabla _ { { \pmb x } _ { j } } \hat { g } _ { j } ( { \pmb x } ) .
+$$
+
+The key insight here is again the usage of integration by parts: after expanding the $\ell _ { 2 }$ loss objective, the cross term can be rewritten as $\bar { \mathbb { E } } _ { q } \left[ \hat { \pmb { g } } ( \pmb { x } ) ^ { \top } \nabla _ { \pmb { x } } \log \bar { \pmb { q } } ( \mathbf { \bar { x } } ) \right] = - \mathbb { E } _ { q } \left[ \langle \nabla , \hat { \pmb { g } } ( \mathbf { \bar { x } } ) \rangle \right]$ , if assuming the boundary condition (4) for $\hat { \pmb { g } }$ (see (6)). The optimum of (16) is referred as the score matching gradient estimator. The $\ell _ { 2 }$ objective (15) is also called Fisher divergence (Johnson, 2004) which is a special case of KSD (11) by selecting $\begin{array} { r } { K ( \pmb { x } , \pmb { x } ^ { \prime } ) = \delta _ { \pmb { x } = \pmb { x } ^ { \prime } } . } \end{array}$ . Thus the Stein gradient estimator can be viewed as a generalisation of the score matching estimator.
+
+The comparison between the two estimators is more complicated. Certainly by the Cauchy-Schwarz inequality the Fisher divergence is stronger than KSD in terms of detecting convergence (Liu et al., 2016). However it is difficult to perform direct gradient estimation by minimising the Fisher divergence, since (i) the Dirac kernel is non-differentiable so that it is impossible to rewrite the divergence in a similar form to (12), and (ii) the transformation to (16) involves computing $\nabla _ { \pmb { x } } \hat { \pmb { g } } ( \pmb { x } )$ . So one needs to propose a parametric approximation to $\mathbf { G }$ and then optimise the associated parameters accordingly, and indeed Sasaki et al. (2014) and Strathmannby first approximating the log density up to a constant as $\begin{array} { r } { \log \hat { q } ( \pmb { x } ) : = \sum _ { k = 1 } ^ { K } a _ { k } \mathcal { K } ( \pmb { x } , \pmb { x } ^ { k } ) + C } \end{array}$ lution, then minimising (16) to obtain the coefficients and constructing the gradient estimator as
+
+$$
+\hat { \mathbf { G } } _ { i \cdot } ^ { \mathrm { s c o r e } } = \sum _ { k = 1 } ^ { K } \hat { a } _ { k } ^ { \mathrm { s c o r e } } \nabla _ { \pmb { x } ^ { i } } { K } ( { \pmb { x } } ^ { i } , { \pmb { x } } ^ { k } ) .
+$$
+
+Therefore the usage of parametric estimation can potentially remove the advantage of using a stronger divergence. Conversely, the proposed Stein gradient estimator (9) is non-parametric in that it directly optimises over functions evaluated at locations $\{ \pmb { x } _ { k } \} _ { k = 1 } ^ { K }$ . This brings in two key advantages over the score matching gradient estimator: (i) it removes the approximation error due to the use of restricted family of parametric approximations and thus can be potentially more accurate; (ii) it has a much simpler and ubiquitous form that applies to any kernel satisfying the boundary condition, whereas the score matching estimator requires tedious derivations for different kernels repeatedly (see appendix).
+
+In terms of computation speed, since in most of the cases the computation of the score matching gradient estimator also involves kernel matrix inversions, both estimators are of the same order of complexity, which is $\mathcal { O } ( K ^ { 3 } + K ^ { 2 } d )$ (kernel matrix computation plus inversion). Low-rank approximations such as the Nystrom method (Smola & Sch ¨ okopf, 2000; Williams & Seeger, 2001) can ¨ enable speed-up, but this is not investigated in the paper. Again we note here that kernel-based gradient estimators can still be faster than e.g. the DAE estimator since no double-loop optimisation is required. Certainly it is possible to apply early-stopping for the inner-loop DAE fitting. However the resulting gradient approximation might be very poor, which leads to unstable training and poorly fitted implicit distributions.
+
+# 3.4 ADDING PREDICTIVE POWER
+
+Though providing potentially more accurate approximations, the non-parametric estimator (9) has no predictive power as described so far. Crucially, many tasks in machine learning require predicting gradient functions at samples drawn from distributions other than $q$ , for example, in MLE $q ( { \pmb x } )$ corresponds to the model distribution which is learned using samples from the data distribution instead. To address this issue, we derive two predictive estimators, one generalised from the nonparametric estimator and the other minimises KSD using parametric approximations.
+
+Predictions using the non-parametric estimator. Let us consider an unseen datum $\textbf {  { y } }$ . If $\textbf {  { y } }$ is sampled from $q$ , then one can also apply the non-parametric estimator (9) for gradient approximation, given the observed data ${ \bf X } = \{ { \pmb x } ^ { \mathrm { i } } , . . . , { \pmb x } ^ { K } \} \sim \dot { { \boldsymbol q } } ^ { }$ . Concretely, if writing $\hat { \pmb g } ( \pmb y ) \overset { } { \approx } \nabla _ { \pmb y } \log \hat { q } ( \pmb y ) \in \mathbb R ^ { d \times 1 }$ then the non-parametric Stein gradient estimator computed on $\mathbf { X } \cup \{ y \}$ is
+
+$$
+\begin{array} { r } { \left[ \hat { g } ( y ) ^ { \mathrm { T } } \right] = - ( \mathbf { K } ^ { * } + \eta I ) ^ { - 1 } \left[ \nabla _ { y } K ( y , y ) + \sum _ { k = 1 } ^ { K } \nabla _ { x ^ { k } } K ( y , x ^ { k } ) \right] , \quad \mathbf { K } ^ { * } = \left[ \mathbf { K } _ { y y } \quad \mathbf { K } _ { y \mathbf { X } } \right] , } \\ { \left. \nabla , \mathbf { K } \right. + \nabla _ { y } K ( \cdot , y ) \qquad \quad \mathbf { K } ^ { * } = \left[ \mathbf { K } _ { \mathbf { X } y } \quad \mathbf { K } \right] , } \end{array}
+$$
+
+with $\nabla _ { \pmb { y } } K ( \cdot , \pmb { y } )$ denoting a $K \times d$ matrix with rows $\nabla _ { \pmb { y } } K ( \pmb { x } ^ { k } , \pmb { y } )$ , and $\nabla _ { \pmb { y } } K ( \pmb { y } , \pmb { y } )$ only differentiates through the second argument. Then we demonstrate in the appendix that, by simple matrix calculations and assuming a translation invariant kernel, we have (with column vector $\mathbf { 1 } \in \mathbb { R } ^ { K \times 1 }$ ):
+
+$$
+\begin{array} { r l } & { \nabla _ { y } \log q ( \pmb { y } ) ^ { \operatorname { T } } \approx - \left( \mathbf { K } _ { y y } + \eta - \mathbf { K } _ { y \mathbf { X } } ( \mathbf { K } + \eta \mathbf { I } ) ^ { - 1 } \mathbf { K } _ { \mathbf { X } y } \right) ^ { - 1 } } \\ & { \qquad \left( \mathbf { K } _ { y \mathbf { X } } \hat { \mathbf { G } } _ { V } ^ { \mathrm { { S t e i n } } } - \left( \mathbf { K } _ { y \mathbf { X } } ( \mathbf { K } + \eta \mathbf { I } ) ^ { - 1 } + \mathbf { 1 } ^ { \operatorname { T } } \right) \nabla _ { y } \mathcal { K } ( \cdot , y ) \right) . } \end{array}
+$$
+
+In practice one would store the computed gradient $\hat { \mathbf { G } } _ { V } ^ { \mathrm { S t e i n } }$ , the kernel matrix inverse $( \mathbf { K } + \eta \mathbf { I } ) ^ { - 1 }$ and $\eta$ as the “parameters” of the predictive estimator. For a new observation $\mathbf { \nabla } _ { \mathbf { \boldsymbol { y } } } \sim \mathbf { \nabla } _ { p }$ in general, one can “pretend” $\textbf {  { y } }$ is a sample from $q$ and apply the above estimator as well. The approximation quality depends on the similarity between $q$ and $p$ , and we conjecture here that this similarity measure, if can be described, is closely related to the KSD.
+
+Fitting a parametric estimator using KSD. The non-parametric predictive estimator could be computationally demanding. Setting aside the cost of fitting the “parameters”, in prediction the time complexity for the non-parametric estimator is $\mathcal { O } ( K ^ { 2 } + K d )$ . Also storing the “parameters” needs $\mathcal O ( K d )$ memory for $\hat { \mathbf { G } } _ { V } ^ { \mathrm { S t e i n } }$ . These costs make the non-parametric estimator undesirable for high-dimensional data, since in order to obtain accurate predictions it often requires $K$ scaling with $d$ as well. To address this, one can also minimise the KSD using parametric approximations, in a similar way as to derive the score matching estimator in Section 3.3.2. More precisely, we define a parametric approximation in a similar fashion as (17), and in the appendix we show that if the RBF kernel is used for both the KSD and the parametric approximation, then the linear coefficients $\pmb { a } = ( a _ { 1 } , . . . , a _ { K } ) ^ { \mathrm { T } }$ can be calculated analytically: $\hat { \pmb { a } } _ { V } ^ { \mathrm { S t e i n } } = \bar { ( \pmb { \Lambda } } + \eta \mathbf { I } ) ^ { - 1 } \pmb { b }$ , where
+
+$$
+\begin{array} { r l } & { \mathbf { A } = \mathbb { X } \odot ( \mathbf { K } \mathbf { K } \mathbf { K } ) + \mathbf { K } ( \mathbf { K } \odot \mathbb { X } ) \mathbf { K } - ( ( \mathbf { K } \mathbf { K } ) \odot \mathbb { X } ) \mathbf { K } - \mathbf { K } ( ( \mathbf { K } \mathbf { K } ) \odot \mathbb { X } ) , } \\ & { \mathbf { \Phi } b = ( \mathbf { K } \mathrm { d i a g } ( \mathbb { X } ) \mathbf { K } + ( \mathbf { K } \mathbf { K } ) \odot \mathbb { X } - \mathbf { K } ( \mathbf { K } \odot \mathbb { X } ) - ( \mathbf { K } \odot \mathbb { X } ) \mathbf { K } ) \mathbf { 1 } , } \end{array}
+$$
+
+with $\mathbb { X }$ the “gram matrix” that has elements $\mathbb X _ { i j } = ( { \pmb x } ^ { i } ) ^ { \mathrm T } { \pmb x } ^ { j }$ . Then for an unseen observation $\mathbf { \mu } _ { \mathbf { \mu } _ { y } \sim }$ $p$ the gradient approximation returns $\nabla _ { \pmb { y } } \log q ( \pmb { y } ) \approx ( \hat { \pmb { a } } _ { V } ^ { \mathrm { S t e i n } } ) ^ { \mathrm { T } } \nabla _ { \pmb { y } } \mathcal { K } ( \cdot , \pmb { y } )$ . In this case one only maintains the linear coefficients $\hat { \pmb { a } } _ { V } ^ { \mathrm { S t e i n } }$ and computes a linear combination in prediction, which takes $\mathcal O ( K )$ memory and $\mathcal O ( K d )$ time and therefore is computationally cheaper than the non-parametric prediction model (27).
+
+# 4 APPLICATIONS
+
+We present some case studies that apply the gradient estimators to implicit models. Detailed settings (architecture, learning rate, etc.) are presented in the appendix. Implementation is released at https://github.com/YingzhenLi/SteinGrad.
+
+# 4.1 SYNTHETIC EXAMPLE: HAMILTONIAN FLOW WITH APPROXIMATE GRADIENTS
+
+We first consider a simple synthetic example to demonstrate the accuracy of the proposed gradient estimator. More precisely we consider the kernel induced Hamiltonian flow (not an exact sampler) (Strathmann et al., 2015) on a 2-dimensional banana-shaped object: $\mathbf { \hat { x } } \ \sim \ B ( \mathbf { x } ; b \ = \ 0 . 0 3 , \bar { \upsilon } \ =$ $1 0 0 ) \Leftrightarrow x _ { 1 } \sim \mathcal { N } ( x _ { 1 } ; 0 , v ) , x _ { 2 } = \epsilon + b ( x _ { 1 } ^ { 2 } - v ) , \epsilon \sim \mathcal { N } ( \epsilon ; 0 , 1 )$ . The approximate Hamiltonian flow is constructed using the same operator as in Hamiltonian Monte Carlo (HMC) (Neal et al., 2011), except that the exact score function $\nabla _ { \pmb { x } } \log B ( \pmb { x } )$ is replaced by the approximate gradients. We still use the exact target density to compute the rejection step as we mainly focus on testing the accuracy of the gradient estimators. We test both versions of the predictive Stein gradient estimator (see section 3.4) since we require the particles of parallel chains to be independent with each other. We fit the gradient estimators on $K = 2 0 0$ training datapoints from the target density. The bandwidth of the RBF kernel is computed by the median heuristic and scaled up by a scalar between [1, 5]. All three methods are simulated for $T = 2 , 0 0 0$ iterations, share the same initial locations that are constructed by target distribution samples plus Gaussian noises of standard deviation 2.0, and the results are averaged over 200 parallel chains.
+
+We visualise the samples and some MCMC statistics in Figure 2. In general all the resulting Hamiltonian flows are HMC-like, which give us the confidence that the gradient estimators extrapolate reasonably well at unseen locations. However all of these methods have trouble exploring the extremes, because at those locations there are very few or even no training data-points. Indeed we found it necessary to use large (but not too large) bandwidths, in order to both allow exploration of those extremes, and ensure that the corresponding test function is not too smooth. In terms of quantitative metrics, the acceptance rates are reasonably high for all the gradient estimators, and the KSD estimates (across chains) as a measure of sample quality are also close to that computed on HMC samples. The returned estimates of $\mathbb { E } [ x _ { 1 } ]$ are close to zero which is the ground true value. We found that the non-parametric Stein gradient estimator is more sensitive to hyper-parameters of the dynamics, e.g. the stepsize of each HMC step. We believe a careful selection of the kernel (e.g. those with long tails) and a better search for the hyper-parameters (for both the kernel and the dynamics) can further improve the sample quality and the chain mixing time, but this is not investigated here.
+
+![](images/bba07b55fa6904729682848bba8c5a5e2202a58574d1f75da2568b525bbfeed3.jpg)  
+Figure 2: Kernel induced Hamiltonian flow compared with HMC. Top: samples generated from the dynamics, training data (in cyan), an the trajectory of a particle for $T = 1$ to 200 starting at the star location (in yellow). Bottom: statistics computed during simulations. See main text for details.
+
+# 4.2 META-LEARNING OF APPROXIMATE POSTERIOR SAMPLERS FOR BAYESIAN NNS
+
+One of the recent focuses on meta-learning has been on learning optimisers for training deep neural networks, e.g. see (Andrychowicz et al., 2016). Could analogous goals be achieved for approximate inference? In this section we attempt to learn an approximate posterior sampler for Bayesian neural networks (Bayesian NNs, BNNs) that generalises to unseen datasets and architectures. A more detailed introduction of Bayesian neural networks is included in the appendix, and in a nutshell, we consider a binary classification task: $p ( y = 1 | x , \pmb { \theta } ) = \mathrm { s i g m o i d } ( \mathrm { N N } _ { \pmb { \theta } } ( \pmb { x } ) )$ , $p _ { 0 } ( \pmb { \theta } ) = \mathcal { N } ( \pmb { \theta } ; \mathbf { 0 } , \mathbf { I } )$ . After observing the training data $\boldsymbol { \mathcal { D } } = \{ ( \boldsymbol { x } _ { n } , y _ { n } ) \} _ { n = 1 } ^ { N }$ , we first obtain the approximate posterior $\begin{array} { r } { q _ { \phi } ( \pmb { \theta } ) \approx p ( \pmb { \theta } | \mathcal { D } ) \propto p _ { 0 } ( \pmb { \theta } ) \prod _ { n = 1 } ^ { N } p ( y _ { n } | \pmb { x } _ { n } , \pmb { \theta } ) } \end{array}$ , then approximate the predictive distribution for a new observation as $\begin{array} { r } { p ( y ^ { * } = 1 | x ^ { * } , \mathcal { D } ) \approx \frac { 1 } { K } \sum _ { k = 1 } ^ { K } p ( y ^ { * } = 1 | x ^ { * } , \pmb { \theta } ^ { k } ) , \pmb { \theta } ^ { k } \sim q _ { \phi } ( \pmb { \theta } ) . } \end{array}$ In this task we define an implicit approximate posterior distribution $q _ { \phi } ( \pmb \theta )$ as the following stochastic normalising flow (Rezende & Mohamed, 2015) $\pmb { \theta } _ { t + 1 } = \pmb { f } ( \pmb { \theta } _ { t } , \nabla _ { t } , \pmb { \epsilon } _ { t } )$ : given the current location $\theta _ { t }$ and the mini-batch data $\{ ( \pmb { x } _ { m } , y _ { m } ) \} _ { m = 1 } ^ { M }$ , the update for the next step is
+
+$$
+\begin{array} { r l r } & { \pmb { \theta } _ { t + 1 } = \pmb { \theta } _ { t } + \zeta \Delta _ { \phi } ( \pmb { \theta } _ { t } , \nabla _ { t } ) + \pmb { \sigma } _ { \phi } ( \pmb { \theta } _ { t } , \nabla _ { t } ) \odot \epsilon _ { t } , \quad \epsilon _ { t } \sim \mathcal { N } ( \epsilon ; \mathbf { 0 } , \mathbf { I } ) , } & \\ & { \nabla _ { t } = \nabla _ { \pmb { \theta } _ { t } } \left[ \frac { N } { M } \displaystyle \sum _ { m = 1 } ^ { M } \log p \big ( y _ { m } | \pmb { x } _ { m } , \pmb { \theta } _ { t } \big ) + \log p _ { 0 } ( \pmb { \theta } _ { t } ) \right] . } & \end{array}
+$$
+
+The coordinates of the noise standard deviation $\sigma _ { \phi } ( \theta _ { t } , \nabla _ { t } )$ and the moving direction $\Delta _ { \phi } ( \theta _ { t } , \nabla _ { t } )$ are parametrised by a coordinate-wise neural network. If properly trained, this neural network will learn the best combination of the current location and gradient information, and produce approximate posterior samples efficiently on different probabilistic modelling tasks. Here we propose using the variational inference objective (2) computed on the samples $\{ \pmb \theta _ { t } ^ { k } \}$ to learn the variational parameters $\phi$ . Since in this case the gradient of the log joint distribution can be computed analytically, we only approximate the gradient of the entropy term $\mathbb { H } [ q ]$ as in (3), with the exact score function replaced by the presented gradient estimators. We report the results using the non-parametric Stein gradient estimator as we found it works better than the parametric version. The RBF kernel is applied for gradient estimation, with the hyper-parameters determined by a grid search on the bandwidth $\sigma ^ { 2 } \overset { \smile } { \in } \{ 0 . 2 5 , 1 . 0 , 4 . 0 , 1 0 . 0$ , median trick} and $\eta \in \{ 0 . 1 , 0 . 5 , 1 . 0 , 2 . 0 \}$ .
+
+We briefly describe the test protocol. We take from the UCI repository (Lichman, 2013) six binary classification datasets (australian, breast, crabs, ionosphere, pima, sonar), train an approximate sampler on crabs with a small neural network that has one 20-unit hidden layer with $R e L U$ activation, and generalise to the remaining datasets with a bigger network that has 50 hidden units and uses sigmoid activation. We use ionosphere as the validation set to tune $\zeta$ . The remaining 4 datasets are further split into $40 \%$ training subset for simulating samples from the approximate sampler, and $60 \%$ test subsets for evaluating the sampler’s performance.
+
+Figure 3 presents the (negative) test log-likelihood (LL), classification error, and an estimate of the KSD U-statistic $\tilde { S _ { U } ^ { 2 } } ( \tilde { p ( \pmb { \theta } | \mathcal { D } ) } , q ( \pmb { \theta } ) )$ (with data sub-sampling) over 5 splits of each test dataset. Besides the gradient estimators we also compare with two baselines: an approximate posterior sampler trained by maximum a posteriori (MAP), and stochastic gradient Langevin dynamics (SGLD)
+
+![](images/acf7f7eec3f062f3f3268dc6c9364e38d43410d9404f6402f27ccd8ac48c052c.jpg)  
+Figure 3: Generalisation performances for trained approximate posterior samplers.
+
+(Welling & Teh, 2011) evaluated on the test datasets directly. In summary, SGLD returns best results in KSD metric. The Stein approach performs equally well or a little better than SGLD in terms of test-LL and test error. The KDE method is slightly worse and is close to MAP, indicating that the KDE estimator does not provide a very informative gradient for the entropy term. Surprisingly the score matching estimator method produces considerably worse results (except for breast dataset), even after carefully tuning the bandwidth and the regularisation parameter $\eta$ . Future work should investigate the usage of advanced recurrent neural networks such as an LSTM (Hochreiter & Schmidhuber, 1997), which is expected to return better performance.
+
+# 4.3 TOWARDS ADDRESSING MODE COLLAPSE IN GANS USING ENTROPY REGULARISATION
+
+GANs are notoriously difficult to train in practice. Besides the instability of gradient-based minimax optimisation which has been partially addressed by many recent proposals (Salimans et al., 2016; Arjovsky et al., 2017; Berthelot et al., 2017), they also suffer from mode collapse. We propose adding an entropy regulariser to the GAN generator loss. Concretely, assume the generative model $p _ { \pmb { \theta } } ( \pmb { x } )$ is implicitly defined by $\pmb { x } = f _ { \pmb { \theta } } ( z ) , z \sim p _ { 0 } ( z )$ , then the generator’s loss is defined by
+
+$$
+\tilde { \mathcal { I } } _ { \mathrm { g e n } } ( \pmb { \theta } ) = \mathcal { I } _ { \mathrm { g e n } } ( \pmb { \theta } ) - \alpha \mathbb { H } [ p _ { \pmb { \theta } } ( \pmb { x } ) ] ,
+$$
+
+where $\mathcal { I } _ { \mathrm { g e n } } ( \pmb { \theta } )$ is the original loss function for the generator from any GAN algorithm and $\alpha$ is a hyper-parameter. In practice (the gradient of) (21) is estimated using Monte Carlo.
+
+We empirically investigate the entropy regularisation idea on the very recently proposed boundary equilibrium GAN (BEGAN) (Berthelot et al., 2017) method using (continuous) MNIST, and we refer to the appendix for the detailed mathematical set-up. In this case the non-parametric V-statistic Stein gradient estimator is used. We use a convolutional generative network and a convolutional auto-encoder and select the hyper-parameters of BEGAN $\bar { \gamma } \in \lbrace 0 . 3 , 0 . 5 , 0 . 7 \rbrace$ , $\alpha \in [ 0 , 1 ]$ and $\lambda =$ 0.001. The Epanechnikov kernel $\begin{array} { r } { K ( \pmb { x } , \pmb { x } ^ { \prime } ) : = \frac { 1 } { d } \sum _ { j = 1 } ^ { d } ( 1 - ( x _ { j } - x _ { j } ^ { \prime } ) ^ { 2 } ) } \end{array}$ is used as the pixel values lie in a unit interval (see appendix for the expression of the score matching estimator), and to ensure the boundary condition we clip the pixel values into range $[ 1 0 ^ { - 8 } , 1 - 1 0 ^ { - 8 } ]$ . The generated images are visualised in Figure 4. BEGAN without the entropy regularisation fails to generate diverse samples even when trained with learning rate decay. The other three images clearly demonstrate the benefit of the entropy regularisation technique, with the Stein approach obtaining the highest diversity without compromising visual quality.
+
+We further consider four metrics to assess the trained models quantitatively. First 500 samples are generated for each trained model, then we compute their nearest neighbours in the training set using $l _ { 1 }$ distance, and obtain a probability vector $\mathbf { p }$ by averaging over these neighbour images’ label vectors. In Figure 5 we depict the entropy of $\mathbf { p }$ (top left), averaged $l _ { 1 }$ distances to the nearest neighbour (top right), and the difference between the largest and smallest elements in $\mathbf { p }$ (bottom right). The error bars are obtained by 5 independent runs. These results demonstrate that the Stein approach performs significantly better than the other two, in that it learns a better generative model not only faster but also in a more stable way. Interestingly the KDE approach achieves the lowest average $l _ { 1 }$ distance to nearest neighbours, possibly because it tends to memorise training examples. We next train a fully connected network $\pi ( \boldsymbol { y } | \boldsymbol { x } )$ on MNIST that achieves $9 8 . 1 6 \%$ text accuracy, and compute on the generated images an empirical estimate of the inception score (Salimans et al., 2016) $\mathbb { E } _ { p ( \pmb { x } ) } [ \mathrm { K L } [ \pi ( \pmb { \bar { y } } | \pmb { x } ) | | \pi ( \pmb { y } ) ] ]$ with $\pi ( \pmb { y } ) \overset { \cdot } { = } \mathbb { E } _ { p ( \pmb { x } ) } [ \pi ( \pmb { y } | \pmb { x } ) ]$ (bottom left panel). High inception score indicates that the generate images tend to be both realistic looking and diverse, and again the Stein approach out-performs the others on this metric by a large margin.
+
+![](images/711c07aaa71533b03397353e9ad56fce9f23a20871bf052140a2ebe313f973cd.jpg)  
+Figure 4: Visualisation of generated images from trained BEGAN models.
+
+![](images/ae918fada81b3ecc38c292fbb7e2cc1cd5bcc0e7605ea4731056d367bd6ae0fe.jpg)  
+Figure 5: Quantitative evaluation on entropy regularised BEGAN. The higher the better for the LHS panels and the other way around for the RHS ones. See main text for details.
+
+Concerning computation speed, all the three methods are of the same order: 10.20s/epoch for KDE, 10.85s/epoch for Score, and 10.30s/epoch for Stein.1 This is because $K < d$ (in the experiments $K ~ = ~ 1 0 0$ and $d \ = \ 7 8 4$ ) so that the complexity terms are dominated by kernel computations $( \mathcal { O } ( K ^ { 2 } d ) )$ required by all the three methods. Also for a comparison, the original BEGAN method without entropy regularisation runs for 9.05s/epoch. Therefore the main computation cost is dominated by the optimisation of the discriminator/generator, and the proposed entropy regularisation can be applied to many GAN frameworks with little computational burden.
+
+# 5 CONCLUSIONS AND FUTURE WORK
+
+We have presented the Stein gradient estimator as a novel generalisation to the score matching gradient estimator. With a focus on learning implicit models, we have empirically demonstrated the efficacy of the proposed estimator by showing how it opens the door to a range of novel learning tasks: approximating gradient-free MCMC, meta-learning for approximate inference, and unsupervised learning for image generation. Future work will expand the understanding of gradient estimators in both theoretical and practical aspects. Theoretical development will compare both the V-statistic and U-statistic Stein gradient estimators and formalise consistency proofs. Practical work will improve the sample efficiency of kernel estimators in high dimensions and develop fast yet accurate approximations to matrix inversion. It is also interesting to investigate applications of gradient approximation methods to training implicit generative models without the help of discriminators. Finally it remains an open question that how to generalise the Stein gradient estimator to non-kernel settings and discrete distributions.
+
+# ACKNOWLEDGEMENT
+
+We thank Marton Havasi, Jiri Hron, David Janz, Qiang Liu, Maria Lomeli, Cuong Viet Nguyen and Mark Rowland for their comments and helps on the manuscript. We also acknowledge the anonymous reviewers for their review. Yingzhen Li thanks Schlumberger Foundation FFTF fellowship. Richard E. Turner thanks Google and EPSRC grants EP/M0269571 and EP/L000776/1.
+
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+# A SCORE MATCHING ESTIMATOR: REMARKS AND DERIVATIONS
+
+In this section we provide more discussions and analytical solutions for the score matching estimator. More specifically, we will derive the linear coefficient $\pmb { a } = ( a _ { 1 } , . . . , a _ { K } )$ for the case of the Epanechnikov kernel.
+
+# A.1 SOME REMARKS ON SCORE MATCHING
+
+Remark. It has been shown in Sarel ¨ a & Valpola (2005); Alain & Bengio (2014) that de-noising auto- ¨ encoders (DAEs) (Vincent et al., 2008), once trained, can be used to compute the score function approximately. Briefly speaking, a DAE learns to reconstruct a datum $_ { \textbf { \em x } }$ from a corrupted input $\tilde { \mathbf { x } } = x { + } \sigma \epsilon , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ by minimising the mean square error. Then the optimal DAE can be used to approximate the score function as $\begin{array} { r } { \nabla _ { \pmb { x } } \log p ( \pmb { x } ) \approx \frac { 1 } { \sigma ^ { 2 } } ( \mathrm { { D A E ^ { * } } } ( \pmb { x } ) - \pmb { x } ) } \end{array}$ . Sonderby et al. (2017) applied this idea to train an implicit model for image super-resolution, providing some promising results in some metrics. However applying similar ideas to variational inference can be computationally expensive, because the estimation of $\nabla _ { z } \log { q ( z | x ) }$ is a sub-routine for VI which is repeatedly required. Therefore in the paper we deploy kernel machines that allow analytical solutions to the score matching estimator in order to avoid double loop optimisation.
+
+Remark. As a side note, score matching can also be used to learn the parameters of an unnormalised density. In this case the target distribution $q$ would be the data distribution and $\hat { q }$ is often a Boltzmann distribution with intractable partition function. As a parameter estimation technique, score matching is also related to contrastive divergence (Hinton, 2002), pseudo likelihood estimation (Hyvarinen, 2006), and DAEs (Vincent, 2011; Alain & Bengio, 2014). Generalisations of score ¨ matching methods are also presented in e.g. Lyu (2009); Marlin et al. (2010).
+
+# A.2 THE RBF KERNEL CASE
+
+The derivations for the RBF kernel case is referred to (Strathmann et al., 2015), and for com$\begin{array} { r } { \log \hat { q } ( \pmb { x } ) = \sum _ { k = 1 } ^ { K } a _ { k } \mathcal { K } ( \pmb { x } , \pmb { x } ^ { k } ) + C } \end{array}$ e parametric approximation iel uses bandwidth parameter $\sigma$ defined as. then the $\hat { \pmb { a } } ^ { \mathrm { s c o r e } } = ( \pmb { \Sigma } + \eta \mathbf { I } ) ^ { - 1 } \pmb { v } ,$
+
+$$
+v = \sum _ { i = 1 } ^ { d } \left[ \sigma ^ { 2 } \mathbf { K } \mathbf { 1 } - \left( \mathbf { K } ( \mathbf { x } _ { i } \odot \mathbf { x } _ { i } ) + \mathrm { { d i a g } } ( \mathbf { x } _ { i } ) \mathbf { K } \mathbf { 1 } - 2 { \mathrm { d i a g } } ( \mathbf { x } _ { i } ) \mathbf { K } \mathbf { x } _ { i } \right) \right] ,
+$$
+
+$$
+\pmb { \Sigma } = \sum _ { i = 1 } ^ { d } \left[ \mathrm { d i a g } ( \mathbf { x } _ { i } ) \mathbf { K } - \mathbf { K } \mathrm { d i a g } ( \mathbf { x } _ { i } ) \right] \left[ \mathbf { K } \mathrm { d i a g } ( \mathbf { x } _ { i } ) - \mathrm { d i a g } ( \mathbf { x } _ { i } ) \mathbf { K } \right] ,
+$$
+
+$$
+\mathbf { x } _ { i } = ( x _ { i } ^ { 1 } , x _ { i } ^ { 2 } , . . . , x _ { i } ^ { K } ) ^ { \mathrm { T } } \in \mathbb { R } ^ { K \times 1 } .
+$$
+
+# A.3 THE EPANECHNIKOV KERNEL CASE
+
+The Epanechnikov kernel is defined as $\begin{array} { r } { \mathcal { K } ( \pmb { x } , \pmb { x } ^ { \prime } ) = \frac { 1 } { d } \sum _ { i = 1 } ^ { d } ( 1 - ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } ) } \end{array}$ , where the first and second order gradients w.r.t. $x _ { i }$ is
+
+$$
+\nabla _ { x _ { i } } K ( { \pmb x } , { \pmb x } ^ { \prime } ) = \frac { 2 } { d } ( x _ { i } ^ { \prime } - x _ { i } ) , \quad \nabla _ { x _ { i } } \nabla _ { x _ { i } } K ( { \pmb x } , { \pmb x } ^ { \prime } ) = - \frac { 2 } { d } .
+$$
+
+Thus the score matching objective with $\begin{array} { r } { \log \hat { q } ( \pmb { x } ) = \sum _ { k = 1 } ^ { K } a _ { k } \mathcal { K } ( \pmb { x } , \pmb { x } ^ { k } ) + C } \end{array}$ is reduced to
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { F } ( \pmb { a } ) = \frac { 1 } { K } \sum _ { j = 1 } ^ { K } \left[ | | \sum _ { k = 1 } ^ { K } a _ { k } \frac { 2 } { d } ( \pmb { x } ^ { k } - \pmb { x } ^ { j } ) | | _ { 2 } ^ { 2 } - 2 \sum _ { k = 1 } ^ { K } a _ { k } \frac { 2 } { d } d \right] } \\ { \displaystyle \qquad = \frac { 4 } { K } \sum _ { j = 1 } ^ { K } \left[ \frac { 1 } { d ^ { 2 } } \sum _ { k = 1 } ^ { K } \sum _ { k ^ { \prime } = 1 } ^ { K } a _ { k } a _ { k ^ { \prime } } ( \pmb { x } ^ { k } - \pmb { x } ^ { j } ) ^ { \mathrm { T } } ( \pmb { x } ^ { k ^ { \prime } } - \pmb { x } ^ { j } ) - \pmb { a } ^ { \mathrm { T } } \pmb { 1 } \right] } \\ { \displaystyle \qquad : = 4 ( \pmb { a } ^ { \mathrm { T } } \pmb { a } - \pmb { a } ^ { \mathrm { T } } \pmb { 1 } ) , } \end{array}
+$$
+
+with the matrix elements
+
+$$
+\Sigma _ { k k ^ { \prime } } = \frac { 1 } { d ^ { 2 } } \left[ ( \pmb { x } ^ { k } ) ^ { \mathrm { T } } \pmb { x } ^ { k ^ { \prime } } + \frac { 1 } { K } \sum _ { j = 1 } ^ { K } \left( | | \pmb { x } ^ { j } | | _ { 2 } ^ { 2 } - ( \pmb { x } ^ { k } + \pmb { x } ^ { k ^ { \prime } } ) ^ { \mathrm { T } } \pmb { x } ^ { j } \right) \right] .
+$$
+
+Define the “gram matrix” $\mathbb X _ { i j } = ( \pmb x ^ { i } ) ^ { \mathrm T } \pmb x ^ { j }$ , we write the matrix form of $\pmb { \Sigma }$ as
+
+$$
+\Sigma = \frac { 1 } { d ^ { 2 } } \left[ \mathbb { X } + \frac { 1 } { K } \left( \mathrm { T r } ( \mathbb { X } ) - 2 \mathbb { X } \mathbf { 1 } \mathbf { 1 } ^ { \mathrm { T } } \right) \right] .
+$$
+
+Thus with an $l _ { 2 }$ regulariser, the fitted coefficients are
+
+$$
+\hat { \pmb { a } } ^ { \mathrm { s c o r e } } = \frac { d ^ { 2 } } { 2 } \left[ \mathbb { X } + \frac { 1 } { K } \left( \mathrm { T r } ( \mathbb { X } ) - 2 \mathbb { X } { \bf 1 1 } ^ { \mathrm { T } } \right) + \eta { \bf I } \right] ^ { - 1 } { \bf 1 } .
+$$
+
+B STEIN GRADIENT ESTIMATOR: DERIVATIONS
+
+B.1 DIRECT MINIMISATION OF KSD V-STATISTIC AND U-STATISTIC
+
+The V-statistic of KSD is the following: given samples $\pmb { x } ^ { k } \sim q , k = 1 , . . . , K$ and recall ${ \bf K } _ { j l } = { \bf \Lambda } $ $\mathcal { K } ( \pmb { x } ^ { j } , \pmb { x } ^ { l } )$
+
+$$
+\mathfrak { S } _ { V } ^ { 2 } ( \mathfrak { q } , \hat { \mathfrak { q } } ) = \frac { 1 } { K ^ { 2 } } \sum _ { j = 1 } ^ { K } \sum _ { l = 1 } ^ { K } \left[ \hat { g } ( \mathfrak { x } ^ { j } ) ^ { \mathrm { T } } \mathbf { K } _ { j l } \hat { g } ( \mathfrak { x } ^ { l } ) + \hat { g } ( \mathfrak { x } ^ { j } ) ^ { \mathrm { T } } \nabla _ { \mathfrak { x } ^ { l } } \mathbf { K } _ { j l } + \nabla _ { \mathfrak { x } ^ { j } } \mathbf { K } _ { j l } ^ { \mathrm { T } } \hat { g } ( \mathfrak { x } ^ { l } ) + \mathrm { T r } ( \nabla _ { \mathfrak { x } ^ { j } , \mathfrak { x } ^ { l } } \mathbf { K } _ { j l } ) \right] .
+$$
+
+The last term $\nabla _ { \pmb { x } ^ { j } , \pmb { x } ^ { l } } \mathbf { K } _ { j l }$ will be ignored as it does not depend on the approximation $\hat { \pmb { g } }$ . Using matrix notations defined in the main text, readers can verify that the $\mathrm { V } .$ -statistic can be computed as
+
+$$
+\mathcal { S } _ { V } ^ { 2 } ( q , \hat { q } ) = \frac { 1 } { K ^ { 2 } } \mathrm { T r } ( \mathbf { K } \hat { \mathbf { G } } \hat { \mathbf { G } } ^ { \mathrm { T } } + 2 \langle \nabla , \mathbf { K } \rangle \hat { \mathbf { G } } ^ { \mathrm { T } } ) + C .
+$$
+
+Using the cyclic invariance of matrix trace leads to the desired result in the main text. The U-statistic of KSD removes terms indexed by $j = l$ in (23), in which the matrix form is
+
+$$
+\mathcal { S } _ { U } ^ { 2 } ( q , \hat { q } ) = \frac { 1 } { K ( K - 1 ) } \mathrm { T r } ( ( { \bf K } - \mathrm { d i a g } ( { \bf K } ) ) \hat { \bf G } \hat { \bf G } ^ { \mathrm { T } } + 2 ( \langle \nabla , { \bf K } \rangle - \nabla \mathrm { d i a g } ( { \bf K } ) ) \hat { \bf G } ^ { \mathrm { T } } ) + C .
+$$
+
+with the $j$ th row of $\nabla \mathrm { d i a g } ( \mathbf { K } )$ defined as $\nabla _ { \pmb { x } ^ { j } } \mathcal { K } ( \pmb { x } ^ { j } , \pmb { x } ^ { j } )$ . For most translation invariant kernels this extra term $\nabla \mathrm { d i a g } ( \mathbf { K } ) = \mathbf { 0 }$ , thus the optimal solution of $\hat { \mathbf { G } }$ by minimising KSD U-statistic is
+
+$$
+\hat { \mathbf { G } } _ { U } ^ { \mathrm { S t e i n } } = - ( \mathbf { K } - \mathrm { d i a g } ( \mathbf { K } ) + \eta \mathbf { I } ) ^ { - 1 } \langle \nabla , \mathbf { K } \rangle .
+$$
+
+# B.2 DERIVING THE NON-PARAMETRIC PREDICTIVE ESTIMATOR
+
+Let us consider an unseen datum $\textbf {  { y } }$ . If $\textbf {  { y } }$ is sampled from the $q$ distribution, then one can also apply the non-parametric estimator (9) for gradient approximations, given the observed data $\mathbf { X } =$ $\{ \bar { \pmb { x } } ^ { 1 } , . . . , \pmb { x } ^ { K } \} \stackrel { - } { \sim } q$ . Concretely, if writing $\bar { \pmb g } ( \pmb y ) \approx \bar { \nabla _ { \pmb y } } \log q ( \pmb y ) \in \bar { \mathbb R ^ { d \times 1 } }$ then the non-parametric Stein gradient estimator (using $\mathrm { V } .$ -statistic) is
+
+$$
+\begin{array} { r } { \left[ \hat { g } ( y ) ^ { \mathrm { T } } \right] = - ( \mathbf { K } ^ { * } + \eta I ) ^ { - 1 } \left[ \nabla _ { y } K ( y , y ) + \sum _ { k = 1 } ^ { K } \nabla _ { x ^ { k } } K ( y , x ^ { k } ) \right] , \quad \mathbf { K } ^ { * } = \left[ \mathbf { K } _ { y y } \quad \mathbf { K } _ { y \mathbf { X } } \right] , } \\ { \left. \nabla , \mathbf { K } \right. + \nabla _ { y } K ( \cdot , y ) \qquad \quad \mathbf { K } ^ { * } = \left[ \mathbf { K } _ { \mathbf { X } y } \quad \mathbf { K } \right] , } \end{array}
+$$
+
+with $\nabla _ { \pmb { y } } K ( \cdot , \pmb { y } )$ denoting a $K \times d$ matrix with rows $\nabla _ { \pmb { y } } K ( \pmb { x } ^ { k } , \pmb { y } )$ , and $\nabla _ { \pmb { y } } K ( \pmb { y } , \pmb { y } )$ only differentiates through the second argument. Thus by simple matrix calculations, we have:
+
+$$
+\begin{array} { l } { \nabla _ { y } \log q ( { y } ) ^ { \mathrm { T } } \approx - \left( \mathbf K _ { y y } + \eta - \mathbf K _ { y \mathbf X } ( \mathbf K + \eta \mathbf I ) ^ { - 1 } \mathbf K _ { \mathbf X { y } } \right) ^ { - 1 } } \\ { \displaystyle \qquad \left( \nabla _ { y } K ( y , y ) + \sum _ { k = 1 } ^ { K } \nabla _ { x ^ { k } } K ( y , x ^ { k } ) + \mathbf K _ { y \mathbf X } \hat { \mathbf G } _ { V } ^ { \mathrm { S t e i n } } - \mathbf K _ { y \mathbf X } ( \mathbf K + \eta \mathbf I ) ^ { - 1 } \nabla _ { y } K ( \cdot , y ) \right) . } \end{array}
+$$
+
+For translation invariant kernels, typically $\nabla _ { \pmb { y } } \mathcal { K } ( \pmb { y } , \pmb { y } ) = \mathbf { 0 }$ , and more conveniently,
+
+$$
+\nabla _ { \pmb { x } ^ { k } } \mathcal { K } ( \pmb { y } , \pmb { x } ^ { k } ) = \nabla _ { \pmb { x } ^ { k } } ( \pmb { x } ^ { k } - \pmb { y } ) \nabla _ { ( \pmb { x } ^ { k } - \pmb { y } ) } \mathcal { K } ( \pmb { x } ^ { k } - \pmb { y } ) = - \nabla _ { \pmb { y } } \mathcal { K } ( \pmb { x } ^ { k } , \pmb { y } ) .
+$$
+
+Thus equation (27) can be further simplified to (with column vector $\mathbf { 1 } \in \mathbb { R } ^ { K \times 1 }$ )
+
+$$
+\begin{array} { r l } & { \nabla _ { y } \log q ( \pmb { y } ) ^ { \operatorname { T } } \approx - \left( \mathbf { K } _ { y y } + \eta - \mathbf { K } _ { y \mathbf { X } } ( \mathbf { K } + \eta \mathbf { I } ) ^ { - 1 } \mathbf { K } _ { \mathbf { X } y } \right) ^ { - 1 } } \\ & { \qquad \left( \mathbf { K } _ { y \mathbf { X } } \hat { \mathbf { G } } _ { V } ^ { \mathrm { { S t e i n } } } - \left( \mathbf { K } _ { y \mathbf { X } } ( \mathbf { K } + \eta \mathbf { I } ) ^ { - 1 } + \mathbf { 1 } ^ { \operatorname { T } } \right) \nabla _ { y } \mathcal { K } ( \cdot , y ) \right) . } \end{array}
+$$
+
+The solution for the U-statistic case can be derived accordingly which we omit here.
+
+B.3 PARAMETRIC STEIN GRADIENT ESTIMATOR WITH THE RBF KERNEL
+
+We define a parametric approximation in a similar way as for the score matching estimator:
+
+$$
+\log \hat { q } ( \pmb { x } ) : = \sum _ { k = 1 } ^ { K } a _ { k } K ( \pmb { x } , \pmb { x } ^ { k } ) + C , \quad K ( \pmb { x } , \pmb { x } ^ { \prime } ) = \exp \left[ - \frac { 1 } { 2 \sigma ^ { 2 } } | | \pmb { x } - \pmb { x } ^ { \prime } | | _ { 2 } ^ { 2 } \right] .
+$$
+
+Now we show the optimal solution of $\pmb { a } = ( a _ { 1 } , . . . , a _ { K } ) ^ { \mathrm { T } }$ by minimising (23). To simplify derivations we assume the approximation and KSD use the same kernel. First note that the gradient of the RBF kernel is
+
+$$
+\nabla _ { \pmb { x } } K ( \pmb { x } , \pmb { x } ^ { \prime } ) = \frac { 1 } { \sigma ^ { 2 } } K ( \pmb { x } , \pmb { x } ^ { \prime } ) ( \pmb { x } ^ { \prime } - \pmb { x } ) .
+$$
+
+Substituting (30) into (23):
+
+$$
+S _ { V } ^ { 2 } ( q , \hat { q } ) = C + \pmb { \mathscr { s } } + 2 \pmb { \mathscr { s } } ,
+$$
+
+$$
+\ P \bullet \mathrm { = } \frac { 1 } { K ^ { \mathrm { 2 } } } \sum _ { k = 1 } ^ { K } \sum _ { k ^ { \prime } = 1 } ^ { K } \sum _ { j = 1 } ^ { K } \sum _ { l = 1 } ^ { K } a _ { k } a _ { k ^ { \prime } } \mathbf { K } _ { k j } \mathbf { K } _ { j l } \mathbf { K } _ { l k ^ { \prime } } \frac { 1 } { \sigma ^ { 4 } } ( { \pmb x } ^ { k } - { \pmb x } ^ { j } ) ^ { \mathrm { T } } ( { \pmb x } ^ { k ^ { \prime } } - { \pmb x } ^ { l } ) ,
+$$
+
+$$
+\pmb { \diamond } = \frac { 1 } { K ^ { 2 } } \sum _ { k = 1 } ^ { K } \sum _ { j = 1 } ^ { K } \sum _ { l = 1 } ^ { K } a _ { k } \mathbf { K } _ { k j } \mathbf { K } _ { j l } \frac { 1 } { \sigma ^ { 4 } } ( \pmb { x } ^ { k } - \pmb { x } ^ { j } ) ^ { \mathrm { T } } ( \pmb { x } ^ { j } - \pmb { x } ^ { l } ) .
+$$
+
+We first consider summing the $j , l$ indices in $\clubsuit$ . Recall the “gram matrix” $\mathbb X _ { i j } = ( { \pmb x } ^ { i } ) ^ { \mathrm T } { \pmb x } ^ { j }$ , the inner product term in $\clubsuit$ can be expressed as ${ \mathbb X } _ { k k ^ { \prime } } + { \mathbb X } _ { j l } - { \mathbb X } _ { k l } - { \mathbb X } _ { j k ^ { \prime } }$ . Thus the summation over $j , l$ can be re-written as
+
+$$
+\begin{array} { r l } {  { \boldsymbol { \Lambda } : = \sum _ { j = 1 } ^ { K } \sum _ { l = 1 } ^ { K } \mathbf { K } _ { k j } \mathbf { K } _ { j l } \mathbf { K } _ { l k ^ { \prime } } ( \mathbb { X } _ { k k ^ { \prime } } + \mathbb { X } _ { j l } - \mathbb { X } _ { k l } - \mathbb { X } _ { j k ^ { \prime } } ) } \quad } & { } \\ & { = \mathbb { X } \odot ( \mathbf { K } \mathbf { K } \mathbf { K } ) + \mathbf { K } ( \mathbf { K } \odot \mathbb { X } ) \mathbf { K } - ( ( \mathbf { K } \mathbf { K } ) \odot \mathbb { X } ) \mathbf { K } - \mathbf { K } ( ( \mathbf { K } \mathbf { K } ) \odot \mathbb { X } ) . } \end{array}
+$$
+
+And thus $\begin{array} { r } { \pmb { \mathscr { s } } = \frac { 1 } { \sigma ^ { 4 } } \pmb { a } ^ { \mathrm { T } } \pmb { \Lambda } \pmb { a } } \end{array}$ . Similarly the summation over $j , l$ in $\spadesuit$ can be simplified into
+
+$$
+\begin{array} { r l } & { - \pmb { b } : = \displaystyle \sum _ { j = 1 } ^ { K } \sum _ { l = 1 } ^ { K } \mathbf { K } _ { k j } \mathbf { K } _ { j l } ( \mathbb { X } _ { k j } + \mathbb { X } _ { j l } - \mathbb { X } _ { k l } - \mathbb { X } _ { j j } ) } \\ & { \quad \quad = \ : - \ : ( \mathbf { K } \mathrm { d i a g } ( \mathbb { X } ) \mathbf { K } + ( \mathbf { K } \mathbf { K } ) \odot \mathbb { X } - \mathbf { K } ( \mathbf { K } \odot \mathbb { X } ) - ( \mathbf { K } \odot \mathbb { X } ) \mathbf { K } ) \mathbf { 1 } , } \end{array}
+$$
+
+which leads to $\begin{array} { r } { \pmb { \langle \mathscr { s } \rangle } = - \frac { 1 } { \sigma ^ { 4 } } \pmb { a } ^ { \mathrm { T } } \pmb { b } } \end{array}$ . Thus minimising $S _ { V } ^ { 2 } ( q , \hat { q } )$ plus an $l _ { 2 }$ regulariser returns the Stein estimator $\hat { \pmb { a } } _ { V } ^ { \mathrm { S t e i n } }$ in the main text.
+
+Similarly we can derive the solution for KSD U-statistic minimisation. The $\mathrm { U }$ statistic can also be represented in quadratic form $S _ { U } ^ { 2 } ( q , \hat { q } ) = C + \tilde { \tilde { \mathbf { \eta } } } + 2 \tilde { \tilde { \mathbf { \eta } } }$ , with $\tilde { \mathbf { A } } = \spadesuit$ and
+
+$$
+\tilde { \mathbf { a } } = \mathbf { a } - \frac { 1 } { K ^ { 2 } } \sum _ { k = 1 } ^ { K } \sum _ { k ^ { \prime } = 1 } ^ { K } \sum _ { j = 1 } ^ { K } a _ { k } a _ { k ^ { \prime } } \mathbf { K } _ { k j } \mathbf { K } _ { j j } \mathbf { K } _ { j k ^ { \prime } } \frac { 1 } { \sigma ^ { 4 } } \big ( \mathbb { X } _ { k k ^ { \prime } } + \mathbb { X } _ { j j } - \mathbb { X } _ { k j } - \mathbb { X } _ { j k ^ { \prime } } \big ) .
+$$
+
+Summing over the $j$ indices for the second term, we have
+
+$$
+\begin{array} { r l } {  { \sum _ { j = 1 } ^ { K } { \mathbf { K } } _ { k j } \mathbf { K } _ { j j } \mathbf { K } _ { j k ^ { \prime } } ( { \mathbb { X } } _ { k k ^ { \prime } } + { \mathbb { X } } _ { j j } - { \mathbb { X } } _ { k j } - { \mathbb { X } } _ { j k ^ { \prime } } ) } \quad } & { } \\ & { = { \mathbb { X } } \odot ( { \mathbf { K } } \mathrm { d i a g } ( { \mathbf { K } } ) { \mathbf { K } } ) + { \mathbf { K } } \mathrm { d i a g } ( { \mathbf { K } } \odot { \mathbb { X } } ) { \mathbf { K } } - ( ( { \mathbf { K } } \mathrm { d i a g } ( { \mathbf { K } } ) ) \odot { \mathbb { X } } ) { \mathbf { K } } - { \mathbf { K } } ( ( \mathrm { d i a g } ( { \mathbf { K } } ) { \mathbf { K } } ) \odot { \mathbb { X } } ) . } \end{array}
+$$
+
+Working through the analogous derivations reveals that $\hat { \pmb { a } } _ { U } ^ { \mathrm { S t e i n } } = ( \tilde { \pmb { \Lambda } } + \eta { \bf I } ) ^ { - 1 } { \pmb { b } }$ , with
+
+$$
+\begin{array} { r l } & { \tilde { \mathbf { A } } = \mathbb { X } \odot ( \mathbf { K } ( \mathbf { K } - \mathrm { d i a g } ( \mathbf { K } ) ) \mathbf { K } ) + \mathbf { K } ( ( \mathbf { K } \odot \mathbb { X } ) - \mathrm { d i a g } ( \mathbf { K } \odot \mathbb { X } ) ) \mathbf { K } } \\ & { \qquad - \left( ( \mathbf { K } ( \mathbf { K } - \mathrm { d i a g } ( \mathbf { K } ) ) ) \odot \mathbb { X } \right) \mathbf { K } - \mathbf { K } ( ( ( \mathbf { K } - \mathrm { d i a g } ( \mathbf { K } ) ) \mathbf { K } ) \odot \mathbb { X } ) . } \end{array}
+$$
+
+# C MORE DETAILS ON THE EXPERIMENTS
+
+We describe the detailed experimental set-up in this section. All experiments use Adam optimiser (Kingma & Ba, 2015) with standard parameter settings.
+
+# C.1 APPROXIMATE POSTERIOR SAMPLER EXPERIMENTS
+
+We start by reviewing Bayesian neural networks with binary classification as a running example. In this task, a normal deep neural network is constructed to predict $y = f _ { \theta } ( { \pmb x } )$ , and the neural network is parameterised by a set of weights (and bias vectors which we omit here for simplicity) $\pmb { \theta } = \{ \mathbf { W } ^ { l } \} _ { l = 1 } ^ { L }$ . In the Bayesian framework these network weights are treated as random variables, and a prior distribution, e.g. Gaussian, is also attached to them: $p _ { 0 } ( \pmb { \theta } ) = \mathcal { N } ( \pmb { \theta } ; \mathbf { 0 } , \mathbf { I } )$ . The likelihood function of $\pmb \theta$ is then defined as
+
+$$
+p ( y = 1 | \mathbf { x } , \pmb { \theta } ) = \mathrm { s i g m o i d } ( \mathrm { N N } _ { \pmb { \theta } } ( \pmb { x } ) ) ,
+$$
+
+and $p ( y = 0 | \mathbf { x } , \pmb { \theta } ) = 1 - p ( y = 1 | \mathbf { x } , \pmb { \theta } )$ accordingly. One can show that the usage of Bernoulli distribution here corresponds to applying cross entropy loss for training.
+
+After framing the deep neural network as a probabilistic model, a Bayesian approach would find the posterior of the network weights $p ( \pmb \theta | \mathcal { D } )$ and use the uncertainty information encoded in it for future predictions. By Bayes’ rule, the exact posterior is
+
+$$
+p ( \pmb \theta | \mathcal { D } ) \propto p _ { 0 } ( \pmb \theta ) \prod _ { n = 1 } ^ { N } p ( y _ { n } | \pmb x _ { n } , \pmb \theta ) ,
+$$
+
+and the predictive distribution for a new input $\pmb { x } ^ { * }$ is
+
+$$
+p ( y ^ { \ast } = 1 | \pmb { x } ^ { \ast } , \mathcal { D } ) = \int p ( y ^ { \ast } = 1 | \pmb { x } ^ { \ast } , \pmb { \theta } ) p ( \pmb { \theta } | \mathcal { D } ) d \pmb { \theta } .
+$$
+
+Again the exact posterior is intractable, and approximate inference would fit an approximate posterior distribution $q _ { \phi } ( \pmb \theta )$ parameterised by the variational parameters $\phi$ to the exact posterior, and then use it to compute the (approximate) predictive distribution.
+
+$$
+p ( y ^ { \ast } = 1 | x ^ { \ast } , \mathcal { D } ) \approx \int p ( y ^ { \ast } = 1 | x ^ { \ast } , \pmb { \theta } ) q _ { \phi } ( \pmb { \theta } ) d \pmb { \theta } .
+$$
+
+Since in practice analytical integration for neural network weights is also intractable, the predictive distribution is further approximated by Monte Carlo:
+
+$$
+p ( y ^ { * } = 1 | x ^ { * } , \mathcal { D } ) \approx \frac { 1 } { K } \sum _ { k = 1 } ^ { K } p ( y ^ { * } = 1 | x ^ { * } , \pmb { \theta } ^ { k } ) , \quad \pmb { \theta } ^ { k } \sim q _ { \phi } ( \pmb { \theta } ) .
+$$
+
+Now it remains to fit the approximate posterior $q _ { \phi } ( \pmb \theta )$ , and in the experiment the approximate posterior is implicitly constructed by a stochastic flow. For the training task, we use a one hidden layer neural network with 20 hidden units to compute the noise variance and the moving direction of the next update. In a nutshell it takes the ith coordinate of the current position and the gradient $\pmb { \theta } _ { t } ( i ) , \bar { \nabla _ { t } } ( i )$ as the inputs, and output the corresponding coordinate of the moving direction $\Delta _ { \phi } ( \theta _ { t } , \nabla _ { t } ) ( i )$ and the noise variance $\sigma _ { \phi } ( \pmb { \theta } _ { t } , \nabla _ { t } ) ( i )$ . Softplus non-linearity is used for the hidden layer and to compute the noise variance we apply ReLU activation to ensure non-negativity. The step-size $\zeta$ is selected as 1e-5 which is tuned on the KDE approach. For SGLD step-size 1e-5 also returns overall good results.
+
+The training process is the following. We simulate the approximate sampler for 10 transitions and sum over the variational lower-bounds computed on the samples of every step. Concretely, the maximisation objective is
+
+$$
+\mathcal { L } ( \phi ) = \sum _ { t = 1 } ^ { T } \mathcal { L } _ { \mathrm { V I } } ( q _ { t } ) ,
+$$
+
+where $T = 1 0 0$ and $q _ { t } ( \pmb \theta )$ is implicitly defined by the marginal distribution of $\theta _ { t }$ that is dependent on $\phi$ . In practice the variational lower-bound ${ \mathcal { L } } _ { \mathrm { V I } } ( q _ { t } )$ is further approximated by Monte Carlo and data sub-sampling:
+
+$$
+\mathcal { L } _ { \mathrm { V I } } ( q _ { t } ) \approx \frac { N } { M } \sum _ { m = 1 } ^ { M } \log p ( y _ { m } | x _ { m } , \pmb { \theta } _ { t } ) + \log p _ { 0 } ( \pmb { \theta } _ { t } ) - \log q _ { t } ( \pmb { \theta } _ { t } ) .
+$$
+
+The MAP baseline considers an alternative objective function by removing the $\log q _ { t } ( \pmb \theta _ { t } )$ term from the above MC-VI objective.
+
+Truncated back-propagation is applied for every 10 steps in order to avoid vanishing/exploding gradients. The simulated samples at time $T$ are stored to initialise the Markov chain for the next iteration, and for every 50 iterations we restart the simulation by randomly sampling the locations from the prior. Early stopping is applied using the validation dataset, and the learning rate is set to 0.001, the number of epochs is set to 500.
+
+We perform hyper-parameter search for the kernel, i.e. a grid search on the bandwidth $\sigma ^ { 2 } \in $ $\{ 0 . 2 \bar { 5 } , 1 . 0 , 4 . 0 , \bar { 1 0 } . 0$ , median trick} and $\eta \in \{ 0 . 1 , 0 . 5 , 1 . 0 , 2 . 0 \}$ . We found the median heuristic is sufficient for the KDE and Stein approaches. However, we failed to obtain desirable results using the score matching estimator with median heuristics, and for other settings the score matching approach underperforms when compared to KDE and Stein methods.
+
+# C.2 BEGAN EXPERIMENTS
+
+In this section we describe the experimental details of the BEGAN experiment, but first we introduce the mathematical idea and discuss how the entropy regulariser is applied.
+
+Assume the generator is implicitly defined: $\pmb { x } \sim p _ { \pmb \theta } ( \pmb { x } )  \pmb { x } = \pmb { f _ { \pmb \theta } } ( z ) , z \sim p _ { 0 } ( z )$ . In BEGAN the discriminator is defined as an auto-encoder $D _ { \varphi } ( \pmb { x } )$ that reconstructs the input $_ { \textbf { \em x } }$ . After selecting a ratio parameter $\gamma > 0$ , a control rate $\beta _ { 0 }$ initialised at 0, and a “learning rate” $\lambda > 0$ for the control rate, the loss functions for the generator $\pmb { x } = \pmb { f _ { \theta } } ( z ) , z \sim p _ { 0 } ( z )$ and the discriminator are:
+
+$$
+\begin{array} { r l } & { \mathcal { I } ( \pmb { x } ) = | | D _ { \varphi } ( \pmb { x } ) - \pmb { x } | | , \quad | | \cdot | | = | | \cdot | | _ { 2 } ^ { 2 } \mathrm { o r } | \cdot | | _ { 1 } , } \\ & { \mathcal { I } _ { \mathtt { g e n } } ( \pmb { \theta } ; \pmb { \varphi } ) = \mathcal { I } ( \pmb { f } _ { \pmb { \theta } } ( z ) ) , \quad z \sim p _ { 0 } ( z ) } \\ & { \mathcal { I } _ { \mathrm { d i s } } ( \pmb { \varphi } ; \pmb { \theta } ) = \mathcal { I } ( \pmb { x } ) - \beta _ { t } \mathcal { I } _ { \mathtt { g e n } } ( \pmb { \theta } ; \pmb { \varphi } ) , \quad \pmb { x } \sim \mathcal { D } } \\ & { \beta _ { t + 1 } = \beta _ { t } + \lambda ( \gamma \mathcal { I } ( \pmb { x } ) - \mathcal { I } ( \pmb { f } _ { \pmb { \theta } } ( z ) ) ) . } \end{array}
+$$
+
+The main idea behind BEGAN is that, as the reconstruction loss $\mathcal { I } ( \cdot )$ is approximately Gaussian distributed, with $\gamma = 1$ the discriminator loss $\mathcal { T } _ { \mathrm { d i s } }$ is (approximately) proportional to the Wasserstein distance between loss distributions induced by the data distribution $p _ { \mathcal { D } } ( \pmb { x } )$ and the generator $p _ { \pmb { \theta } } ( \pmb { x } )$ . In practice it is beneficial to maintain the equilibrium $\gamma \mathbb { E } _ { p _ { \mathcal { D } } } \left[ \mathcal { I } ( \pmb { x } ) \right] = \mathbb { E } _ { p _ { \theta } } \left[ \mathcal { I } ( \pmb { x } ) \right]$ through the optimisation procedure described in (34) that is motivated by proportional control theory. This approach effectively stabilises training, however it suffers from catastrophic mode collapsing problem (see the left most panel in Figure 4). To address this issue, we simply subtract an entropy term from the generator’s loss function, i.e.
+
+$$
+\tilde { \mathcal { I } } _ { \mathrm { g e n } } ( \pmb { \theta } ; \varphi ) = \mathcal { I } _ { \mathrm { g e n } } ( \pmb { \theta } ; \varphi ) - \alpha \mathbb { H } [ p _ { \pmb { \theta } } ] ,
+$$
+
+where the rest of the optimisation objectives remains as in (34). This procedure would maintain the equilibrium $\gamma \mathbb { E } _ { p _ { D } } \left[ \mathcal { \bar { I } } ( \pmb { x } ) \right] = \mathbb { E } _ { p _ { \theta } } \left[ \mathcal { I } ( \pmb { x } ) \right] - \alpha \mathbb { H } [ p ]$ . We approximate the gradient $\nabla _ { \pmb { \theta } } \mathbb { H } [ p _ { \pmb { \theta } } ]$ using the estimators presented in the main text. For the purpose of updating the control rate $\beta _ { t }$ two strategies are considered to approximate the contribution of the entropy term. Given $K$ samples $\pmb { x } ^ { 1 } , . . . , \pmb { x } ^ { k } \sim p _ { \pmb { \theta } } ( \pmb { x } )$ , The first proposal considers a plug-in estimate of the entropy term with a KDE estimate of $p _ { \pmb { \theta } } ( \pmb { x } )$ , which is consistent with the KDE estimator but not necessary with the other two (as they use kernelsproxy of the entropy loss $\log p _ { \theta } ( { \pmb x } )$ $\nabla _ { \pmb { x } } \log p _ { \pmb { \theta } } ( \pmb { x } ) )$ . The second onerated samples es a and $\begin{array} { r } { - \mathbb { H } [ p ] \approx \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \nabla _ { \pmb { x } ^ { k } } \log p _ { \pmb { \theta } } ( \pmb { x } ^ { k } ) ^ { \mathrm { T } } \pmb { x } ^ { k } } \end{array}$ $\{ \boldsymbol { x } ^ { k } \}$ $\nabla _ { \pmb { x } ^ { k } } \log p _ { \pmb { \theta } } ( \pmb { x } ^ { k } )$ approximated by the gradient estimator in use.
+
+In the experiment, we construct a deconvolutional net for the generator and a convolutional autoencoder for the discriminator. The convolutional encoder consists of 3 convolutional layers with filter width 3, stride 2, and number of feature maps [32, 64, 64]. These convolutional layers are followed by two fully connected layers with [512, 64] units. The decoder and the generative net have a symmetric architecture but with stride convolutions replaced by deconvolutions. ReLU activation function is used for all layers except the last layer of the generator, which uses sigmoid non-linearity. The reconstruction loss in use is the squared $\ell _ { 2 }$ norm $| | \cdot | | _ { 2 } ^ { 2 }$ . The randomness $p _ { 0 } ( z )$ is selected as uniform distribution in [-1, 1] as suggested in the original paper (Berthelot et al., 2017). The minibatch size is set to $K = 1 0 0$ . Learning rate is initialised at 0.0002 and decayed by 0.9 every 10 epochs, which is tuned on the KDE model. The selected $\gamma$ and $\alpha$ values are: for KDE estimator approach $\gamma = 0 . 3 , \alpha \gamma = 0 . 0 5$ , for score matching estimator approach $\gamma = 0 . 3 , \alpha \gamma = 0 . 1$ , and for Stein approach $\gamma = 0 . 5$ and $\alpha \gamma = 0 . 3$ . The presented results use the KDE plug-in estimator for the entropy estimates (used to tune $\beta$ ) for the KDE and score matching approaches. Initial experiments found that for the Stein approach, using the KDE entropy estimator works slightly worse than the proxy loss, thus we report results using the proxy loss. An advantage of using the proxy loss is that it directly relates to the approximate gradient. Furthermore we empirically observe that the performance of the Stein approach is much more robust to the selection of $\gamma$ and $\alpha$ when compared to the other two methods.

parse/train/SJxbHkrKDH/SJxbHkrKDH.md

@@ -0,0 +1,434 @@
+# EVOLUTIONARY POPULATION CURRICULUM FORSCALING MULTI-AGENT REINFORCEMENT LEARNING
+
+Qian Long∗   
+CMU   
+qianlong@cs.cmu.edu   
+Zihan Zhou∗   
+SJTU   
+footoredo@sjtu.edu.cn
+
+Abhibav Gupta CMU, Facebook AI Research abhinavg@cs.cmu.edu
+
+Fei Fang   
+CMU   
+feif@cs.cmu.edu   
+Yi Wu†   
+OpenAI   
+jxwuyi@openai.com
+
+Xiaolong Wang† UCSD xiw012@ucsd.edu
+
+# ABSTRACT
+
+In multi-agent games, the complexity of the environment can grow exponentially as the number of agents increases, so it is particularly challenging to learn good policies when the agent population is large. In this paper, we introduce Evolutionary Population Curriculum (EPC), a curriculum learning paradigm that scales up MultiAgent Reinforcement Learning (MARL) by progressively increasing the population of training agents in a stage-wise manner. Furthermore, EPC uses an evolutionary approach to fix an objective misalignment issue throughout the curriculum: agents successfully trained in an early stage with a small population are not necessarily the best candidates for adapting to later stages with scaled populations. Concretely, EPC maintains multiple sets of agents in each stage, performs mix-and-match and fine-tuning over these sets and promotes the sets of agents with the best adaptability to the next stage. We implement EPC on a popular MARL algorithm, MADDPG, and empirically show that our approach consistently outperforms baselines by a large margin as the number of agents grows exponentially. The source code and videos can be found at https://sites.google.com/view/epciclr2020/.
+
+# 1 INTRODUCTION
+
+Most real-world problems involve interactions between multiple agents and the problem becomes significantly harder when there exist complex cooperation and competition among agents. Inspired by the tremendous success of deep reinforcement learning (RL) in single-agent applications, such as Atari games (Mnih et al., 2013), robotics manipulation (Levine et al., 2016), and navigation (Zhu et al., 2017; Wu et al., 2018; Yang et al., 2019), it has become a popular trend to apply deep RL techniques into multi-agent applications, including communication (Foerster et al., 2016; Sukhbaatar et al., 2016; Mordatch & Abbeel, 2018), traffic light control (Wu et al., 2017), physical combats (Bansal et al., 2018), and video games (Liu et al., 2019; OpenAI, 2018).
+
+A fundamental challenge for multi-agent reinforcement learning (MARL) is that, as the number of agents increases, the problem becomes significantly more complex and the variance of policy gradients can grow exponentially (Lowe et al., 2017). Despite the advances on tackling this challenge via actor-critic methods (Lowe et al., 2017; Foerster et al., 2018), which utilize decentralized actors and centralized critics to stabilize training, recent works still scale poorly and are mostly restricted to less than a dozen agents. However, many real-world applications involve a moderately large population of agents, such as algorithmic trading (Wellman et al., 2005), sport team competition (Hausknecht & Stone, 2015), and humanitarian assistance and disaster response (Meier, 2015), where one agent should collaborate and/or compete with all other agents. When directly applying the existing MARL algorithms to complex games with a large number of agents, as we will show in Sec. 5.3, the agents may fail to learn good strategies and end up with little interaction with other agents even when collaboration is significantly beneficial. Yang et al. (2018) proposed a provably-converged meanfield formulation to scale up the actor-critic framework by feeding the state information and the average value of nearby agents’ actions to the critic. However, this formulation strongly relies on the assumption that the value function for each agent can be well approximated by the mean of local pairwise interactions. This assumption often does not hold when the interactions between agents become complex, leading to a significant drop in the performance.
+
+In this paper, we propose a general learning paradigm called Evolutionary Population Curriculum (EPC), which allows us to scale up the number of agents exponentially. The core idea of EPC is to progressively increase the population of agents throughout the training process. Particularly, we divide the learning procedure into multiple stages with increasing number of agents in the environment. The agents first learn to play in simpler scenarios with less agents and then leverage these experiences to gradually adapt to later stages with more agents and ultimately our desired population.
+
+There are two key components in our curriculum learning paradigm. To process the varying number of agents during the curriculum procedure, the policy/critic needs to be population-invariant. So, we choose a self-attention (Vaswani et al., 2017) based architecture which can generalize to an arbitrary number of agents with a fixed number of parameters. More importantly, we introduce an evolutionary selection process, which helps address the misalignment of learning goals across stages and improves the agents’ performance in the target environment. Intuitively, our within-stage MARL training objective only incentivizes agents to overfit a particular population in the current stage. When moving towards a new stage with a larger population, the successfully trained agents may not adapt well to the scaled environment. To mitigate this issue, we maintain multiple sets of agents in each stage, evolve them through cross-set mix-and-match and parallel MARL fine-tuning in the scaled environment, and select those with better adaptability to the next stage.
+
+EPC is RL-algorithm agnostic and can be potentially integrated with most existing MARL algorithms. In this paper, we illustrate the empirical benefits of EPC by implementing it on a popular MARL algorithm, MADDPG (Lowe et al., 2017), and experimenting on three challenging environments, including a predator-prey-style individual survival game, a mixed cooperative-and-competitive battle game, and a fully cooperative food collection game. We show that EPC outperforms baseline approaches by a large margin on all these environments as the number of agents grows even exponentially. We also demonstrate that our method can improve the stability of the training procedure.
+
+# 2 RELATED WORK
+
+Multi-Agent Reinforcement Learning: It has been a long history in applying RL to multi-agent games (Littman, 1994; Shoham et al., 2003; Panait & Luke, 2005; Wright et al., 2019). Recently, deep RL techniques have been applied into the multi-agent scenarios to solve complex Markov games and great algorithmic advances have been achieved. Foerster et al. (2016) and He et al. (2016) explored a multi-agent variant of deep Q-learning; Peng et al. (2017) studied a fully centralized actor-critic variant; Foerster et al. (2018) developed a decentralized multi-agent policy gradient algorithm with a centralized baseline; Lowe et al. (2017) proposes the MADDPG algorithm which extended DDPG to the multi-agent setting with decentralized policies and centralized Q functions. Our population curriculum approach is a general framework for scaling MARL which can be potentially combined with any of these algorithms. Particularly, we implement our method on top of the MADDPG algorithm in this paper and take different MADDPG variants as baselines in experiments. There are also other works studying large-scale MARL recently (Lin et al., 2018; Jiang & Lu, 2018; Yang et al., 2018; Suarez et al., 2019), which typically simplify the problem by weight sharing and taking only local observations. We consider a much more general setting with global observations and unsharedweight agents. Additionally, our approach is a general learning paradigm which is complementary to the specific techniques proposed in these works.
+
+Attention-Based Policy Architecture: Attention mechanism is widely used in RL policy representation to capture object level information (Duan et al., 2017; Wang et al., 2018), represent relations (Zambaldi et al., 2018; Malysheva et al., 2018; Yang et al., 2019) and extract communication channels (Jiang & Lu, 2018). Iqbal & Sha (2019) use an attention-based critic. In our work, we utilize an attention module in both policy and critic, inspired by the transformer architecture (Vaswani et al., 2017), for the purpose of generalization to an arbitrary number of input entities.
+
+Curriculum Learning: Curriculum learning can be tracked back to Elman (1993), and its core idea is to “start small”: learn the easier aspects of the task first and then gradually increase the task difficulty. It has been extended to deep neural networks on both vision and language tasks (Bengio et al., 2009) and much beyond: Karras et al. (2017) propose to progressively increase the network capacity for synthesizing high quality images; Murali et al. (2018) apply a curriculum over the control space for robotic manipulation tasks; several works (Wu & Tian, 2016; Florensa et al., 2017; Sukhbaatar et al., 2017; Wang et al., 2019) have proposed to first train RL agents on easier goals and switch to harder ones later. Baker et al. (2019) show that multi-agent self-play can also lead to autocurricula in open-ended environments. In our paper, we propose to progressively increase the number of the agents as a curriculum for better scaling multi-agent reinforcement learning.
+
+Evolutionary Learning: Evolutionary algorithms, originally inspired by Darwin’s natural selection, has a long history (Back & Schwefel, 1993), which trains a population of agents in parallel, and ¨ let them evolve via crossover, mutation and selection processes. Recently, evolutionary algorithms have been applied to learn deep RL policies with various aims, such as to enhance training scalability (Salimans et al., 2017), to tune hyper-parameters (Jaderberg et al., 2017), to evolve intrinsic dense rewards (Jaderberg et al., 2018), to learn a neural loss for better generalization (Houthooft et al., 2018), to obtain diverse samples for faster off-policy learning (Khadka & Tumer, 2018), and to encourage exploration (Conti et al., 2018). Leveraging this insight, we apply evolutionary learning to better scale MARL: we train several groups of agents in each curriculum stage and keep evolving them to larger populations for the purpose of better adaptation towards the desired population scale and improved training stability. Czarnecki et al. (2018) proposed a similar evolutionary mix-and-match training paradigm to progressively increase agent capacity, i.e., larger action spaces and more parameters. Their work considers a fixed environment with an increasingly more complex agent and utilizes the traditional parameter crossover and mutation during evolution. By contrast, we focus on scaling MARL, namely an increasingly more complex environment with a growing number of agents. More importantly, we utilize MARL fine-tuning as an implicit mutation operator rather than the classical way of mutating parameters, which is more efficient, guided and applicable to even a very small number of evolution individuals. A similar idea of using learning for mutation is also considered by Gangwani & Peng (2018) in the single-agent setting.
+
+# 3 BACKGROUND
+
+Markov Games: We consider a multi-agent Markov decision processes (MDPs) (Littman, 1994). Such an $N$ -agent Markov game is defined by state space $s$ of the game, action spaces $\mathcal { A } _ { 1 } , . . . , \mathcal { A } _ { N }$ and observation spaces $\mathcal { O } _ { 1 } , . . . , \mathcal { O } _ { N }$ for each agent. Each agent $i$ receives a private observation correlated with the state $\mathbf { o } _ { i } : { \mathcal { S } } \mapsto { \mathcal { O } } _ { i }$ and produces an action by a stochastic policy $\pmb { \pi } _ { \pmb { \theta } _ { i } } : \mathcal { O } _ { i } \times \mathcal { A } _ { i } \mapsto [ 0 , 1 ]$ parameterized by $\theta _ { i }$ . Then the next states are produced according to the transition function $\tau$ : $\mathcal { S } \times \mathcal { A } _ { 1 } \times . . . \times \mathcal { A } _ { N } \mapsto \mathcal { S }$ . The initial state is determined by a distribution $\rho : { \cal S } \mapsto [ 0 , 1 ]$ . Each agent $i$ obtains rewards as a function of the state and its action $r _ { i } : S \times \mathcal { A } _ { i } \mapsto \mathbb { R }$ , and aims to maximize its own expected return $\begin{array} { r } { R _ { i } = \sum _ { t = 0 } ^ { T } \gamma ^ { t } r _ { i } ^ { t } ( s ^ { t } , a _ { i } ^ { t } ) } \end{array}$ , where $\gamma$ is a discount factor and $T$ is the time horizon. To minimize notation, we omit subscript of policy when there is no ambiguity.
+
+Multi-Agent Deep Deterministic Policy Gradient (MADDPG): MADDPG (Lowe et al., 2017) is a multi-agent variant of the deterministic policy gradient algorithm (Silver et al., 2014). It learns a centralized $\mathrm { \bf Q }$ function for each agent which conditions on global state information to resolve the non-stationary issue. Consider $N$ agents with deterministic policies $\pmb { \mu } = \{ \pmb { \mu } _ { 1 } , . . . , \pmb { \mu } _ { N } \}$ where $\pmb { \mu } _ { i } : \mathcal { O } _ { i } \mapsto \mathcal { A } _ { i }$ is parameterized by $\theta _ { i }$ . The policy gradient for agent $i$ is:
+
+$$
+\nabla _ { \boldsymbol { \theta } _ { i } } J ( \boldsymbol { \theta } _ { i } ) = \mathbb { E } _ { \mathbf { x } , a \sim \mathcal { D } } [ \nabla _ { \boldsymbol { \theta } _ { i } } \mu _ { i } ( o _ { i } ) \nabla _ { a _ { i } } Q _ { i } ^ { \mu } ( \mathbf { x } , a _ { 1 } , . . . , a _ { N } ) | _ { a _ { i } = \mu _ { i } ( o _ { i } ) } ] ,
+$$
+
+Here $\mathcal { D }$ denotes the replay buffer while $Q _ { i } ^ { \pmb { \mu } } ( { \bf x } , a _ { 1 } , . . . , a _ { N } )$ is a centralized action-value function for agent $i$ that takes the actions of all agents, $a _ { 1 } , \dots , a _ { N }$ and the state information $\mathbf { x }$ (i.e., $\mathbf { x } =$ $\left( o _ { 1 } , . . . , o _ { N } \right)$ or simply $\mathbf { x } = s$ if $s$ is available). Let $\mathbf { x } ^ { \prime }$ denote the next state from the environment transition. The replay buffer $\mathcal { D }$ contains experiences in the form of tuples $( \mathbf { x } , \mathbf { x } ^ { \prime } , a _ { 1 } , \ldots , a _ { N } , r _ { 1 } , \ldots , r _ { N } )$ Suppose the centralized critic $Q _ { i } ^ { \mu }$ is parameterized by $\phi _ { i }$ . Then it is updated via:
+
+$$
+\begin{array} { r } { \mathcal { L } ( \boldsymbol { \phi } _ { i } ) = \mathbb { E } _ { \mathbf { x } , a , r , \mathbf { x } ^ { \prime } } [ ( Q _ { i } ^ { \mu } ( \mathbf { x } , a _ { 1 } , \dots , a _ { N } ) - y ) ^ { 2 } ] , \quad y = r _ { i } + \gamma Q _ { i } ^ { \mu ^ { \prime } } ( \mathbf { x } ^ { \prime } , a _ { 1 } ^ { \prime } , \dots , a _ { N } ^ { \prime } ) \big | _ { a _ { j } ^ { \prime } = \mu _ { j } ^ { \prime } ( o _ { j } ) } , } \end{array}
+$$
+
+where $\boldsymbol { \mu } ^ { \prime } = \{ \pmb { \mu } _ { \theta _ { 1 } ^ { \prime } } , . . . , \pmb { \mu } _ { \theta _ { N } ^ { \prime } } \}$ is the set of target policies with delayed parameters $\theta _ { i } ^ { \prime }$ . Note that the centralized critic is only used during training. At execution time, each policy $\pmb { \mu } _ { \theta _ { i } }$ remains decentralized and only takes local observation $o _ { i }$ .
+
+# 4 EVOLUTIONARY POPULATION CURRICULUM
+
+In this section, we will first describe the base network architecture with the self-attention mechanism (Vaswani et al., 2017) which allows us to incorporate a flexible number of agents during training. Then we will introduce the population curriculum paradigm and the evolutionary selection process.
+
+![](images/093992fce52d59fed8fcd540a0f8bcee9ca8c0cffad4a0073640b1e5aa0f6948.jpg)  
+Figure 1: Our population-invariant Q function: (a) utilizes the attention mechanism to combine embeddings from different observation-action encoder $f _ { i }$ ; (b) is a detailed description for $f _ { i }$ , which also utilizes an attention module to combine $M$ different entities in one observation.
+
+# 4.1 POPULATION-INVARIANT ARCHITECTURE
+
+We describe our choice of architecture based on the MADDPG algorithm (Lowe et al., 2017), which is population-invariant in the sense that both the Q function and the policy can take in an arbitrary number of input entities. We first introduce the Q function (Fig. 1) and then the policy.
+
+We adopt the decentralized execution framework, so each agent has its own Q function and policy network. Particularly for agent $i$ , its centralized Q function is represented as follows:
+
+$$
+Q _ { i } ^ { \mu } ( \mathbf { x } , a _ { 1 } , \ldots , a _ { N } ) = h _ { i } ( [ g _ { i } ( f _ { i } ( o _ { i } , a _ { i } ) ) , v _ { i } ] ) , { \mathrm { ~ w h e r e ~ } } v _ { i } = \mathrm { a t t e n t i o n } ( f _ { i } ( o _ { j } , a _ { j } ) \forall j \neq i )
+$$
+
+Here $f _ { i } ( o _ { j } , a _ { j } )$ is an observation-action encoder (the green box in Fig. 1(a)) which takes in the observation $o _ { j }$ and the action $a _ { j }$ from agent $j$ , and outputs the agent embedding of agent $j ; v _ { i }$ denotes the global attention embedding (the orange box in Fig. 1(a)) over all the agent embeddings. We will explain $v _ { i }$ and $f _ { i }$ later. $g _ { i }$ is a 1-layer fully connected network processing the embedding of the ith agent’s own observation and action. $h _ { i }$ is a 2-layer fully connected network that takes the concatenation of the output of $g _ { i }$ and the global attention embedding $v _ { i }$ and outputs the final $\mathrm { Q }$ value.
+
+Attention Embedding $v _ { i }$ : We define the attention embedding $v _ { i }$ by a weighted sum of each agent’s embedding $f _ { i } ( o _ { j } , a _ { j } )$ for $j \neq i$ :
+
+$$
+v _ { i } = \sum _ { j \neq i } \alpha _ { i , j } f _ { i } ( o _ { j } , a _ { j } )
+$$
+
+The coefficient $\alpha _ { i , j }$ is computed by
+
+$$
+\alpha _ { i , j } = \frac { \exp { ( \beta _ { i , j } ) } } { \sum _ { j \neq i } \exp { ( \beta _ { i , j } ) } } , \quad \beta _ { i , j } = f _ { i } ^ { T } ( o _ { i } , a _ { i } ) \boldsymbol { W } _ { \psi } ^ { T } \boldsymbol { W } _ { \phi } f _ { i } ( o _ { j } , a _ { j } )
+$$
+
+where $W _ { \psi }$ and $W _ { \phi }$ are parameters to learn. $\beta _ { i , j }$ computes the correlation between the embeddings of agent $i$ and every other agent $j$ via an inner product. $\alpha _ { i , j }$ is then obtained by normalizing $\beta _ { i , j }$ by a softmax function. Since we represent the observations and actions of other agents with a weighted mean $v _ { i }$ from Eq. 4, we can model the interactions between agent $i$ and an arbitrary number of other agents, which allows us to easily increase the number of agents in our curriculum training paradigm.
+
+Observation-Action Encoder $f _ { i }$ : We now define the structure of $f _ { i } ( o _ { j } , a _ { j } )$ (Fig. 1(b)). Note that the observation of agent $j$ , $o _ { j }$ , also includes many entities, i.e., states of all visible agents and objects in the game. Suppose $o _ { j }$ contains $M$ entities, i.e., $o _ { j } = [ o _ { j , 1 } , \dotsc , o _ { j , M } ]$ . $M$ may also vary as the agent population scales over training procedure or simply during an episode when some agents die. Thus, we apply another attention module to combine these entity observations together in a similar way to how $v _ { i }$ is computed (Eq. 4, 5).
+
+In more details, we first apply an entity encoder for each entity type to obtain entity embeddings of all the entities within that type. For example, in $o _ { j }$ , we can have embeddings for agent entities (green boxes in Fig. 1(b)) and landmark/object entities (purple boxes in Fig. 1(b)). Then we apply an attention module over each entity type by attending the entity embedding of agent $j$ to all the entities of this type to obtain an attended type embedding (the orange box in Fig. 1(b)). Next, we concatenate all the type embeddings together with the entity embedding of agent $j$ as well as its action embedding. Finally, this concatenated vector is forwarded to a fully connected layer to generate the output of $f _ { i } ( o _ { j } , a _ { j } )$ . Note that in the overall critic network of agent $i$ , the same encoder $f _ { i }$ is applied to every observation-action pair so that the network can maintain a fixed size of parameters even when the number of agents increases significantly.
+
+Policy Network: The policy network $\pmb { \mu } _ { i } ( o _ { i } )$ has a similar structure as the observation-action encoder $f _ { i } ( o _ { i } , a _ { i } )$ , which uses an attention module over the entities of each type in the observation $o _ { i }$ to adapt to the changing population during training. The only difference in this network is that the action $a _ { i }$ is not included in the input. Notably, we do not share parameters between the Q function and the policy.
+
+# 4.2 POPULATION CURRICULUM
+
+We propose to progressively scale the number of agents in MARL with a curriculum. Before combining with the evolutionary selection process, we first introduce a simpler version, the vanilla population curriculum (PC), where we perform the following stage-wise procedure: (i) the initial stage starts with MARL training over a small number of agents using MADDPG and our populationinvariant architecture; (ii) we start a new stage and double1 the number of agents by cloning each of the existing agents; (iii) apply MADDPG training on this scaled population until convergence; (iv) if the desired number of agents is not reached, go back to step (ii).
+
+Mathematically, given $N$ trained agents with parameters $\pmb { \theta } = \{ \theta _ { 1 } , . . . , \theta _ { N } \}$ from the previous stage, we want to increase the number of the agents to $2 N$ with new parameters $\tilde { \pmb { \theta } } = \{ \tilde { \theta } _ { 1 } , . . . , \tilde { \theta } _ { N } , . . . , \tilde { \theta } _ { 2 N } \}$ for the next stage . In this vanilla version of population curriculum, we simply initialize $\tilde { \pmb { \theta } }$ by setting $\tilde { \theta } _ { i } \gets \theta _ { i }$ and $\widetilde { \theta } _ { N + i }  \theta _ { i }$ , and then continue MADDPG training on $\tilde { \pmb { \theta } }$ to get the final policies for the new stage. Although $\tilde { \theta _ { i } }$ and $\tilde { \theta } _ { N + i }$ are both initialized from $\theta _ { i }$ , as training proceeds, they will converge to different policies since these policies are trained in a decentralized manner in MADDPG.
+
+# 4.3 EVOLUTIONARY SELECTION
+
+Introducing new agents by directly cloning existing ones from the previous stage has a clear limitation: the policy parameters suitable for the previous environment are not necessarily the best initialization for the current stage as the population is scaled up. In the purpose of better performance in the final game with our desired population, we need to promote agents with better adaptation abilities during early stages of training.
+
+Therefore, we propose an evolutionary selection process to facilitate the agents’ scaling adaption ability during the curriculum procedure. Instead of training a single set of agents, we maintain $K$ parallel sets of agents in each stage, and perform crossover, mutation and selection among them for the next stage. This is the last piece in our proposed Evolutionary Population Curriculum (EPC) paradigm, which is essentially population curriculum enhanced by the evolutionary selection process.
+
+Specifically, we assume the agents in the multi-agent game have $\Omega$ different roles. Agents in the same role have the same action set and reward structure. For example, we have $\Omega = 2$ roles in a predator-prey game, namely predators and prey, and $\Omega = 1$ role of agents for a fully cooperative game with homogeneous agents. For notation conciseness, we assume there are $N _ { 1 }$ agents of role 1, namely $A _ { 1 } = \{ \pmb { \mu } _ { 1 } , . . . , \pmb { \mu } _ { N _ { 1 } } \}$ ; $N _ { 2 }$ agents of role 2, namely $A _ { 2 } = \{ \pmb { \mu } _ { N _ { 1 } + 1 } , . . . , \pmb { \mu } _ { N _ { 1 } + N _ { 2 } } \}$ , and so on. In each stage, we keep $K$ parallel sets for each role of agents, denoted by $A _ { i } ^ { ( 1 ) } , \ldots , A _ { i } ^ { ( K ) }$ for role $i$ , and take a 3-step procedure, i.e., mix-and-match (crossover), MARL fine-tuning (mutation) and selection, as follows to evolve these $K$ parallel sets of agents for the next stage.
+
+Mix-and-Match (Crossover): In the beginning of a curriculum stage, we scale the population of agents from $N$ to $2 N$ . Note that we have $K$ parallel agent sets of size $N _ { i }$ for role $i$ , namely A(1)i , . $A _ { i } ^ { ( 1 ) } , \ldots , A _ { i } ^ { ( K ) }$ . We first perform a mix-and-match over these parallel sets within every role $i$ : for each set A(j), we pair it with all the $K$ sets of the same role, which leads to $K ( K + 1 ) / 2$ new scaled agent sets of size $2 N _ { i }$ . Given these scaled sets of agents, we then perform another mix-and-match across all the $\Omega$ roles: we pick one scaled set for each role and combine these $\Omega$ selected sets to produce a scaled game with $2 N$ agents. For example, in the case of $\Omega = 2$ , we can pick one agent set A(k1)1 from the first role and another agent set $A _ { 2 } ^ { ( k _ { 2 } ) }$ from the second role to form a scaled game.
+
+Thus, there are $C _ { \mathrm { m a x } } = \left( K ( K + 1 ) / 2 \right) ^ { \Omega }$ different combinations in total through this mix-and-match process. We sample $C$ games from these combinations for mutation in the next step. Since we are mixing parallel sets of agents, this process can be considered as the crossover operator in standard evolutionary algorithms.
+
+MARL Fine-Tuning (Mutation): In standard evolutionary algorithms, mutations are directly performed on the parameters, which is inefficient in high-dimensional spaces and typically requires a large amount of mutants to achieve sufficient diversity for evolution. Instead, here we adopt MARL fine-tuning in each curriculum stage (step (iii) in vanilla PC) as our guided mutation operator, which naturally and efficiently explores effective directions in the parameter space. Meanwhile, due to the training variance, MARL also introduces randomness which benefits the overall diversity of the evolutionary process. Concretely, we apply parallel MADDPG training on each of the $C$ scaled games generated from the mix-and-match step and obtain $C$ mutated sets of agents for each role.
+
+Selection: Among these $C$ mutated sets of agents for each role, only the best $K$ mutants can survive. In the case of $\Omega = 1$ , the fitness score of a set of agents is computed as their average reward after MARL training. In other cases when $\Omega \geq 2$ , given a particular mutated set of agents of a specific role, we randomly generate games for this set of agents and other mutated sets from different agent roles. We take its average reward from these randomly generated games as the fitness score for this mutated set. We pick the top- $K$ scored sets of agents in each role to advance to the next curriculum stage.
+
+# Algorithm 1: Evolutionary Population Curriculum
+
+Data: environment $E ( N , \{ A _ { i } \} _ { 1 \leq i \leq \Omega } )$ with $N$ agents of $\Omega$ roles, desired population $N _ { d }$ , initial population $N _ { 0 }$ , evolution size $K$ , mix-and-match size $C$   
+Result: a set of $N _ { d }$ best policies   
+$N \gets N _ { 0 }$ ;   
+initialize $K$ parallel agent sets $A _ { i } ^ { ( 1 ) } , \ldots , A _ { i } ^ { ( K ) }$ for each role $1 \leq i \leq \Omega$ ;   
+initial parallel MARL training on $K$ games, $E ( N , \{ A _ { i } ^ { ( j ) } \} _ { 1 \leq i \leq \Omega } )$ for $1 \leq j \leq K$ ;   
+while $N < N _ { d }$ do $N \gets 2 \times N$ ; for $1 \leq j \leq C$ do $\lfloor$ for each role $1 \leq i \leq \Omega \colon j _ { 1 } , j _ { 2 } \gets \mathrm { u n i f } ( 1 , K ) , ~ \tilde { A } _ { i } ^ { ( j ) } \gets A _ { i } ^ { ( j _ { 1 } ) } + A _ { i } ^ { ( j _ { 2 } ) }$ (mix-and-match); MARL training in parallel on $E ( N , \{ \tilde { A } _ { i } ^ { ( j ) } \} _ { 1 \leq i \leq \Omega } )$ for $1 \leq j \leq C$ (guided mutation) ; for role $1 \leq i \leq \Omega$ do for $1 \leq j \leq C$ do $S _ { i } ^ { ( \bar { j } ) } \stackrel { \smile } {  } { \mathbb { E } } _ { k _ { t \neq i } \sim [ 1 , C ] } [ \mathrm { a v g . r e w a r d s o n } E ( N , \{ \tilde { A } _ { 1 } ^ { ( k _ { 1 } ) } , \dots , \tilde { A } _ { i } ^ { ( j ) } , \dots , \tilde { A } _ { \Omega } ^ { ( k _ { \Omega } ) } \} ) ]$ (fitness); $A _ { i } ^ { ( 1 ) } , \ldots , A _ { i } ^ { ( K ) } \gets \mathrm { t o p } { - } K$ w.r.t. $S _ { i }$ from $\tilde { A } _ { i } ^ { ( 1 ) } , \ldots , \tilde { A } _ { i } ^ { ( C ) }$ (selection);
+
+return the best set of agents in each role, i.e., $\{ A _ { i } ^ { ( k _ { i } ^ { \star } ) } | k _ { i } ^ { \star } \in [ 1 , K ] \ \forall 1 \leq i \leq \Omega \} \colon$ ;
+
+Overall Algorithm: Finally, when the desired population is achieved, we take the best set of agents in each role based on their last fitness scores as the output. We conclude the detailed steps of EPC in Alg. 1. Note that in the first curriculum stage, we just train $K$ parallel games without mix-and-match or mutation. So, EPC simply selects the best from the $K$ initial sets in the first stage while the evolutionary selection process only takes effect starting from the second stage. We emphasize that although we evolve multiple sets of agents in each stage, the three operators, mix-and-match, MARL fine-tuning and selection, are all perfectly parallel. Thus, the evolutionary selection process only introduces little influence on the overall training time. Lastly, EPC is an RL-algorithm-agnostic learning paradigm that can be potentially integrated with any MARL algorithm other than MADDPG.
+
+# 5 EXPERIMENT
+
+We experiment on three challenging environments, including a predatory-prey-style Grassland game, a mixed-cooperative-and-competitive Adversarial Battle game and a fully cooperative Food Collection game. We compare EPC with multiple baseline methods on these environments with different scales of agent populations and show consistently large gains over the baselines. In the following, we will first introduce the environments and the baselines, and then both qualitative and quantitative performances of different methods on all three environments.
+
+# 5.1 ENVIRONMENTS
+
+All these environments are built on top of the particle-world environment (Mordatch & Abbeel, 2018) where agents take actions in discrete timesteps in a continous 2D world.
+
+Grassland: In this game, we have $\Omega \ = \ 2$ roles of agents, $N _ { S }$ sheep and $N _ { W }$ wolves, where sheep moves twice as fast as wolves. We also have a fixed amount of $L$ grass pellets (food for sheep) as green landmarks (Fig. 2a). A wolf will be rewarded when it collides with (eats) a sheep,
+
+![](images/24e6b7437d63af3c204e1fdc3e21a6e3d6ca8187435e82821283ac9a9da897d4.jpg)  
+Figure 2: Environment Visualizations
+
+and the (eaten) sheep will obtain a negative reward and becomes inactive (dead). A sheep will be rewarded when it comes across a grass pellet and the grass will be collected and respawned in another random position. Note that in this survival game, each individual agent has its own reward and does not share rewards with others.
+
+Adversarial Battle: This scenario consists of $L$ units of resources as green landmarks and two teams of agents (i.e., $\Omega = 2$ for each team) competing for the resources (Fig. 2b). Both teams have the same number of agents $( N _ { 1 } = N _ { 2 }$ ). When an agent collects a unit of resource, the resource will be respawned and all the agents in its team will receive a positive reward. Furthermore, if there are more than two agents from team 1 collide with one agent from team 2, the whole team 1 will be rewarded while the trapped agent from team 2 will be deactivated (dead) and the whole team 2 will be penalized, and vice versa.
+
+Food Collection: This game has $N$ food locations and $N$ fully cooperative agents $\Omega = 1 \times$ ). The agents need to collaboratively occupy as many food locations as possible within the game horizon (Fig. 2c). Whenever a food is occupied by any agent, the whole team will get a reward of $6 / N$ in that timestep for that food. The more food occupied, the more rewards the team will collect.
+
+In addition, we introduce collision penalties as well as auxiliary shaped rewards for each agent in each game for easier training. All the environments are fully observable so that each agent needs to handle a lot of entities and react w.r.t. the global state. More environment details are in Appx. A.
+
+# 5.2 METHODS AND METRIC
+
+We evaluate the following approaches in our experiments: (1) the MADDPG algorithm (Lowe et al., 2017) with its original architecture (MADDPG); (2) the provably-converged mean-field algorithm (Yang et al., 2018) (mean-field); (3) the MADDPG algorithm with our population-invariant architecture (Att-MADDPG); (4) the vanilla population curriculum without evolutionary selection (vanilla-PC); and (5) our proposed EPC approach (EPC). For EPC parameters, we choose $K = 2$ for Grassland and Adversarial Battle and $K = 3$ for Food Collection; for the mix-and-match size $C$ , we simply set it $C _ { \mathrm { m a x } }$ and enumerate all possible mix-and-match combinations instead of random sampling. All the baseline methods are trained until the same amount of accumulative episodes as EPC took. More training details can be found in Appx. B.
+
+For Grassland and Adversarial Battle with $\Omega = 2$ , we evaluate the performance of different methods by competing their trained agents against our EPC trained agents. Specifically, in Grassland, we let sheep trained by each approach compete with the wolves from EPC and collect the average sheep reward as the evaluation metric for sheep. Similarly, we take the same measurement for wolves from each method. In Adversarial Battle, since two teams are symmetric, we just evaluate the shared reward of one team trained by each baseline against another team by EPC as the metric. For Food Collection with $\Omega = 1$ , since it is fully cooperative, we take the team reward for each method as the evaluation metric. In addition, for better visualization, we plot the normalized scores by normalizing the rewards of different methods between 0 and 1 in each scale for each game. More evaluation details are in Appx. C.
+
+# 5.3 QUALITATIVE RESULTS
+
+In Grassland, as the number of wolves goes up, it becomes increasingly more challenging for sheep to survive; meanwhile, as the sheep become more intelligent, the wolves will be incentivized to be more aggressive accordingly. In Fig. 3, we illustrate two representative matches for competition, including one using the MADDPG sheep against the EPC wolves (Fig. 3a), and the other between the EPC sheep and the MADDPG wolves (Fig. 3b). From Fig. 3a, we can observe that the MADDPG sheep can be easily eaten up by the EPC wolves (note that dark circle means the sheep is eaten). On the other hand, in Fig. 3b, we can see that the EPC sheep learns to eat the grass and avoid the wolves at the same time.
+
+![](images/223ac5dab1c175bc806b98e6d1e31b9b70e2c3c0554197d641200100c99c1760.jpg)
+
+![](images/7376a966ff95eb5db154ad9aa83eee6220f22869d5760b9e508f4b27aa2dc26c.jpg)  
+(a) MADDPG sheep vs EPC wolves (b) MADDPG wolves vs EPC sheep Figure 3: Example matches between EPC and MADDPG trained agents in Grassland
+
+![](images/95a8287041fa44173e285dd951c6bfbfbb28a9224d26eb93736f12999dcd645e.jpg)  
+Figure 4: Adversarial Battle: dark particles are dead agents.
+
+![](images/9c0a25c31414a42411d7b4ec11e490d99524f9dfc5529ea1456d8a37d564f272.jpg)  
+Figure 5: Food Collection
+
+In Adversarial Battle, we visualize two matches in Fig. 4 with one over agents by EPC (Fig. 4a) and the other over agents by MADDPG (Fig. 4b). We can clearly see the collaborations between the EPC agents: although the agents are initially spread over the environment, they learn to quickly gather as a group to protect themselves from being killed. While for the MADDPG agents, their behavior shows little incentives to cooperate or compete — these agents stay in their local regions throughout the episode and only collect resources or kill enemies very infrequently.
+
+In Food Collection (Fig. 5), the EPC agents in Fig. 5a learn to spread out and occupy as many food as possible to maximize the team rewards. While only one agent among the MADDPG agents in Fig. 5b successfully occupies a food in the episode.
+
+# 5.4 QUANTITATIVE RESULTS
+
+# Quantitative Results in Grassland
+
+In the Grassland game, we perform curriculum training by starting with 3 sheep and 2 wolves, and gradually increase the population of agents. We denote a game with $N _ { S }$ sheep and $N _ { W }$ wolves by “scale $N _ { S ^ { - } } N _ { W }$ ”. We start with scale 3-2 and gradually increase the game size to scales 6-4, 12-8 and finally 24-16.For the two curriculum learning approach, vanilla-PC and EPC, we train over $1 0 ^ { 5 }$ episodes in the first curriculum stage (scale 3-2) and fine-tune the agents with $5 \times 1 0 ^ { 4 }$ episodes after mix-and-match in each of the following stage. For other methods that train the agents from scratch, we take the same accumulative training iterations as the curriculum methods for a fair comparison.
+
+Main Results: We report the performance of different methods for each game scale in Fig. 6a. Overall, there are little differences between the mean-field approach and the original MADDPG algorithm while the using the population-invariant architecture (i.e., Att-MADDPG) generally boosts the performance of MADDPG. For the method with population curriculum, vanilla-PC performs almost the same as training from scratch (Att-MADDPG) when the number of agents in the environment is small (i.e., 6-4) but the performance gap becomes much more significant when the population further grows (i.e., 12-18 and 24-16). For our proposed EPC method, it consistently outperforms all the baselines across all the scales. Particularly, in the largest scale 24-16, EPC sheep receive $1 0 \mathrm { x }$ more rewards than the best baseline sheep without curriculum training.
+
+Detailed Statistics: Besides rewards, we also compute the statistics of sheep to understand how the trained sheep behave in the game. We perform competitions between sheep trained by different methods against the EPC wolves and measure the average number of total grass pellets eaten per episode, i.e, #grass eaten, and the average percentage of sheep that survive until the end of an episode, i.e., survival rate, in Fig. 6b. We can observe that as the population increases, it becomes increasingly harder for sheep to survive while EPC trained sheep remain a high survival rate even on the largest scale. Moreover, as more sheep in the game, EPC trained sheep consistently learn to eat more grass even under the strong pressure from wolves. In contrast, the amount of eaten grass of MADDPG approach (i.e., Att-MADDPG) drastically decreases when the number of wolves becomes large.
+
+# Quantitative Results in Adversarial Battle
+
+![](images/27ec6c79dc58012bdf0b0d8a3c43208bfd6a13b74c3fc0d482c7d261d8041a1e.jpg)  
+Figure 6: Results in Grassland. In part (a), we show the normalized scores of wolves and sheep trained by different methods when competing with EPC sheep and EPC wolves respectively. In part (b), we measure the sheep statistics over different scales $\mathbf { \dot { x } }$ -axis), including the average number of total grass pellets eaten per episode (left) and the average percentage of sheep that survive until the end of episode (right). EPC trained agents (yellow) are consistently better than any baseline method.
+
+In this game, we evaluate on environments with different sizes of agent population $N$ , denoted by scale $N _ { 1 } – N _ { 2 }$ where $N _ { 1 } = N _ { 2 } =$ $\bar { N } / 2$ . We start the curriculum from scale 4-4 and the increase the population size to scale 8-8 $N = 1 6$ ) and finally 16-16 $N = 3 2$ ). Both vanilla-PC and EPC take $5 \times 1 0 ^ { 4 }$ training episodes in the first stage and then $2 \times 1 0 ^ { 4 }$ episodes in the following two curriculum stages. We report the normalized scores of different methods in Fig. 7, where agents trained by EPC outperforms all the baseline methods increasingly more significant as the agent population grows.
+
+# Quantitative Results in Food Collection
+
+![](images/a5ec3320b5797a63d2419f5b17f6bd6ef8cdb45a064214a96d2e2c2b71f33076.jpg)  
+Figure 7: Adversarial Battle
+
+In this game, we begin curriculum training with $N = 3$ , namely 3 agents and 3 food locations, and progressively increase the population size $N$ to 6, 12 and finally 24. Both vanilla-PC and EPC perform training on $5 \times 1 0 ^ { 4 }$ episodes on the first stage of $N = 3$ and then $2 \times 1 0 ^ { \overline { { 4 } } }$ episodes in each of the following curriculum stage. We report the normalized scores for all the methods in Fig. 8, where EPC is always the best among all the approaches with a clear margin. Note that the performance of the original MADDPG and the meanfield approach drops drastically as the population size $N$ increases. Particularly, the mean-field approach performs even worse than the original MADDPG method. We believe this is because in this game, the agents must act according to the global team state collaboratively, which means the local approximation assumption in the mean-field approach does not hold clearly.
+
+![](images/f70986fc2a2058a05b780148d32f36862c5672218cbd862eefe5756cd2c3b85c.jpg)  
+Figure 8: Food Collection
+
+# Ablative Analysis
+
+Stability Analysis: The evolutionary selection process in EPC not only leads to better final performances but also stabilizes the training procedure. We validate the stability of EPC by computing the variance over 3 training seeds for the same experiment and comparing with the variance of vanilla-PC, which is also obtained from 3 training seeds. Specifically, we pick the second stage of curriculum learning and visualize the variance of agent scores throughout the stage of training. These scores are computed by competing against the final policy trained by EPC. We perform analysis on all the 3 environments: Grassland with scale 6-4 (Fig. 9a), Adversarial Battle with scale 8-8 (Fig. 9b) and Food Collection with scale 6 (Fig. 9c). We can observe that the variance of EPC is much smaller than vanilla-PC in different games.
+
+Convergence Analysis: To illustrate that the self-attention based policies trained from a smaller scale is able to well adapt to a larger scale via fine-tuning, we pick a particular mutant by EPC in the second curriculum stage and visualize its learning curve throughout fine-tuning for all the environments, Grassland (Fig. 9d), Adversarial Battle (Fig. 9e) and Food Collection (Fig. 9f). The scores are computed in the same way as the stability analysis. By comparing to MADDPG and Att-MADDPG, which train policies from scratch, we can see that EPC starts learning with a much higher score, continues to improve during fine-tuning and quickly converges to a better solution. Note that all baselines are in fact trained much longer. The full convergence curves are in App. D.1.
+
+Generalization: We investigate whether the learned policies can generalize to a different test environment with even a larger scale than the training ones. To do so, we take the best polices trained by different methods on the largest population and directly apply these policies to a new environment with a doubled population by self-cloning. We evaluate in all the environments with EPC, vanilla-PC and Att-MADDPG and measure the normalized scores of different methods, which is computed in the same way as the fitness score. In all cases, we observe a large advantage of EPC over the other two methods, indicating the better generalization ability for policies trained by EPC.
+
+![](images/10de18c8599cba73aa5c05edfdbffa4e5cffd36cb0afcd13096be8716675afd4.jpg)  
+Figure 9: Ablation analysis on the second curriculum stage in all the games over 3 different training seeds. Stability comparison (top) in (a), (b) and (c): We observe EPC has much less variance comparing to vanilla-PC. Normalized scores during fine-tuning (bottom) in (d), (e) and (f): This illustrates that EPC can successfully transfer the agents trained with a smaller population to a larger population by fine-tuning.
+
+![](images/a6e38641f77a8b32c6b9abb93d8524efa1b23b0448d5ab5582bc0f09f2730fb0.jpg)  
+Figure 10: Environment Generalization: We take the agents trained on the largest scale and test on an environment with twice the population. We perform experiments on all the games and show that EPC also advances the agents’ generalizability.
+
+# 6 CONCLUSION
+
+In this paper, we propose to scale multi-agent reinforcement learning by using curriculum learning over the agent population with evolutionary selection. Our approach has shown significant improvements over baselines not only in the performance but also the training stability. Given these encouraging results on different environments, we believe our method is general and can potentially benefit scaling other MARL algorithms. We also hope that learning with a large population of agents can also lead to the emergence of swarm intelligence in environments with simple rules in the future.
+
+# ACKNOWLEDGMENT
+
+This research is supported in part by ONR MURI N000141612007, ONR Young Investigator to AG. FF is also supported in part by NSF grant IIS-1850477, a research grant from Lockheed Martin, and the U.S. Army Combat Capabilities Development Command Army Research Laboratory Cooperative Agreement Number W911NF-13-2-0045 (ARL Cyber Security CRA). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the funding agencies. We also sincerely thank Bowen Baker and Ingmar Kanitscheider from OpenAI for valuable suggestions and comments.
+
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+
+# A ENVIRONMENT DETAILS
+
+In the Grassland game, sheep gets $+ 2$ reward when he eats the grass, -5 reward when eaten by wolf. The wolf get $^ { + 5 }$ reward when eats a sheep. We also shape the reward by distance, sheep will get less negative reward when it is closer to grass and wolf will get less negative reward when it is closer to sheep. This game is adapted from the original Predator-prey game in the MADDPG paper (Lowe et al., 2017) by introducing grass and allowing agent to die.
+
+In the Adversarial Battle game, agent will get $+ 1$ reward when he eats the food, $- 6$ reward when killed by other agents. If $N$ agents kill an enemy, they will be rewarded $+ 6 / N$ . We shape the reward by distance. Agent will receive less negative rewards when it is closer to other agents and grass. We want to encourage collision within agents and also will be easier for them to learn to eat. This game is adapted from the mean-field MARL paper (Yang et al., 2018) by converting it from a grid world to particle-world, introducing food and only allowing 2-agent cooperative killing.
+
+In the Food Collection game, there are $N$ agents and $N$ food locations. Each agent will get a shared $+ 6 / N$ reward per timestep when one food is occupied by any agent. If one agent gets collision with another, all of the agents will get a punish of $- 6 / N$ . We shape the reward by distance. Agents will receive less negative rewards when it gets closer to the food. Since the number of agents and food are equal, we want to avoid the collision within agents and let the agents to learn to occupy as many food as possible. This is exactly the same game as the Cooperative Navigation game in the MADDPG paper. We slightly change the reward function to ensure it is bounded w.r.t. arbitrary number of agents.
+
+We use the normalized reward as the score during evaluation. For a particular game with a particular scale, we first collect the reward for each type of agents, namely the average reward of each individual of that type without the shaped rewards. Then we re-scale the collected rewards by considering the lowest reward among all methods as score 0 and highest reward as score 1.
+
+# B TRAINING DETAILS
+
+We follow all the hyper-parameters in the original MADDPG paper (Lowe et al., 2017) for both EPC and all the baseline methods considered. Particularly, we use the Adam optimizer with learning rate 0.01, $\beta _ { 1 } = 0 . 9$ , $\beta _ { 2 } = 0 . 9 9 9$ and $\varepsilon = 1 0 ^ { - 8 }$ across all experiments. $\tau = 0 . 0 1$ is set for target network update and $\gamma = 0 . 9 5$ is used as discount factor. We also use a replay buffer of size $1 0 ^ { 6 }$ and we update the network parameters after every 100 samples. The batch size is 1024. All the baseline methods are trained for a number of episodes that equals the accumulative number of episodes that EPC has taken.
+
+We set $K \ : = \ : 2$ in all the games during training except that $K \ : = \ : 3$ in the food collection game due to computational constraints. During EPC training, in the grassland game, we train the scale of 3 sheep 2 wolf for 100000 episodes. We train another 50000 episodes every time the agents number doubles. In the adversarial battle game and food collection game, we train the first scale for 50000 episodes. We train another 20000 episodes every time the agents number doubles.
+
+In the grassland game, the entity types are the agent itself, other sheep, other wolf and food. We thus have four types of entity encoders for each of those entity types. In the adversarial battle game, Similar to grassland game, the entity types are agent itself, other teammates, enemies and food. We also have four types of entity encoders for each of those entity types. Since there is only one group in the food collection game, the entity types are agent itself, other teammates and food. We thus have three entity encoders in our network.
+
+# C EVALUATION DETAILS
+
+To evaluate the agents trained in the environment with $\Omega = 2$ , we make two roles of agents trained with different approaches compete against each other. Each competition is simulated for 10000 episodes. The average normalized reward over the 10000 episodes will be used as the competition score for each side. Note that in our experiments, we let all the methods compete against our EPC approach for evaluation. For adversarial battle game, we take the average score of two teams as the model’s final evaluation score, since the two teams in this game are completely symmetric.
+
+In the food collection game, since there is only one role, we simply simulate the model for 10000 episodes. The average normalized reward over the 10000 episodes will be used as the score of the model.
+
+# D ADDITIONAL DETAILS ON EXPERIMENT RESULTS
+
+# D.1 FULL TRAINING CURVES FOR BASELINES
+
+All the baseline methods are trained for a number of episodes that equals the accumulative number of episodes that EPC has taken.
+
+The purpose of Figure 9e,9e,9f is simply showing the transfer performance, i.e., the initialization produced by EPC from the previous stage is effective and can indeed leverage past experiences to warm-start. The $\mathbf { X }$ -axis of
+
+the plot was shrunk for visualization purpose. Here we illustrate the complete convergence curve of baselines, i.e., ATT-MADDPG and MADDPG, in Figure 11a,11b,11c for the 3 games respective. Although Att-MADDPG takes a much longer time to learn, its performance is still far worse than EPC.
+
+![](images/7ad0517d06250593c39fc1f7fe0c29605484bee30b01e016035364f31af19d2c.jpg)  
+Figure 11: Full learning curves on the second curriculum stage in all the games. EPC fine-tunes the policies obtained from the previous stage while MADDPG and Att-MADDPG are trained from scratch for a much longer time.
+
+# D.2 RAW REWARD NUMBERS OF EVALUATION RESULTS
+
+In this section, we provide the actual rewards without normalization when comparing all the baselines with EPC.   
+These scores are corresponding to the histograms reported in Figure 6a, 7, 8.
+
+Grassland game, wolf rewards, corresponding to wolf in Figure 6a:
+
+<table><tr><td rowspan=1 colspan=1>scale</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>3-2</td><td rowspan=1 colspan=1>0.596</td><td rowspan=1 colspan=1>0.877</td><td rowspan=1 colspan=1>0.8145</td><td rowspan=1 colspan=1>0.8145</td><td rowspan=1 colspan=1>1.407</td></tr><tr><td rowspan=1 colspan=1>6-4</td><td rowspan=1 colspan=1>3.7735</td><td rowspan=1 colspan=1>0.9515</td><td rowspan=1 colspan=1>2.5905</td><td rowspan=1 colspan=1>2.001</td><td rowspan=1 colspan=1>3.7735</td></tr><tr><td rowspan=1 colspan=1>12-8</td><td rowspan=1 colspan=1>3.1915</td><td rowspan=1 colspan=1>3.385</td><td rowspan=1 colspan=1>10.2125</td><td rowspan=1 colspan=1>9.974</td><td rowspan=1 colspan=1>14.377</td></tr><tr><td rowspan=1 colspan=1>24-16</td><td rowspan=1 colspan=1>14.482</td><td rowspan=1 colspan=1>18.272</td><td rowspan=1 colspan=1>32.8945</td><td rowspan=1 colspan=1>47.6365</td><td rowspan=1 colspan=1>61.4245</td></tr></table>
+
+Grassland game, sheep rewards, corresponding to sheep in Figure 6a:
+
+<table><tr><td rowspan=1 colspan=1>scale</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>3-2</td><td rowspan=1 colspan=1>-4.0026</td><td rowspan=1 colspan=1>-3.9947</td><td rowspan=1 colspan=1>2.66</td><td rowspan=1 colspan=1>2.66</td><td rowspan=1 colspan=1>8.3846</td></tr><tr><td rowspan=1 colspan=1>6-4</td><td rowspan=1 colspan=1>-20.2494</td><td rowspan=1 colspan=1>-20.5107</td><td rowspan=1 colspan=1>0.9892</td><td rowspan=1 colspan=1>1.1804</td><td rowspan=1 colspan=1>10.1455</td></tr><tr><td rowspan=1 colspan=1>12-8</td><td rowspan=1 colspan=1>-52.863</td><td rowspan=1 colspan=1>-53.6338</td><td rowspan=1 colspan=1>-42.4801</td><td rowspan=1 colspan=1>-11.3736</td><td rowspan=1 colspan=1>3.3774</td></tr><tr><td rowspan=1 colspan=1>24-16</td><td rowspan=1 colspan=1>-119.1327</td><td rowspan=1 colspan=1>-118.5668</td><td rowspan=1 colspan=1>-111.0656</td><td rowspan=1 colspan=1>-70.1981</td><td rowspan=1 colspan=1>-44.1031</td></tr></table>
+
+Adversarial Battle game, rewards of team 1, corresponding to Figure 7:
+
+Food Collection game, team rewards, corresponding to Figure 8:   
+
+<table><tr><td rowspan=1 colspan=1>scale</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>4-4</td><td rowspan=1 colspan=1>7.51555</td><td rowspan=1 colspan=1>5.81165</td><td rowspan=1 colspan=1>22.6357</td><td rowspan=1 colspan=1>22.6357</td><td rowspan=1 colspan=1>26.86355</td></tr><tr><td rowspan=1 colspan=1>8-8</td><td rowspan=1 colspan=1>0.6692</td><td rowspan=1 colspan=1>-0.58115</td><td rowspan=1 colspan=1>43.7801</td><td rowspan=1 colspan=1>46.89595</td><td rowspan=1 colspan=1>65.75585</td></tr><tr><td rowspan=1 colspan=1>16-16</td><td rowspan=1 colspan=1>-46.6398</td><td rowspan=1 colspan=1>-35.5978</td><td rowspan=1 colspan=1>28.8336</td><td rowspan=1 colspan=1>109.4406</td><td rowspan=1 colspan=1>189.69775</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>scale</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>55.06</td><td rowspan=1 colspan=1>42.74</td><td rowspan=1 colspan=1>61.6488</td><td rowspan=1 colspan=1>61.6488</td><td rowspan=1 colspan=1>64.822</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>17.01</td><td rowspan=1 colspan=1>3.37</td><td rowspan=1 colspan=1>49.3626</td><td rowspan=1 colspan=1>58.0014</td><td rowspan=1 colspan=1>63.7004</td></tr><tr><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>6.32</td><td rowspan=1 colspan=1>6.735</td><td rowspan=1 colspan=1>49.45755</td><td rowspan=1 colspan=1>52.3625</td><td rowspan=1 colspan=1>59.54</td></tr><tr><td rowspan=1 colspan=1>24</td><td rowspan=1 colspan=1>10.346075</td><td rowspan=1 colspan=1>7.830975</td><td rowspan=1 colspan=1>33.435</td><td rowspan=1 colspan=1>49.998025</td><td rowspan=1 colspan=1>59.47035</td></tr></table>
+
+# D.3 PAIRWISE COMPETITION RESULTS BETWEEN ALL METHODS IN COMPETITIVE GAMES
+
+For visualization purpose, we only illustrate the scores of the competitions between baselines and EPC in the main paper. Here we provide the complete competition rewards between every pair of methods in both Grassland and Adversarial Battle with the largest population of agents as follows.
+
+Here show the wolf rewards in Grassland with scale 24-16. For wolves trained by each approach, we compare them against the sheep by all the methods. EPC wolves always have the highest rewards as in the bottom row. Correspondingly, when different wolves compete against EPC sheep, they always obtain the lowest rewards as in the rightmost column.
+
+<table><tr><td rowspan=1 colspan=1>sheepwolf</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean-field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>Vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>66.914</td><td rowspan=1 colspan=1>67.0945</td><td rowspan=1 colspan=1>66.34</td><td rowspan=1 colspan=1>23.048</td><td rowspan=1 colspan=1>14.482</td></tr><tr><td rowspan=1 colspan=1>mean-field</td><td rowspan=1 colspan=1>75.7655</td><td rowspan=1 colspan=1>74.23</td><td rowspan=1 colspan=1>74.7375</td><td rowspan=1 colspan=1>28.3705</td><td rowspan=1 colspan=1>18.272</td></tr><tr><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>103.22</td><td rowspan=1 colspan=1>103.326</td><td rowspan=1 colspan=1>98.07</td><td rowspan=1 colspan=1>49.557</td><td rowspan=1 colspan=1>32.8945</td></tr><tr><td rowspan=1 colspan=1>Vanilla-PC</td><td rowspan=1 colspan=1>110.333</td><td rowspan=1 colspan=1>111.3735</td><td rowspan=1 colspan=1>101.8975</td><td rowspan=1 colspan=1>64.53</td><td rowspan=1 colspan=1>47.6365</td></tr><tr><td rowspan=1 colspan=1>EPC</td><td rowspan=1 colspan=1>120.9025</td><td rowspan=1 colspan=1>121.4325</td><td rowspan=1 colspan=1>115.956</td><td rowspan=1 colspan=1>82.381</td><td rowspan=1 colspan=1>61.4245</td></tr></table>
+
+Here show the sheep rewards in Grassland with scale 24-16. For sheep trained by each approach, we compete them against the all different wolves. EPC sheep always have the highest rewards as in the last row. Correspondingly, when competing different sheep against EPC wolves, the rewards are always the lowest as in the rightmost column.
+
+<table><tr><td rowspan=1 colspan=1>wolfsheep</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean-field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>Vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>-63.5096</td><td rowspan=1 colspan=1>-72.5443</td><td rowspan=1 colspan=1>-100.4636</td><td rowspan=1 colspan=1>-107.825</td><td rowspan=1 colspan=1>-119.1327</td></tr><tr><td rowspan=1 colspan=1>mean-field</td><td rowspan=1 colspan=1>-63.7089</td><td rowspan=1 colspan=1>-71.0714</td><td rowspan=1 colspan=1>-100.6304</td><td rowspan=1 colspan=1>-108.8917</td><td rowspan=1 colspan=1>-118.5668</td></tr><tr><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>-62.9416</td><td rowspan=1 colspan=1>-71.3339</td><td rowspan=1 colspan=1>-95.0522</td><td rowspan=1 colspan=1>-99.1207</td><td rowspan=1 colspan=1>-111.0656</td></tr><tr><td rowspan=1 colspan=1>Vanilla-PC</td><td rowspan=1 colspan=1>-5.7086</td><td rowspan=1 colspan=1>-7.8011</td><td rowspan=1 colspan=1>-31.9936</td><td rowspan=1 colspan=1>-49.5186</td><td rowspan=1 colspan=1>-70.1981</td></tr><tr><td rowspan=1 colspan=1>EPC</td><td rowspan=1 colspan=1>9.2135</td><td rowspan=1 colspan=1>10.5892</td><td rowspan=1 colspan=1>-6.9846</td><td rowspan=1 colspan=1>-27.3475</td><td rowspan=1 colspan=1>-44.1031</td></tr></table>
+
+Here we show the rewards of team 1 in Adversarial Battle with scale 16-16. For agents trained by each approach, we compare them as team 1 against all different methods as team 2. When EPC agents as team 1, no matter which opponent is, they always get the highest rewards as in the last row. When other methods compete against EPC, the obtained rewards are always the lowest as in the rightmost column.
+
+<table><tr><td rowspan=1 colspan=1>comparedreported</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean-field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>Vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>61.4555</td><td rowspan=1 colspan=1>17.1591</td><td rowspan=1 colspan=1>2.8033</td><td rowspan=1 colspan=1>-29.9242</td><td rowspan=1 colspan=1>-46.6398</td></tr><tr><td rowspan=1 colspan=1>mean-field</td><td rowspan=1 colspan=1>104.41315</td><td rowspan=1 colspan=1>59.01315</td><td rowspan=1 colspan=1>27.9004</td><td rowspan=1 colspan=1>11.1891</td><td rowspan=1 colspan=1>-35.5978</td></tr><tr><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>146.9829</td><td rowspan=1 colspan=1>117.3804</td><td rowspan=1 colspan=1>81.702</td><td rowspan=1 colspan=1>18.57425</td><td rowspan=1 colspan=1>28.8336</td></tr><tr><td rowspan=1 colspan=1>Vanilla-PC</td><td rowspan=1 colspan=1>202.9163</td><td rowspan=1 colspan=1>155.2805</td><td rowspan=1 colspan=1>174.91965</td><td rowspan=1 colspan=1>123.7212</td><td rowspan=1 colspan=1>109.4406</td></tr><tr><td rowspan=1 colspan=1>EPC</td><td rowspan=1 colspan=1>339.63255</td><td rowspan=1 colspan=1>318.3223</td><td rowspan=1 colspan=1>256.8464</td><td rowspan=1 colspan=1>198.3621</td><td rowspan=1 colspan=1>189.69775</td></tr></table>
+
+# D.4 VARIANCE OF PERFORMANCE EVALUATIONS
+
+We present the performance of all approaches in all three games with the largest scale. We train all the approaches with 3 different seeds and show the normalized scores with variance as following. We can see that EPC not only gives better results but also much smaller variance.
+
+E ADDITIONAL EXPERIMENTS ON THE ORIGINAL PREDATOR-PREY GAME Grassland is adapted from the Predator-prey game introduced by the original MADDPG paper (Lowe et al., 2017). To further validate our empirical results, we additionally study the performances of different algorithms on the unmodified Predator-prey game as follows.
+
+![](images/e0d570ad33e4661bc0897105082828623035242e35e446afc2e07eff4b7d26c9.jpg)  
+(a) Normalized scores with variances in Grassland in scale 24-16
+
+![](images/5636794dc9508a62df3bdc91c579a2bd12b03d89f153f9458ba4899e112580e0.jpg)  
+(b) Normalized scores with variances in Adversarial battle in scale 16-16
+
+![](images/6e5045416ab4949e110acc38daa49e0ea899f2e5e133e823194214af5c1c67e3.jpg)  
+(c) Normalized scores with variances in Food Collection with 24 agents
+
+We first report the normalized score in Fig. 13 by comparing all the methods against EPC. EPC is consistently better than all methods in all the scales.
+
+![](images/9da201823fa2e419fb4a5093d6efce0dd5da78e7e7bcb0cceceefa37b0d18294.jpg)  
+Figure 13: Normalized scores in the original Predator-prey game
+
+Besides, we also report the raw reward numbers when competing against EPC. Since the Predator-prey game is a zero-sum game, we simply report the predator rewards (the prey reward is exactly the negative value).
+
+<table><tr><td rowspan=1 colspan=1>scale</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>3-2</td><td rowspan=1 colspan=1>10.405</td><td rowspan=1 colspan=1>7.39</td><td rowspan=1 colspan=1>8.675</td><td rowspan=1 colspan=1>8.675</td><td rowspan=1 colspan=1>12.662</td></tr><tr><td rowspan=1 colspan=1>6-4</td><td rowspan=1 colspan=1>31.458</td><td rowspan=1 colspan=1>25.529</td><td rowspan=1 colspan=1>31.747</td><td rowspan=1 colspan=1>45.155</td><td rowspan=1 colspan=1>54.546</td></tr><tr><td rowspan=1 colspan=1>12-8</td><td rowspan=1 colspan=1>74.939</td><td rowspan=1 colspan=1>64.377</td><td rowspan=1 colspan=1>133.261</td><td rowspan=1 colspan=1>200.638</td><td rowspan=1 colspan=1>214.328</td></tr></table>
+
+Furthermore, we also demonstrate the results of full pairwise competition between every two methods for scale 12-8 below. Consistently, we can see that EPC predators always have the highest scores as in the last row. When competing against EPC prey, the lowest rewards are observed.
+
+<table><tr><td rowspan=1 colspan=1>preypredator</td><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>mean-field</td><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>Vanilla-PC</td><td rowspan=1 colspan=1>EPC</td></tr><tr><td rowspan=1 colspan=1>MADDPG</td><td rowspan=1 colspan=1>229.887</td><td rowspan=1 colspan=1>249.4</td><td rowspan=1 colspan=1>247.423</td><td rowspan=1 colspan=1>122.878</td><td rowspan=1 colspan=1>74.939</td></tr><tr><td rowspan=1 colspan=1>mean-field</td><td rowspan=1 colspan=1>207.022</td><td rowspan=1 colspan=1>210.838</td><td rowspan=1 colspan=1>228.469</td><td rowspan=1 colspan=1>107.811</td><td rowspan=1 colspan=1>64.377</td></tr><tr><td rowspan=1 colspan=1>Att-MADDPG</td><td rowspan=1 colspan=1>569.862</td><td rowspan=1 colspan=1>532.611</td><td rowspan=1 colspan=1>373.979</td><td rowspan=1 colspan=1>204.743</td><td rowspan=1 colspan=1>133.261</td></tr><tr><td rowspan=1 colspan=1>Vanilla-PC</td><td rowspan=1 colspan=1>758.293</td><td rowspan=1 colspan=1>737.067</td><td rowspan=1 colspan=1>521.486</td><td rowspan=1 colspan=1>303.86</td><td rowspan=1 colspan=1>200.638</td></tr><tr><td rowspan=1 colspan=1>EPC</td><td rowspan=1 colspan=1>827.298</td><td rowspan=1 colspan=1>764.417</td><td rowspan=1 colspan=1>519.319</td><td rowspan=1 colspan=1>299.505</td><td rowspan=1 colspan=1>214.328</td></tr></table>

parse/train/SJzRZ-WCZ/SJzRZ-WCZ.md

@@ -0,0 +1,439 @@
+# LATENT SPACE ODDITY: ON THE CURVATURE OF DEEP GENERATIVE MODELS
+
+Georgios Arvanitidis, Lars Kai Hansen, Søren Hauberg Technical University of Denmark, Section for Cognitive Systems {gear,lkai,sohau}@dtu.dk
+
+# ABSTRACT
+
+Deep generative models provide a systematic way to learn nonlinear data distributions through a set of latent variables and a nonlinear “generator” function that maps latent points into the input space. The nonlinearity of the generator implies that the latent space gives a distorted view of the input space. Under mild conditions, we show that this distortion can be characterized by a stochastic Riemannian metric, and we demonstrate that distances and interpolants are significantly improved under this metric. This in turn improves probability distributions, sampling algorithms and clustering in the latent space. Our geometric analysis further reveals that current generators provide poor variance estimates and we propose a new generator architecture with vastly improved variance estimates. Results are demonstrated on convolutional and fully connected variational autoencoders, but the formalism easily generalizes to other deep generative models.
+
+# 1 INTRODUCTION
+
+Deep generative models (Goodfellow et al., 2014; Kingma & Welling, 2014; Rezende et al., 2014) model the data distribution of observations $\mathbf { x } \in \mathcal { X }$ through corresponding latent variables $\mathbf { z } \in { \mathcal { Z } }$ and a stochastic generator function $f : { \mathcal { Z } } \to { \mathcal { X } }$ as
+
+$$
+\mathbf { x } = f ( \mathbf { z } ) .
+$$
+
+Using reasonably low-dimensional latent variables and highly flexible generator functions allows these models to efficiently represent a useful distribution over the underlying data manifold. These approaches have recently attracted a lot of attention, as deep neural networks are suitable generators which lead to the impressive performance of current variational autoencoders (VAEs) (Kingma & Welling, 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014).
+
+Consider the left panel of Fig. 1, which shows the latent representations of digits 0 and 1 from MNIST under a VAE. Three latent points are highlighted: one point (A) far away from the class boundary, and two points (B, C) near the boundary, but on opposite sides. Points B and C near the boundary seem to be very close to each other, while the third is far away from the others. Intuitively, we would hope that points from the same class (A and B) are closer to each other than to members of other classes (C), but this is seemingly not the case. In this paper, we argue this seemed conclusion is incorrect and only due to a misinterpretation of the latent space — in fact points $A$ and $B$ are closer to each other than to $c$ in the latent representation. Correcting this misinterpretation not only improves our understanding of generative models, but also improves interpolations, clusterings, latent probability distributions, sampling algorithms, interpretability and more.
+
+In general, latent space distances lack physical units (making them difficult to interpret) and are sensitive to specifics of the underlying neural nets. It is therefore more robust to consider infinitesimal distances along the data manifold in the input space. Let $\mathbf { z }$ be a latent point and let $\Delta \mathbf { z } _ { 1 }$ and $\Delta { \bf z } _ { 2 }$ be infinitesimals, then we can compute the squared distance
+
+$$
+\left\| f ( \mathbf { z } + \Delta \mathbf { z } _ { 1 } ) - f ( \mathbf { z } + \Delta \mathbf { z } _ { 2 } ) \right\| _ { 2 } ^ { 2 } = ( \Delta \mathbf { z } _ { 1 } - \Delta \mathbf { z } _ { 2 } ) ^ { \mathsf { T } } \left( \mathbf { J } _ { \mathbf { z } } ^ { \mathsf { T } } \mathbf { J } _ { \mathbf { z } } \right) ( \Delta \mathbf { z } _ { 1 } - \Delta \mathbf { z } _ { 2 } ) , \quad \mathbf { J } _ { \mathbf { z } } = \frac { \partial f } { \partial \mathbf { z } } \bigg | _ { \mathbf { z } = \mathbf { z } } ,
+$$
+
+using Taylor’s Theorem. This implies that the natural distance function in $\mathcal { Z }$ changes locally as it is governed by the local Jacobian. Mathematically, the latent space should not then be seen as a linear Euclidean space, but rather as a curved space. The right panel of Fig. 1 provides an example of the implications of this curvature. The figure shows synthetic data from two classes, and the corresponding latent representation of the data. The background color of the latent space corresponds to $\mathrm { \sqrt { d e t } } ( \mathbf { J _ { z } ^ { \mathsf { T } } } \mathbf { J _ { z } } )$ , which can be seen as a measure of the local distortion of the latent space. We interpolate two points from the same class by walking along the connecting straight line (red); in the right panel, we show points along this straight line which have been mapped by the generator to the input space. Since the generator defines a surface in the input space, we can alternatively seek the shortest curve along this surface that connects the two points; this is perhaps the most natural choice of interpolant. We show this shortest curve in green. From the center panel it is evident that the natural interpolant is rather different from the straight line. This is due to the distortion of the latent space, which is the topic of the present paper.
+
+![](images/4f6ab73930c0c558bca7cac815f1ab4cf01c5f356a658bc9c3f933e9ccea2d5e.jpg)  
+Figure 1: Left: An example of how latent space distances do not reflect actual data distances. Right: Shortest paths on the surface spanned by the generator do not correspond to straight lines in the latent space, as is assumed by the Euclidean metric.
+
+Outline. In Sec. 2 we briefly present the VAE as a representative instance of generative models. In Sec. 3 we connect generative models with their underlying geometry, and in Sec. 4 we argue that a stochastic Riemannian metric is naturally induced in the latent space by the generator. This metric enables us to compute length-minimizing curves and corresponding distances. This analysis, however, reveals that the traditional variance approximations in VAEs are rather poor and misleading; we propose a solution in Sec. 4.1. In Sec. 5 we demonstrate how the resulting view of the latent space improves latent interpolations, gives rise to more meaningful latent distributions, clusterings and more. We discuss related work in Sec. 6 and conclude the paper with an outlook in Sec. 7.
+
+# 2 THE VARIATIONAL AUTOENCODERS ACTING AS THE GENERATOR
+
+The variational autoencoder (VAE) proposed by Kingma & Welling (2014) is a simple yet powerful generative model which consists of two parts: (1) an inference network or recognition network (encoder) learns the latent representation (codes) of the data in the input space $\chi = \mathbb { R } ^ { D }$ ; and (2) the generator (decoder) learns how to reconstruct the data from these latent space codes in $\mathcal { Z } = \mathbb { R } ^ { d }$ .
+
+Formally, a prior distribution is defined for the latent representations $p ( \mathbf { z } ) = \mathcal { N } ( \mathbf { 0 } , \mathbb { I } _ { d } )$ , and there exists a mapping function $\mu _ { \theta } : \mathcal { Z } \to \mathcal { X }$ that generates a surface in $\mathcal { X }$ . Moreover, we assume that another function $\sigma _ { \theta } : \mathcal { Z } \to \mathbb { R } _ { + } ^ { D }$ captures the error (or uncertainty) between the actual data observation $\mathbf { x } \in \mathcal { X }$ and its reconstruction as $\mathbf { x } = \pmb { \mu } _ { \boldsymbol { \theta } } ( \mathbf { z } ) + \pmb { \sigma } _ { \boldsymbol { \theta } } \odot \pmb { \epsilon } ,$ , where $\epsilon \sim \mathcal { N } ( \mathbf { 0 } , \mathbb { I } _ { D } )$ and $\odot$ is the Hadamard (element-wise) product. Then the likelihood is naturally defined as $p _ { \boldsymbol { \theta } } ( \mathbf { x } \mid \mathbf { z } ) =$ $\mathcal { N } ( \mathbf { x } \mid \pmb { \mu } _ { \boldsymbol { \theta } } ( \mathbf { z } ) , \mathbb { I } _ { D } \pmb { \sigma } _ { \boldsymbol { \theta } } ^ { 2 } ( \mathbf { z } ) )$ . The flexible functions $\mu _ { \theta }$ and $\pmb { \sigma } \theta$ are usually deep neural networks with parameters $\theta$ .
+
+However, the corresponding posterior distribution $p _ { \theta } ( \mathbf { z } \mid \mathbf { x } )$ is unknown, as the marginal likelihood $p ( \mathbf { x } )$ is intractable. Hence, the posterior is approximated using a variational distribution $q _ { \phi } ( \mathbf { z } \mid \mathbf { x } ) =$ $\mathcal { N } ( \mathbf { z } \mid \pmb { \mu } _ { \phi } ( \mathbf { x } ) , \ \mathbb { I } _ { d } \pmb { \sigma } _ { \phi } ^ { 2 } ( \mathbf { x } ) )$ , where the functions $\mu _ { \phi } : \mathcal { X } \to \mathcal { Z }$ and $\pmb { \sigma } _ { \phi } : \mathcal { X }  \mathbb { R } _ { + } ^ { d }$ are again deep neural networks with parameters $\phi$ . Since the generator (decoder) is a composition of linear maps and activation functions, its smoothness is based solely on the chosen activation functions.
+
+The optimal parameters $\theta$ and $\phi$ are found by maximizing the evidence lower bound (ELBO) of the marginal likelihood $p ( \mathbf { x } )$ as
+
+$$
+\{ \theta ^ { * } , \phi ^ { * } \} = \underset { \theta , \phi } { \mathrm { a r g m a x } } \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ \log ( p _ { \theta } ( \mathbf { x } | \mathbf { z } ) ) ] - \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } ) ) ,
+$$
+
+where the bound follows from Jensen’s inequality. The optimization is based on variations of gradient descent using the reparametrization trick (Kingma $\&$ Welling, 2014; Rezende et al., 2014). Further improvements have been proposed that provide more flexible posterior approximations (Rezende & Mohamed, 2015; Kingma et al., 2016) or tighter lower bound (Burda et al., 2016). In this paper, we consider the standard VAE for simplicity. The optimization problem in Eq. 3 is difficult since poor reconstructions by $\mu _ { \theta }$ can be explained by increasing the corresponding variance $\sigma _ { \theta } ^ { 2 }$ . A common trick, which we also follow, is to optimize $\mu _ { \theta }$ while keeping $\sigma _ { \theta } ^ { 2 }$ constant, and then finally optimize for the variance $\sigma _ { \theta } ^ { 2 }$ .
+
+# 3 SURFACES AS THE FOUNDATION OF GENERATIVE MODELS
+
+Mathematically, a deterministic generative model ${ \bf x } = f ( { \bf z } )$ can be seen as a surface model (Gauss, 1827) if the generator $f$ is sufficiently smooth. Here, we briefly review the basic concepts on surfaces, as they form the mathematical foundation of this work.
+
+Intuitively, a surface is a smoothly-connected set of points embedded in $\mathcal { X }$ . When we want to make computations on a surface, it is often convenient to parametrize the surface by a low-dimensional (latent) variable $\mathbf { z }$ along with an appropriate function $f : { \mathcal { Z } } \to$ $\mathcal { X }$ . We let $d = \dim ( { \mathcal { Z } } )$ denote the intrinsic dimensionality of the surface, while $D \ = \ \dim ( { \mathcal { X } } )$ is the dimensionality of the input space. If we consider a smooth (latent) curve $\gamma _ { t } ^ { - } : [ \bar { 0 } , 1 ] \ \to \ \mathcal { Z }$ , then it has length $\begin{array} { r } { \int _ { 0 } ^ { 1 } \| \dot { \gamma } _ { t } \| \mathrm { d } t } \end{array}$ , where $\dot { \gamma } _ { t } = \mathrm { d } \gamma _ { t } / \mathrm { d } t$ denotes the velocity of the curve. In practice, the low-dimensional parametrization $\mathcal { Z }$ often lacks a principled meaningful metric, so we measure lengths in input space by mapping the curve through $f$ ,
+
+![](images/afdcd561f0b146263dabebec35098f3e72223fbc41c96c86ec27fc4a512e1cf6.jpg)  
+Figure 2: The Jacobian $\mathbf { J }$ of a nonlinear function $f$ provides a local basis in the input space, while $\sqrt { \operatorname* { d e t } ( \mathbf { J } \boldsymbol { \tau } \mathbf { J } ) }$ measures the volume of an infinitesimal region.
+
+$$
+\operatorname { L e n g t h } [ f ( \gamma _ { t } ) ] = \int _ { 0 } ^ { 1 } \left\| { \dot { f } } ( \gamma _ { t } ) \right\| _ { 2 } \mathrm { d } t = \int _ { 0 } ^ { 1 } \left\| \mathbf { J } _ { \gamma _ { t } } { \dot { \gamma } } _ { t } \right\| _ { 2 } \mathrm { d } t , \qquad \mathbf { J } _ { \gamma _ { t } } = \frac { \partial f } { \partial \mathbf { z } } \bigg | _ { \mathbf { z } = \gamma _ { t } }
+$$
+
+where the last step follows from the chain rule. This implies that the length of a curve $\gamma _ { t }$ along the surface can be computed directly in the latent space using the (locally defined) norm
+
+$$
+\| \mathbf { J } _ { \gamma } \dot { \gamma } \| _ { 2 } = \sqrt { ( \mathbf { J } _ { \gamma } \dot { \gamma } ) ^ { \intercal } ( \mathbf { J } _ { \gamma } \dot { \gamma } ) } = \sqrt { \dot { \gamma } ^ { \intercal } ( \mathbf { J } _ { \gamma } ^ { \intercal } \mathbf { J } _ { \gamma } ) \dot { \gamma } } = \sqrt { \dot { \gamma } ^ { \intercal } \mathbf { M } _ { \gamma } \dot { \gamma } } .
+$$
+
+Here, ${ \bf M } _ { \gamma } = { \bf J } _ { \gamma } ^ { \top } { \bf J } _ { \gamma }$ is a symmetric positive definite matrix, which acts akin to a local Mahalanobis distance measure. This gives rise to the definition of a Riemannian metric, which represents a smoothly changing inner product structure.
+
+Definition 1. A Riemannian metric $\mathbf { M } : \mathcal { Z }  \mathbb { R } ^ { d \times d }$ is a smooth function that assigns a symmetric positive definite matrix to any point in $\mathcal { Z }$ .
+
+It should be clear that if the generator function $f$ is sufficiently smooth, then ${ { \bf { M } } _ { \gamma } }$ in Eq. 5 is a Riemannian metric.
+
+When defining distances across a given surface, it is meaningful to seek the shortest curve connecting two points. Then a distance can be defined as the length of this curve. The shortest curve connecting points $\mathbf { z } _ { 0 }$ and $\mathbf { z } _ { 1 }$ is by (trivial) definition
+
+$$
+\boldsymbol \gamma _ { t } ^ { ( \mathrm { s h o r t e s t } ) } = \underset { \boldsymbol \gamma _ { t } } { \mathrm { a r g m i n L e n g t h } } [ f ( \boldsymbol \gamma _ { t } ) ] , \qquad \boldsymbol \gamma _ { 0 } = \mathbf z _ { 0 } , \boldsymbol \gamma _ { 1 } = \mathbf z _ { 1 } .
+$$
+
+A classic result of differential geometry (do Carmo, 1992) is that solutions to this optimization problem satisfy the following system of ordinary differential equations (ODEs)
+
+$$
+\ddot { \gamma } _ { t } = - \frac { 1 } { 2 } \mathbf { M } _ { \gamma _ { t } } ^ { - 1 } \left[ 2 \big ( \mathbb { I } _ { d } \otimes \dot { \gamma } _ { t } ^ { \intercal } \big ) \frac { \partial \mathrm { v e c } \big [ \mathbf { M } _ { \gamma _ { t } } \big ] } { \partial \gamma _ { t } } \dot { \gamma } _ { t } - \frac { \partial \mathrm { v e c } \big [ \mathbf { M } _ { \gamma _ { t } } \big ] ^ { \intercal } } { \partial \gamma _ { t } } ( \dot { \gamma } _ { t } \otimes \dot { \gamma } _ { t } ) \right] ,
+$$
+
+where vec[·] stacks the columns of a matrix into a vector and $\otimes$ is the Kronecker product. For completeness, we provide a derivation of this result in Appendix A. Shortest curves can then be computed by solving the ODEs numerically; our implementation uses $_ { \mathrm { b v p } 5 \mathrm { c } }$ from Matlab.
+
+# 4 THE GEOMETRY OF STOCHASTIC GENERATORS
+
+In the previous section, we considered deterministic generators $f$ to provide relevant background information. We now extend these results to the stochastic case; in particular we consider
+
+$$
+f ( \mathbf { z } ) = \pmb { \mu } ( \mathbf { z } ) + \pmb { \sigma } ( \mathbf { z } ) \odot \epsilon , \qquad \pmb { \mu } : \mathscr { Z }  \pmb { \chi } , ~ \pmb { \sigma } : \mathscr { Z }  \mathbb { R } _ { + } ^ { D } , ~ \epsilon \sim \mathcal { N } ( \mathbf { 0 } , \mathbb { I } _ { D } ) .
+$$
+
+This is the generator driving VAEs and related models. For our purposes, we will call $\mu ( \cdot )$ the mean function and $\sigma ^ { 2 } ( \cdot )$ the variance function.
+
+Following the discussion from the previous section, it is natural to consider the Riemannian metric $\mathbf { M _ { z } } = \mathbf { J _ { z } ^ { \intercal } } \mathbf { J _ { z } }$ in the latent space. Since the generator is now stochastic, this metric also becomes stochastic, which complicates analysis. The following results, however, simplify matters.
+
+Theorem 1. If the stochastic generator in Eq. 8 has mean and variance functions that are at least twice differentiable, then the expected metric equals
+
+$$
+\overline { { \mathbf { M } } } _ { \mathbf { z } } = \mathbb { E } _ { p ( \boldsymbol { \epsilon } ) } [ \mathbf { M } _ { \mathbf { z } } ] = \left( \mathbf { J } _ { \mathbf { z } } ^ { ( \boldsymbol { \mu } ) } \right) ^ { \intercal } \left( \mathbf { J } _ { \mathbf { z } } ^ { ( \boldsymbol { \mu } ) } \right) + \left( \mathbf { J } _ { \mathbf { z } } ^ { ( \boldsymbol { \sigma } ) } \right) ^ { \intercal } \left( \mathbf { J } _ { \mathbf { z } } ^ { ( \boldsymbol { \sigma } ) } \right) ,
+$$
+
+where $\mathbf { J _ { z } ^ { ( \mu ) } }$ and $\mathbf { J } _ { \mathbf { z } } ^ { ( \sigma ) }$ are the Jacobian matrices of $\mu ( \cdot )$ and $\sigma ( \cdot )$
+
+Proof. See Appendix B.
+
+Remark 1. By Definition $I$ , the metric tensor must change smoothly, which implies that the Jacobians must be smooth functions as well. This is easily ensured with activation functions for the neural networks that are $\mathcal { C } ^ { 2 }$ differentiable, e.g. tanh $( \cdot ) ,$ , sigmoid(·), and softplus(·).
+
+Theorem 2 (Due to Tosi et al. (2014)). The variance of the metric under the $L _ { 2 }$ measure vanishes when the data dimension goes to infinity, i.e. $\begin{array} { r } { \operatorname* { l i m } _ { D \to \infty } \mathrm { V a r } \left( \mathbf { M } _ { \mathbf { z } } \right) = 0 } \end{array}$ .
+
+Theorem 2 suggests that the (deterministic) expected metric $\overline { { \mathbf { M } } } _ { \mathbf { z } }$ is a good approximation to the underlying stochastic metric when the data dimension is large. We make this approximation, which allows us to apply the theory of deterministic generators.
+
+This expected metric has a particularly appealing form, where the two terms capture the distortion of the mean and the variance functions respectively. In particular, the variance term $( \mathbf { J _ { z } ^ { ( \sigma ) } } ) ^ { \mathsf { T } } ( \mathbf { J _ { z } ^ { ( \sigma ) } } )$ will be large in regions of the latent space, where the generator has large variance. This implies that induced distances will be large in regions of the latent space where the generator is highly uncertain, such that shortest paths will tend to avoid these regions. These paths will then tend to follow the data in the latent space, c.f. Fig. 3. It is worth stressing, that no learning is needed to compute this metric: it only consists of terms that can be derived directly from the generator.
+
+![](images/954871399527a17cead87d658d94f6a7dfe5fa2172e6093b0a13ea64a1641026.jpg)  
+Figure 3: Example shortest paths and distances.
+
+# 4.1 ENSURING PROPER GEOMETRY THROUGH MEANINGFUL VARIANCE FUNCTIONS
+
+Theorem 1 informs us about how the geometry of the generative model depends on both the mean and the variance of the generator. Assuming successful training of the generator, we can expect to have good estimates of the geometry in regions near the data. But what happens in regions further away from the data? In general, the mean function cannot be expected to give useful extrapolations to such regions, so it is reasonable to require that the generator has high variance in regions that are not near the data. In practice, the neural net used to represent the variance function is only trained in regions where data is available, which implies that variance estimates are extrapolated to regions with no data. As neural nets tend to extrapolate poorly, practical variance estimates tend to be arbitrarily poor in regions without data.
+
+![](images/fa99796e62b2e36f02959898009e95b045fb0cd0c77b64f431b6792bd61c83be.jpg)  
+Figure 4: From left to right: training data in $\mathcal { X }$ , latent representations in $\mathcal { Z }$ , the standard deviation $\begin{array} { r } { \log ( \sum _ { j = 1 } ^ { D } \sigma _ { j } ( { \mathbf z } ) ) } \end{array}$ for the standard variance network, and the proposed solution.
+
+Figure 4 illustrates this problem. The first two panels show the data and its corresponding latent representations (here both input and latent dimensions are 2 to ease illustration). The third panel shows the variance function under a standard architecture, deep multilayer perceptron with softplus nonlinearity for the output layer. It is evident that variance estimates in regions without data are not representative of either uncertainty or error of the generative process; sometimes variance is high, sometimes it is low. From a probabilistic modeling point-of-view, this is disheartening. An informal survey of publicly available VAE implementations also reveals that it is common to enforce a constant unit variance everywhere; this is further disheartening.
+
+For our purposes, we need well-behaved variance functions to ensure a well-behaved geometry, but reasonable variance estimates are of general use. Here, as a general strategy, we propose to model the inverse variance with a network that extrapolates towards zero. This at least ensures that variances are large in regions without data. Specifically, we model the precision as $\begin{array} { r } { \beta _ { \psi } ( \mathbf { z } ) = \frac { 1 } { \pmb { \sigma } _ { \psi } ^ { 2 } ( \mathbf { z } ) } } \end{array}$ , where all operations are element-wise. Then, we model this precision with a radial basis function $( R B F )$ neural network (Que & Belkin, 2016). Formally this is written
+
+$$
+\begin{array} { r } { \beta _ { \psi } ( { \bf z } ) = { \bf W } { \bf v } ( { \bf z } ) + \boldsymbol { \zeta } , \quad \mathrm { w i t h } \quad v _ { k } ( { \bf z } ) = \exp \left( - \lambda _ { k } \left. { \bf z } - { \bf c } _ { k } \right. _ { 2 } ^ { 2 } \right) , k = 1 , \dots , K , } \end{array}
+$$
+
+where $\psi$ are all parameters, $\mathbf { W } \in \mathbb { R } _ { > 0 } ^ { D \times K }$ are the positive weights of the n ork (positivity ensures a positive precision), and $\lambda _ { k }$ are the centers and the bandwidth of the $K$ and $\zeta \to 0$ is a vector of positive constants to prevent division by zero. It is easy to see that with this approach the variance of the generator increases with the distance to the centers. The rightmost panel of Fig. 4 shows an estimated variance function, which indeed has the desired property that variance is large outside the data support. Further, note the increased variance between the two clusters, which captures that even interpolating between clusters comes with a level of uncertainty. In Appendix C we also demonstrate that this simple variance model improves the marginal likelihood $p ( \mathbf { x } )$ on held-out data.
+
+Training the variance network amounts to fitting the RBF network. Assuming we have already trained the inference network (Sec. 2), we can encode the training data, and use $k$ -means to estimate the RBF centers. Then, an estimate for the bandwidths of each kernel can be computed as
+
+$$
+\lambda _ { k } = \frac { 1 } { 2 } \left( a \frac { 1 } { | \mathcal { C } _ { k } | } \sum _ { \mathbf { z } _ { j } \in \mathcal { C } _ { k } } \left\| \mathbf { z } _ { j } - \mathbf { c } _ { k } \right\| _ { 2 } \right) ^ { - 2 }
+$$
+
+where the hyper-parameter $a \in \mathbb { R } _ { + }$ controls the curvature of the Riemannian metric, i.e. how fast it changes based on the uncertainty. Since the mean function of the generator is already trained, the weights of the RBF can be found using projected gradient descent to ensure positive weights.
+
+One visualization of the distortion of the latent space relative to the input space is the geometric volume measure $\sqrt { \operatorname* { d e t } ( \mathbf { M } _ { \mathbf { z } } ) }$ , which captures the volume of an infinitesimal area in the input space. Figure 5 shows this volume measure for both standard variance functions as well as our proposed RBF model. We see the trend of the data, unlike the standard model.
+
+![](images/c7075f8077cba97a8ced072d15552c951119f42cbb37ef44a5700aff838d8091.jpg)  
+Figure 5: Comparison of (log) measures of standard (top) and proposed (bottom) variances.
+
+# 5 EMPIRICAL RESULTS
+
+We demonstrate the usefulness of the geometric view of the latent space with several experiments. Model and implementation details can be found in Appendix D. In all experiments we first train a VAE and then use the induced Riemannian metric.
+
+# 5.1 MEANINGFUL DISTANCES
+
+First we seek to quantify if the induced Riemannian distance in the latent space is more useful than the usual Euclidean distance. For this we perform basic $k$ -means clustering under the two metrics. We construct 3 sets of MNIST digits, using 1000 random samples for each digit. We train a VAE for
+
+<table><tr><td>Digits</td><td>Linear</td><td>Riemannian</td></tr><tr><td>{0,1,2}</td><td>77.57(±0.87)%</td><td>94.28(±1.14)%</td></tr><tr><td>{3,4,7}</td><td>77.80(±0.91)%</td><td>89.54(±1.61)%</td></tr><tr><td>{5,6,9}</td><td>64.93(±0.96)%</td><td>81.13(±2.52)%</td></tr></table>
+
+Table 1: The $F$ -measure results for $k$ -means.
+
+each set, and then subdivide each into 10 sub-sets, and performed $k$ -means clustering under both distances. One example result is shown in Fig. 6. Here it is evident that, since the latent points roughly follow a unit Gaussian, there is little structure to be discovered by the Euclidean $k$ -means, and consequently it performs poorly. The Riemannian clustering is remarkably accurate. Summary statistics across all subsets are provided in Table 1, which shows the established $F$ -measure for clustering accuracy. Again, the Riemannian metric significantly improves clustering. This implies that the underlying Riemannian distance is more useful than its Euclidean counterpart.
+
+![](images/d806b849a937f63ce95cada43c48d104ccea6c002aa6302cf87a7bb9d7b7e0eb.jpg)  
+Figure 6: The result of $k$ -means comparing the distance measures. For the decision boundaries we used 7-NN classification.
+
+# 5.2 INTERPOLATIONS
+
+Next, we investigate whether the Riemannian metric gives more meaningful interpolations. First, we train a VAE for the digits 0 and 1 from MNIST. The upper left panel of Fig. 7 shows the latent space with the Riemannian measure as background color, together with two interpolations. Images generated by both Riemannian and Euclidean interpolations are shown in the bottom of Fig. 7. The Euclidean interpolations seem to have a very abrupt change when transitioning from one class to another. The Riemannian interpolant gives smoother changes in the generated images. The topright panel of the figure shows the auto-correlation of images along the interpolants; again we see a very abrupt change in the Euclidean interpolant, while the Riemannian is significantly smoother. We also train a convolutional VAE on frames from a video. Figure 8 shows the corresponding latent space and some sample interpolations. As before, we see more smooth changes in generated images when we take the Riemannian metric into account.
+
+# 5.3 LATENT PROBABILITY DISTRIBUTIONS
+
+We have seen strong indications that the Riemannian metric gives a more meaningful view of the latent space, which may also improve probability distributions in the latent space. A relevant candidate distribution is the locally adaptive normal distribution (LAND) (Arvanitidis et al., 2016)
+
+$$
+\mathrm { L A N D } ( { \bf z } \mid { \pmb \mu } , { \pmb \Sigma } ) \propto \exp \left( - \frac { 1 } { 2 } \mathrm { d i s t } _ { { \pmb \Sigma } } ^ { 2 } ( { \bf z } , { \pmb \mu } ) \right) ,
+$$
+
+![](images/7c6c7604b2beef99c6958e92dc2f972869e6161897c59d460e8cbd4c5540d244.jpg)  
+Figure 7: Left: the latent space with example interpolants. Right: auto-correlations of Riemannian (top) and Euclidean (bottom) samples along the curves of the left panel. Bottom: decoded images along Euclidean (top rows) and Riemannian (bottom rows) interpolants.
+
+![](images/1a1c6b5de149bfc16c6c0898140b356140a1b13ca85c6283cfaa6b15b9031579.jpg)  
+Figure 8: Left: the latent space and geodesic interpolants. Right: samples comparing Euclidean (top row) with Riemannian (bottom row) interpolation. Corresponding videos can be found here.
+
+where $\mathrm { d i s t } _ { \Sigma }$ is the Riemannian extension of Mahalanobis distance. We fit a mixture of two LANDs to the MNIST data from Sec. 5.2 alongside a mixture of Euclidean normal distributions. The first column of Fig. 9 shows the density functions of the two mixture models. Only the Riemannian model reveals the underlying clusters. We then sample 40 points from each component of these generative models1 (center column of the figure). We see that the Riemannian model generates high-quality samples, whereas the Euclidean model generates several samples in regions where the generator is not trained and therefore produces blurry images. Finally, the right column of Fig. 9 shows all pairwise distances between the latent points under both Riemannian and Euclidean distances. Again, we see that the geometric view clearly reveals the underlying clusters.
+
+# 5.4 RANDOM WALK ON THE DATA MANIFOLD
+
+Finally, we consider random walks over the data manifold, which is a common tool for exploring latent spaces. To avoid the walk drifting outside the data support, practical implementations artificially restrict the walk to stay inside the $[ - 1 , 1 ] ^ { d }$ hypercube. Here, we consider unrestricted Brownian motion under both the Euclidean and Riemannian metric. We perform this random walk in the latent space of the convolutional VAE from Sec. 5.2. Figure 10 shows example walks, while Fig. 11 shows generated images (video here). While the Euclidean random walk moves freely, the Riemannian walk stays within the support of the data. This is explained in the left panel of Fig. 10, which shows that the variance term in the Riemannian metric creates a “wall” around the data, which the random walk will only rarely cross. These “walls” also force shortest paths to follow the data.
+
+![](images/577991ffae81ea5905686dbc7921ef66cea4290c0a2a878688a739565cdce305.jpg)  
+Figure 9: From left to right: the mixture models, generated samples, and pairwise distances. Top row corresponds to the Riemannian model and bottom row to the Euclidean model.
+
+![](images/a5f5f765ad4230246a37dbc4b51550a3462afbf7f547424e14ddfc378ba24947.jpg)  
+Figure 10: Left: the measure in the latent space. Right: the random walks.
+
+![](images/ca933ecdb20184b747d5f8fde4b873b1da41d8ad609bc99fed8eed4945dbc9fb.jpg)  
+Figure 11: The comparison of the random walks, at the steps 200, 300, 3000, 4000 and 5000.
+
+# 6 RELATED WORK
+
+Generative models. This unsupervised learning category attracted a lot of attention, especially, due to the advances on the deep neural networks. We have considered VAEs (Kingma & Welling, 2014; Rezende et al., 2014), but the ideas extend to similar related models. These include extensions that provide more flexible approximate posteriors (Rezende & Mohamed, 2015; Kingma et al., 2016). GANs (Goodfellow et al., 2014) also fall in this category, as these models have an explicit generator. While the inference network is not a necessary component in the GAN model, it has been shown that incorporating it improves overall performance (Donahue et al., 2017; Dumoulin et al., 2017). The same thoughts hold for approaches that transform the latent space through a sequence of bijective functions (Dinh et al., 2017)
+
+Geometry in neural networks. Bengio et al. (2013) discuss the importance of geometry in neural networks as a tool to understand local generalization. For instance, the Jacobian matrix is a measure of smoothness for a function that interpolates a surface to the given data. This is exactly the implication in (Rifai et al., 2011), where the norm of the Jacobian acts as a regularizer for the deterministic autoencoder. Recently, Kumar et al. (2017) used the Jacobian to inject invariances in a classifier.
+
+Riemannian Geometry. Like the present paper, Tosi et al. (2014) derive a suitable Riemannian metric in Gaussian process (GP) latent variable models (Lawrence, 2005), but the computational complexity of GPs causes practical concerns. Unlike works that explicitly learn a Riemannian metric (Hauberg et al., 2012; Peltonen et al., 2004), our metric is fully derived from the generator and requires no extra learning once the generator is available.
+
+# 7 DISCUSSION AND FURTHER EXTENSIONS
+
+The geometric interpretation of representation learning is that the latent space is a compressed and flattened version of the data manifold. We show that the actual geometry of the data manifold can be more complex than it first appears.
+
+Here we have initiated the study of proper geometries for generative models. We showed that the latent space not only provides a low-dimensional representation of the data manifold, but at the same time, can reveal the underlying geometrical structure. We proposed a new variance network for the generator, which provides meaningful uncertainty estimates while regularizing the geometry. The new detailed understanding of the geometry provides us with more relevant distance measures, as demonstrated by the fact that a $k$ -means clustering, on these distances, is better aligned with the ground truth label structure than a clustering based on conventional Euclidean distances. We also found that the new distance measure produces smoother interpolation, and when training Riemannian “LAND” mixture models based on the new geometry, the components aligned much better with the ground truth group structure. Finally, inspired by the recent interest in sequence generation by random walks in latent space, we found that geometrically informed random walks stayed on the manifold for much longer runs than sequences based on Euclidean random walks.
+
+The presented analysis easily extends to sophisticated generative models, where the latent space will be potentially endowed with more flexible nonlinear structures. This directly implies particularly interesting geometrical models. An obvious question is: can the geometry of the latent space play a role while we learn the generative model? Either way, we believe that this geometric perspective provides a new way of thinking and further interpreting the generative models, while at the same time it encourages development of new nonlinear models in the representation space.
+
+# ACKNOWLEDGMENTS
+
+LKH is supported by Innovation Fund Denmark / the Danish Center for Big Data Analytics Driven Innovation. SH was supported by a research grant (15334) from VILLUM FONDEN. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement $\boldsymbol { \mathrm { n ^ { \circ } } }$ 757360). We gratefully acknowledge the support of the NVIDIA Corporation with the donation of the used Titan Xp GPU.
+
+# REFERENCES
+
+Georgios Arvanitidis, Lars Kai Hansen, and Søren Hauberg. A Locally Adaptive Normal Distribution. In Advances in Neural Information Processing Systems (NIPS), 2016.
+
+Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation Learning: A Review and New Perspectives. IEEE Trans. Pattern Anal. Mach. Intell., 35(8):1798–1828, August 2013.
+
+Yuri Burda, Roger B. Grosse, and Ruslan Salakhutdinov. Importance Weighted Autoencoders. In International Conference on Learning Representations (ICLR), 2016.
+
+Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using Real NVP. In International Conference on Learning Representations (ICLR), 2017.
+
+M.P. do Carmo. Riemannian Geometry. Mathematics (Boston, Mass.). Birkhauser, 1992. ¨
+
+Jeff Donahue, Philipp Krhenbhl, and Trevor Darrell. Adversarial Feature Learning. In International Conference on Learning Representations (ICLR), 2017.
+
+Vincent Dumoulin, Ishmael Belghazi, Ben Poole, Alex Lamb, Martin Arjovsky, Olivier Mastropietro, and Aaron Courville. Adversarially Learned Inference. In International Conference on Learning Representations (ICLR), 2017.
+
+Carl Friedrich Gauss. Disquisitiones generales circa superficies curvas. Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores, VI:99–146, 1827.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative Adversarial Nets. In Advances in Neural Information Processing Systems (NIPS), 2014.
+
+Søren Hauberg, Oren Freifeld, and Michael J. Black. A Geometric Take on Metric Learning. In Advances in Neural Information Processing Systems (NIPS) 25, pp. 2033–2041, 2012.
+
+Diederik P Kingma and Max Welling. Auto-Encoding Variational Bayes. In Proceedings of the 2nd International Conference on Learning Representations (ICLR), 2014.
+
+Diederik P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improved Variational Inference with Inverse Autoregressive Flow. In Advances in Neural Information Processing Systems (NIPS), 2016.
+
+Abhishek Kumar, Prasanna Sattigeri, and Tom Fletcher. Improved Semi-supervised Learning with Gans using Manifold Invariances. In Advances in Neural Information Processing Systems (NIPS). 2017.
+
+Neil Lawrence. Probabilistic non-linear principal component analysis with Gaussian process latent variable models. Journal of machine learning research, 6(Nov):1783–1816, 2005.
+
+Jaakko. Peltonen, Arto Klami, and Samuel Kaski. Improved learning of riemannian metrics for exploratory analysis. Neural Networks, 17(8):1087–1100, 2004.
+
+Qichao Que and Mikhail Belkin. Back to the future: Radial basis function networks revisited. In Artificial Intelligence and Statistics (AISTATS), 2016.
+
+Danilo Rezende and Shakir Mohamed. Variational Inference with Normalizing Flows. In Proceedings of the 32nd International Conference on Machine Learning (ICML), 2015.
+
+Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic Backpropagation and Approximate Inference in Deep Generative Models. In Proceedings of the 31st International Conference on Machine Learning (ICML), 2014.
+
+Salah Rifai, Pascal Vincent, Xavier Muller, Xavier Glorot, and Yoshua Bengio. Contractive AutoEncoders: Explicit Invariance During Feature Extraction. In Proceedings of the 28th International Conference on Machine Learning (ICML), 2011.
+
+Alessandra Tosi, Søren Hauberg, Alfredo Vellido, and Neil D. Lawrence. Metrics for Probabilistic Geometries. In The Conference on Uncertainty in Artificial Intelligence (UAI), July 2014.
+
+# A THE DERIVATION OF THE GEODESIC DIFFERENTIAL EQUATION
+
+The shortest path between two points $\mathbf { x } , \mathbf { y } \in \mathcal { M }$ on a Riemannian manifold $\mathcal { M }$ is found by optimizing the functional
+
+$$
+\gamma _ { t } ^ { ( \mathrm { s h o r t e s t } ) } = \underset { \gamma _ { t } } { \arg \operatorname* { m i n } } \int _ { 0 } ^ { 1 } \sqrt { \langle \dot { \gamma } _ { t } , \mathbf { M } _ { \gamma _ { t } } \dot { \gamma } _ { t } \rangle } \mathrm { d } t , \quad \gamma ( 0 ) = \mathbf { x } , \gamma ( 1 ) = \mathbf { y }
+$$
+
+where $\gamma _ { t } : [ 0 , 1 ] \to \mathcal { M }$ and $\begin{array} { r } { \dot { \gamma } _ { t } ~ = ~ \frac { \partial \gamma _ { t } } { \partial t } } \end{array}$ ∂γt∂t . The minima of this problem can be found instead by optimizing the curve energy (do Carmo, 1992), so the functional becomes
+
+$$
+\gamma _ { t } ^ { ( \mathrm { s h o r t e s t } ) } = \underset { \gamma _ { t } } { \operatorname { a r g m i n } } \int _ { 0 } ^ { 1 } \langle \dot { \gamma } _ { t } , \mathbf { M } _ { \gamma _ { t } } \dot { \gamma } _ { t } \rangle \mathrm { d } t , \quad \gamma ( 0 ) = \mathbf { x } , \gamma ( 1 ) = \mathbf { y } .
+$$
+
+The inner product can be written explicitly as
+
+$$
+L ( \gamma _ { t } , \dot { \gamma } _ { t } , \mathbf { M } _ { \gamma _ { t } } ) = \langle \dot { \gamma } _ { t } , \mathbf { M } _ { \gamma _ { t } } \dot { \gamma } _ { t } \rangle = \sum _ { i = 1 } ^ { d } \sum _ { j = 1 } ^ { d } \dot { \gamma } _ { t } ^ { ( i ) } \cdot \dot { \gamma } _ { t } ^ { ( j ) } \cdot M _ { \gamma _ { t } } ^ { ( i j ) } = ( \dot { \gamma } _ { t } \otimes \dot { \gamma } _ { t } ) ^ { \top } \mathrm { v e c } [ \mathbf { M } _ { \gamma _ { t } } ]
+$$
+
+where the index in the parenthesis represents the corresponding element in the vector or matrix. In the derivation the $\otimes$ is the usual Kronecker product and the $\mathrm { v e c } [ \cdot ]$ stacks the column of a matrix into a vector. We find the minimizers by the Euler-Lagrange equation
+
+$$
+{ \frac { \partial L } { \partial \gamma _ { t } } } = { \frac { \partial } { \partial t } } { \frac { \partial L } { \partial { \dot { \gamma } } _ { t } } }
+$$
+
+where
+
+$$
+\frac { \partial } { \partial t } \frac { \partial L } { \partial \dot { \gamma } _ { t } } = \frac { \partial } { \partial t } \frac { \partial \langle \dot { \gamma } _ { t } , \mathbf { M } _ { \gamma _ { t } } \dot { \gamma } _ { t } \rangle } { \partial \dot { \gamma } _ { t } } = \frac { \partial } { \partial t } \left( 2 \cdot \mathbf { M } _ { \gamma _ { t } } \dot { \gamma } _ { t } \right) = 2 \left[ \frac { \partial \mathbf { M } _ { \gamma _ { t } } } { \partial t } \dot { \gamma } _ { t } + \mathbf { M } _ { \gamma _ { t } } \ddot { \gamma } _ { t } \right] .
+$$
+
+Since the term
+
+$$
+\frac { \partial \mathbf { M } _ { \gamma _ { t } } } { \partial t } = \left[ \begin{array} { c c c } { \frac { \partial M _ { \gamma _ { t } } ^ { ( 1 1 ) } } { \partial t } } & { \ldots } & { \frac { \partial M _ { \gamma _ { t } } ^ { ( 1 D ) } } { \partial t } } \\ { \frac { \partial M _ { \gamma _ { t } } ^ { ( 2 1 ) } } { \partial t } } & { \ldots } & { \frac { \partial M _ { \gamma _ { t } } ^ { ( 2 D ) } } { \partial t } } \\ { \vdots } & { \ddots } & { \vdots } \\ { \frac { \partial M _ { \gamma } ^ { ( D 1 ) } } { \partial t } } & { \ldots } & { \frac { \partial M _ { \gamma _ { t } } ^ { ( D D ) } } { \partial t } } \end{array} \right] = \left[ \begin{array} { c c c c } { \frac { \partial M _ { \gamma _ { t } } ^ { ( 1 1 ) } \mathsf { T } } { \partial \gamma _ { t } } \dot { \gamma } _ { t } } & { \ldots } & { \frac { \partial M _ { \gamma _ { t } } ^ { ( 1 D ) } } { \partial \gamma _ { t } } \dot { \gamma } _ { t } } \\ { \frac { \partial M _ { \gamma _ { t } } ^ { ( 2 1 ) } \mathsf { T } } { \partial \gamma _ { t } } \dot { \gamma } _ { t } } & { \ldots } & { \frac { \partial M _ { \gamma } ^ { ( 2 D ) } } { \partial \gamma _ { t } } \dot { \gamma } _ { t } } \\ { \vdots } & { \ddots } & { \vdots } \\ { \frac { \partial M _ { \gamma _ { t } } ^ { ( D 1 ) } \mathsf { T } } { \partial \gamma _ { t } } \dot { \gamma } _ { t } } & { \ldots } & { \frac { \partial M _ { \gamma _ { t } } ^ { ( D D ) } } { \partial \gamma _ { t } } \dot { \gamma } _ { t } } \end{array} \right]
+$$
+
+we can write the right hand side of the Eq. 16 as
+
+$$
+\frac { \partial } { \partial t } \frac { \partial L } { \partial \dot { \gamma } _ { t } } = 2 \left[ ( \mathbb { I } _ { d } \otimes \dot { \gamma } _ { t } ^ { \mathsf { T } } ) \frac { \partial \mathrm { v e c } \left[ \mathbf { M } _ { \gamma _ { t } } \right] } { \partial \gamma _ { t } } \dot { \gamma } _ { t } + \mathbf { M } _ { \gamma _ { t } } \ddot { \gamma } _ { t } \right] .
+$$
+
+The left hand side term of the Eq. 16 is equal to
+
+$$
+\frac { \partial L } { \partial \gamma _ { t } } = \frac { \partial } { \partial \gamma _ { t } } \left( ( \dot { \gamma } _ { t } \otimes \dot { \gamma } _ { t } ) ^ { \top } \mathrm { v e c } \big [ \mathbf { M } _ { \gamma _ { t } } \big ] \right) = ( \dot { \gamma } _ { t } \otimes \dot { \gamma } _ { t } ) ^ { \top } \frac { \partial \mathrm { v e c } \big [ \mathbf { M } _ { \gamma _ { t } } \big ] } { \partial \gamma _ { t } } .
+$$
+
+The final system of $2 ^ { \mathrm { n d } }$ order ordinary differential equations is
+
+$$
+\ddot { \gamma } _ { t } = - \frac { 1 } { 2 } \mathbf { M } _ { \gamma _ { t } } ^ { - 1 } \left[ 2 \cdot ( \mathbb { I } _ { d } \otimes \dot { \gamma } _ { t } ^ { \intercal } ) \frac { \partial \mathrm { v e c } \big [ \mathbf { M } _ { \gamma _ { t } } \big ] } { \partial \gamma _ { t } } \dot { \gamma } _ { t } - \frac { \partial \mathrm { v e c } \big [ \mathbf { M } _ { \gamma _ { t } } \big ] ^ { \intercal } } { \partial \gamma _ { t } } ( \dot { \gamma } _ { t } \otimes \dot { \gamma } _ { t } ) \right] .
+$$
+
+B THE DERIVATION OF THE RIEMANNIAN METRIC
+
+The proof of Theorem 1.
+
+Proof. As we introduced in Eq. 8 the stochastic generator is
+
+$$
+f ( \mathbf { z } ) = \pmb { \mu } ( \mathbf { z } ) + \pmb { \sigma } ( \mathbf { z } ) \odot \epsilon , \qquad \pmb { \mu } : \mathscr { Z }  \pmb { \chi } , ~ \pmb { \sigma } : \mathscr { Z }  \mathbb { R } _ { + } ^ { D } , ~ \epsilon \sim \mathcal { N } ( \mathbf { 0 } , \mathbb { I } _ { D } ) .
+$$
+
+Thus, we can compute the corresponding Jacobian as follows
+
+$$
+\begin{array} { c } { { \displaystyle \frac { \partial f ( { \bf z } ) } { \partial { \bf z } } = { \bf J } _ { \bf z } = [ \begin{array} { c c c } { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( 1 ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } & { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( 1 ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } & { { \bf \cdots } } & { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( 1 ) } } { \partial z _ { { \bf z } } } } } \\ { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( 2 ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } & { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( 2 ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } & { { \bf \cdots } } & { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( 2 ) } } { \partial z _ { { \bf z } } } } } \\ { { \vdots } } & { { \vdots } } & { { \ddots } } & { { \vdots } } \\ { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( D ) } } { \partial z _ { { \bf z } } } } } & { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( D ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } & { { \bf \cdots } } & { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( D ) } } { \partial z _ { { \bf z } } } } } \end{array} ] _ { D ^ { \times d } } } } \\   = [ \begin{array} { c c c } { { \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( 1 ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } & { { \displaystyle \frac { \partial \mu _ { { \bf z } } ^ { ( 1 ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } & { { \bf \cdots } } & { { \displaystyle \frac { \partial \mu _ { { \bf z } } ^ { ( 1 ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } \\ { { \displaystyle \frac { \partial \mu _ { { \bf z } } ^ { ( 2 ) } } { \partial z _ { { \bf z } } ^ { 2 } } } } &   \displaystyle \frac { \partial f _ { { \bf z } } ^ { ( 1 ) } }  \partial z _ { { \bf z } } ^  \end{array} \end{array}
+$$
+
+$$
+\mathbf { S } _ { i } = \left[ \begin{array} { c c c c } { \frac { \partial \sigma _ { z } ^ { ( 1 ) } } { \partial z _ { i } } } & { 0 } & { \cdots } & { 0 } \\ { 0 } & { \frac { \partial \sigma _ { z } ^ { ( 2 ) } } { \partial z _ { i } } } & { \cdots } & { 0 } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { 0 } & { 0 } & { \cdots } & { \frac { \partial \sigma _ { z } ^ { ( D ) } } { \partial z _ { i } } } \end{array} \right] _ { D \times D } , i = 1 , \ldots , d
+$$
+
+and the resulting “random” metric in the latent space is $\mathbf { M _ { z } } = \mathbf { J _ { z } ^ { \intercal } } \mathbf { J _ { z } }$ . The randomness is due to the random variable $\epsilon$ , and thus, we can compute the expectation
+
+$$
+\begin{array} { r l } & { \mathbf { M _ { z } } = \mathbb { E } _ { p ( \epsilon ) } [ \mathbf { M _ { z } } ] = \mathbb { E } _ { p ( \epsilon ) } [ ( \mathbf { A } + \mathbf { B } ) ^ { \mathsf { T } } ( \mathbf { A } + \mathbf { B } ) ] = \mathbb { E } _ { p ( \epsilon ) } [ \mathbf { A ^ { \mathsf { T } } A } + \mathbf { A ^ { \mathsf { T } } B } + \mathbf { B ^ { \mathsf { T } } A } + \mathbf { B ^ { \mathsf { T } } B } ] . } \end{array}
+$$
+
+Using the linearity of expectation we get that
+
+$$
+\begin{array} { r } { \mathbb { E } _ { p ( \epsilon ) } [ \mathbf { A } ^ { \top } \mathbf { B } ] = \mathbb { E } _ { p ( \epsilon ) } \left[ \mathbf { A } ^ { \top } [ \mathbf { S } _ { 1 } \epsilon , \mathbf { S } _ { 2 } \epsilon , \cdot \cdot , \mathbf { S } _ { d } \epsilon ] \right] = \mathbf { A } ^ { \top } [ \mathbf { S } _ { 1 } \mathbb { E } _ { p ( \epsilon ) } [ \epsilon ] ^ { * } , \cdot \cdot , \mathbf { 0 } ] = 0 } \end{array}
+$$
+
+because $\mathbb { E } _ { p ( \epsilon ) } [ \epsilon ] = \mathbf { 0 }$ . The other term
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { p ( \epsilon ) } [ \mathbf { B } ^ { \mathsf { T } } \mathbf { B } ] = \mathbb { E } _ { p ( \epsilon ) } \left( \left[ \begin{array} { c } { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { 1 } } \\ { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { 2 } } \\ { \vdots } \\ { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { d } } \end{array} \right] _ { d \times D } \quad [ \mathbf { S } _ { 1 } \epsilon , \mathbf { S } _ { 2 } \epsilon , \cdots , \mathbf { S } _ { d } \epsilon ] \right) } \\ & { \quad \quad \quad \quad = \mathbb { E } _ { p ( \epsilon ) } \left( \left[ \begin{array} { c c c c } { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { 1 } \mathbf { S } _ { 1 } \epsilon } & { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { 1 } \mathbf { S } _ { 2 } \epsilon } & { \cdots } & { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { 1 } \mathbf { S } _ { d } \epsilon } \\ { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { 2 } \mathbf { S } _ { 1 } \epsilon } & { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { 2 } \mathbf { S } _ { 2 } \epsilon } & { \cdots } & { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { 2 } \mathbf { S } _ { d } \epsilon } \\ { \vdots } & { \vdots } & { \vdots } \\ { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { d } \mathbf { S } _ { 1 } \epsilon } & { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { d } \mathbf { S } _ { 2 } \epsilon } & { \cdots } & { \epsilon ^ { \mathsf { T } } \mathbf { S } _ { d } \mathbf { S } _ { d } \epsilon } \end{array} \right] \right) } \end{array}
+$$
+
+with
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { p ( \epsilon ) } \left[ \epsilon ^ { \mathsf { T } } \mathbf { S } _ { i } \mathbf { S } _ { j } \epsilon \right] = \mathbb { E } _ { p ( \epsilon ) } \left[ \left( \epsilon _ { 1 } \frac { \partial \sigma _ { z } ^ { ( 1 ) } } { \partial z _ { i } } , \epsilon _ { 2 } \frac { \partial \sigma _ { z } ^ { ( 2 ) } } { \partial z _ { i } } , \cdots , \epsilon _ { D } \frac { \partial \sigma _ { z } ^ { ( D ) } } { \partial z _ { i } } \right) \left( \begin{array} { c } { \epsilon _ { 1 } \frac { \partial \sigma _ { z } ^ { ( 1 ) } } { \partial z _ { j } } } \\ { \epsilon _ { 2 } \frac { \partial \sigma _ { z } ^ { ( 2 ) } } { \partial z _ { j } } } \\ { \vdots } \\ { \epsilon _ { D } \frac { \partial \sigma _ { z } ^ { ( D ) } } { \partial z _ { j } } } \end{array} \right) \right] } \\ & { \quad = \mathbb { E } _ { p ( \epsilon ) } \left[ \epsilon _ { 1 } ^ { 2 } \left( \frac { \partial \sigma _ { z } ^ { ( 1 ) } } { \partial z _ { i } } \frac { \partial \sigma _ { z } ^ { ( 1 ) } } { \partial z _ { j } } \right) + \epsilon _ { 2 } ^ { 2 } \left( \frac { \partial \sigma _ { z } ^ { ( 2 ) } } { \partial z _ { i } } \frac { \partial \sigma _ { z } ^ { ( 2 ) } } { \partial z _ { j } } \right) + \cdots \epsilon _ { D } ^ { 2 } \left( \frac { \partial \sigma _ { z } ^ { ( D ) } } { \partial z _ { i } } \frac { \partial \sigma _ { z } ^ { ( D ) } } { \partial z _ { j } } \right) \right] } \end{array}
+$$
+
+$$
+\begin{array} { r } { = d i a g ( \mathbf { S } _ { i } ) ^ { \mathsf { T } } d i a g ( \mathbf { S } _ { j } ) , } \end{array}
+$$
+
+because $\mathbb { E } _ { p ( \epsilon ) } [ \epsilon _ { i } ^ { 2 } ] = 1 , \forall i = 1 , \dots , D$ .
+
+The matrix $\mathbf { A } = \mathbf { J } _ { \mathbf { z } } ^ { ( \mu ) }$ and for the variance network
+
+$$
+\mathbf { J } _ { \mathbf { z } } ^ { ( \sigma ) } = \left[ \begin{array} { c c c c } { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( 1 ) } } { \partial z _ { 1 } } } & { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( 1 ) } } { \partial z _ { 2 } } } & { \cdot \cdot \cdot } & { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( 1 ) } } { \partial z _ { d } } } \\ { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( 2 ) } } { \partial z _ { 1 } } } & { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( 2 ) } } { \partial z _ { 2 } } } & { \cdot \cdot \cdot } & { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( 2 ) } } { \partial z _ { d } } } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( D ) } } { \partial z _ { 1 } } } & { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( D ) } } { \partial z _ { 2 } } } & { \cdot \cdot \cdot } & { \frac { \partial \sigma _ { \mathbf { z } } ^ { ( D ) } } { \partial z _ { d } } } \end{array} \right]
+$$
+
+it is easy to see that $\mathbb { E } _ { p ( \epsilon ) } [ \mathbf { B } ^ { \intercal } \mathbf { B } ] = \left( \mathbf { J } _ { \mathbf { z } } ^ { ( \sigma ) } \right) ^ { \intercal } \mathbf { J } _ { \mathbf { z } } ^ { ( \sigma ) }$ . So the expectation of the induced Riemannian metric in the latent space by the generator is
+
+$$
+\bar { \mathbf { M _ { z } } } = \left( \mathbf { J _ { z } ^ { ( \mu ) } } \right) ^ { \mathsf { T } } \mathbf { J _ { z } ^ { ( \mu ) } } + \left( \mathbf { J _ { z } ^ { ( \sigma ) } } \right) ^ { \mathsf { T } } \mathbf { J _ { z } ^ { ( \sigma ) } }
+$$
+
+which concludes the proof.
+
+# C INFLUENCE OF VARIANCE ON THE MARGINAL LIKELIHOOD
+
+We trained a VAE on the digits 0 and 1 of the MNIST scaled to $[ - 1 , 1 ]$ . We randomly split the data to $9 0 \%$ training and $1 0 \%$ test data, ensuring balanced classes. First, we only trained the encoder and the mean function of the decoder. Then, keeping these fixed, we trained two variance functions: one based on standard deep neural network architecture, and the other using our proposed RBF model. Clearly, we have two generators with the same mean function, but different variance functions. Below we present the architectures for the standard neural networks. For the RBF model we used 32 centers and $a = 1$ .
+
+<table><tr><td>Encoder/Decoder</td><td>Layer 1</td><td>Layer 2</td><td>Layer 3</td><td></td></tr><tr><td></td><td>64,(softplus)</td><td>32,(softplus)</td><td>d,(linear)</td><td rowspan="5"></td></tr><tr><td></td><td>64,(softplus)</td><td>32, (softplus)</td><td>d,(softplus)</td></tr><tr><td>μ</td><td>32,(softplus)</td><td>64,(softplus)</td><td>D,(tanh)</td></tr><tr><td>00</td><td>32, (softplus)</td><td>64,(softplus)</td><td>D,(softplus)</td></tr></table>
+
+The numbers corresponds to the layer size together with the activation function in parenthesis. Further, the mean and the variance functions share the weights of the first layer. The input space dimension is $D = 7 8 4$ . Then, we computed the marginal likelihood $p ( \mathbf { x } )$ of the test data using Monte Carlo as:
+
+$$
+p ( \mathbf { x } ) = \int _ { \mathbb { Z } } p ( \mathbf { x } | \mathbf { z } ) p ( \mathbf { z } ) \mathrm { d } \mathbf { z } \simeq \frac { 1 } { S } \sum _ { s = 1 } p ( \mathbf { x } | \mathbf { z } _ { s } ) , \quad \mathbf { z } _ { s } \sim p ( \mathbf { z } )
+$$
+
+using $S = 1 0 0 0 0$ samples. The generator with the standard variance function achieved -68.25 mean log-marginal likelihood, while our proposed model -50.34, where the higher the better.
+
+The reason why the proposed RBF model performs better can be easily analyzed. The marginal likelihood under the Monte Carlo estimation is, essentially, a large Gaussian mixture model with equal weights $\textstyle { \frac { 1 } { S } }$ . Each mixture component is defined by the generator through the likelihood $p ( \mathbf { x } | \mathbf { z } ) = \mathcal { N } \left( \mathbf { x } \mid \pmb { \mu } _ { \theta } ( \mathbf { z } ) , \mathbb { I } _ { D } \pmb { \sigma } _ { \theta } ^ { 2 } ( \mathbf { z } ) \right)$ . Considering the variance term, the standard neural network approach is trained on the given data points and the corresponding latent codes. Unfortunately, its behavior is arbitrary in regions where there are not any encoded data. On the other hand our proposed model assigns large variance to these regions, while on the regions where we have latent codes its behavior will be approximately the same with the standard neural network. This implies that the resulting marginal likelihood $p ( \mathbf { x } )$ for the two models are highly similar in regions of high data density, but significantly different elsewhere. The RBF variance model ensures that mixture components in these regions have high variance, whereas the standard architecture assign arbitrary variance. Consequently, the RBF-based $p ( \mathbf { x } )$ assigns minimal density to regions with no data, and, thus, attains higher marginal likelihood elsewhere.
+
+# D IMPLEMENTATION DETAILS FOR THE EXPERIMENTS
+
+Algorithm 1 The training of a VAE that ensures geometry
+
+Output: the estimated parameters of the neural networks $\theta , \phi , \psi$ 1: Train the $\mu _ { \phi } , \pmb { \sigma } _ { \phi } , \pmb { \mu } _ { \theta }$ as in Kingma & Welling (2014), keeping $\sigma _ { \psi }$ fixed. 2: Train the $\sigma _ { \psi }$ as explained in Sec. 4.1.
+
+Details for Experiments 5.1, 5.2 & 5.3. The pixel values of the images are scaled to the interval $[ 0 , 1 ]$ . We use for the functions $\mu _ { \phi } , \pmb { \sigma } _ { \phi } , \pmb { \mu } _ { \theta }$ multilayer perceptron (MLP) deep neural networks, and for the $\beta _ { \psi }$ the proposed RBF model with 64 centers, so $\mathbf { W } \in \mathbb { R } ^ { D \times 6 4 }$ and the parameter $a$ of Eq. 11 is set to 2. We used $L _ { 2 }$ regularization with parameter equal to $1 e ^ { - 5 }$ .
+
+<table><tr><td>Encoder/Decoder</td><td>Layer 1</td><td>Layer 2</td><td>Layer 3</td></tr><tr><td></td><td>64,(tanh)</td><td>32,(tanh)</td><td>d,(linear)</td></tr><tr><td></td><td>64, (tanh)</td><td>32, (tanh)</td><td>d,(softplus)</td></tr><tr><td>μ</td><td>32,(tanh)</td><td>64,(tanh)</td><td>D,(sigmoid)</td></tr></table>
+
+The number corresponds to the size of the layer, and in the parenthesis the activation function. For the encoder, the mean and the variance functions share the weights of the Layer 1. The input space dimension $D = 7 8 4$ . After the training, the geodesics can be computed by solving Eq. 7 numerically. The LAND mixture model is fitted as explained in (Arvanitidis et al., 2016).
+
+Details for Experiments 5.4. In this experiment we used Convolutional Variational AutoEncoders. The pixel values of the images are scaled to the interval [0, 1]. For the $\beta _ { \psi }$ we used the proposed RBF model with 64 centers and the parameter $a$ of Eq. 11 is set to 2.
+
+Considering the variance network during the decoding stage, the RBF generates an image, which represents intuitively the total variance of each pixel for the decoded final image, but in an initial sub-sampled version. Afterwards, this image is passed through a sequence of deconvolution layers, and at the end will represent the variance of every pixel for each RGB channel. However, it is critical that the weights of the filters must be clipped during the training to $\mathbb { R } _ { + }$ to ensure positive variance.
+
+<table><tr><td>Encoder</td><td>Layer 1 (Conv)</td><td>Layer 2 (Conv)</td><td>Layer 3 (MLP)</td><td>Layer 4 (MLP)</td></tr><tr><td>山</td><td>32,3,2,(tanh)</td><td>32,3,2,(tanh)</td><td>1024, (tanh)</td><td>d,(linear)</td></tr><tr><td>o</td><td>32,3,2,(tanh)</td><td>32,3,2,(tanh)</td><td>1024, (tanh)</td><td>d,(softplus)</td></tr></table>
+
+For the convolutional and deconvolutional layers, the first number is the number of applied filters, the second is the kernel size, and third is the stride. Also, for the encoder, the mean and the variance functions share the convolutional layers. We used $L _ { 2 }$ regularization with parameter equal to $1 e ^ { - 5 }$ .
+
+$$
+\begin{array}{c} \frac { \mathrm { D e c o d e r } \mathrm { ~  ~ \cal ~ L ~ } . 1 ( \mathrm { M L P } ) \mathrm { ~  ~ \cal ~ L . ~ } 2 ( \mathrm { M L P } ) \mathrm { ~  ~ \cal ~ L . ~ } 3 ( \mathrm { D E } ) \mathrm { ~  ~ \cal ~ L . ~ } 4 ( \mathrm { D E } ) \mathrm { ~  ~ \cal ~ L . ~ } 5 ( \mathrm { D E } ) \mathrm { ~  ~ \cal ~ L . ~ } 6 ( \mathrm { C O } ) } { \mu _ { \theta } } \frac { D / 4 , ( t ) } { 3 2 , 3 , 2 , ( t ) } \frac { 3 2 , 3 , 2 , ( t ) } { 3 2 , 3 , 2 , ( t ) } \mathrm { ~  ~ \cal ~ 3 . 2 , } 3 , 1 , ( t ) \mathrm { ~  ~ \cal ~ 3 , 3 , 1 , } 3 , 1 , ( s ) \mathrm { ~  ~ \cal ~ 3 , 3 , 1 ~ } , 0 ) \mathrm { ~  ~ \cal ~ O ~ }  \end{array}
+$$
+
+For the decoder, the acronyms $\mathrm { ( D E ) = }$ Deconvolution, $\left( \mathbf { C O } \right) =$ Convolution and $( t )$ , (s) stand for tanh and sigmoid, respectively. Also, $D = w i d t h \times h e i g h t \times$ channels of the images, in our case 64,64,3. For all the convolutions and deconvolutions, the padding is set to same. We used $L _ { 2 }$ regularization with parameter equal to $1 e ^ { - 5 }$ .
+
+<table><tr><td>Decoder</td><td>rLayer1(RBF)</td><td>Layer 2 (Deconv）Layer 3 (Conv)</td><td></td></tr><tr><td>β</td><td>W ∈ R(D/2)×64</td><td>1,3,2 (linear)</td><td>3,3,1 (linear)</td></tr></table>
+
+The Brownian motion over the Riemannian manifold in the latent space is presented in Alg. 2.
+
+# Algorithm 2 Brownian motion on a Riemannian manifold
+
+Input: the starting point $\mathbf { z } \in \mathbb { R } ^ { d \times 1 }$ , stepsize $s$ , number of steps $N _ { s }$ , the metric tensor $\mathbf { M } ( \cdot )$ Output: the random steps Z ∈ RNs×d.
+
+1: for $n = 0$ to $N _ { s }$ do   
+2: $\mathbf { L } , \mathbf { U } = e i g \left( \mathbf { M ( z ) } \right) .$ (L: eigenvalues, U: eigenvectors)   
+3: $\mathbf { v } = \mathbf { U } \mathbf { L } ^ { - \frac { 1 } { 2 } } \boldsymbol { \epsilon } , \qquad \boldsymbol { \epsilon } \sim \mathcal { N } ( \mathbf { 0 } , \mathbb { I } _ { d } )$   
+4: z = z + s · v   
+5: Z(n, :) = z   
+6: end for

parse/train/UoVpP8R2Vn/UoVpP8R2Vn.md

@@ -0,0 +1,349 @@
+# PARP: Prune, Adjust and Re-Prune for Self-Supervised Speech Recognition
+
+Cheng-I Jeff Lai1, Yang Zhang2⇤, Alexander H. Liu1⇤, Shiyu Chang2, 4⇤ Yi-Lun Liao1, Yung-Sung Chuang1, 3, Kaizhi Qian2, Sameer Khurana1 David $\mathbf { C o x } ^ { 2 }$ , James Glass1 1MIT CSAIL, 2MIT-IBM Watson AI Lab, 3National Taiwan University, $^ 4 \mathrm { U C }$ Santa Barbara
+
+clai24@mit.edu
+
+# Abstract
+
+Self-supervised speech representation learning (speech SSL) has demonstrated the benefit of scale in learning rich representations for Automatic Speech Recognition (ASR) with limited paired data, such as wav2vec 2.0. We investigate the existence of sparse subnetworks in pre-trained speech SSL models that achieve even better low-resource ASR results. However, directly applying widely adopted pruning methods such as the Lottery Ticket Hypothesis (LTH) is suboptimal in the computational cost needed. Moreover, we show that the discovered subnetworks yield minimal performance gain compared to the original dense network.
+
+We present Prune-Adjust-Re-Prune (PARP), which discovers and finetunes subnetworks for much better performance, while only requiring a single downstream ASR finetuning run. PARP is inspired by our surprising observation that subnetworks pruned for pre-training tasks need merely a slight adjustment to achieve a sizeable performance boost in downstream ASR tasks. Extensive experiments on lowresource ASR verify (1) sparse subnetworks exist in mono-lingual/multi-lingual pre-trained speech SSL, and (2) the computational advantage and performance gain of PARP over baseline pruning methods.
+
+In particular, on the $1 0 \mathrm { { m i n } }$ Librispeech split without LM decoding, PARP discovers subnetworks from wav2vec 2.0 with an absolute $1 0 . 9 \% / 1 2 . 6 \%$ WER decrease compared to the full model. We further demonstrate the effectiveness of PARP via: cross-lingual pruning without any phone recognition degradation, the discovery of a multi-lingual subnetwork for 10 spoken languages in 1 finetuning run, and its applicability to pre-trained BERT/XLNet for natural language tasks1.
+
+# 1 Introduction
+
+For many low-resource spoken languages in the world, collecting large-scale transcribed corpora is very costly and sometimes infeasible. Inspired by efforts such as the IARPA BABEL program, Automatic Speech Recognition (ASR) trained without sufficient transcribed speech data has been a critical yet challenging research agenda in speech processing [31, 33, 42, 32, 21]. Recently, SelfSupervised Speech Representation Learning (speech SSL) has emerged as a promising pathway toward solving low-resource ASR [84, 25, 110, 6, 29, 127, 55, 27]. Speech SSL involves pre-training a speech representation module on large-scale unlabelled data with a self-supervised learning objective, followed by finetuning on a small amount of supervised transcriptions. Many recent studies have demonstrated the empirical successes of speech SSL on low-resource English and multi-lingual ASR, matching systems trained on fully-supervised settings [6, 29, 127, 4, 126]. Prior research attempts, however, focus on pre-training objectives [84, 25, 110, 72, 57, 74, 71, 73, 55, 22, 27, 17, 129], scaling up speech representation modules [5, 6, 53], pre-training data selections [108, 54, 107, 111, 78], or applications of pre-trained speech representations [26, 62, 94, 28, 63, 29, 76, 120, 64, 117, 112, 44, 4, 86, 59, 66, 2, 56, 102, 15, 30, 20]. In this work, we aim to develop an orthogonal approach that is complementary to these existing speech SSL studies, that achieves 1) lower architectural complexity and 2) higher performance (lower WER) under the same low-resource ASR settings.
+
+Neural network pruning [65, 51, 49, 69], as well as the more recently proposed Lottery Ticket Hypothesis (LTH) $\mathbb { |  \vert 3 9 \| }$ , provide a potential solution that accomplishes both objectives. According to LTH, there exists sparse subnetworks that can achieve the same or even better accuracy than the original dense network. Such phenomena have been successfully observed in various domains: Natural Language Processing (NLP) [123, 19, 88, 80], Computer Vision (CV) [18, 45], and many others. All finding sparse subnetworks with comparable or better performance than the dense network. Given the lack of similar studies on pruning self-supervised ASR, we intend to fill this gap by finding sparse subnetworks within a pre-trained speech SSL that can achieve superior performance to the full pre-trained model on downstream ASR tasks.
+
+However, directly applying widely-adopted pruning methods, such as One-Shot Magnitude Pruning (OMP) and Iterative Magnitude Pruning (IMP) [49, 39], to pre-trained speech SSL suffers from two challenges. First, adopting these methods in the conventional pruning framework is extremely time-consuming for SOTA speech SSL models. OMP and IMP involve more than one round of finetuning on downstream tasks (c.f. Figure $\blacktriangleleft$ , and finetuning for ASR is timeconsuming and computationally demanding2. The second challenge is that we do not observe any performance improvement of the subnetworks over the original dense network with OMP or IMP. Figure 3 shows the WER under low-resource scenarios of the subnetworks identified by OMP (purple line) and IMP (blue dashed line) at different sparsity levels. None of the sparsity levels achieves a visible drop in WER compared to the zero sparsity case, corresponding to the original dense network. These two challenges have prompted us to ask – do there exist sparse subnetworks within pre-trained speech SSL with improved performance on low-resource ASR? How can we discover them efficiently in a single downstream finetuning run?
+
+![](images/5ad30abc480b483eec225c5622e6b8845d1bd60713e2b6d8a5af4e19dca47a3c.jpg)  
+Figure 1: Number of ASR finetuning iterations needed (y-axis) versus target sparsities $\mathbf { \check { X } }$ -axis) for each downstream task/language. Crossreferencing Figure $\textcircled { 3 }$ indicates that IMP requires linearly more compute to match the performance (either sparsity/WER) of PARP.
+
+We propose a magnitude-based unstructured pruning method [41, 11], termed Prune-Adjust-Re-Prune (PARP), for discovering sparse subnetworks within pre-trained speech SSL. PARP consists of the following two steps:
+
+1. Directly prune the SSL pre-trained model at target sparsity, and obtain an initial subnetwork and an initial pruning mask.   
+2. Finetune the initial subnetwork on target downstream task/language. During finetuning, zero out the pruned weights specified by the pruning mask, but allow the weights be updated by gradient descent during backpropogation. After a few number of model updates, re-prune the updated subnetwork at target sparsity again.
+
+Step 1 provides an initial subnetwork that is agnostic to the downstream task, and Step 2 makes learnable adjustments by reviving pruned out weights. A formal and generalized description and its extension are introduced in Section $3 .$ Different from pruning methods in [49, 39], PARP allows pruned-out weights to be revived during finetuning. Although such a high-level idea was introduced in $\lVert \rVert \bigotimes \rVert$ , we provide an alternative insight: despite its flexibility, Step 2 only makes minimal adjustment to the initial subnetwork, and obtaining a good initial subnetwork in Step 1 is the key. We empirically show in Section $\textcircled { 3 }$ that any task-agnostic subnetwork surprisingly provides a good basis for Step 2, suggesting that the initial subnetwork can be cheaply obtained either from a readily available task/language or directly pruning the pre-trained SSL model itself. In addition, this observation allows us to perform cross-lingual pruning (mask transfer) experiments, where the initial subnetwork is obtained via a different language other than the target language.
+
+Our Contributions. We conduct extensive PARP and baseline (OMP and IMP) pruning experiments on low-resource ASR with mono-lingual (pre-trained wav2vec 2.0 [6]) and cross-lingual (pre-trained XLSR-53 [29]) transfer. PARP finds significantly superior speech SSL subnetworks for low-resource
+
+ASR, while only requiring a single pass of downstream ASR finetuning. Due to its simplicity, PARP adds minimal computation overhead to existing SSL downstream finetuning.
+
+• We show that sparse subnetworks exist in pre-trained speech SSL when finetuned for low-resource ASR. In addition, PARP achieves superior results to OMP and IMP across all sparsities, amount of finetuning supervision, pre-trained model scale, and downstream spoken languages. Specifically, on Librispeech $1 0 \mathrm { { m i n } }$ without LM decoding, PARP discovers subnetworks from wav2vec 2.0 with an absolute $1 0 . 9 \% / 1 2 . 6 \%$ WER decrease compared to the full model, without modifying the finetuning hyper-parameters or objective (Section 4.1) • Ablation studies on demonstrating the importance of PARP’s initial subnetwork (Section 4.2) • PARP minimizes phone recognition error increases in cross-lingual mask transfer, where a subnetwork pruned for ASR in one spoken language is adapted for ASR in another language (Section $4 . 3 { \bar { ) } }$ . PARP can also be applied to efficient multi-lingual subnetwork discovery for 10 spoken languages (Section $4 . { \overset { \vartriangle } { 4 } } )$ • Last but not least, we demonstrate PARP’s effectiveness on pre-trained BERT/XLNet, mitigating the cross-task performance degradation reported in BERT-Ticket [19] (Section 4.5)
+
+Significance. Findings of this work not only complement and advance current and future speech SSL for low-resource ASR, but also provide new insights for the rich body of pruning work.
+
+# 2 Preliminaries
+
+# 2.1 Problem Formulation
+
+Consider the low-resource ASR problem, where there is only a small transcribed training set $( x , y ) \in$ $\mathcal { D } _ { l }$ . Here $x$ represents input audio, and $y$ represents output transcription. Subscript $l \in \{ 1 , 2 , \cdots \}$ represents the downstream spoken language identity. Because of the small dataset size, empirical risk minimization generally does not yield good results. Speech SSL instead assumes there is a much larger unannotated dataset $x \in \mathcal { D } _ { 0 }$ . SSL pre-trains a neural network $f ( x ; \theta )$ , where $\theta \in \mathcal { R } ^ { d }$ represents the network parameters and $d$ represents the number of parameters, on some self-supervised objective, and obtains the pre-trained weights $\theta _ { 0 }$ . $f ( x ; \theta _ { 0 } )$ is then finetuned on downstream ASR tasks specified by a downstream loss $\mathcal { L } _ { l } ( \boldsymbol { \theta } )$ , such as CTC, and evaluated on target dataset $\mathcal { D } _ { l }$ .
+
+Our goal is to discover a subnetwork that minimizes downstream ASR WER on $\mathcal { D } _ { l }$ . Formally, denote $m \in \{ 0 , 1 \} ^ { d }$ , as a binary pruning mask for the pre-trained weights $\theta _ { 0 }$ , and $\theta ^ { l }$ as the finetuned weights on $\mathcal { D } _ { l }$ . The ideal pruning method should learn $( m , \theta ^ { l } )$ , such that the subnetwork $f ( x ; m \odot \theta ^ { l } )$ (where $\odot$ is element-wise product) achieves minimal finetuning $\mathcal { L } _ { l } ( \boldsymbol { \theta } )$ loss on $\mathcal { D } _ { l }$ .
+
+# 2.2 Pruning Targets and Settings
+
+We adopted pre-trained speech SSL wav2vec2 and xlsr for the pre-trained initialization $\theta _ { 0 }$ .
+
+wav2vec 2.0 We took wav2vec 2.0 base (wav2vec2-base) and large (wav2vec2-large) pre-trained on Librispeech 960 hours $\textcircled { 6 }$ . During finetuning, a task specific linear layer is added on top of wav2vec2 and jointly finetuned with CTC loss. More details can be found in Appendix 8.
+
+XLSR-53 (xlsr) shares the same architecture, pre-training and finetuning objectives as wav2vec2-large. xlsr is pre-trained on 53 languages sampled from CommonVoice, BABEL, and Multilingual LibriSpeech, totaling for 56k hours of multi-lingual speech data.
+
+We consider three settings where wav2vec2 and xlsr are used as the basis for low-resource ASR:
+
+LSR: Low-Resource English ASR. Mono-lingual pre-training and finetuning – an English pretrained speech SSL such as wav2vec2 is finetuned for low-resource English ASR.
+
+H2L: High-to-Low Resource Transfer for Multi-lingual ASR. Mono-lingual pre-training and multi-lingual finetuning – a speech SSL pre-trained on a high-resource language such as English is finetuned for low-resource multi-lingual ASR.
+
+CSR: Cross-lingual Transfer for Multi-lingual ASR. Multi-lingual pre-training and finetuning – a cross-lingual pretrained speech SSL such as xlsr is finetuned for low-resource multi-lingual ASR.
+
+# 2.3 Subnetwork Discovery in Pre-trained SSL
+
+One obvious solution to the aforementioned problem in Section $2 . 1$ is to directly apply pruning with rewinding to $\theta _ { 0 }$ , which has been successfully applied to pre-trained BERT $\dot { \mathbb { I D } }$ and SimCLR [18].
+
+All pruning methods, including our proposed PARP, are based on Unstructured Magnitude Pruning (UMP) $\mathbb { B 9 } \mathbb { \breve { A 1 } }$ , where weights of the lowest magnitudes are pruned out regardless of the network structure to meet the target sparsity level. We introduce four pruning baselines below, and we also provide results with Random Pruning (RP) [39, 41, 19], where weights in $\theta _ { 0 }$ are randomly eliminated.
+
+Task-Aware Subnetwork Discovery is pruning with target dataset $D _ { l }$ seen in advance, including One-Shot Magnitude Pruning (OMP) and Iterative Magnitude Pruning (IMP). OMP is summarized as:
+
+1. Finetune pretrained weights $\theta _ { 0 }$ on target dataset $\mathcal { D } _ { l }$ to get the finetuned weights $\theta ^ { l }$ .
+
+2. Apply UMP on $\theta ^ { l }$ and retrieve pruning mask $m$
+
+IMP breaks down the above subnetwork discovery phase into multiple iterations – in our case multiple downstream ASR finetunings. Each iteration itself is an OMP with a fraction of the target sparsity pruned. We follow the IMP implementation described in BERT-Ticket $\mathbb { I m }$ , where each iteration prunes out $10 \%$ of the remaining weights. The main bottleneck for OMP and IMP is the computational cost, since multiple rounds of finetunings are required for subnetwork discovery.
+
+Task-Agnostic Subnetwork Discovery refers to pruning without having seen $D _ { l }$ nor $l$ in advance. One instance is applying UMP directly on $\theta _ { 0 }$ without any downstream finetuning to retrieve $m$ , referred to as Magnitude Pruning at Pre-trained Initailizations (MPI). Another case is pruning weights finetuned for a different language $t$ , i.e. applying UMP on $\theta ^ { t }$ for the target language $l$ ; in our study, we refer to this as cross-lingual mask transfer. While these approaches do not require target task finetuning, the discovered subnetworks generally have worse performance than those from OMP or IMP.
+
+The above methods are only for subnetwork discovery via applying pruning mask $m$ on $\theta _ { 0 }$ . The discovered subnetwork $f ( x ; m \odot \theta _ { 0 } )$ needs another downstream finetuning to recover the pruning loss3, i.e. finetune $f ( x ; m \odot \theta _ { 0 } )$ on $D _ { l }$ .
+
+# 3 Method
+
+In this section, we highlight our proposed pruning method, PARP (Section $3 . 1 )$ , its underlying intuition (Section $3 . 2 )$ , and an extension termed PARP-P (Section $\textcircled { 3 . 3 }$ .
+
+# 3.1 Algorithm
+
+We formally describe PARP with the notations from Section 2. A visual overview of PARP is Figure 8.
+
+1: Assume there are $N$ model updates in target task/language $l$ ’s downstream finetuning.   
+2: Take a pre-trained SSL $f ( x ; { \bar { \theta } } _ { 0 } )$ model. Apply task-agnostic subnetwork discovery, such as MPI4 , at target   
+sparsity to obtain initial subnetwork $f ( x ; m _ { 0 } \odot \theta _ { 0 } )$ . Set $m = m _ { 0 }$ and variable .   
+3: repeat   
+4: Zero-out masked-out weights in $\theta _ { n 1 }$ given by $m$ . Lift up $m$ such that whole $\theta _ { n 1 }$ is updatable.   
+5: Train $f ( x ; \theta _ { n 1 } )$ for $_ n$ model updates and obtain $f ( x ; \theta _ { n 2 } )$ .   
+6: Apply UMP on $f ( x ; \theta _ { n 2 } )$ and adjust $m$ accordingly. The adjusted subnetwork is $f ( x ; m \odot \theta _ { n 2 } )$ . Set   
+variable $n _ { 1 } = n _ { 2 }$ .   
+7: until total model updates reach $N$ .   
+8: Return finetuned subnetwork $f ( x ; m \odot \theta _ { N } )$ .
+
+Empirically, we found the choice of $n$ has little impact. In contrast to OMP/IMP/MPI, PARP allows the pruned-out weights to take gradient descent updates. A side benefit of PARP is it jointly discovers and finetunes subnetwork in a single pass, instead of two or more in OMP and IMP.
+
+# 3.2 Obtaining and Adjusting the Initial Subnetwork
+
+PARP achieves superior or comparable pruning results as task-aware subnetwork discovery, while inducing similar computational cost as task-agnostic subnetwork discovery. How does it get the best of both worlds? The key is the discovered subnetworks from task-aware and task-agnostic prunings have high, non-trivial overlaps in LSR, H2L, and CSR. We first define Intersection over Union (IOU) for quantifying subnetworks’ (represented by their pruning masks $m ^ { a }$ and $m ^ { b }$ ) similarity:
+
+$$
+\operatorname { I O U } ( m ^ { a } , m ^ { b } ) \triangleq { \frac { | ( m ^ { a } = 1 ) \cap ( m ^ { b } = 1 ) | } { | ( m ^ { a } = 1 ) \cup ( m ^ { b } = 1 ) | } }
+$$
+
+Take H2L and CSR for instance, Figure $2$ visualizes language pairs’ OMP pruning mask IOUs on wav2vec2 and xlsr. Observe the high overlaps across all pairs, but also the high IOUs with the MPI masks (second to last row). We generalize these observations to the following:
+
+Observation 1 For any sparsity, any amount of finetuning supervision, any pre-training model scale, and any downstream spoken languages, the non-zero ASR pruning masks obtained from task-agnostic subnetwork discovery has high IOUs with those obtained from task-aware subnetwork discovery.
+
+Observation 1 suggests that any task-agnostic subnetwork could sufficiently be a good initial subnetwork in PARP due to the high similarities. In the same instance for H2L and CSR, we could either take MPI on wav2vec2 and xlsr, or take OMP on a different spoken language as the initial subnetworks. Similarly in LSR, we take MPI on wav2vec2 as the initial subnetwork. The underlying message is – the initial subnetwork can be obtained cheaply, without target task finetuning.
+
+Now, because of the high similarity, the initial subnetwork (represented by its pruning mask $m _ { 0 }$ ) needed merely a slight adjustment for the target downstream task. While there are techniques such as dynamic mask adjustment [48], important weights pruning $\mathbb { \underline { { \nabla 7 9 } } } ]$ , and deep rewiring $\mathbb { m }$ , we provide an even simpler alternative suited for our setting. Instead of permanently removing the masked-out weights from the computation graph, PARP merely zeroes them out. Weights that are important for the downstream task (the “important weights”) should emerge with gradient updates; those that are relatively irrelevant should decrease in magnitude, and thus be zero-outed at the end. Doing so circumvents the need of straight-through estimation or additional sparsity loss, see Table 1 of $\dot { \mathbb { Z } } \dot { \mathbb { Z } } \mathbb { I }$
+
+# 3.3 PARP-Progressive (PARP-P)
+
+An extension to PARP is PARP-P, where the second P stands for Progressive. In PARP-P, the initial subnetwork starts at a lower sparsity, and progressively prune up to the target sparsity $s$ in Step 2. The intuition is that despite Observation $\bigtriangledown$ not any subnetwork can be a good initial subnetwork, such as those obtained from RP, or those obtained at very high sparsities in MPI/OMP/IMP. We show later that PARP-P is especially effective in higher sparsity regions, e.g. $90 \%$ for LSR. Note that PARP-P has the same computational cost as PARP, and the only difference is the initial starting sparsity in Step 1.
+
+![](images/d1a8ab0cd8540ceffb7e200ce060ffbed2ea3fda8f98a261908fef21bcbe4cdc.jpg)  
+Figure 2: IOUs over all spoken language pairs’ OMP pruning masks on finetuned wav2vec2 and xlsr. Second to last row is the IOUs between OMP masks and the MPI masks from pre-trained wav2vec2 and xlsr. Here, we show the IOUs at $50 \%$ sparsity, and the rest can be found in Appendix $1 1 .$ Surprisingly at any sparsities, there is a high, non-trivial (c.f. RP in the last row), similarity $( > 9 0 \% )$ ) between all spoken language OMP masks, as well as with the MPI masks. Language IDs are in Appendix 9.
+
+# 4 Experiments and Analysis
+
+# 4.1 Comparing PARP, OMP, and IMP on LSR, H2L, and CSR
+
+Our experimental setup can be found in Appendix $\bigtriangledown .$ We first investigate the existence of sparse subnetworks in speech SSL. Figure $3$ shows the pruning results on LSR. Observe that subnetworks discovered by PARP and PARP-P can achieve $60 \sim 8 0 \%$ sparsities with minimal degradation to the full models. The gap between PARP and other pruning methods also widens as sparsities increase. For instance, Table 1 compares PARP and PARP-P with OMP and IMP at $90 \%$ sparsity, and PARP-P has a $40 \%$ absolute WER reduction. In addition, observe the WER reduction with PARP in the low sparsity regions on the $1 0 \mathrm { { m i n } }$ split in Figure $3$ . The same effect is not seen with OMP, IMP, nor MPI. Table $\nsupseteq$ compares the subnetworks discovered by PARP with the full wav2vec2 and prior work on LSR under the same setting5. Surprisingly, the discovered subnetwork attains an absolute $1 0 . 9 \% / 1 2 . 6 \%$ WER reduction over the full wav2vec2-large. We hypothesize that the performance gains are attributed to pruning out generic, unnecessary weights while preserving important weights, which facilitates training convergence. In other words, PARP provides additional regularization effects to downstream finetuning. We also examined the effectiveness of IMP with different rewinding starting points as studied in $\mathbb { \lVert 4 0 , \rVert 9 3 \rVert }$ , and found rewinding initializations bear minimal effect on downstream ASR. Full rewinding details are in Appendix 10.
+
+![](images/0581ad7d5bca3d4e4c4e360ee99e63a3c6cf1c46b9eaea72b698a7344e6bc8c5.jpg)  
+Figure 3: Comparison of different pruning techniques on LSR (wav2vec2 with $1 0 \mathrm { { m i n } / 1 \mathrm { { h } / 1 0 \mathrm { { h } } } }$ Librispeech finetuning splits). PARP (black line) and PARP-P (black dashed line) are especially effective under ultra-low data regime (e.g. $1 0 \mathrm { { m i n } } )$ and high-sparsity $( 7 0 - 1 0 0 \% )$ regions.
+
+Table 1: WER comparison of pruning LSR: wav2vec2-base at $90 \%$ sparsity with 10h finetuning on Librispeech without LM decoding. At $90 \%$ sparsity, OMP/IMP/MPI perform nearly as bad as RP. sub-finetuning stands for subnetwork finetuning.   
+
+<table><tr><td>Method</td><td>#ASR finetunings</td><td>test clean</td><td>test other</td></tr><tr><td>RP + sub-finetuning</td><td>1</td><td>94.5</td><td>96.4</td></tr><tr><td>MPI + sub-finetuning</td><td>1</td><td>93.6</td><td>96.1</td></tr><tr><td>OMP + sub-finetuning</td><td>2</td><td>92.0</td><td>95.3</td></tr><tr><td>IMP + sub-finetuning</td><td>10</td><td>89.6</td><td>93.9</td></tr><tr><td>PARP (90%→90%)</td><td>1</td><td>83.6</td><td>90.7</td></tr><tr><td>PARP-P 70%→90%</td><td>1</td><td>51.9</td><td>69.1</td></tr><tr><td>60%→80%→90%</td><td>2</td><td>33.6</td><td>53.3</td></tr></table>
+
+Table 2: WER comparison of PARP for LSR with previous speech SSL results on Librispeech $1 0 \mathrm { { m i n } }$ . PARP discovers sparse subnetworks within wav2vec2 with lower WER while adding minimal computational cost to the original ASR finetuning.   
+
+<table><tr><td>Method</td><td>test clean</td><td>test other</td></tr><tr><td>Continuous BERT 国 + LM</td><td>49.5</td><td>66.3</td></tr><tr><td>Discrete BERT 国 + LM</td><td>16.3</td><td>25.2</td></tr><tr><td>wav2vec2-base reported 回</td><td>46.9</td><td>50.9</td></tr><tr><td>wav2vec2-large reported [6</td><td>43.5</td><td>45.3</td></tr><tr><td>wav2vec2-base replicated</td><td>49.3</td><td>53.2</td></tr><tr><td>wav2vec2-large replicated</td><td>46.3</td><td>48.1</td></tr><tr><td>wav2vec2-base w/10% PARP</td><td>38.0</td><td>44.3</td></tr><tr><td>wav2vec2-largew/10%PARP</td><td>33.7</td><td>37.2</td></tr></table>
+
+Next, we examine if the pruning results of LSR transfers to H2L and CSR. Figure $\sharp$ is pruning H2L and CSR with 1h of Dutch $( n l )$ finetuning, and the same conclusion can be extended to other spoken languages. Comparing Figures $\textcircled { 3 }$ and $^ { 4 , }$ we notice that shapes of their pruning curves are different, which can be attributed to the effect of character versus phone predictions. Comparing left and center of Figure $\mathbb { H }$ we show that PARP and OMP reach $50 \%$ sparsity on H2L and $70 \%$ sparsity on CSR with minimal degradations. Furthermore, while PARP is more effective than OMP on H2L for all sparsities, such advantage is only visible in the higher sparsity regions on CSR. Lastly, Table $3$ compares the subnetworks from H2L and CSR with prior work. Even with as high as $90 \%$ sparsities in either settings, subnetworks from PARP and OMP out-performs prior art.
+
+![](images/368fc7de4c84f2fff3c1151872af912baa4191aaba16b87769268efdafa32b71.jpg)  
+Figure 4: Comparison of pruning techniques on H2L & CSR with 1h of Dutch $( n l )$ ASR finetuning. (Left) Pruning H2L (wav2vec2-base $+ n l )$ . (Center) Pruning CSR $( \mathbf { x } 1 \mathbf { s } \mathbf { r } + n l )$ . (Right) Pruning jointly-finetuned wav2vec2-base and xlsr on $n l$ . Trend is consistent for other 9 spoken languages.
+
+Table 3: Comparing subnetworks discovered by OMP and PARP from wav2vec2-base and xlsr with prior work on H2L and CSR. PER is averaged over 10 languages.   
+
+<table><tr><td>Method</td><td>Pre-training</td><td>Sparsity</td><td>avg. PER</td></tr><tr><td>Bottleneck 38</td><td>Babel-1070h</td><td>0%</td><td>44.9</td></tr><tr><td>CPC 图</td><td>LS-100h</td><td>0%</td><td>50.9</td></tr><tr><td>Modified CPC 四</td><td>LS-360h</td><td>0%</td><td>44.5</td></tr><tr><td>wav2vec2-base</td><td>LS-960h</td><td>0%</td><td>18.7</td></tr><tr><td>wav2vec2+ OMP</td><td>LS-960h</td><td>70%</td><td>41.3</td></tr><tr><td>wav2vec2+PARP</td><td>LS-960h</td><td>90%</td><td>40.1</td></tr><tr><td> xlsr reported [29]</td><td>56,000h</td><td>0%</td><td>7.6</td></tr><tr><td>xlsr replicated</td><td>56.000h</td><td>0%</td><td>9.9</td></tr><tr><td>xlsr+OMP</td><td>56.000h</td><td>90%</td><td>33.9</td></tr><tr><td>xlsr+PARP-P</td><td>56,000h</td><td>90%</td><td>22.9</td></tr></table>
+
+![](images/a7694b462a9bd4edc7bebc7f494284e276f249da605557bab0d6e11d793a54a1.jpg)  
+Figure 5: PARP’s final subnetwork and its initial MPI subnetwork exceeds $9 9 . 9 9 \%$ IOU after $20 \%$ sparsity (black line).
+
+# 4.2 How Important is the Initial Subnetwork (Step 1) in PARP?
+
+Obtaining a good initial subnetwork (Step 1) is critical for PARP, as Adjust & Re-Prune (Step 2) is operated on top of it. In this section, we isolate the effect of Step 1 from Step 2 and examine the role of the initial subnetwork in PARP. Figure $\boxed { 6 }$ shows PARP with a random subnetwork from RP, instead of subnetwork from MPI, as the initial subnetwork. PARP with random initial subnetwork performs nearly as bad as RP (grey line), signifying the importance of the initial subnetwork.
+
+Secondly, despite Observation 1, MPI in high sparsity regions (e.g. $90 \%$ in LSR) is not a good initial subnetwork, since the majority of the weights are already pruned out (thus is hard to be recovered from). From Figure 3, PARP performs only on par or even worse than IMP in high sparsity regions. In contrast, PARP-P starts with a relatively lower sparsity (e.g. $60 \%$ or $70 \%$ MPI), and progressively prunes up to the target sparsity. Doing so yields considerable performance gain (up to over $50 \%$ absolute WER reduction). Third, as shown in Figure $\underline { { \boldsymbol { \mathsf { F } } } } ,$ there is ${ > } 9 9 . 9 9 \%$ IOU between the final “adjusted” subnetwork from PARP and its initial MPI subnetwork after $2 0 \%$ sparsity, confirming Step 2 indeed only made minimal “adjustment” to the initial subnetwork.
+
+![](images/517450599b9f945f7458ac2802e6cf1240256e1de40f4600508dd23410512726.jpg)  
+Figure 6: PARP with random (red line) v.s. with MPI (black line) initial subnetworks in LSR.
+
+![](images/b448f34fe2f5b61d0a35a138bcba3b78cf3bdaf96af9e1f464158c795b7e04af.jpg)  
+Transferrability of Language Masks at $5 0 \%$ Sparsity in wav2vec 2.0 with PARP
+
+![](images/663911949f9d167024d6cfa3907289acfb193d6fa8fb47241943d41d0ad56c73.jpg)  
+Transferrability of Language Masks at $5 0 \%$ Sparsity in wav2vec 2.0   
+Figure 7: (Left) Cross-lingual OMP mask transfer with regular subnetwork finetuning. (Right) Cross-lingual OMP mask transfer with PARP. Last rows are RP. Values are relative PER gains over same-language pair transfer (hence the darker the bettter). Both are on H2L with pretrained wav2vec2. The same observation is observed on CSR with pretrained xlsr in Appendix 12.
+
+# 4.3 Are Pruning Masks Transferrable across Spoken Languages?
+
+Is it possible to discover subnetworks with the wrong guidance, and how transferrable are such subnetworks? More concretely, we investigate the transferability of OMP pruning mask discovered from a source language by finetuning its subnetwork on another target language. Such study should shed some insights on the underlying influence of spoken language structure on network pruning – that similar language pairs should be transferrable. From a practical perspective, consider pruning for an unseen new language in H2L, we could deploy the readily available discovered subnetworks and thus save the additional finetuning and memory costs.
+
+In this case, the initial subnetwork of PARP is given by applying OMP on another spoken language. According to Observation $^ { 1 , }$ PARP’s Step 2 is effectively under-going cross-lingual subnetwork adaptation for the target language. Figure $\checkmark$ shows the transferability results on H2L with pre-trained wav2vec2-base. On the left is a subnetwork at $50 \%$ sparsity transfer with regular finetuning that contains subtle language clusters – for example, when finetuning on $r u$ , source masks from es, fr, it, ky, nl induces a much higher PER compare to that from sv-SE, tr, tt, zh-TW. On the right of Figure 7, we show that there is no cross-lingual PER degradation with PARP, supporting our claim above.
+
+# 4.4 Discovering a Single Subnetwork for 10 Spoken Languages
+
+A major downside of pruning pre-trained SSL models for many downstream tasks is the exponential computational and memory costs. In H2L and CSR, the same pruning method needs to be repeatedly re-run for each downstream spoken language at each given sparsity. Therefore, we investigate the possibility of obtaining a single shared subnetwork for all downstream languages. Instead of finetuning separately for each language, we construct a joint phoneme dictionary and finetune wav2vec2 and xlsr on all 10 languages jointly in H2L and CSR. Note that PARP with joint-finetuning can retrieve a shared subnetwork in a single run. The shared subnetwork can then be decoded for each language separately. The right side of Figure 4 illustrates the results.
+
+Comparing joint-finetuning and individual-finetuning, in H2L, we found that the shared subnetwork obtained via OMP has lower PERs between $60 \sim 8 0 \%$ but slightly higher PERs in other sparsity regions; in CSR, the shared subnetwork from OMP has slightly worse PERs at all sparsities. Comparing PARP to OMP in joint-finetuning, we found that while PARP is effective in the individual-finetuning setting (left of Figure $\textcircled { 4 }$ , its shared subnetworks are only slightly better than OMP in both H2L and CSR (right of Figure $\bigoplus$ . The smaller performance gain of PARP over OMP in pruning jointly-finetuned models is expected, since the important weights for each language are disjoint and joint-finetuning may send mixed signal to the adjustment step in PARP (see Figure $\bigtriangledown$ for better illustration).
+
+# 4.5 Does PARP work on Pre-trained BERT/XLNet?
+
+We also analyzed whether Observation $\perp$ holds for pre-trained BERT/XLNet on 9 GLUE tasks. Surprisingly, we found that there are also high $( > 9 8 \% )$ overlaps between the 9 tasks’ IMP pruning masks. Given this observation, we replicated the cross-task subnetwork transfer experiment (take subnetwork found by IMP at task A and finetune it for task B) in BERT-Ticket $\mathbb { \oplus }$ on pre-trained BERT/XLNet with PARP. Table $\boxed { 4 }$ compares PARP (averaged for each target task) to regular finetuning, hinting the applicability of PARP to more pre-trained NLP models and downstream natural language tasks. Detailed scores and figures are in Appendix 13.
+
+![](images/6b33062236588c83677a91b2b658d463d018dbf68d345eb44aa4a62c1b368a57.jpg)  
+Figure 8: Conceptual sketch of pruning the few task-specific important weights in pretrained SSL. (A) Task-aware subnetwork discovery(OMP/IMP) is more effective than task-agnostic pruning (MPI) since it foresees the important weights in advance, via multiple downstream finetunings. $\mathbf { ( B ) }$ PARP starts with an initial subnetwork given by MPI. Observation 1 suggests that the subnetwork is only off by the few important weights, and thus Step 2 revives them by adjusting the initial subnetwork.
+
+# 4.6 Implications
+
+Observation 1 is consistent with the findings of probing large pre-trained NLP models, that pre-trained SSL models are over-parametrized and there exist task-oriented weights/neurons. Figure $2$ implies that these important weights only account for a small part of the pre-trained speech SSL. In fact, a large body of NLP work is dedicated to studying task-oriented weights in pre-trained models. To name a few, $\boxed { 1 3 7 } \boxed { 3 5 } \boxed { 7 } \boxed { 1 1 5 }$ measured, [7, 34, 61] leveraged, [81, 46] visualized, and [105, 36, 13] pruned out these important weights/neurons via probing and quantifying contextualized representations. Based on Observation 1, we can project that these NLP results should in general transfer to speech, see pioneering studies [9, 8, 24, 23]. However, different from them, PARP leverages important weights for UMP on the whole network structure instead of just the contextualized representations.
+
+We could further hypothesize that a good pruning algorithm avoids pruning out task-specific neurons in pre-trained SSL [67, 48, 79], see Figure $8 .$ This hypothesis not only offers an explanation on why PARP is effective in high sparsity regions and cross-lingual mask transfer, it also suggests that an iterative method such as IMP is superior to OMP because IMP gradually avoids pruning out important weights in several iterations, at the cost of more compute6. Finally, we make connections to prior work that showed RP prevail [11, 19, 75, 77, 92] – under a certain threshold and setting, task-specific neurons are less likely to get “accidentally” pruned and thus accuracy is preserved even with RP.
+
+# 5 Related Work
+
+Modern Speech Paradigm and ASR Pruning. As model scale [101, 6, 50, 47, 124, 90, 89, 125, 16, 121, $\bar { \left\lfloor 6 8 \right\rfloor }$ and model pre-training [6, 127, 29, 60, 57, 63, 55, 118, 14, 58, 96, 95, 83, 86, 109] have become the two essential ingredients for obtaining SOTA performance in ASR and other speech tasks, applying and developing various forms of memory-efficient algorithms, such as network pruning, to these large-scale pre-trained models will predictably soon become an indispensable research endeavor. Early work on ASR pruning can be dated back to pruning decoding search spaces [1, 91, 100, 52, 116, 128] and HMM state space $\mathbb { I O 3 } \mathbb { I }$ . Since the seminal work of Yu et al. $\mathbb { \lVert 1 2 2 \rVert }$ , ASR pruning has focused primarily on end-to-end network architecture: [98, 114] applied pruning and quantization to LSTM-based RNN-Transducers, $\lVert \overline { { 8 5 } } \rVert$ applied knowledge distillation to Conformer-based RNN-Transducers, [104, 99, $\textcircled { 7 0 }$ designed efficient architecture/mechanisms for LSTM, Transformer, Conformer-based ASR models, $\pmb { \mathbb { B 2 } }$ applied pruning to Deep Speech, [12] introduced SNR-based probabilistic pruning on LSTM-based CTC model, $\check { \mathbb { B } } 3 \mathbb { I }$ proposed entropyregularizer for LSTM-based ASR model, [119, $\textcircled { 8 7 }$ applied SVD on ASR models’ weight matrices. We emphasize that our work is the first on pruning large self-supervised pre-trained models for low-resource and multi-lingual ASR. In addition, to our knowledge, none of the prior speech pruning work demonstrated the pruned models attain superior performance than its original counterpart.
+
+# 6 Conclusions
+
+We introduce PARP, a simple and intuitive pruning method for self-supervised speech recognition. We conduct extensive experiments on pruning pre-trained wav2vec 2.0 and XLSR-53 under three low-resource settings, demonstrating (1) PARP discovers better subnetworks than baseline pruning methods while requiring a fraction of their computational cost, (2) the discovered subnetworks yields over $10 \%$ WER reduction over the full model, (3) PARP induces minimal cross-lingual subnetwork adaptation errors, (4) PARP can discover a shared subnetwork for multiple spoken languages in one pass, and (5) PARP significantly reduces cross-task adaptation errors of pre-trained BERT/XLNet. Beyond the scope of our study, we aspire PARP as the beginning of many future endeavours on developing more efficient speech SSL models.
+
+Broader Impact. The broader impact of this research work is making speech technologies more accessible in two orthogonal dimensions: (i) extending modern-day speech technology to many under-explored low-resource spoken languages, and (ii) introducing a new and flexible pruning technique to current and future speech SSL frameworks that reduces the computational costs required for adapting (finetuning) them to custom settings. We do not see its potential societal harm.
+
+# Limitations and Future Work
+
+We make clear of the major limitations of our work, and the full list is in Appendix 19. The basis of all the pruning methods in the study is unstructured magnitude weight pruning. Although sparsity is explicitly enforced in the models, we do not suggest that the sparse models are more memory or energy efficient than the original dense models. We do believe that our methodology and results should provide meaningful insights and be easily extended upon to more advanced unstructured or structured pruning methods. We are also curious of the possibility of finetuning or storing modern speech SSL models on local hardware devices.
+
+Results on cross-lingual mask transfer on pre-trained wav2vec 2.0 in Section $\boxed { 4 . 3 }$ is limited to ASR. We do not claim pruning masks to be transferrable across speech tasks (e.g. prune wav2vec2 for speaker ID and transfer for ASR). We provide a pilot cross-task mask transfer study on 3 speech tasks (phone recognition, speaker recognition, slot-filling) in SUPERB [120], and results is in Appendix 16.
+
+We claim PARP could improve the downstream ASR performance over the full wav2vec 2.0, yet we do not claim it as a plug-and-play method into any SOTA ASR pipeline, such as $\mathbb { L } 2 6 \mathbb { I }$ , to get a performance boost. We provide a preliminary experiment on combining PARP and transformer-LM decoding in Appendix $\boxed { 1 5 }$ Nonetheless, due to resource limitations and to isolate the effect of pruning, it remains upon investigations on the complete effects of speech pruning in different setups.
+
+# Acknowledgments
+
+We thank IBM for the donation to MIT of the Satori GPU cluster, and John Cohn for maintaining the cluster. We also thank Lucy Chai, Wei-Ning Hsu, Desh Raj, Shu-wen Leo Yang, Abdelrahman Mohamedm, Erica Cooper, and anonymous reviewers for helpful suggestions and paper editing. This work is part of the low-resource language learning project funded by the MIT-IBM Waston AI Lab.
+
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+        "type": "text",
+        "text": "PARP: Prune, Adjust and Re-Prune for Self-Supervised Speech Recognition ",
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+        "text": "Cheng-I Jeff Lai1, Yang Zhang2⇤, Alexander H. Liu1⇤, Shiyu Chang2, 4⇤ Yi-Lun Liao1, Yung-Sung Chuang1, 3, Kaizhi Qian2, Sameer Khurana1 David $\\mathbf { C o x } ^ { 2 }$ , James Glass1 1MIT CSAIL, 2MIT-IBM Watson AI Lab, 3National Taiwan University, $^ 4 \\mathrm { U C }$ Santa Barbara ",
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+        "type": "text",
+        "text": "clai24@mit.edu ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Self-supervised speech representation learning (speech SSL) has demonstrated the benefit of scale in learning rich representations for Automatic Speech Recognition (ASR) with limited paired data, such as wav2vec 2.0. We investigate the existence of sparse subnetworks in pre-trained speech SSL models that achieve even better low-resource ASR results. However, directly applying widely adopted pruning methods such as the Lottery Ticket Hypothesis (LTH) is suboptimal in the computational cost needed. Moreover, we show that the discovered subnetworks yield minimal performance gain compared to the original dense network. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We present Prune-Adjust-Re-Prune (PARP), which discovers and finetunes subnetworks for much better performance, while only requiring a single downstream ASR finetuning run. PARP is inspired by our surprising observation that subnetworks pruned for pre-training tasks need merely a slight adjustment to achieve a sizeable performance boost in downstream ASR tasks. Extensive experiments on lowresource ASR verify (1) sparse subnetworks exist in mono-lingual/multi-lingual pre-trained speech SSL, and (2) the computational advantage and performance gain of PARP over baseline pruning methods. ",
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+        "type": "text",
+        "text": "In particular, on the $1 0 \\mathrm { { m i n } }$ Librispeech split without LM decoding, PARP discovers subnetworks from wav2vec 2.0 with an absolute $1 0 . 9 \\% / 1 2 . 6 \\%$ WER decrease compared to the full model. We further demonstrate the effectiveness of PARP via: cross-lingual pruning without any phone recognition degradation, the discovery of a multi-lingual subnetwork for 10 spoken languages in 1 finetuning run, and its applicability to pre-trained BERT/XLNet for natural language tasks1. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
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+        "type": "text",
+        "text": "For many low-resource spoken languages in the world, collecting large-scale transcribed corpora is very costly and sometimes infeasible. Inspired by efforts such as the IARPA BABEL program, Automatic Speech Recognition (ASR) trained without sufficient transcribed speech data has been a critical yet challenging research agenda in speech processing [31, 33, 42, 32, 21]. Recently, SelfSupervised Speech Representation Learning (speech SSL) has emerged as a promising pathway toward solving low-resource ASR [84, 25, 110, 6, 29, 127, 55, 27]. Speech SSL involves pre-training a speech representation module on large-scale unlabelled data with a self-supervised learning objective, followed by finetuning on a small amount of supervised transcriptions. Many recent studies have demonstrated the empirical successes of speech SSL on low-resource English and multi-lingual ASR, matching systems trained on fully-supervised settings [6, 29, 127, 4, 126]. Prior research attempts, however, focus on pre-training objectives [84, 25, 110, 72, 57, 74, 71, 73, 55, 22, 27, 17, 129], scaling up speech representation modules [5, 6, 53], pre-training data selections [108, 54, 107, 111, 78], or applications of pre-trained speech representations [26, 62, 94, 28, 63, 29, 76, 120, 64, 117, 112, 44, 4, 86, 59, 66, 2, 56, 102, 15, 30, 20]. In this work, we aim to develop an orthogonal approach that is complementary to these existing speech SSL studies, that achieves 1) lower architectural complexity and 2) higher performance (lower WER) under the same low-resource ASR settings. ",
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+    {
+        "type": "text",
+        "text": "Neural network pruning [65, 51, 49, 69], as well as the more recently proposed Lottery Ticket Hypothesis (LTH) $\\mathbb { |  \\vert 3 9 \\| }$ , provide a potential solution that accomplishes both objectives. According to LTH, there exists sparse subnetworks that can achieve the same or even better accuracy than the original dense network. Such phenomena have been successfully observed in various domains: Natural Language Processing (NLP) [123, 19, 88, 80], Computer Vision (CV) [18, 45], and many others. All finding sparse subnetworks with comparable or better performance than the dense network. Given the lack of similar studies on pruning self-supervised ASR, we intend to fill this gap by finding sparse subnetworks within a pre-trained speech SSL that can achieve superior performance to the full pre-trained model on downstream ASR tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, directly applying widely-adopted pruning methods, such as One-Shot Magnitude Pruning (OMP) and Iterative Magnitude Pruning (IMP) [49, 39], to pre-trained speech SSL suffers from two challenges. First, adopting these methods in the conventional pruning framework is extremely time-consuming for SOTA speech SSL models. OMP and IMP involve more than one round of finetuning on downstream tasks (c.f. Figure $\\blacktriangleleft$ , and finetuning for ASR is timeconsuming and computationally demanding2. The second challenge is that we do not observe any performance improvement of the subnetworks over the original dense network with OMP or IMP. Figure 3 shows the WER under low-resource scenarios of the subnetworks identified by OMP (purple line) and IMP (blue dashed line) at different sparsity levels. None of the sparsity levels achieves a visible drop in WER compared to the zero sparsity case, corresponding to the original dense network. These two challenges have prompted us to ask – do there exist sparse subnetworks within pre-trained speech SSL with improved performance on low-resource ASR? How can we discover them efficiently in a single downstream finetuning run? ",
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+    {
+        "type": "image",
+        "img_path": "images/5ad30abc480b483eec225c5622e6b8845d1bd60713e2b6d8a5af4e19dca47a3c.jpg",
+        "image_caption": [
+            "Figure 1: Number of ASR finetuning iterations needed (y-axis) versus target sparsities $\\mathbf { \\check { X } }$ -axis) for each downstream task/language. Crossreferencing Figure $\\textcircled { 3 }$ indicates that IMP requires linearly more compute to match the performance (either sparsity/WER) of PARP. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose a magnitude-based unstructured pruning method [41, 11], termed Prune-Adjust-Re-Prune (PARP), for discovering sparse subnetworks within pre-trained speech SSL. PARP consists of the following two steps: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Directly prune the SSL pre-trained model at target sparsity, and obtain an initial subnetwork and an initial pruning mask.   \n2. Finetune the initial subnetwork on target downstream task/language. During finetuning, zero out the pruned weights specified by the pruning mask, but allow the weights be updated by gradient descent during backpropogation. After a few number of model updates, re-prune the updated subnetwork at target sparsity again. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Step 1 provides an initial subnetwork that is agnostic to the downstream task, and Step 2 makes learnable adjustments by reviving pruned out weights. A formal and generalized description and its extension are introduced in Section $3 .$ Different from pruning methods in [49, 39], PARP allows pruned-out weights to be revived during finetuning. Although such a high-level idea was introduced in $\\lVert \\rVert \\bigotimes \\rVert$ , we provide an alternative insight: despite its flexibility, Step 2 only makes minimal adjustment to the initial subnetwork, and obtaining a good initial subnetwork in Step 1 is the key. We empirically show in Section $\\textcircled { 3 }$ that any task-agnostic subnetwork surprisingly provides a good basis for Step 2, suggesting that the initial subnetwork can be cheaply obtained either from a readily available task/language or directly pruning the pre-trained SSL model itself. In addition, this observation allows us to perform cross-lingual pruning (mask transfer) experiments, where the initial subnetwork is obtained via a different language other than the target language. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our Contributions. We conduct extensive PARP and baseline (OMP and IMP) pruning experiments on low-resource ASR with mono-lingual (pre-trained wav2vec 2.0 [6]) and cross-lingual (pre-trained XLSR-53 [29]) transfer. PARP finds significantly superior speech SSL subnetworks for low-resource ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "ASR, while only requiring a single pass of downstream ASR finetuning. Due to its simplicity, PARP adds minimal computation overhead to existing SSL downstream finetuning. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• We show that sparse subnetworks exist in pre-trained speech SSL when finetuned for low-resource ASR. In addition, PARP achieves superior results to OMP and IMP across all sparsities, amount of finetuning supervision, pre-trained model scale, and downstream spoken languages. Specifically, on Librispeech $1 0 \\mathrm { { m i n } }$ without LM decoding, PARP discovers subnetworks from wav2vec 2.0 with an absolute $1 0 . 9 \\% / 1 2 . 6 \\%$ WER decrease compared to the full model, without modifying the finetuning hyper-parameters or objective (Section 4.1) • Ablation studies on demonstrating the importance of PARP’s initial subnetwork (Section 4.2) • PARP minimizes phone recognition error increases in cross-lingual mask transfer, where a subnetwork pruned for ASR in one spoken language is adapted for ASR in another language (Section $4 . 3 { \\bar { ) } }$ . PARP can also be applied to efficient multi-lingual subnetwork discovery for 10 spoken languages (Section $4 . { \\overset { \\vartriangle } { 4 } } )$ • Last but not least, we demonstrate PARP’s effectiveness on pre-trained BERT/XLNet, mitigating the cross-task performance degradation reported in BERT-Ticket [19] (Section 4.5) ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Significance. Findings of this work not only complement and advance current and future speech SSL for low-resource ASR, but also provide new insights for the rich body of pruning work. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
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+            363,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Problem Formulation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider the low-resource ASR problem, where there is only a small transcribed training set $( x , y ) \\in$ $\\mathcal { D } _ { l }$ . Here $x$ represents input audio, and $y$ represents output transcription. Subscript $l \\in \\{ 1 , 2 , \\cdots \\}$ represents the downstream spoken language identity. Because of the small dataset size, empirical risk minimization generally does not yield good results. Speech SSL instead assumes there is a much larger unannotated dataset $x \\in \\mathcal { D } _ { 0 }$ . SSL pre-trains a neural network $f ( x ; \\theta )$ , where $\\theta \\in \\mathcal { R } ^ { d }$ represents the network parameters and $d$ represents the number of parameters, on some self-supervised objective, and obtains the pre-trained weights $\\theta _ { 0 }$ . $f ( x ; \\theta _ { 0 } )$ is then finetuned on downstream ASR tasks specified by a downstream loss $\\mathcal { L } _ { l } ( \\boldsymbol { \\theta } )$ , such as CTC, and evaluated on target dataset $\\mathcal { D } _ { l }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to discover a subnetwork that minimizes downstream ASR WER on $\\mathcal { D } _ { l }$ . Formally, denote $m \\in \\{ 0 , 1 \\} ^ { d }$ , as a binary pruning mask for the pre-trained weights $\\theta _ { 0 }$ , and $\\theta ^ { l }$ as the finetuned weights on $\\mathcal { D } _ { l }$ . The ideal pruning method should learn $( m , \\theta ^ { l } )$ , such that the subnetwork $f ( x ; m \\odot \\theta ^ { l } )$ (where $\\odot$ is element-wise product) achieves minimal finetuning $\\mathcal { L } _ { l } ( \\boldsymbol { \\theta } )$ loss on $\\mathcal { D } _ { l }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Pruning Targets and Settings ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We adopted pre-trained speech SSL wav2vec2 and xlsr for the pre-trained initialization $\\theta _ { 0 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "wav2vec 2.0 We took wav2vec 2.0 base (wav2vec2-base) and large (wav2vec2-large) pre-trained on Librispeech 960 hours $\\textcircled { 6 }$ . During finetuning, a task specific linear layer is added on top of wav2vec2 and jointly finetuned with CTC loss. More details can be found in Appendix 8. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "XLSR-53 (xlsr) shares the same architecture, pre-training and finetuning objectives as wav2vec2-large. xlsr is pre-trained on 53 languages sampled from CommonVoice, BABEL, and Multilingual LibriSpeech, totaling for 56k hours of multi-lingual speech data. ",
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+    {
+        "type": "text",
+        "text": "We consider three settings where wav2vec2 and xlsr are used as the basis for low-resource ASR: ",
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+    {
+        "type": "text",
+        "text": "LSR: Low-Resource English ASR. Mono-lingual pre-training and finetuning – an English pretrained speech SSL such as wav2vec2 is finetuned for low-resource English ASR. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "H2L: High-to-Low Resource Transfer for Multi-lingual ASR. Mono-lingual pre-training and multi-lingual finetuning – a speech SSL pre-trained on a high-resource language such as English is finetuned for low-resource multi-lingual ASR. ",
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+    },
+    {
+        "type": "text",
+        "text": "CSR: Cross-lingual Transfer for Multi-lingual ASR. Multi-lingual pre-training and finetuning – a cross-lingual pretrained speech SSL such as xlsr is finetuned for low-resource multi-lingual ASR. ",
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+    {
+        "type": "text",
+        "text": "2.3 Subnetwork Discovery in Pre-trained SSL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "One obvious solution to the aforementioned problem in Section $2 . 1$ is to directly apply pruning with rewinding to $\\theta _ { 0 }$ , which has been successfully applied to pre-trained BERT $\\dot { \\mathbb { I D } }$ and SimCLR [18]. ",
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+    {
+        "type": "text",
+        "text": "All pruning methods, including our proposed PARP, are based on Unstructured Magnitude Pruning (UMP) $\\mathbb { B 9 } \\mathbb { \\breve { A 1 } }$ , where weights of the lowest magnitudes are pruned out regardless of the network structure to meet the target sparsity level. We introduce four pruning baselines below, and we also provide results with Random Pruning (RP) [39, 41, 19], where weights in $\\theta _ { 0 }$ are randomly eliminated. ",
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+    {
+        "type": "text",
+        "text": "Task-Aware Subnetwork Discovery is pruning with target dataset $D _ { l }$ seen in advance, including One-Shot Magnitude Pruning (OMP) and Iterative Magnitude Pruning (IMP). OMP is summarized as: ",
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+    {
+        "type": "text",
+        "text": "1. Finetune pretrained weights $\\theta _ { 0 }$ on target dataset $\\mathcal { D } _ { l }$ to get the finetuned weights $\\theta ^ { l }$ . ",
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+        "type": "text",
+        "text": "2. Apply UMP on $\\theta ^ { l }$ and retrieve pruning mask $m$ ",
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+    {
+        "type": "text",
+        "text": "IMP breaks down the above subnetwork discovery phase into multiple iterations – in our case multiple downstream ASR finetunings. Each iteration itself is an OMP with a fraction of the target sparsity pruned. We follow the IMP implementation described in BERT-Ticket $\\mathbb { I m }$ , where each iteration prunes out $10 \\%$ of the remaining weights. The main bottleneck for OMP and IMP is the computational cost, since multiple rounds of finetunings are required for subnetwork discovery. ",
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+    {
+        "type": "text",
+        "text": "Task-Agnostic Subnetwork Discovery refers to pruning without having seen $D _ { l }$ nor $l$ in advance. One instance is applying UMP directly on $\\theta _ { 0 }$ without any downstream finetuning to retrieve $m$ , referred to as Magnitude Pruning at Pre-trained Initailizations (MPI). Another case is pruning weights finetuned for a different language $t$ , i.e. applying UMP on $\\theta ^ { t }$ for the target language $l$ ; in our study, we refer to this as cross-lingual mask transfer. While these approaches do not require target task finetuning, the discovered subnetworks generally have worse performance than those from OMP or IMP. ",
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+        "type": "text",
+        "text": "The above methods are only for subnetwork discovery via applying pruning mask $m$ on $\\theta _ { 0 }$ . The discovered subnetwork $f ( x ; m \\odot \\theta _ { 0 } )$ needs another downstream finetuning to recover the pruning loss3, i.e. finetune $f ( x ; m \\odot \\theta _ { 0 } )$ on $D _ { l }$ . ",
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+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we highlight our proposed pruning method, PARP (Section $3 . 1 )$ , its underlying intuition (Section $3 . 2 )$ , and an extension termed PARP-P (Section $\\textcircled { 3 . 3 }$ . ",
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+    {
+        "type": "text",
+        "text": "3.1 Algorithm ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We formally describe PARP with the notations from Section 2. A visual overview of PARP is Figure 8. ",
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+    },
+    {
+        "type": "text",
+        "text": "1: Assume there are $N$ model updates in target task/language $l$ ’s downstream finetuning.   \n2: Take a pre-trained SSL $f ( x ; { \\bar { \\theta } } _ { 0 } )$ model. Apply task-agnostic subnetwork discovery, such as MPI4 , at target   \nsparsity to obtain initial subnetwork $f ( x ; m _ { 0 } \\odot \\theta _ { 0 } )$ . Set $m = m _ { 0 }$ and variable .   \n3: repeat   \n4: Zero-out masked-out weights in $\\theta _ { n 1 }$ given by $m$ . Lift up $m$ such that whole $\\theta _ { n 1 }$ is updatable.   \n5: Train $f ( x ; \\theta _ { n 1 } )$ for $_ n$ model updates and obtain $f ( x ; \\theta _ { n 2 } )$ .   \n6: Apply UMP on $f ( x ; \\theta _ { n 2 } )$ and adjust $m$ accordingly. The adjusted subnetwork is $f ( x ; m \\odot \\theta _ { n 2 } )$ . Set   \nvariable $n _ { 1 } = n _ { 2 }$ .   \n7: until total model updates reach $N$ .   \n8: Return finetuned subnetwork $f ( x ; m \\odot \\theta _ { N } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Empirically, we found the choice of $n$ has little impact. In contrast to OMP/IMP/MPI, PARP allows the pruned-out weights to take gradient descent updates. A side benefit of PARP is it jointly discovers and finetunes subnetwork in a single pass, instead of two or more in OMP and IMP. ",
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+    {
+        "type": "text",
+        "text": "3.2 Obtaining and Adjusting the Initial Subnetwork ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "PARP achieves superior or comparable pruning results as task-aware subnetwork discovery, while inducing similar computational cost as task-agnostic subnetwork discovery. How does it get the best of both worlds? The key is the discovered subnetworks from task-aware and task-agnostic prunings have high, non-trivial overlaps in LSR, H2L, and CSR. We first define Intersection over Union (IOU) for quantifying subnetworks’ (represented by their pruning masks $m ^ { a }$ and $m ^ { b }$ ) similarity: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ade60d5dea32bf3c0e93ea40adc82323075ae0a146e08752fe886c23a4224bc3.jpg",
+        "text": "$$\n\\operatorname { I O U } ( m ^ { a } , m ^ { b } ) \\triangleq { \\frac { | ( m ^ { a } = 1 ) \\cap ( m ^ { b } = 1 ) | } { | ( m ^ { a } = 1 ) \\cup ( m ^ { b } = 1 ) | } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Take H2L and CSR for instance, Figure $2$ visualizes language pairs’ OMP pruning mask IOUs on wav2vec2 and xlsr. Observe the high overlaps across all pairs, but also the high IOUs with the MPI masks (second to last row). We generalize these observations to the following: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Observation 1 For any sparsity, any amount of finetuning supervision, any pre-training model scale, and any downstream spoken languages, the non-zero ASR pruning masks obtained from task-agnostic subnetwork discovery has high IOUs with those obtained from task-aware subnetwork discovery. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Observation 1 suggests that any task-agnostic subnetwork could sufficiently be a good initial subnetwork in PARP due to the high similarities. In the same instance for H2L and CSR, we could either take MPI on wav2vec2 and xlsr, or take OMP on a different spoken language as the initial subnetworks. Similarly in LSR, we take MPI on wav2vec2 as the initial subnetwork. The underlying message is – the initial subnetwork can be obtained cheaply, without target task finetuning. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Now, because of the high similarity, the initial subnetwork (represented by its pruning mask $m _ { 0 }$ ) needed merely a slight adjustment for the target downstream task. While there are techniques such as dynamic mask adjustment [48], important weights pruning $\\mathbb { \\underline { { \\nabla 7 9 } } } ]$ , and deep rewiring $\\mathbb { m }$ , we provide an even simpler alternative suited for our setting. Instead of permanently removing the masked-out weights from the computation graph, PARP merely zeroes them out. Weights that are important for the downstream task (the “important weights”) should emerge with gradient updates; those that are relatively irrelevant should decrease in magnitude, and thus be zero-outed at the end. Doing so circumvents the need of straight-through estimation or additional sparsity loss, see Table 1 of $\\dot { \\mathbb { Z } } \\dot { \\mathbb { Z } } \\mathbb { I }$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.3 PARP-Progressive (PARP-P) ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "An extension to PARP is PARP-P, where the second P stands for Progressive. In PARP-P, the initial subnetwork starts at a lower sparsity, and progressively prune up to the target sparsity $s$ in Step 2. The intuition is that despite Observation $\\bigtriangledown$ not any subnetwork can be a good initial subnetwork, such as those obtained from RP, or those obtained at very high sparsities in MPI/OMP/IMP. We show later that PARP-P is especially effective in higher sparsity regions, e.g. $90 \\%$ for LSR. Note that PARP-P has the same computational cost as PARP, and the only difference is the initial starting sparsity in Step 1. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/d1a8ab0cd8540ceffb7e200ce060ffbed2ea3fda8f98a261908fef21bcbe4cdc.jpg",
+        "image_caption": [
+            "Figure 2: IOUs over all spoken language pairs’ OMP pruning masks on finetuned wav2vec2 and xlsr. Second to last row is the IOUs between OMP masks and the MPI masks from pre-trained wav2vec2 and xlsr. Here, we show the IOUs at $50 \\%$ sparsity, and the rest can be found in Appendix $1 1 .$ Surprisingly at any sparsities, there is a high, non-trivial (c.f. RP in the last row), similarity $( > 9 0 \\% )$ ) between all spoken language OMP masks, as well as with the MPI masks. Language IDs are in Appendix 9. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments and Analysis ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Comparing PARP, OMP, and IMP on LSR, H2L, and CSR ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our experimental setup can be found in Appendix $\\bigtriangledown .$ We first investigate the existence of sparse subnetworks in speech SSL. Figure $3$ shows the pruning results on LSR. Observe that subnetworks discovered by PARP and PARP-P can achieve $60 \\sim 8 0 \\%$ sparsities with minimal degradation to the full models. The gap between PARP and other pruning methods also widens as sparsities increase. For instance, Table 1 compares PARP and PARP-P with OMP and IMP at $90 \\%$ sparsity, and PARP-P has a $40 \\%$ absolute WER reduction. In addition, observe the WER reduction with PARP in the low sparsity regions on the $1 0 \\mathrm { { m i n } }$ split in Figure $3$ . The same effect is not seen with OMP, IMP, nor MPI. Table $\\nsupseteq$ compares the subnetworks discovered by PARP with the full wav2vec2 and prior work on LSR under the same setting5. Surprisingly, the discovered subnetwork attains an absolute $1 0 . 9 \\% / 1 2 . 6 \\%$ WER reduction over the full wav2vec2-large. We hypothesize that the performance gains are attributed to pruning out generic, unnecessary weights while preserving important weights, which facilitates training convergence. In other words, PARP provides additional regularization effects to downstream finetuning. We also examined the effectiveness of IMP with different rewinding starting points as studied in $\\mathbb { \\lVert 4 0 , \\rVert 9 3 \\rVert }$ , and found rewinding initializations bear minimal effect on downstream ASR. Full rewinding details are in Appendix 10. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0581ad7d5bca3d4e4c4e360ee99e63a3c6cf1c46b9eaea72b698a7344e6bc8c5.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of different pruning techniques on LSR (wav2vec2 with $1 0 \\mathrm { { m i n } / 1 \\mathrm { { h } / 1 0 \\mathrm { { h } } } }$ Librispeech finetuning splits). PARP (black line) and PARP-P (black dashed line) are especially effective under ultra-low data regime (e.g. $1 0 \\mathrm { { m i n } } )$ and high-sparsity $( 7 0 - 1 0 0 \\% )$ regions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/b5909bd5408a72df3a27c95b58c24b94ee65cb54fecb2928e1c5da743d77f6dc.jpg",
+        "table_caption": [
+            "Table 1: WER comparison of pruning LSR: wav2vec2-base at $90 \\%$ sparsity with 10h finetuning on Librispeech without LM decoding. At $90 \\%$ sparsity, OMP/IMP/MPI perform nearly as bad as RP. sub-finetuning stands for subnetwork finetuning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>#ASR finetunings</td><td>test clean</td><td>test other</td></tr><tr><td>RP + sub-finetuning</td><td>1</td><td>94.5</td><td>96.4</td></tr><tr><td>MPI + sub-finetuning</td><td>1</td><td>93.6</td><td>96.1</td></tr><tr><td>OMP + sub-finetuning</td><td>2</td><td>92.0</td><td>95.3</td></tr><tr><td>IMP + sub-finetuning</td><td>10</td><td>89.6</td><td>93.9</td></tr><tr><td>PARP (90%→90%)</td><td>1</td><td>83.6</td><td>90.7</td></tr><tr><td>PARP-P 70%→90%</td><td>1</td><td>51.9</td><td>69.1</td></tr><tr><td>60%→80%→90%</td><td>2</td><td>33.6</td><td>53.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/95a680c68e44169a3371b0ef9598ed2f4f625366ddde816736783d65c435c4d4.jpg",
+        "table_caption": [
+            "Table 2: WER comparison of PARP for LSR with previous speech SSL results on Librispeech $1 0 \\mathrm { { m i n } }$ . PARP discovers sparse subnetworks within wav2vec2 with lower WER while adding minimal computational cost to the original ASR finetuning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>test clean</td><td>test other</td></tr><tr><td>Continuous BERT 国 + LM</td><td>49.5</td><td>66.3</td></tr><tr><td>Discrete BERT 国 + LM</td><td>16.3</td><td>25.2</td></tr><tr><td>wav2vec2-base reported 回</td><td>46.9</td><td>50.9</td></tr><tr><td>wav2vec2-large reported [6</td><td>43.5</td><td>45.3</td></tr><tr><td>wav2vec2-base replicated</td><td>49.3</td><td>53.2</td></tr><tr><td>wav2vec2-large replicated</td><td>46.3</td><td>48.1</td></tr><tr><td>wav2vec2-base w/10% PARP</td><td>38.0</td><td>44.3</td></tr><tr><td>wav2vec2-largew/10%PARP</td><td>33.7</td><td>37.2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Next, we examine if the pruning results of LSR transfers to H2L and CSR. Figure $\\sharp$ is pruning H2L and CSR with 1h of Dutch $( n l )$ finetuning, and the same conclusion can be extended to other spoken languages. Comparing Figures $\\textcircled { 3 }$ and $^ { 4 , }$ we notice that shapes of their pruning curves are different, which can be attributed to the effect of character versus phone predictions. Comparing left and center of Figure $\\mathbb { H }$ we show that PARP and OMP reach $50 \\%$ sparsity on H2L and $70 \\%$ sparsity on CSR with minimal degradations. Furthermore, while PARP is more effective than OMP on H2L for all sparsities, such advantage is only visible in the higher sparsity regions on CSR. Lastly, Table $3$ compares the subnetworks from H2L and CSR with prior work. Even with as high as $90 \\%$ sparsities in either settings, subnetworks from PARP and OMP out-performs prior art. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/368fc7de4c84f2fff3c1151872af912baa4191aaba16b87769268efdafa32b71.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of pruning techniques on H2L & CSR with 1h of Dutch $( n l )$ ASR finetuning. (Left) Pruning H2L (wav2vec2-base $+ n l )$ . (Center) Pruning CSR $( \\mathbf { x } 1 \\mathbf { s } \\mathbf { r } + n l )$ . (Right) Pruning jointly-finetuned wav2vec2-base and xlsr on $n l$ . Trend is consistent for other 9 spoken languages. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/a91fad9ca9fb13bc5b629abbb1cfd01530cd8bc871339c8c108f870e8db01460.jpg",
+        "table_caption": [
+            "Table 3: Comparing subnetworks discovered by OMP and PARP from wav2vec2-base and xlsr with prior work on H2L and CSR. PER is averaged over 10 languages. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Pre-training</td><td>Sparsity</td><td>avg. PER</td></tr><tr><td>Bottleneck 38</td><td>Babel-1070h</td><td>0%</td><td>44.9</td></tr><tr><td>CPC 图</td><td>LS-100h</td><td>0%</td><td>50.9</td></tr><tr><td>Modified CPC 四</td><td>LS-360h</td><td>0%</td><td>44.5</td></tr><tr><td>wav2vec2-base</td><td>LS-960h</td><td>0%</td><td>18.7</td></tr><tr><td>wav2vec2+ OMP</td><td>LS-960h</td><td>70%</td><td>41.3</td></tr><tr><td>wav2vec2+PARP</td><td>LS-960h</td><td>90%</td><td>40.1</td></tr><tr><td> xlsr reported [29]</td><td>56,000h</td><td>0%</td><td>7.6</td></tr><tr><td>xlsr replicated</td><td>56.000h</td><td>0%</td><td>9.9</td></tr><tr><td>xlsr+OMP</td><td>56.000h</td><td>90%</td><td>33.9</td></tr><tr><td>xlsr+PARP-P</td><td>56,000h</td><td>90%</td><td>22.9</td></tr></table>",
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+        "image_caption": [
+            "Figure 5: PARP’s final subnetwork and its initial MPI subnetwork exceeds $9 9 . 9 9 \\%$ IOU after $20 \\%$ sparsity (black line). "
+        ],
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+        "type": "text",
+        "text": "4.2 How Important is the Initial Subnetwork (Step 1) in PARP? ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Obtaining a good initial subnetwork (Step 1) is critical for PARP, as Adjust & Re-Prune (Step 2) is operated on top of it. In this section, we isolate the effect of Step 1 from Step 2 and examine the role of the initial subnetwork in PARP. Figure $\\boxed { 6 }$ shows PARP with a random subnetwork from RP, instead of subnetwork from MPI, as the initial subnetwork. PARP with random initial subnetwork performs nearly as bad as RP (grey line), signifying the importance of the initial subnetwork. ",
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+        "type": "text",
+        "text": "Secondly, despite Observation 1, MPI in high sparsity regions (e.g. $90 \\%$ in LSR) is not a good initial subnetwork, since the majority of the weights are already pruned out (thus is hard to be recovered from). From Figure 3, PARP performs only on par or even worse than IMP in high sparsity regions. In contrast, PARP-P starts with a relatively lower sparsity (e.g. $60 \\%$ or $70 \\%$ MPI), and progressively prunes up to the target sparsity. Doing so yields considerable performance gain (up to over $50 \\%$ absolute WER reduction). Third, as shown in Figure $\\underline { { \\boldsymbol { \\mathsf { F } } } } ,$ there is ${ > } 9 9 . 9 9 \\%$ IOU between the final “adjusted” subnetwork from PARP and its initial MPI subnetwork after $2 0 \\%$ sparsity, confirming Step 2 indeed only made minimal “adjustment” to the initial subnetwork. ",
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+        "image_caption": [
+            "Figure 6: PARP with random (red line) v.s. with MPI (black line) initial subnetworks in LSR. "
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+        "image_caption": [
+            "Transferrability of Language Masks at $5 0 \\%$ Sparsity in wav2vec 2.0 with PARP "
+        ],
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+        "image_caption": [
+            "Transferrability of Language Masks at $5 0 \\%$ Sparsity in wav2vec 2.0 ",
+            "Figure 7: (Left) Cross-lingual OMP mask transfer with regular subnetwork finetuning. (Right) Cross-lingual OMP mask transfer with PARP. Last rows are RP. Values are relative PER gains over same-language pair transfer (hence the darker the bettter). Both are on H2L with pretrained wav2vec2. The same observation is observed on CSR with pretrained xlsr in Appendix 12. "
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+        "type": "text",
+        "text": "4.3 Are Pruning Masks Transferrable across Spoken Languages? ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Is it possible to discover subnetworks with the wrong guidance, and how transferrable are such subnetworks? More concretely, we investigate the transferability of OMP pruning mask discovered from a source language by finetuning its subnetwork on another target language. Such study should shed some insights on the underlying influence of spoken language structure on network pruning – that similar language pairs should be transferrable. From a practical perspective, consider pruning for an unseen new language in H2L, we could deploy the readily available discovered subnetworks and thus save the additional finetuning and memory costs. ",
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+        "type": "text",
+        "text": "In this case, the initial subnetwork of PARP is given by applying OMP on another spoken language. According to Observation $^ { 1 , }$ PARP’s Step 2 is effectively under-going cross-lingual subnetwork adaptation for the target language. Figure $\\checkmark$ shows the transferability results on H2L with pre-trained wav2vec2-base. On the left is a subnetwork at $50 \\%$ sparsity transfer with regular finetuning that contains subtle language clusters – for example, when finetuning on $r u$ , source masks from es, fr, it, ky, nl induces a much higher PER compare to that from sv-SE, tr, tt, zh-TW. On the right of Figure 7, we show that there is no cross-lingual PER degradation with PARP, supporting our claim above. ",
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+    {
+        "type": "text",
+        "text": "4.4 Discovering a Single Subnetwork for 10 Spoken Languages ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A major downside of pruning pre-trained SSL models for many downstream tasks is the exponential computational and memory costs. In H2L and CSR, the same pruning method needs to be repeatedly re-run for each downstream spoken language at each given sparsity. Therefore, we investigate the possibility of obtaining a single shared subnetwork for all downstream languages. Instead of finetuning separately for each language, we construct a joint phoneme dictionary and finetune wav2vec2 and xlsr on all 10 languages jointly in H2L and CSR. Note that PARP with joint-finetuning can retrieve a shared subnetwork in a single run. The shared subnetwork can then be decoded for each language separately. The right side of Figure 4 illustrates the results. ",
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+    {
+        "type": "text",
+        "text": "Comparing joint-finetuning and individual-finetuning, in H2L, we found that the shared subnetwork obtained via OMP has lower PERs between $60 \\sim 8 0 \\%$ but slightly higher PERs in other sparsity regions; in CSR, the shared subnetwork from OMP has slightly worse PERs at all sparsities. Comparing PARP to OMP in joint-finetuning, we found that while PARP is effective in the individual-finetuning setting (left of Figure $\\textcircled { 4 }$ , its shared subnetworks are only slightly better than OMP in both H2L and CSR (right of Figure $\\bigoplus$ . The smaller performance gain of PARP over OMP in pruning jointly-finetuned models is expected, since the important weights for each language are disjoint and joint-finetuning may send mixed signal to the adjustment step in PARP (see Figure $\\bigtriangledown$ for better illustration). ",
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+    {
+        "type": "text",
+        "text": "4.5 Does PARP work on Pre-trained BERT/XLNet? ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We also analyzed whether Observation $\\perp$ holds for pre-trained BERT/XLNet on 9 GLUE tasks. Surprisingly, we found that there are also high $( > 9 8 \\% )$ overlaps between the 9 tasks’ IMP pruning masks. Given this observation, we replicated the cross-task subnetwork transfer experiment (take subnetwork found by IMP at task A and finetune it for task B) in BERT-Ticket $\\mathbb { \\oplus }$ on pre-trained BERT/XLNet with PARP. Table $\\boxed { 4 }$ compares PARP (averaged for each target task) to regular finetuning, hinting the applicability of PARP to more pre-trained NLP models and downstream natural language tasks. Detailed scores and figures are in Appendix 13. ",
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+        "type": "text",
+        "text": "",
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+    {
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+        "img_path": "images/6b33062236588c83677a91b2b658d463d018dbf68d345eb44aa4a62c1b368a57.jpg",
+        "image_caption": [
+            "Figure 8: Conceptual sketch of pruning the few task-specific important weights in pretrained SSL. (A) Task-aware subnetwork discovery(OMP/IMP) is more effective than task-agnostic pruning (MPI) since it foresees the important weights in advance, via multiple downstream finetunings. $\\mathbf { ( B ) }$ PARP starts with an initial subnetwork given by MPI. Observation 1 suggests that the subnetwork is only off by the few important weights, and thus Step 2 revives them by adjusting the initial subnetwork. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.6 Implications ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Observation 1 is consistent with the findings of probing large pre-trained NLP models, that pre-trained SSL models are over-parametrized and there exist task-oriented weights/neurons. Figure $2$ implies that these important weights only account for a small part of the pre-trained speech SSL. In fact, a large body of NLP work is dedicated to studying task-oriented weights in pre-trained models. To name a few, $\\boxed { 1 3 7 } \\boxed { 3 5 } \\boxed { 7 } \\boxed { 1 1 5 }$ measured, [7, 34, 61] leveraged, [81, 46] visualized, and [105, 36, 13] pruned out these important weights/neurons via probing and quantifying contextualized representations. Based on Observation 1, we can project that these NLP results should in general transfer to speech, see pioneering studies [9, 8, 24, 23]. However, different from them, PARP leverages important weights for UMP on the whole network structure instead of just the contextualized representations. ",
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+    {
+        "type": "text",
+        "text": "We could further hypothesize that a good pruning algorithm avoids pruning out task-specific neurons in pre-trained SSL [67, 48, 79], see Figure $8 .$ This hypothesis not only offers an explanation on why PARP is effective in high sparsity regions and cross-lingual mask transfer, it also suggests that an iterative method such as IMP is superior to OMP because IMP gradually avoids pruning out important weights in several iterations, at the cost of more compute6. Finally, we make connections to prior work that showed RP prevail [11, 19, 75, 77, 92] – under a certain threshold and setting, task-specific neurons are less likely to get “accidentally” pruned and thus accuracy is preserved even with RP. ",
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+    {
+        "type": "text",
+        "text": "5 Related Work ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Modern Speech Paradigm and ASR Pruning. As model scale [101, 6, 50, 47, 124, 90, 89, 125, 16, 121, $\\bar { \\left\\lfloor 6 8 \\right\\rfloor }$ and model pre-training [6, 127, 29, 60, 57, 63, 55, 118, 14, 58, 96, 95, 83, 86, 109] have become the two essential ingredients for obtaining SOTA performance in ASR and other speech tasks, applying and developing various forms of memory-efficient algorithms, such as network pruning, to these large-scale pre-trained models will predictably soon become an indispensable research endeavor. Early work on ASR pruning can be dated back to pruning decoding search spaces [1, 91, 100, 52, 116, 128] and HMM state space $\\mathbb { I O 3 } \\mathbb { I }$ . Since the seminal work of Yu et al. $\\mathbb { \\lVert 1 2 2 \\rVert }$ , ASR pruning has focused primarily on end-to-end network architecture: [98, 114] applied pruning and quantization to LSTM-based RNN-Transducers, $\\lVert \\overline { { 8 5 } } \\rVert$ applied knowledge distillation to Conformer-based RNN-Transducers, [104, 99, $\\textcircled { 7 0 }$ designed efficient architecture/mechanisms for LSTM, Transformer, Conformer-based ASR models, $\\pmb { \\mathbb { B 2 } }$ applied pruning to Deep Speech, [12] introduced SNR-based probabilistic pruning on LSTM-based CTC model, $\\check { \\mathbb { B } } 3 \\mathbb { I }$ proposed entropyregularizer for LSTM-based ASR model, [119, $\\textcircled { 8 7 }$ applied SVD on ASR models’ weight matrices. We emphasize that our work is the first on pruning large self-supervised pre-trained models for low-resource and multi-lingual ASR. In addition, to our knowledge, none of the prior speech pruning work demonstrated the pruned models attain superior performance than its original counterpart. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We introduce PARP, a simple and intuitive pruning method for self-supervised speech recognition. We conduct extensive experiments on pruning pre-trained wav2vec 2.0 and XLSR-53 under three low-resource settings, demonstrating (1) PARP discovers better subnetworks than baseline pruning methods while requiring a fraction of their computational cost, (2) the discovered subnetworks yields over $10 \\%$ WER reduction over the full model, (3) PARP induces minimal cross-lingual subnetwork adaptation errors, (4) PARP can discover a shared subnetwork for multiple spoken languages in one pass, and (5) PARP significantly reduces cross-task adaptation errors of pre-trained BERT/XLNet. Beyond the scope of our study, we aspire PARP as the beginning of many future endeavours on developing more efficient speech SSL models. ",
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+    {
+        "type": "text",
+        "text": "Broader Impact. The broader impact of this research work is making speech technologies more accessible in two orthogonal dimensions: (i) extending modern-day speech technology to many under-explored low-resource spoken languages, and (ii) introducing a new and flexible pruning technique to current and future speech SSL frameworks that reduces the computational costs required for adapting (finetuning) them to custom settings. We do not see its potential societal harm. ",
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+    {
+        "type": "text",
+        "text": "Limitations and Future Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We make clear of the major limitations of our work, and the full list is in Appendix 19. The basis of all the pruning methods in the study is unstructured magnitude weight pruning. Although sparsity is explicitly enforced in the models, we do not suggest that the sparse models are more memory or energy efficient than the original dense models. We do believe that our methodology and results should provide meaningful insights and be easily extended upon to more advanced unstructured or structured pruning methods. We are also curious of the possibility of finetuning or storing modern speech SSL models on local hardware devices. ",
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+    {
+        "type": "text",
+        "text": "Results on cross-lingual mask transfer on pre-trained wav2vec 2.0 in Section $\\boxed { 4 . 3 }$ is limited to ASR. We do not claim pruning masks to be transferrable across speech tasks (e.g. prune wav2vec2 for speaker ID and transfer for ASR). We provide a pilot cross-task mask transfer study on 3 speech tasks (phone recognition, speaker recognition, slot-filling) in SUPERB [120], and results is in Appendix 16. ",
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+    {
+        "type": "text",
+        "text": "We claim PARP could improve the downstream ASR performance over the full wav2vec 2.0, yet we do not claim it as a plug-and-play method into any SOTA ASR pipeline, such as $\\mathbb { L } 2 6 \\mathbb { I }$ , to get a performance boost. We provide a preliminary experiment on combining PARP and transformer-LM decoding in Appendix $\\boxed { 1 5 }$ Nonetheless, due to resource limitations and to isolate the effect of pruning, it remains upon investigations on the complete effects of speech pruning in different setups. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We thank IBM for the donation to MIT of the Satori GPU cluster, and John Cohn for maintaining the cluster. We also thank Lucy Chai, Wei-Ning Hsu, Desh Raj, Shu-wen Leo Yang, Abdelrahman Mohamedm, Erica Cooper, and anonymous reviewers for helpful suggestions and paper editing. This work is part of the low-resource language learning project funded by the MIT-IBM Waston AI Lab. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "type": "text",
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parse/train/rsRq--gsiE/rsRq--gsiE.md

@@ -0,0 +1,246 @@
+# Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction
+
+Dominik Stöger Katholische Universität Eichstätt-Ingolstadt 85072 Eichstätt, Germany Dominik.Stoeger@ku.de
+
+Mahdi Soltanolkotabi University of Southern California Los Angeles, CA 90089 soltanol@usc.edu
+
+# Abstract
+
+Recently there has been significant theoretical progress on understanding the convergence and generalization of gradient-based methods on nonconvex losses with overparameterized models. Nevertheless, many aspects of optimization and generalization and in particular the critical role of small random initialization are not fully understood. In this paper, we take a step towards demystifying this role by proving that small random initialization followed by a few iterations of gradient descent behaves akin to popular spectral methods. We also show that this implicit spectral bias from small random initialization, which is provably more prominent for overparameterized models, also puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well. Concretely, we focus on the problem of reconstructing a low-rank matrix from a few measurements via a natural nonconvex formulation. In this setting, we show that the trajectory of the gradient descent iterations from small random initialization can be approximately decomposed into three phases: (I) a spectral or alignment phase where we show that that the iterates have an implicit spectral bias akin to spectral initialization allowing us to show that at the end of this phase the column space of the iterates and the underlying low-rank matrix are sufficiently aligned, (II) a saddle avoidance/refinement phase where we show that the trajectory of the gradient iterates moves away from certain degenerate saddle points, and (III) a local refinement phase where we show that after avoiding the saddles the iterates converge quickly to the underlying low-rank matrix. Underlying our analysis are insights for the analysis of overparameterized nonconvex optimization schemes that may have implications for computational problems beyond low-rank reconstruction.
+
+# 1 Introduction
+
+Many contemporary problems in machine learning and signal estimation spanning deep learning to low-rank matrix reconstruction involve fitting nonlinear models to training data. Despite tremendous empirical progress, theoretical understanding of these problems poses two fundamental challenges. First, from an optimization perspective, fitting these models often requires solving highly nonconvex optimization problems and except for a few special cases, it is not known how to provably find globally or approximately optimal solutions. Yet simple heuristics such as running (stochastic) gradient descent from (typically) small random initialization is surprisingly effective at finding globally optimal solutions. A second generalization challenge is that many modern learning models including neural network architectures are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. It is well understood that in this overparameterized regime, these large models are highly expressive and have the capacity to (over)fit arbitrary training datasets including pure noise. Mysteriously however overparameterized models trained via simple algorithms such as (stochastic) gradient descent when initialized at random continue to predict well or generalize on yet unseen test data. In particular, it has been noted in a number of works that for many modern machine learning architectures, the scale of initialization is important for the generalization/test behavior [1, 2]. It has been noted that stronger generalization performance is typically observed for a smaller scale initialization. Indeed, small random initialization followed by (stochastic) gradient descent iterative updates is arguably the most widely used learning algorithm in modern machine learning and signal estimation.
+
+There has been a large number of exciting results aimed at demystifying both the optimization and generalization aspects over the past few years. We will elaborate on these results in detail in the supplementary, however, we would like to briefly mention the common techniques and their existing limitations. On the optimization front a large body of work has emerged on providing guarantees for nonconvex optimization which can roughly be put into two categories: (I) smart initialization+local convergence and (II) landscape analysis $^ +$ saddle escaping algorithms. Approaches in (I) focus on showing local convergence of local search techniques from carefully designed spectral initializations [3, 4, 5, 6, 7, 8, 9, 10]. Approaches in (II) focus on showing that in some cases the optimization landscape is benign in the sense that all local minima are global (no spurious local minima) and the saddle points have a direction of strict negative curvature (strict saddle) [11]. Then specialized truncation or saddle escaping algorithms such as trust region, cubic regularization [12, 13], or noisy (stochastic) gradient-based methods [14, 15, 16, 17] are deployed to provably find a global optimum. Both approaches fail to fully explain the typical behavior of local search techniques in practice. Indeed, for many nonconvex problems local search techniques or simple variants, when initialized at random, quickly converge to globally optimal solutions without getting stuck in local optima/saddles without the need for sophisticated initialization or saddle escaping heuristics. We note that while for differentiable losses eventual convergence to local minimizers is known from a random initialization [18] on problems of the form (II), these results cannot rule out exponentially slow cases in the worst-case [19]. Indeed, it has been argued that in general a more granular analysis of the trajectory of gradient descent beyond the landscape may be necessary [20]. For example, some recent advances has been made by analysing the trajectory of gradient descent using a leave-one-out analysis for the phase retrieval problem [21].
+
+Similarly, there has been a lot of exciting progress on the generalization front, especially for neural networks. Specific to generalization capabilities of gradient-based approaches these results broadly fall into two categories: (a) the first category is based on a linearization principle which characterizes the performance of nonlinear models such as neural networks by comparing it to a linearized kernel problem around the initialization (a.k.a. Neural Tangent Kernels) [22, 23, 24, 25, 26, 27, 28]. This has often been referred to as ”lazy training". (b) the second category is based on a continuous limit analysis in the limit of width going to infinity and learning rate going to zero (mean-field analysis) [29, 30, 31, 32, 33]. However, these existing analyses contain many idealized and nonrealistic assumptions (e.g. requiring large, random initialization in (a), which typically leads to worse generalization than what is observed in practice, or unrealistically large widths in (b)) and therefore cannot fully explain the success of overparameterized models or serve as a guiding principle for practitioners [34].
+
+Despite the aforementioned exciting recent theoretical progress many aspects of optimization and generalization and in particular the role of random initialization remains mysterious. This leads us to the main challenge of this paper
+
+Why is small random initialization combined with gradient descent updates so effective at finding globally optimal models that generalize well despite the nonconvex nature of the optimization landscape or model overparameterization?
+
+In this paper we wish to take a step towards addressing the above challenge by demystifying the critical role of small random initialization in gradient-based approaches. Specifically we show that
+
+Small random initialization followed by a few iterations of gradient descent behaves akin to spectral initialization.
+
+By that, we mean more precisely, that if the initialization is chosen small enough, then in the initial stage of the training, gradient descent implicitly behaves like spectral initialization techniques such as those commonly used in techniques based on the method of moments. This implicit spectral bias of gradient descent from random initialization puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well for overparameterized models. We also show that with small random initialization this implicit spectral bias phenomenon is more prominent for more overparameterized models in the sense that it materializes after fewer iterations. This intriguing phenomenon is depicted in Figure 1 in the context of a low-rank reconstruction problem. This figure clearly demonstrates that the first few iterations of gradient descent starting from a small random initialization are virtually identical to that of running power iterations (a popular algorithm to find the spectral initialization, see, e.g. [35]).
+
+![](images/529cc7c147b35826e3f6274037ac6cf0cce6109d6a6d365cdfdb021513bce9d5.jpg)  
+Figure 1: Gradient descent from small random initialization is akin to spectral initialization. The left figure depicts the empirical landscape of a low-rank matrix reconstruction problem with the two green circles depicting the two global minima and the white circle the saddle point at the origin. In this figure, we also depict the trajectory of the gradient descent iterations (magenta) together with the power method based on a popular spectral initialization technique (blue). Both gradient descent and power method use the same small initialization near the origin. We see that in the early stage, the two trajectories are almost the same. The figure on the right depicts the angle between the gradient descent (magenta)/power method (blue) iterates and a popular spectral initialization technique, denoted by $\theta _ { G D }$ and $\theta _ { P }$ respectively. This figure clearly demonstrates that for the first iterations these angles are practically the same further confirming that the initial trajectory of gradient descent and power methods are similar. See Section 5 for further detail on the experimental setup. (In this figure we have used $r = r _ { \star } = 1 .$ )
+
+Concretely we focus on the problem of low-rank matrix recovery, which appears in many different application areas such as recommendation systems, phase retrieval, and quantum tomography [36]. Here, our goal is to recover a low-rank matrix of the form $X X ^ { T }$ from a few linear measurements. We consider a natural, non-convex approach based on matrix factorization, where we minimize the loss function via gradient descent. In this paper, we show that, regardless of the amount of overparameterization used, for small random initialization vanilla gradient descent will always converge towards the low-rank solution. This holds as long as the measurement operator obeys a popular restricted isometry property [37].
+
+Our analysis consists of three phases. The first phase is the aforementioned spectral or alignment phase where we show gradient descent from small random initialization behaves akin to spectral initialization, which is a key insight of this paper. Indeed, we show that the first few gradient descent iterates can be accurately approximated by power method iterates. Next, we show that after this first spectral or alignment phase, gradient descent enters a second phase, which we refer to as saddle avoidance phase. In this phase, we show that the trajectory of the gradient iterates moves away from degenerate saddle points, while the iterates maintain almost the same effective rank as $X X ^ { T }$ . In the third phase, the local refinement phase, we show that the iterates approximately converge towards the underlying low-rank matrix $X X ^ { T }$ with a geometric rate up to a certain error floor which depends on the initialization scale. In particular, by decreasing the scale of initialization this error threshold can be made arbitrarily small. While in this paper our main focus is on low-rank matrix reconstruction, we believe that our analysis holds more generally for a variety of contemporary machine learning and signal estimation tasks including neural networks.
+
+Finally we note that while a similar setting has already been studied in [38], our analysis goes beyond it in many important ways. For example, our result holds for any amount of overparameterization and allows for arbitrarily small initialization. Maybe most importantly, we study the spectral phase phenomenon at initialization.
+
+# 2 Low-rank matrix recovery via non-convex optimization
+
+As mentioned earlier in this paper we focus on reconstructing a (possibly overparameterized) Positive Semidefinite (PSD) low rank matrix from a few measurements. In this problem, given $m$ observations of the form
+
+$$
+y _ { i } = \left. A _ { i } , X X ^ { T } \right. = \operatorname { T r } \left( A _ { i } X X ^ { T } \right) \qquad { \mathrm { ~ } } i = 1 , \ldots , m ,
+$$
+
+we wish to reconstruct the unknown matrix $X X ^ { T }$ . Here, $X \in \mathbb { R } ^ { n \times r _ { \star } }$ with $1 \leq r _ { \star } \leq n$ is a factor of the unknown matrix and $\left\{ A _ { i } \right\} _ { i = 1 } ^ { m }$ are known symmetric measurement matrices. A common approach to solving this problem is via minimizing the loss function
+
+$$
+\operatorname* { m i n } _ { \bar { U } \in \mathbb { R } ^ { n \times r } } f ( \bar { U } ) : = \operatorname* { m i n } _ { \bar { U } \in \mathbb { R } ^ { n \times r } } \frac { 1 } { 4 m } \sum _ { i = 1 } ^ { m } \left( y _ { i } - \langle A _ { i } , \bar { U } \bar { U } ^ { T } \rangle \right) ^ { 2 } ,
+$$
+
+with $r \geq r _ { \star }$ . More compactly one can rewrite the optimization problem above in the form
+
+$$
+\operatorname* { m i n } _ { \bar { U } \in \mathbb { R } ^ { n \times r } } f ( \bar { U } ) : = \operatorname* { m i n } _ { \bar { U } \in \mathbb { R } ^ { n \times r } } \frac { 1 } { 4 } \left. A \left( \bar { U } \bar { U } ^ { T } - X X ^ { T } \right) \right. _ { \ell _ { 2 } } ^ { 2 } ,
+$$
+
+where $\mathcal { A } : \mathbb { R } ^ { n \times n } \longrightarrow \mathbb { R } ^ { m }$ is the measurement operator defined by $\begin{array} { r } { [ \boldsymbol { \mathcal { A } } \left( Z \right) ] _ { i } : = \frac { 1 } { \sqrt { m } } \big \langle \boldsymbol { A } _ { i } , Z \big \rangle } \end{array}$ .
+
+In order to solve the minimization problem (2) we run gradient descent iterations starting from (often small) random initialization. More specifically,
+
+$$
+\begin{array} { r l } & { U _ { t + 1 } = U _ { t } - \mu \nabla f \left( U _ { t } \right) = U _ { t } + \mu \mathcal { A } ^ { * } \left[ y - \mathcal { A } \left( U _ { t } U _ { t } ^ { T } \right) \right] U _ { t } } \\ & { \qquad = U _ { t } + \mu \left[ \left( \mathcal { A } ^ { * } \mathcal { A } \right) \left( X X ^ { T } - U _ { t } U _ { t } ^ { T } \right) \right] U _ { t } . } \end{array}
+$$
+
+where we have set $U _ { 0 } = \alpha U$ is the initialization matrix, $A ^ { * }$ denotes the adjoint operator of $\mathcal { A }$ and $y = \left( y _ { i } \right) _ { i = 1 } ^ { m } \in \mathbb { R } ^ { m }$ = A denotes the measurement vector. Here, $U \in \mathbb { R } ^ { n \times r }$ Ais a typically random matrix =which represents the form of the initialization and $\alpha > 0$ is a scaling parameter.
+
+There are two challenges associated with analyzing such randomly initialized gradient descent updates. The first is an optimization challenge. Since $f$ is non-convex it is a priori not clear whether gradient descent converges to a global optimum or whether it gets stuck in a local minima and/or saddle. The second challenge is that of generalization. This is particularly pronounced in the overparameterized scenario where the number of parameters are larger than the number of data points i.e. $r n \geq m$ . In this case, there are infinitely many $\bar { U }$ such that $\check { f ( U ) } = 0$ , but $\| \bar { U } \bar { U } ^ { T } - X X ^ { T } \| _ { F }$ is arbitrarily large (see, e.g., [39, Proposition 1]). That is, even if gradient descent converges to a global optimum, i.e. $f \left( { \bar { U } } \right) { \bar { = } } 0$ , it is a priori not clear whether it has found the low-rank solution $X X ^ { T }$ (see also Figure 5).
+
+# 3 Main results
+
+In this section, we present our main results. Stating these results requires a couple of simple definitions.   
+The first definition concerns the measurement operator $\mathcal { A }$ .
+
+Definition 3.1 (Restricted Isometry Property (RIP)). The measurement operator $\mathcal { A } : \mathbb { R } ^ { n \times n } \longrightarrow \mathbb { R } ^ { m }$ satisfies RIP of rank $r$ with constant $\delta > 0$ , if it holds for all matrices $Z$ of rank at most $r$
+
+$$
+\left( 1 - \delta \right) \left\| Z \right\| _ { F } ^ { 2 } \leq \left\| A \left( Z \right) \right\| _ { \ell _ { 2 } } ^ { 2 } \leq \left( 1 + \delta \right) \left\| Z \right\| _ { F } ^ { 2 } .
+$$
+
+We note that for a Gaussian measurement operator $A ^ { 1 }$ , RIP of rank $r$ and constant $\delta > 0$ holds with high probability, if the number of observations satisfies $m \gtrsim n r / \delta ^ { 2 }$ [37, 40].
+
+The second definition concerns the condition number of the factor $X$ .
+
+Definition 3.2 (condition number). We denote the condition number of $X \in \mathbb { R } ^ { n \times r _ { \star } }$ by $\kappa : = { \frac { \| X \| } { \sigma _ { r , \star } ( X ) } }$ , where $\sigma _ { r _ { \star } }$ $( X )$ denotes $r _ { \star }$ -th largest singular value of $X$ .
+
+With these definitions in place we are now ready to state our main results. Due to space limitations in the main paper we focus on the case of $r \geq 2 r _ { \star }$ . With refer the reader to the supplementary for results covering all $r \geq r _ { \star }$ including two special cases: (1) the fully overparameterized case, i.e., $r = n$ along with comparisons with existing work, in this case [38], and (2) the scenario that $U$ has the same number of parameters as $X$ , i.e., $r = r _ { \star }$ .
+
+Theorem 3.3. Let $X \in \mathbb { R } ^ { n \times r _ { * } }$ and assume we have m measurements of the low rank matrix $X X ^ { T }$ of the form $y = \mathcal { A } \left( X X ^ { T } \right)$ with $\mathcal { A }$ the measurement operator. We assume $\mathcal { A }$ satisfies the restricted isometry property for all matrices of rank at most $2 r _ { \star } + 1$ with constant $\delta \le c \kappa ^ { - 4 } { r _ { \star } } ^ { - 1 / 2 }$ . To reconstruct $X X ^ { T }$ from the measurements we fit a model of the form $\bar { U } \mapsto \mathcal { A } \left( \bar { U } \bar { U } ^ { T } \right)$ with $\bar { U } \in \mathbb { R } ^ { n \times r }$ via running gradient descent iterations of the form $U _ { t + 1 } = U _ { t } - \mu \nabla f \left( U _ { t } \right)$ on the objective (2) with a step size obeying $\mu \leq c \kappa ^ { - 4 } \| X \| ^ { - 2 }$ . Here, the initialization is given by $U _ { 0 } = \alpha U$ , where $U \in \mathbb { R } ^ { n \times r }$ has i.i.d. entries distributed as $\mathcal { N } \left( 0 , 1 / \sqrt { r } \right)$ . Furthermore, we assume $r \geq 2 r _ { \star }$ and that the scale of initialization fulfills
+
+$$
+\alpha \lesssim \operatorname* { m i n } \left\{ \frac { \left( \operatorname* { m i n } \left\{ r ; n \right\} \right) ^ { 1 / 4 } } { \kappa ^ { 1 / 2 } n ^ { 3 / 4 } } \left( 2 \kappa ^ { 2 } \sqrt { \frac { n } { \operatorname* { m i n } \left\{ r ; n \right\} } } \right) ^ { - 6 \kappa ^ { 2 } } ; \frac { 1 } { \kappa ^ { 7 } n } \right\} \| X \| .
+$$
+
+Then, after
+
+$$
+\hat { t } \lesssim \frac { 1 } { \mu \sigma _ { \mathrm { m i n } } \left( \boldsymbol X \right) ^ { 2 } } \ln \left( \frac { C _ { 1 } n \kappa } { \operatorname* { m i n } \left\{ \boldsymbol r ; n \right\} } \cdot \operatorname* { m a x } \left\{ 1 ; \frac { \kappa r _ { \star } } { \operatorname* { m i n } \left\{ \boldsymbol r ; n \right\} - r _ { \star } } \right\} \cdot \frac { \| \boldsymbol X \| } { \alpha } \right)
+$$
+
+iterations we have that
+
+$$
+\frac { \| U _ { \widehat { t } } U _ { \widehat { t } } ^ { T } - X X ^ { T } \| _ { F } } { \| X \| ^ { 2 } } \lesssim \frac { n ^ { 2 } \kappa ^ { 8 1 / 1 6 } r _ { \star } ^ { 1 / 8 } } { \left( \operatorname* { m i n } \left\{ r ; n \right\} \right) ^ { 1 5 / 1 6 } } \cdot \frac { \alpha ^ { 2 1 / 1 6 } } { \| X \| ^ { 2 1 / 1 6 } } ,
+$$
+
+holds with probability at least $1 - C e ^ { - \tilde { c } r }$ . Here, $c , \tilde { c } , C , C _ { 1 } > 0$ are fixed numerical constants.
+
+Note that the test error $\| U _ { \hat { t } } U _ { \hat { t } _ { \cdot } } ^ { T } - X X ^ { T } \| _ { F } ^ { 2 }$ can be made arbitrarily small by choosing the scale of initialization $\alpha$ small enough. In particular, the dependence of the test error on $\alpha$ is polynomial and the dependence of the number of iterations on $\alpha$ is logarithmic, which means that reducing the test error by scaling down $\alpha$ introduces only modest additional computational cost. Hence, as long as the rank at most $2 r _ { \star } + 1$ RIP with constant $\delta \le c \kappa ^ { - 4 } { r _ { \star } } ^ { - 1 / 2 }$ holds, gradient descent converges to a point in the proximity of the low-rank solution, whenever the initialization is chosen small enough regardless of the choice of $r$ . This holds even when the model is overparameterized i.e. $r n \gg m$ and the optimization problem has many global optima many of which do not obey $U U ^ { T } \approx X X ^ { T }$ . This result thus further demonstrates that when initialized with a small random initialization gradient descent has an implicit bias towards solutions of low-rank or small nuclear norm. This is in sharp contrast to Neural Tangent Kernel (NTK)-based theory for low-rank matrix recovery (see [23, Section 4.2]) which will not approximately recover the ground truth matrix $X X ^ { T }$ due to the larger scale of initialization required when using that technique.
+
+As discussed in Section 2, the restricted isometry property holds with high probability for a sample complexity $m \gtrsim n r _ { \star } ^ { 2 } \kappa ^ { 8 }$ for Gaussian measurement matrices. Up to constants, this sample complexity is optimal in $n$ , while it is sub-optimal in $r _ { \star }$ and $\kappa$ compared to approaches based on nuclear-norm minimization (see, e.g., [37]). While there is numerical evidence that the true scaling of $m$ in $r _ { \star }$ should also be linear in the non-convex case [41], we note that the optimal dependence of the sample complexity on $r _ { \star }$ is a major open problem in the field, as the sample complexities in all theoretical results for non-convex approaches in the literature scale at least quadratically in $r _ { \star }$ .
+
+Interpretation: Recall from Section 1 that our convergence analysis can be divided into three phases: the spectral phase, the saddle avoidance phase, and the local refinement phase. As it will become clear from the proofs in the supplementary when $r \geq 2 r _ { \star }$ the bound on the number of iterations can
+
+be decomposed as follows
+
+$$
+\begin{array} { r l } { \hat { t } \lesssim \frac { 1 } { \mu \sigma _ { \operatorname* { m i n } } ( X ) ^ { 2 } } \Bigg [ \ln ( 2 \kappa ^ { 2 } \sqrt { \frac { n } { \operatorname* { m i n } \{ r ; n \} } } ) + } &  \ : \underbrace { \ln ( \frac { \sigma _ { \operatorname* { m i n } } ( X ) } { \alpha } ) } _ { \displaystyle \mu \sigma _ { \operatorname* { m i n } } ( X ) ; \frac { 1 } { \operatorname* { m i n } \{ r ; n \} - r _ { \star } } \} \frac { \| X \| } { \alpha } \Bigg ) \Bigg ] . } \end{array}
+$$
+
+Phase III: local refinement phase
+
+First, we note that the duration of all three phases scales inversely with $\sigma _ { \mathrm { m i n } } \left( X \right) ^ { 2 }$ . This is due to the fact that in all three phases the dynamics associated the smallest singular value of $X$ is the slowest one and hence needs the most time to complete.
+
+In the spectral phase, the eigenvectors corresponding to the leading $r _ { \star }$ eigenvalues of $U _ { t } U _ { t } ^ { T }$ become aligned with the eigenvectors corresponding to the leading $r _ { \star }$ eigenvalues of $\mathcal { A } ^ { \ast } \mathcal { A } \left( X X ^ { T } \right)$ . We observe in (6) that in the spectral phase increasing $r$ , i.e. the amount of parameters, decreases the number of iterations in this phase. As we will explain in the supplementary, the reason is that increasing $r$ decreases the angle between the column space of the initialization $U _ { 0 }$ and the span of the eigenvectors corresponding to the leading $r _ { \star }$ eigenvalues of $\mathcal { A } ^ { \ast } \mathcal { A } \left( X X ^ { T } \right)$ used in spectral initialization. As a consequence, gradient descent needs fewer iterations to align these two subspaces.
+
+In the saddle avoidance phase (Phase II), $\sigma _ { r _ { \star } } \left( U _ { t } \right)$ , the $r _ { \star }$ th largest singular value of $U _ { t }$ , grows geometrically until it is on the order of $\sigma _ { \mathrm { m i n } } \left( X \right)$ . Hence, this duration depends on the ratio between the $\sigma _ { \mathrm { m i n } } \left( X \right)$ and the the scale of initialization $\alpha$ . This is clearly reflected in the upper bound on the number of needed iterations in equation (6).
+
+In Phase III, the local refinement phase, the matrix $U _ { t } U _ { t } ^ { T }$ converges towards $X X ^ { T }$ . In particular, at iteration $\hat { t }$ the test error obeys (5). We observe that a smaller $\alpha$ allows for a smaller test error in (5) but per (6) this higher accuracy is achieved with a modest increase in the required iterations.
+
+# 4 A glimpse of our analysis
+
+In our proofs, we show that the trajectory of the gradient descent iterations can be approximately decomposed into three phases: (I) a spectral or alignment phase where we show that gradient descent from random initialization behaves akin to spectral initialization allowing us to show that at the end of this phase the column spaces of the iterates $U _ { t }$ and the ground truth matrix $X$ are sufficiently aligned, (II) a saddle avoidance phase, where we show that the trajectory of the gradient iterates move away from certain degenerate saddle points , and (III) a refinement phase, where the product of the gradient descent iterates $U _ { t } U _ { t } ^ { T }$ converges quickly to the underlying low-rank matrix $X X ^ { T }$ . The latter result holds up to a small error that is commensurate with the scale of the initialization and tends to zero as the scale of the initialization goes to zero.
+
+To formalize the above, we use $L$ and $L _ { t }$ to denote the subspaces spanned by the eigenvectors corresponding to the $r _ { \star }$ largest eigenvalues of the matrix $\mathcal { A } ^ { \ast } \bar { \mathcal { A } } \left( X X ^ { \bar { T } } \right)$ , and $U _ { t } U _ { t } ^ { T }$ , respectively. Moreover, for a subspace $L$ of dimension $r _ { \star }$ we use $V _ { L } \in \mathbb { R } ^ { n \times r _ { \star } }$ to denote an orthonormal matrix whose columns span the subspace $L$ . Note that $L$ is the subspace, which is obtained by commonly used spectral methods. Using this notation, Figure 2a depicts the three phases described above.
+
+In the spectral phase we will prove that we can approximate the iterate $U _ { t }$ by
+
+$$
+U _ { t } \approx \underbrace { \big ( \mathbf { I d } + \mu \mathbf { \mathcal { A } } ^ { * } \mathbf { \mathcal { A } } \left( X X ^ { T } \right) \big ) } _ { = : Z _ { t } } ^ { t } U _ { 0 } = Z _ { 1 } ^ { t } U _ { 0 } : = \tilde { U } _ { t } .
+$$
+
+We note that the matrix $Z _ { 1 } \ = \ \mathrm { I d } + \mu \mathcal { A } ^ { * } \mathcal { A } \left( X X ^ { T } \right)$ is the basis for the commonly used spectral initialization, where typically a factorization of the rank $r _ { * }$ approximation of this matrix is used as the initialization [6, 5, 42]. Therefore, the approximation (7) suggests that gradient descent iterates modulo the normalization are akin to running power method on $Z _ { 1 }$ . Hence, we expect that at the end of the spectral phase the subspace $L _ { t }$ to be closely aligned with the subspace $L$ , i.e. the subspace obtained by commonly used spectral initialization techniques. In particular, this also implies that $L _ { t }$ is also aligned with the subspace $X$ . Figure 2b clearly illustrates that the first few iterations of gradient descent behave essentially identical to the power method, confirming our intuition.
+
+The description of the second and third phase is more elaborate and technical in nature and we defer to the supplementary for a more detailed and intuitive explanation. However, to give a brief description, in these two phases we will decompose the iterates $U _ { t }$ into the sum of two matrices, a "signal" matrix of rank $r _ { \star }$ and a "noise" matrix of rank at most $r - r _ { \star }$ . In Phase (II) we will prove that the smallest singular value of the signal term, which is approximately the same as $\sigma _ { r _ { \star } }$ $\left( U _ { t } \right)$ grows, whereas the spectral norm of the noise matrix grows at a much slower rate. We will also show that in this phase the columns of the signal term stay approximately aligned with the span of the matrix $X$ . As soon as the smallest singular value of the signal term of $U _ { t }$ is approximately at the same order than the smallest singular value of $X$ we enter Phase (III). In that Phase we provide a local convergence argument, which shows that the signal term of $U _ { t }$ converges towards $X$ (up to a rotation), whereas the noise term stays small.
+
+![](images/1bd5cd3f84f3b10d56dfbc66dd9d4cf467ad43f5bf6f2cea7b25b7f27b08514f.jpg)  
+Figure 2: (a) Depiction of the three phases of convergence. This figure demonstrates that the convergence analysis can be divided into three phases: (I) spectral/alignment phase; (II) saddle avoidance phase and (III) the refinement phase. We see that in the first phase the first $r _ { \star }$ eigenvectors of $U _ { t } U _ { t } ^ { T }$ rapidly learn the subspace corresponding to the first $r _ { \star }$ eigenvectors of $\mathcal { A } ^ { \ast } \mathcal { A } \left( X X ^ { T } \right)$ , i.e. the angle $\| V _ { L ^ { \bot } } ^ { T } V _ { L _ { t } } \|$ becomes small. The $r _ { \star }$ th largest singular value of $U _ { t }$ is still small in this phase and the (normalized) test error $\| U _ { t } U _ { t } - X X ^ { T } \| _ { F } ^ { 2 } / \| X X ^ { T } \| _ { F } ^ { 2 }$ has not decreased yet. In Phase (II), however, we see that $\sigma _ { r _ { \star } }$ $\left( U _ { t } \right)$ is growing, whereas the loss begins to decrease in this phase and the subspaces stay aligned. In Phase (III) we see that the test error is converging towards 0 rapidly, meaning that $U _ { t } U _ { t } ^ { T }$ converges to $X X ^ { T }$ . Consequently, $\sigma _ { r _ { \star } } \left( U _ { t } \right) / \sigma _ { r _ { \star } } \left( X \right)$ converges to 1 (red curve). We also see that in this phase the angle $\| V _ { L ^ { \bot } } ^ { T } V _ { L _ { t } } \|$ grows again, until it reaches a certain threshold. This is because in this phase the top $r _ { \star }$ eigenvalues of $U _ { t } U _ { t } ^ { T }$ become aligned with the eigenvectors of $X X ^ { T }$ . (b) Depiction of the spectral alignment phase: in the first few iterations, gradient descent with small initialization behaves like a power method. Denote by ${ \tilde { L } } _ { t }$ the subspace spanned by the eigenvectors corresponding to the $r _ { \star }$ largest eigenvalues of the matrix $\widetilde { U _ { t } } \widetilde { U _ { t } } ^ { T }$ . Analogously as before denote by $V _ { \tilde { L } _ { t } }$ an orthonormal matrix, whose columns span the subspace ${ \tilde { L } } _ { t }$ . In this figure, we observe that in the first iterations $U _ { t }$ and $\widetilde { U } _ { t }$ learn the subspace $L$ at almost exactly the same rate.
+
+# 5 Numerical experiments
+
+In this section, we perform several numerical experiments to corroborate our theoretical results.
+
+Experimental setup. For the experiments we set the ground truth matrix $X \in \mathbb { R } ^ { n \times r _ { \star } }$ to be a random orthogonal matrix with $n = 2 0 0$ and $r _ { \star } = 5$ . Moreover, we use $m = 1 0 n { r _ { \star } } = 5 0 n$ random Gaussian measurements. The initialization $U$ is chosen as in Theorem 3.3 and we use a step size of $\mu = 1 / 4$ which is consistent with these theorems. We note that while all experimental depictions are based on a single trial, in line with the NeurIPS guidelines we have drawn these curves multiple times (not depicted) and the behavior of the plots do not change.
+
+Depiction of the three phases and the role of overparameterization. In our first experiment, we want to examine how increasing the number of parameters via increasing the number of the columns $r$ of the matrix $U _ { t } \in \mathbb { R } ^ { n \times r }$ , affects the spectral phase. To this aim we set the scale of initialization to $\alpha = 1 / \left( 7 0 n ^ { 2 } \right)$ . Let $L$ denote the subspace spanned by the eigenvectors corresponding to the leading $r _ { \star }$ singular values of $\mathcal { A } ^ { \ast } \mathcal { A } ( X X ^ { T } )$ and $L _ { t }$ denotes the subspace spanned by the left-singular vectors corresponding to the largest $r _ { \star }$ singular values of $U _ { t }$ .
+
+![](images/665e23a4f08b841b53ae6d28270bed70053f954399ac8df02b5f30441b348f05.jpg)  
+Figure 3: Impact of different levels of overparameterization on (a) the angle $\| V _ { L ^ { \bot } } ^ { T } V _ { L _ { t } } \|$ and (b) the $r _ { \star }$ th largest singular value, (c) the trajectory of the (normalized) test error $\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \| _ { F } / \| X X ^ { T } \| _ { F }$ .
+
+Spectral phase and alignment under different levels of overparameterization. First, we examine how the angle between these two subspaces (i.e. $\| V _ { L ^ { \perp } } ^ { T } V _ { L _ { t } } \| ,$ ) changes in the first few iterations. We depict the results for different $r$ in Figure 3a. We see that in the first few iterations, i.e. in the spectral phase, this angle converges towards zero. This confirms the main conclusion of this paper that the first few iterations of gradient descent from small random initialization indeed behaves akin to running power method for spectral initialization. This experiment also shows that changing the number of columns $r$ of $U _ { t }$ has an interesting effect on the spectral phase. In particular, increasing $r$ allows the gradient descent algorithm to learn the subspace $L$ with fewer iterations, i.e. $\| V _ { L ^ { \bot } } ^ { T } V _ { L _ { t } } \|$ becomes small with fewer iterations. This is in accordance with our theory for $r _ { \star } \le r \le n$ (see, for example, the first summand on the right-hand side of equation (6)), where we show that more overparameterization allows gradient descent to leave the spectral phase earlier. Interestingly, this improvement continues to hold even when increasing $r$ beyond $n$ allowing for even faster convergence of $\| V _ { L ^ { \bot } } ^ { T } V _ { L _ { t } } \|$ . This holds even though in this case the rank of $U _ { 0 }$ is still not larger than $n$ . One potential explanation for this phenomenon might be that for such a choice of $r$ the matrix $U _ { 0 }$ is better conditioned.
+
+Growth of $\sigma _ { r _ { \star } } \left( U _ { t } \right)$ and saddle avoidance. In Figure 3b we depict how $\sigma _ { r _ { \star } } \left( U _ { t } \right)$ grows during the training for different choices of $r$ . We see that the curves look similar, although for smaller $r$ the growth phase sets in at a slightly later time. This is due to the fact that for smaller $r$ , as we have seen in Figure 3a, Phase I, the spectral phase takes longer to complete.
+
+Evolution of the test error and the refinement phase. Similarly, in Figure 3c we depict how the (normalized) test error $\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \| _ { F } / \| X X ^ { T } \| _ { F }$ evolves during the training for different choices of $r$ . We observe that for smaller $r$ the third phase sets in slightly later. Again, this is due to the fact that for smaller $r$ the spectral phase takes slightly longer to complete (see inequality (6)).
+
+Test error under different scales of initialization. In the next experiment, we focus on understanding how the scale of initialization $\alpha$ affects the generalization error $\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \| _ { F } ^ { 2 }$
+
+![](images/f37264bb67eb12df04f997c4ce59e16a703a16cebf79bcfc5db995012edcdb80.jpg)  
+Figure 4: Relative test erro r ∥UtUTt −XXT ∥FT for different scales of initialization α .
+
+![](images/064c23755d50858b141e0a12e3b8fb4f590d169b4eaadbef14b71d65606dd375.jpg)  
+Figure 5: Change of test error $\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \| _ { F } ^ { 2 }$ and train error $f \left( U _ { t } \right)$ for (a) small and (b) large $\alpha$ during training.
+
+For that, we set $r = 1 8 0$ and run gradient descent with for different choices of $\alpha$ . We stop as soon as the training error becomes small $\dot { ( } f \left( U _ { t } \right) \le 0 . 5 \cdot 1 0 ^ { - 9 } )$ . We depict the results in Figure 4. We see that the test error decreases as $\alpha$ decreases. In particular, this figure indicates that the test error depends polynomially on the scale of initialization $\alpha$ . This is in line with our theory, where we also show that the test error decreases at least with the rate $\alpha ^ { 2 1 / 1 6 }$ (see inequality (5) in Theorem 3.3).
+
+Change of test and train error during training. In the next experiment, we set $r = 1 8 0$ and examine how the test error $\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \| _ { F } ^ { 2 }$ and the train error $f \left( U _ { t } \right)$ changes throughout training and, in particular, how this depends on the scale of initialization. To this aim, we run gradient descent with $\mathrm { \dot { 4 } \cdot 1 0 ^ { 5 } }$ iterations. We see that for a small scale of initialization, $\alpha = 1 0 ^ { - 3 }$ , which is the scenario studied in this paper, both test error and train error decrease throughout training.
+
+We observe that in the beginning, as described our theory, both test and train error decrease rapidly. After that the decrease of both test and train error slows down significantly. Moreover, the train error converges towards zero, in contrast to the test error. One reason for the slow convergence in this phase might be that $U _ { t }$ is ill-conditioned in the sense that $\sigma _ { r _ { \star } }$ $( U _ { t } W _ { t } )$ is much larger than $\Vert U _ { t } W _ { t , \bot } \Vert$ ⋆It is an interesting future research direction to extend our theory to this part of the training.
+
+For large scale of initialization $\alpha = 0 . 5$ , we observe a very different behaviour. We see that the train error converges with linear rate until machine precision is reached. However, the test error barely changes throughout the training. This scale of initialization corresponds to the lazy training regime [34], where the parameters stay close to the initialization during the training. We depict the results in Figure 5.
+
+Number of iterations until convergence: In the last experiment, we set $\alpha = 1 0 ^ { - 3 }$ and examine how many iterations are needed until the test error $\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \| _ { F } ^ { 2 }$ falls below a certain threshold of $1 0 ^ { - 4 }$ for different values of $r$ obeying $5 \leq r \leq 3 0$ . For each choice of $r$ we run the experiment ten times and then average the number of iterations for each choice of $r$ . The results are depicted in Figure 6. We observe that increasing the number of columns $r$ from 5 to 10, i.e., a small amount of overparameterization, decreases the number of iterations needed. After that the number of iterations needed stays roughly constant. This observation is in line with Figure 3, where we have seen that overparameterization leads to fast decrease of the test error in the spectral phase (with diminishing speedup as $r$ becomes larger and larger) without affecting the other two phases.
+
+![](images/708720b79c80fa2eb96e1f8e6541f2b603ed1c1fcec3c248b501d23e83e1bffe.jpg)  
+Figure 6: Number of iterations required for the test error to fall below $1 0 ^ { - 4 }$ for different levels of overparameterization.
+
+# 6 Conclusion and Broader Impact
+
+In this paper we focused on demystifying the role of initialization when training overparameterized models by showing that small random initialization followed by a few iterations of gradient descent behaves akin to popular spectral methods. We also show that this implicit spectral bias from small random initialization, which is provably more prominent for overparameterized models, also puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well.
+
+We think that our results give rise to a number of interesting future research directions. For example, one could extend our results to scenarios where the measurement matrices are more structured such as in matrix completion [43] or in blind deconvolution [44]. Moreover, while our main results, e.g. Theorem 3.3 do require early stopping, our simulations (e.g. Figure 5a) indicate that early stopping is not needed. It would be interesting to examine whether we can remove the early stopping requirement. It is also an interesting future avenue to examine whether the quadratic dependence of the sample complexity $m$ on $r _ { \star }$ in our results is really needed.
+
+Moreover, while in this paper our main focus was on low-rank matrix reconstruction, we believe that our analysis holds more generally for a variety of contemporary overparameterized machine learning and signal estimation tasks including neural network training. This is a tantalizing future research direction.
+
+Despite being theoretical/foundational in nature our results have potential for broader practical impact. In particular, low rank reconstruction problems are an important component of many recommender engines and our insights may guide better algorithm and systems designs for such engines. More broadly, training overparameterized models using stochastic GD starting from small random initialization is the work-horse of modern learning ncluding deep learning and our insights may in the long term help enable more efficient/reliable training with a smaller carbon footprint and improved test accuracy. As with other technologies such insights may potentially also be used nefariously.
+
+# Acknowledgments and Disclosure of Funding
+
+M.S. is supported by the Packard Fellowship in Science and Engineering, a Sloan Research Fellowship in Mathematics, an NSF-CAREER under award #1846369, the Air Force Office of Scientific Research Young Investigator Program (AFOSR-YIP) under award #FA9550-18-1-0078, DARPA Learning with Less Labels (LwLL) and FastNICS programs, and NSF-CIF awards #1813877 and #2008443.
+
+References   
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+        "text": "Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction ",
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+        "text": "Dominik Stöger Katholische Universität Eichstätt-Ingolstadt 85072 Eichstätt, Germany Dominik.Stoeger@ku.de ",
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+        "text": "Mahdi Soltanolkotabi University of Southern California Los Angeles, CA 90089 soltanol@usc.edu ",
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+        "text": "Abstract ",
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+        "text": "Recently there has been significant theoretical progress on understanding the convergence and generalization of gradient-based methods on nonconvex losses with overparameterized models. Nevertheless, many aspects of optimization and generalization and in particular the critical role of small random initialization are not fully understood. In this paper, we take a step towards demystifying this role by proving that small random initialization followed by a few iterations of gradient descent behaves akin to popular spectral methods. We also show that this implicit spectral bias from small random initialization, which is provably more prominent for overparameterized models, also puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well. Concretely, we focus on the problem of reconstructing a low-rank matrix from a few measurements via a natural nonconvex formulation. In this setting, we show that the trajectory of the gradient descent iterations from small random initialization can be approximately decomposed into three phases: (I) a spectral or alignment phase where we show that that the iterates have an implicit spectral bias akin to spectral initialization allowing us to show that at the end of this phase the column space of the iterates and the underlying low-rank matrix are sufficiently aligned, (II) a saddle avoidance/refinement phase where we show that the trajectory of the gradient iterates moves away from certain degenerate saddle points, and (III) a local refinement phase where we show that after avoiding the saddles the iterates converge quickly to the underlying low-rank matrix. Underlying our analysis are insights for the analysis of overparameterized nonconvex optimization schemes that may have implications for computational problems beyond low-rank reconstruction. ",
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+        "text": "1 Introduction ",
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+        "text": "Many contemporary problems in machine learning and signal estimation spanning deep learning to low-rank matrix reconstruction involve fitting nonlinear models to training data. Despite tremendous empirical progress, theoretical understanding of these problems poses two fundamental challenges. First, from an optimization perspective, fitting these models often requires solving highly nonconvex optimization problems and except for a few special cases, it is not known how to provably find globally or approximately optimal solutions. Yet simple heuristics such as running (stochastic) gradient descent from (typically) small random initialization is surprisingly effective at finding globally optimal solutions. A second generalization challenge is that many modern learning models including neural network architectures are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. It is well understood that in this overparameterized regime, these large models are highly expressive and have the capacity to (over)fit arbitrary training datasets including pure noise. Mysteriously however overparameterized models trained via simple algorithms such as (stochastic) gradient descent when initialized at random continue to predict well or generalize on yet unseen test data. In particular, it has been noted in a number of works that for many modern machine learning architectures, the scale of initialization is important for the generalization/test behavior [1, 2]. It has been noted that stronger generalization performance is typically observed for a smaller scale initialization. Indeed, small random initialization followed by (stochastic) gradient descent iterative updates is arguably the most widely used learning algorithm in modern machine learning and signal estimation. ",
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+        "text": "There has been a large number of exciting results aimed at demystifying both the optimization and generalization aspects over the past few years. We will elaborate on these results in detail in the supplementary, however, we would like to briefly mention the common techniques and their existing limitations. On the optimization front a large body of work has emerged on providing guarantees for nonconvex optimization which can roughly be put into two categories: (I) smart initialization+local convergence and (II) landscape analysis $^ +$ saddle escaping algorithms. Approaches in (I) focus on showing local convergence of local search techniques from carefully designed spectral initializations [3, 4, 5, 6, 7, 8, 9, 10]. Approaches in (II) focus on showing that in some cases the optimization landscape is benign in the sense that all local minima are global (no spurious local minima) and the saddle points have a direction of strict negative curvature (strict saddle) [11]. Then specialized truncation or saddle escaping algorithms such as trust region, cubic regularization [12, 13], or noisy (stochastic) gradient-based methods [14, 15, 16, 17] are deployed to provably find a global optimum. Both approaches fail to fully explain the typical behavior of local search techniques in practice. Indeed, for many nonconvex problems local search techniques or simple variants, when initialized at random, quickly converge to globally optimal solutions without getting stuck in local optima/saddles without the need for sophisticated initialization or saddle escaping heuristics. We note that while for differentiable losses eventual convergence to local minimizers is known from a random initialization [18] on problems of the form (II), these results cannot rule out exponentially slow cases in the worst-case [19]. Indeed, it has been argued that in general a more granular analysis of the trajectory of gradient descent beyond the landscape may be necessary [20]. For example, some recent advances has been made by analysing the trajectory of gradient descent using a leave-one-out analysis for the phase retrieval problem [21]. ",
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+        "text": "Similarly, there has been a lot of exciting progress on the generalization front, especially for neural networks. Specific to generalization capabilities of gradient-based approaches these results broadly fall into two categories: (a) the first category is based on a linearization principle which characterizes the performance of nonlinear models such as neural networks by comparing it to a linearized kernel problem around the initialization (a.k.a. Neural Tangent Kernels) [22, 23, 24, 25, 26, 27, 28]. This has often been referred to as ”lazy training\". (b) the second category is based on a continuous limit analysis in the limit of width going to infinity and learning rate going to zero (mean-field analysis) [29, 30, 31, 32, 33]. However, these existing analyses contain many idealized and nonrealistic assumptions (e.g. requiring large, random initialization in (a), which typically leads to worse generalization than what is observed in practice, or unrealistically large widths in (b)) and therefore cannot fully explain the success of overparameterized models or serve as a guiding principle for practitioners [34]. ",
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+        "type": "text",
+        "text": "Despite the aforementioned exciting recent theoretical progress many aspects of optimization and generalization and in particular the role of random initialization remains mysterious. This leads us to the main challenge of this paper ",
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+        "type": "text",
+        "text": "Why is small random initialization combined with gradient descent updates so effective at finding globally optimal models that generalize well despite the nonconvex nature of the optimization landscape or model overparameterization? ",
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+        "text": "In this paper we wish to take a step towards addressing the above challenge by demystifying the critical role of small random initialization in gradient-based approaches. Specifically we show that ",
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+        "type": "text",
+        "text": "Small random initialization followed by a few iterations of gradient descent behaves akin to spectral initialization. ",
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+        "type": "text",
+        "text": "By that, we mean more precisely, that if the initialization is chosen small enough, then in the initial stage of the training, gradient descent implicitly behaves like spectral initialization techniques such as those commonly used in techniques based on the method of moments. This implicit spectral bias of gradient descent from random initialization puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well for overparameterized models. We also show that with small random initialization this implicit spectral bias phenomenon is more prominent for more overparameterized models in the sense that it materializes after fewer iterations. This intriguing phenomenon is depicted in Figure 1 in the context of a low-rank reconstruction problem. This figure clearly demonstrates that the first few iterations of gradient descent starting from a small random initialization are virtually identical to that of running power iterations (a popular algorithm to find the spectral initialization, see, e.g. [35]). ",
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+            "Figure 1: Gradient descent from small random initialization is akin to spectral initialization. The left figure depicts the empirical landscape of a low-rank matrix reconstruction problem with the two green circles depicting the two global minima and the white circle the saddle point at the origin. In this figure, we also depict the trajectory of the gradient descent iterations (magenta) together with the power method based on a popular spectral initialization technique (blue). Both gradient descent and power method use the same small initialization near the origin. We see that in the early stage, the two trajectories are almost the same. The figure on the right depicts the angle between the gradient descent (magenta)/power method (blue) iterates and a popular spectral initialization technique, denoted by $\\theta _ { G D }$ and $\\theta _ { P }$ respectively. This figure clearly demonstrates that for the first iterations these angles are practically the same further confirming that the initial trajectory of gradient descent and power methods are similar. See Section 5 for further detail on the experimental setup. (In this figure we have used $r = r _ { \\star } = 1 .$ ) "
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+    {
+        "type": "text",
+        "text": "Concretely we focus on the problem of low-rank matrix recovery, which appears in many different application areas such as recommendation systems, phase retrieval, and quantum tomography [36]. Here, our goal is to recover a low-rank matrix of the form $X X ^ { T }$ from a few linear measurements. We consider a natural, non-convex approach based on matrix factorization, where we minimize the loss function via gradient descent. In this paper, we show that, regardless of the amount of overparameterization used, for small random initialization vanilla gradient descent will always converge towards the low-rank solution. This holds as long as the measurement operator obeys a popular restricted isometry property [37]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our analysis consists of three phases. The first phase is the aforementioned spectral or alignment phase where we show gradient descent from small random initialization behaves akin to spectral initialization, which is a key insight of this paper. Indeed, we show that the first few gradient descent iterates can be accurately approximated by power method iterates. Next, we show that after this first spectral or alignment phase, gradient descent enters a second phase, which we refer to as saddle avoidance phase. In this phase, we show that the trajectory of the gradient iterates moves away from degenerate saddle points, while the iterates maintain almost the same effective rank as $X X ^ { T }$ . In the third phase, the local refinement phase, we show that the iterates approximately converge towards the underlying low-rank matrix $X X ^ { T }$ with a geometric rate up to a certain error floor which depends on the initialization scale. In particular, by decreasing the scale of initialization this error threshold can be made arbitrarily small. While in this paper our main focus is on low-rank matrix reconstruction, we believe that our analysis holds more generally for a variety of contemporary machine learning and signal estimation tasks including neural networks. ",
+        "bbox": [
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+        "type": "text",
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+        "type": "text",
+        "text": "Finally we note that while a similar setting has already been studied in [38], our analysis goes beyond it in many important ways. For example, our result holds for any amount of overparameterization and allows for arbitrarily small initialization. Maybe most importantly, we study the spectral phase phenomenon at initialization. ",
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+        "type": "text",
+        "text": "2 Low-rank matrix recovery via non-convex optimization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As mentioned earlier in this paper we focus on reconstructing a (possibly overparameterized) Positive Semidefinite (PSD) low rank matrix from a few measurements. In this problem, given $m$ observations of the form ",
+        "bbox": [
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+        "img_path": "images/17c5154753add6a12af58954611ee8af9e198530eb03c2ac9baab2ddfc5c65c4.jpg",
+        "text": "$$\ny _ { i } = \\left. A _ { i } , X X ^ { T } \\right. = \\operatorname { T r } \\left( A _ { i } X X ^ { T } \\right) \\qquad { \\mathrm { ~ } } i = 1 , \\ldots , m ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "we wish to reconstruct the unknown matrix $X X ^ { T }$ . Here, $X \\in \\mathbb { R } ^ { n \\times r _ { \\star } }$ with $1 \\leq r _ { \\star } \\leq n$ is a factor of the unknown matrix and $\\left\\{ A _ { i } \\right\\} _ { i = 1 } ^ { m }$ are known symmetric measurement matrices. A common approach to solving this problem is via minimizing the loss function ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/ddc90aa63d2d53676d4778f3b6fb06592c1d61b6cbc5a010ccf4f32051ac0010.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\bar { U } \\in \\mathbb { R } ^ { n \\times r } } f ( \\bar { U } ) : = \\operatorname* { m i n } _ { \\bar { U } \\in \\mathbb { R } ^ { n \\times r } } \\frac { 1 } { 4 m } \\sum _ { i = 1 } ^ { m } \\left( y _ { i } - \\langle A _ { i } , \\bar { U } \\bar { U } ^ { T } \\rangle \\right) ^ { 2 } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "with $r \\geq r _ { \\star }$ . More compactly one can rewrite the optimization problem above in the form ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/4c5da8d1b41552068c101dcc19f1dd19603b7a22028caa88ab2d7e7beeaa52fb.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\bar { U } \\in \\mathbb { R } ^ { n \\times r } } f ( \\bar { U } ) : = \\operatorname* { m i n } _ { \\bar { U } \\in \\mathbb { R } ^ { n \\times r } } \\frac { 1 } { 4 } \\left. A \\left( \\bar { U } \\bar { U } ^ { T } - X X ^ { T } \\right) \\right. _ { \\ell _ { 2 } } ^ { 2 } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { A } : \\mathbb { R } ^ { n \\times n } \\longrightarrow \\mathbb { R } ^ { m }$ is the measurement operator defined by $\\begin{array} { r } { [ \\boldsymbol { \\mathcal { A } } \\left( Z \\right) ] _ { i } : = \\frac { 1 } { \\sqrt { m } } \\big \\langle \\boldsymbol { A } _ { i } , Z \\big \\rangle } \\end{array}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to solve the minimization problem (2) we run gradient descent iterations starting from (often small) random initialization. More specifically, ",
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+        "img_path": "images/5212d41b3fe1c490fdb557b77937af01bb0bfbd1876c21f70af82f85e2f5db8c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { U _ { t + 1 } = U _ { t } - \\mu \\nabla f \\left( U _ { t } \\right) = U _ { t } + \\mu \\mathcal { A } ^ { * } \\left[ y - \\mathcal { A } \\left( U _ { t } U _ { t } ^ { T } \\right) \\right] U _ { t } } \\\\ & { \\qquad = U _ { t } + \\mu \\left[ \\left( \\mathcal { A } ^ { * } \\mathcal { A } \\right) \\left( X X ^ { T } - U _ { t } U _ { t } ^ { T } \\right) \\right] U _ { t } . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where we have set $U _ { 0 } = \\alpha U$ is the initialization matrix, $A ^ { * }$ denotes the adjoint operator of $\\mathcal { A }$ and $y = \\left( y _ { i } \\right) _ { i = 1 } ^ { m } \\in \\mathbb { R } ^ { m }$ = A denotes the measurement vector. Here, $U \\in \\mathbb { R } ^ { n \\times r }$ Ais a typically random matrix =which represents the form of the initialization and $\\alpha > 0$ is a scaling parameter. ",
+        "bbox": [
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+        "type": "text",
+        "text": "There are two challenges associated with analyzing such randomly initialized gradient descent updates. The first is an optimization challenge. Since $f$ is non-convex it is a priori not clear whether gradient descent converges to a global optimum or whether it gets stuck in a local minima and/or saddle. The second challenge is that of generalization. This is particularly pronounced in the overparameterized scenario where the number of parameters are larger than the number of data points i.e. $r n \\geq m$ . In this case, there are infinitely many $\\bar { U }$ such that $\\check { f ( U ) } = 0$ , but $\\| \\bar { U } \\bar { U } ^ { T } - X X ^ { T } \\| _ { F }$ is arbitrarily large (see, e.g., [39, Proposition 1]). That is, even if gradient descent converges to a global optimum, i.e. $f \\left( { \\bar { U } } \\right) { \\bar { = } } 0$ , it is a priori not clear whether it has found the low-rank solution $X X ^ { T }$ (see also Figure 5). ",
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+        "type": "text",
+        "text": "3 Main results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we present our main results. Stating these results requires a couple of simple definitions.   \nThe first definition concerns the measurement operator $\\mathcal { A }$ . ",
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+        "text": "Definition 3.1 (Restricted Isometry Property (RIP)). The measurement operator $\\mathcal { A } : \\mathbb { R } ^ { n \\times n } \\longrightarrow \\mathbb { R } ^ { m }$ satisfies RIP of rank $r$ with constant $\\delta > 0$ , if it holds for all matrices $Z$ of rank at most $r$ ",
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+        "text": "$$\n\\left( 1 - \\delta \\right) \\left\\| Z \\right\\| _ { F } ^ { 2 } \\leq \\left\\| A \\left( Z \\right) \\right\\| _ { \\ell _ { 2 } } ^ { 2 } \\leq \\left( 1 + \\delta \\right) \\left\\| Z \\right\\| _ { F } ^ { 2 } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We note that for a Gaussian measurement operator $A ^ { 1 }$ , RIP of rank $r$ and constant $\\delta > 0$ holds with high probability, if the number of observations satisfies $m \\gtrsim n r / \\delta ^ { 2 }$ [37, 40]. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The second definition concerns the condition number of the factor $X$ . ",
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+    {
+        "type": "text",
+        "text": "Definition 3.2 (condition number). We denote the condition number of $X \\in \\mathbb { R } ^ { n \\times r _ { \\star } }$ by $\\kappa : = { \\frac { \\| X \\| } { \\sigma _ { r , \\star } ( X ) } }$ , where $\\sigma _ { r _ { \\star } }$ $( X )$ denotes $r _ { \\star }$ -th largest singular value of $X$ . ",
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+        "type": "text",
+        "text": "With these definitions in place we are now ready to state our main results. Due to space limitations in the main paper we focus on the case of $r \\geq 2 r _ { \\star }$ . With refer the reader to the supplementary for results covering all $r \\geq r _ { \\star }$ including two special cases: (1) the fully overparameterized case, i.e., $r = n$ along with comparisons with existing work, in this case [38], and (2) the scenario that $U$ has the same number of parameters as $X$ , i.e., $r = r _ { \\star }$ . ",
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+        "type": "text",
+        "text": "Theorem 3.3. Let $X \\in \\mathbb { R } ^ { n \\times r _ { * } }$ and assume we have m measurements of the low rank matrix $X X ^ { T }$ of the form $y = \\mathcal { A } \\left( X X ^ { T } \\right)$ with $\\mathcal { A }$ the measurement operator. We assume $\\mathcal { A }$ satisfies the restricted isometry property for all matrices of rank at most $2 r _ { \\star } + 1$ with constant $\\delta \\le c \\kappa ^ { - 4 } { r _ { \\star } } ^ { - 1 / 2 }$ . To reconstruct $X X ^ { T }$ from the measurements we fit a model of the form $\\bar { U } \\mapsto \\mathcal { A } \\left( \\bar { U } \\bar { U } ^ { T } \\right)$ with $\\bar { U } \\in \\mathbb { R } ^ { n \\times r }$ via running gradient descent iterations of the form $U _ { t + 1 } = U _ { t } - \\mu \\nabla f \\left( U _ { t } \\right)$ on the objective (2) with a step size obeying $\\mu \\leq c \\kappa ^ { - 4 } \\| X \\| ^ { - 2 }$ . Here, the initialization is given by $U _ { 0 } = \\alpha U$ , where $U \\in \\mathbb { R } ^ { n \\times r }$ has i.i.d. entries distributed as $\\mathcal { N } \\left( 0 , 1 / \\sqrt { r } \\right)$ . Furthermore, we assume $r \\geq 2 r _ { \\star }$ and that the scale of initialization fulfills ",
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+        "text": "$$\n\\alpha \\lesssim \\operatorname* { m i n } \\left\\{ \\frac { \\left( \\operatorname* { m i n } \\left\\{ r ; n \\right\\} \\right) ^ { 1 / 4 } } { \\kappa ^ { 1 / 2 } n ^ { 3 / 4 } } \\left( 2 \\kappa ^ { 2 } \\sqrt { \\frac { n } { \\operatorname* { m i n } \\left\\{ r ; n \\right\\} } } \\right) ^ { - 6 \\kappa ^ { 2 } } ; \\frac { 1 } { \\kappa ^ { 7 } n } \\right\\} \\| X \\| .\n$$",
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+        "type": "text",
+        "text": "Then, after ",
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+        "text": "$$\n\\hat { t } \\lesssim \\frac { 1 } { \\mu \\sigma _ { \\mathrm { m i n } } \\left( \\boldsymbol X \\right) ^ { 2 } } \\ln \\left( \\frac { C _ { 1 } n \\kappa } { \\operatorname* { m i n } \\left\\{ \\boldsymbol r ; n \\right\\} } \\cdot \\operatorname* { m a x } \\left\\{ 1 ; \\frac { \\kappa r _ { \\star } } { \\operatorname* { m i n } \\left\\{ \\boldsymbol r ; n \\right\\} - r _ { \\star } } \\right\\} \\cdot \\frac { \\| \\boldsymbol X \\| } { \\alpha } \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "iterations we have that ",
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+    {
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+        "text": "$$\n\\frac { \\| U _ { \\widehat { t } } U _ { \\widehat { t } } ^ { T } - X X ^ { T } \\| _ { F } } { \\| X \\| ^ { 2 } } \\lesssim \\frac { n ^ { 2 } \\kappa ^ { 8 1 / 1 6 } r _ { \\star } ^ { 1 / 8 } } { \\left( \\operatorname* { m i n } \\left\\{ r ; n \\right\\} \\right) ^ { 1 5 / 1 6 } } \\cdot \\frac { \\alpha ^ { 2 1 / 1 6 } } { \\| X \\| ^ { 2 1 / 1 6 } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "holds with probability at least $1 - C e ^ { - \\tilde { c } r }$ . Here, $c , \\tilde { c } , C , C _ { 1 } > 0$ are fixed numerical constants. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that the test error $\\| U _ { \\hat { t } } U _ { \\hat { t } _ { \\cdot } } ^ { T } - X X ^ { T } \\| _ { F } ^ { 2 }$ can be made arbitrarily small by choosing the scale of initialization $\\alpha$ small enough. In particular, the dependence of the test error on $\\alpha$ is polynomial and the dependence of the number of iterations on $\\alpha$ is logarithmic, which means that reducing the test error by scaling down $\\alpha$ introduces only modest additional computational cost. Hence, as long as the rank at most $2 r _ { \\star } + 1$ RIP with constant $\\delta \\le c \\kappa ^ { - 4 } { r _ { \\star } } ^ { - 1 / 2 }$ holds, gradient descent converges to a point in the proximity of the low-rank solution, whenever the initialization is chosen small enough regardless of the choice of $r$ . This holds even when the model is overparameterized i.e. $r n \\gg m$ and the optimization problem has many global optima many of which do not obey $U U ^ { T } \\approx X X ^ { T }$ . This result thus further demonstrates that when initialized with a small random initialization gradient descent has an implicit bias towards solutions of low-rank or small nuclear norm. This is in sharp contrast to Neural Tangent Kernel (NTK)-based theory for low-rank matrix recovery (see [23, Section 4.2]) which will not approximately recover the ground truth matrix $X X ^ { T }$ due to the larger scale of initialization required when using that technique. ",
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+        "type": "text",
+        "text": "As discussed in Section 2, the restricted isometry property holds with high probability for a sample complexity $m \\gtrsim n r _ { \\star } ^ { 2 } \\kappa ^ { 8 }$ for Gaussian measurement matrices. Up to constants, this sample complexity is optimal in $n$ , while it is sub-optimal in $r _ { \\star }$ and $\\kappa$ compared to approaches based on nuclear-norm minimization (see, e.g., [37]). While there is numerical evidence that the true scaling of $m$ in $r _ { \\star }$ should also be linear in the non-convex case [41], we note that the optimal dependence of the sample complexity on $r _ { \\star }$ is a major open problem in the field, as the sample complexities in all theoretical results for non-convex approaches in the literature scale at least quadratically in $r _ { \\star }$ . ",
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+        "type": "text",
+        "text": "Interpretation: Recall from Section 1 that our convergence analysis can be divided into three phases: the spectral phase, the saddle avoidance phase, and the local refinement phase. As it will become clear from the proofs in the supplementary when $r \\geq 2 r _ { \\star }$ the bound on the number of iterations can ",
+        "bbox": [
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+        "type": "text",
+        "text": "be decomposed as follows ",
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+    {
+        "type": "equation",
+        "img_path": "images/b57371d4b5693f30180964b2443fc9d0f31df5de1252696993d7f6734b3cf624.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\hat { t } \\lesssim \\frac { 1 } { \\mu \\sigma _ { \\operatorname* { m i n } } ( X ) ^ { 2 } } \\Bigg [ \\ln ( 2 \\kappa ^ { 2 } \\sqrt { \\frac { n } { \\operatorname* { m i n } \\{ r ; n \\} } } ) + } &  \\ : \\underbrace { \\ln ( \\frac { \\sigma _ { \\operatorname* { m i n } } ( X ) } { \\alpha } ) } _ { \\displaystyle \\mu \\sigma _ { \\operatorname* { m i n } } ( X ) ; \\frac { 1 } { \\operatorname* { m i n } \\{ r ; n \\} - r _ { \\star } } \\} \\frac { \\| X \\| } { \\alpha } \\Bigg ) \\Bigg ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Phase III: local refinement phase ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "First, we note that the duration of all three phases scales inversely with $\\sigma _ { \\mathrm { m i n } } \\left( X \\right) ^ { 2 }$ . This is due to the fact that in all three phases the dynamics associated the smallest singular value of $X$ is the slowest one and hence needs the most time to complete. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the spectral phase, the eigenvectors corresponding to the leading $r _ { \\star }$ eigenvalues of $U _ { t } U _ { t } ^ { T }$ become aligned with the eigenvectors corresponding to the leading $r _ { \\star }$ eigenvalues of $\\mathcal { A } ^ { \\ast } \\mathcal { A } \\left( X X ^ { T } \\right)$ . We observe in (6) that in the spectral phase increasing $r$ , i.e. the amount of parameters, decreases the number of iterations in this phase. As we will explain in the supplementary, the reason is that increasing $r$ decreases the angle between the column space of the initialization $U _ { 0 }$ and the span of the eigenvectors corresponding to the leading $r _ { \\star }$ eigenvalues of $\\mathcal { A } ^ { \\ast } \\mathcal { A } \\left( X X ^ { T } \\right)$ used in spectral initialization. As a consequence, gradient descent needs fewer iterations to align these two subspaces. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In the saddle avoidance phase (Phase II), $\\sigma _ { r _ { \\star } } \\left( U _ { t } \\right)$ , the $r _ { \\star }$ th largest singular value of $U _ { t }$ , grows geometrically until it is on the order of $\\sigma _ { \\mathrm { m i n } } \\left( X \\right)$ . Hence, this duration depends on the ratio between the $\\sigma _ { \\mathrm { m i n } } \\left( X \\right)$ and the the scale of initialization $\\alpha$ . This is clearly reflected in the upper bound on the number of needed iterations in equation (6). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Phase III, the local refinement phase, the matrix $U _ { t } U _ { t } ^ { T }$ converges towards $X X ^ { T }$ . In particular, at iteration $\\hat { t }$ the test error obeys (5). We observe that a smaller $\\alpha$ allows for a smaller test error in (5) but per (6) this higher accuracy is achieved with a modest increase in the required iterations. ",
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+    {
+        "type": "text",
+        "text": "4 A glimpse of our analysis ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our proofs, we show that the trajectory of the gradient descent iterations can be approximately decomposed into three phases: (I) a spectral or alignment phase where we show that gradient descent from random initialization behaves akin to spectral initialization allowing us to show that at the end of this phase the column spaces of the iterates $U _ { t }$ and the ground truth matrix $X$ are sufficiently aligned, (II) a saddle avoidance phase, where we show that the trajectory of the gradient iterates move away from certain degenerate saddle points , and (III) a refinement phase, where the product of the gradient descent iterates $U _ { t } U _ { t } ^ { T }$ converges quickly to the underlying low-rank matrix $X X ^ { T }$ . The latter result holds up to a small error that is commensurate with the scale of the initialization and tends to zero as the scale of the initialization goes to zero. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To formalize the above, we use $L$ and $L _ { t }$ to denote the subspaces spanned by the eigenvectors corresponding to the $r _ { \\star }$ largest eigenvalues of the matrix $\\mathcal { A } ^ { \\ast } \\bar { \\mathcal { A } } \\left( X X ^ { \\bar { T } } \\right)$ , and $U _ { t } U _ { t } ^ { T }$ , respectively. Moreover, for a subspace $L$ of dimension $r _ { \\star }$ we use $V _ { L } \\in \\mathbb { R } ^ { n \\times r _ { \\star } }$ to denote an orthonormal matrix whose columns span the subspace $L$ . Note that $L$ is the subspace, which is obtained by commonly used spectral methods. Using this notation, Figure 2a depicts the three phases described above. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the spectral phase we will prove that we can approximate the iterate $U _ { t }$ by ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/a5ffbedecb3f3b13b318a41b6e6d0ddb14fd617b694e76cee586e407e1d80034.jpg",
+        "text": "$$\nU _ { t } \\approx \\underbrace { \\big ( \\mathbf { I d } + \\mu \\mathbf { \\mathcal { A } } ^ { * } \\mathbf { \\mathcal { A } } \\left( X X ^ { T } \\right) \\big ) } _ { = : Z _ { t } } ^ { t } U _ { 0 } = Z _ { 1 } ^ { t } U _ { 0 } : = \\tilde { U } _ { t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We note that the matrix $Z _ { 1 } \\ = \\ \\mathrm { I d } + \\mu \\mathcal { A } ^ { * } \\mathcal { A } \\left( X X ^ { T } \\right)$ is the basis for the commonly used spectral initialization, where typically a factorization of the rank $r _ { * }$ approximation of this matrix is used as the initialization [6, 5, 42]. Therefore, the approximation (7) suggests that gradient descent iterates modulo the normalization are akin to running power method on $Z _ { 1 }$ . Hence, we expect that at the end of the spectral phase the subspace $L _ { t }$ to be closely aligned with the subspace $L$ , i.e. the subspace obtained by commonly used spectral initialization techniques. In particular, this also implies that $L _ { t }$ is also aligned with the subspace $X$ . Figure 2b clearly illustrates that the first few iterations of gradient descent behave essentially identical to the power method, confirming our intuition. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The description of the second and third phase is more elaborate and technical in nature and we defer to the supplementary for a more detailed and intuitive explanation. However, to give a brief description, in these two phases we will decompose the iterates $U _ { t }$ into the sum of two matrices, a \"signal\" matrix of rank $r _ { \\star }$ and a \"noise\" matrix of rank at most $r - r _ { \\star }$ . In Phase (II) we will prove that the smallest singular value of the signal term, which is approximately the same as $\\sigma _ { r _ { \\star } }$ $\\left( U _ { t } \\right)$ grows, whereas the spectral norm of the noise matrix grows at a much slower rate. We will also show that in this phase the columns of the signal term stay approximately aligned with the span of the matrix $X$ . As soon as the smallest singular value of the signal term of $U _ { t }$ is approximately at the same order than the smallest singular value of $X$ we enter Phase (III). In that Phase we provide a local convergence argument, which shows that the signal term of $U _ { t }$ converges towards $X$ (up to a rotation), whereas the noise term stays small. ",
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+        "image_caption": [
+            "Figure 2: (a) Depiction of the three phases of convergence. This figure demonstrates that the convergence analysis can be divided into three phases: (I) spectral/alignment phase; (II) saddle avoidance phase and (III) the refinement phase. We see that in the first phase the first $r _ { \\star }$ eigenvectors of $U _ { t } U _ { t } ^ { T }$ rapidly learn the subspace corresponding to the first $r _ { \\star }$ eigenvectors of $\\mathcal { A } ^ { \\ast } \\mathcal { A } \\left( X X ^ { T } \\right)$ , i.e. the angle $\\| V _ { L ^ { \\bot } } ^ { T } V _ { L _ { t } } \\|$ becomes small. The $r _ { \\star }$ th largest singular value of $U _ { t }$ is still small in this phase and the (normalized) test error $\\| U _ { t } U _ { t } - X X ^ { T } \\| _ { F } ^ { 2 } / \\| X X ^ { T } \\| _ { F } ^ { 2 }$ has not decreased yet. In Phase (II), however, we see that $\\sigma _ { r _ { \\star } }$ $\\left( U _ { t } \\right)$ is growing, whereas the loss begins to decrease in this phase and the subspaces stay aligned. In Phase (III) we see that the test error is converging towards 0 rapidly, meaning that $U _ { t } U _ { t } ^ { T }$ converges to $X X ^ { T }$ . Consequently, $\\sigma _ { r _ { \\star } } \\left( U _ { t } \\right) / \\sigma _ { r _ { \\star } } \\left( X \\right)$ converges to 1 (red curve). We also see that in this phase the angle $\\| V _ { L ^ { \\bot } } ^ { T } V _ { L _ { t } } \\|$ grows again, until it reaches a certain threshold. This is because in this phase the top $r _ { \\star }$ eigenvalues of $U _ { t } U _ { t } ^ { T }$ become aligned with the eigenvectors of $X X ^ { T }$ . (b) Depiction of the spectral alignment phase: in the first few iterations, gradient descent with small initialization behaves like a power method. Denote by ${ \\tilde { L } } _ { t }$ the subspace spanned by the eigenvectors corresponding to the $r _ { \\star }$ largest eigenvalues of the matrix $\\widetilde { U _ { t } } \\widetilde { U _ { t } } ^ { T }$ . Analogously as before denote by $V _ { \\tilde { L } _ { t } }$ an orthonormal matrix, whose columns span the subspace ${ \\tilde { L } } _ { t }$ . In this figure, we observe that in the first iterations $U _ { t }$ and $\\widetilde { U } _ { t }$ learn the subspace $L$ at almost exactly the same rate. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 Numerical experiments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we perform several numerical experiments to corroborate our theoretical results. ",
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+    {
+        "type": "text",
+        "text": "Experimental setup. For the experiments we set the ground truth matrix $X \\in \\mathbb { R } ^ { n \\times r _ { \\star } }$ to be a random orthogonal matrix with $n = 2 0 0$ and $r _ { \\star } = 5$ . Moreover, we use $m = 1 0 n { r _ { \\star } } = 5 0 n$ random Gaussian measurements. The initialization $U$ is chosen as in Theorem 3.3 and we use a step size of $\\mu = 1 / 4$ which is consistent with these theorems. We note that while all experimental depictions are based on a single trial, in line with the NeurIPS guidelines we have drawn these curves multiple times (not depicted) and the behavior of the plots do not change. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Depiction of the three phases and the role of overparameterization. In our first experiment, we want to examine how increasing the number of parameters via increasing the number of the columns $r$ of the matrix $U _ { t } \\in \\mathbb { R } ^ { n \\times r }$ , affects the spectral phase. To this aim we set the scale of initialization to $\\alpha = 1 / \\left( 7 0 n ^ { 2 } \\right)$ . Let $L$ denote the subspace spanned by the eigenvectors corresponding to the leading $r _ { \\star }$ singular values of $\\mathcal { A } ^ { \\ast } \\mathcal { A } ( X X ^ { T } )$ and $L _ { t }$ denotes the subspace spanned by the left-singular vectors corresponding to the largest $r _ { \\star }$ singular values of $U _ { t }$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/665e23a4f08b841b53ae6d28270bed70053f954399ac8df02b5f30441b348f05.jpg",
+        "image_caption": [
+            "Figure 3: Impact of different levels of overparameterization on (a) the angle $\\| V _ { L ^ { \\bot } } ^ { T } V _ { L _ { t } } \\|$ and (b) the $r _ { \\star }$ th largest singular value, (c) the trajectory of the (normalized) test error $\\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \\| _ { F } / \\| X X ^ { T } \\| _ { F }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Spectral phase and alignment under different levels of overparameterization. First, we examine how the angle between these two subspaces (i.e. $\\| V _ { L ^ { \\perp } } ^ { T } V _ { L _ { t } } \\| ,$ ) changes in the first few iterations. We depict the results for different $r$ in Figure 3a. We see that in the first few iterations, i.e. in the spectral phase, this angle converges towards zero. This confirms the main conclusion of this paper that the first few iterations of gradient descent from small random initialization indeed behaves akin to running power method for spectral initialization. This experiment also shows that changing the number of columns $r$ of $U _ { t }$ has an interesting effect on the spectral phase. In particular, increasing $r$ allows the gradient descent algorithm to learn the subspace $L$ with fewer iterations, i.e. $\\| V _ { L ^ { \\bot } } ^ { T } V _ { L _ { t } } \\|$ becomes small with fewer iterations. This is in accordance with our theory for $r _ { \\star } \\le r \\le n$ (see, for example, the first summand on the right-hand side of equation (6)), where we show that more overparameterization allows gradient descent to leave the spectral phase earlier. Interestingly, this improvement continues to hold even when increasing $r$ beyond $n$ allowing for even faster convergence of $\\| V _ { L ^ { \\bot } } ^ { T } V _ { L _ { t } } \\|$ . This holds even though in this case the rank of $U _ { 0 }$ is still not larger than $n$ . One potential explanation for this phenomenon might be that for such a choice of $r$ the matrix $U _ { 0 }$ is better conditioned. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Growth of $\\sigma _ { r _ { \\star } } \\left( U _ { t } \\right)$ and saddle avoidance. In Figure 3b we depict how $\\sigma _ { r _ { \\star } } \\left( U _ { t } \\right)$ grows during the training for different choices of $r$ . We see that the curves look similar, although for smaller $r$ the growth phase sets in at a slightly later time. This is due to the fact that for smaller $r$ , as we have seen in Figure 3a, Phase I, the spectral phase takes longer to complete. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Evolution of the test error and the refinement phase. Similarly, in Figure 3c we depict how the (normalized) test error $\\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \\| _ { F } / \\| X X ^ { T } \\| _ { F }$ evolves during the training for different choices of $r$ . We observe that for smaller $r$ the third phase sets in slightly later. Again, this is due to the fact that for smaller $r$ the spectral phase takes slightly longer to complete (see inequality (6)). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Test error under different scales of initialization. In the next experiment, we focus on understanding how the scale of initialization $\\alpha$ affects the generalization error $\\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \\| _ { F } ^ { 2 }$ ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/f37264bb67eb12df04f997c4ce59e16a703a16cebf79bcfc5db995012edcdb80.jpg",
+        "image_caption": [
+            "Figure 4: Relative test erro r ∥UtUTt −XXT ∥FT for different scales of initialization α . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "image",
+        "img_path": "images/064c23755d50858b141e0a12e3b8fb4f590d169b4eaadbef14b71d65606dd375.jpg",
+        "image_caption": [
+            "Figure 5: Change of test error $\\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \\| _ { F } ^ { 2 }$ and train error $f \\left( U _ { t } \\right)$ for (a) small and (b) large $\\alpha$ during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For that, we set $r = 1 8 0$ and run gradient descent with for different choices of $\\alpha$ . We stop as soon as the training error becomes small $\\dot { ( } f \\left( U _ { t } \\right) \\le 0 . 5 \\cdot 1 0 ^ { - 9 } )$ . We depict the results in Figure 4. We see that the test error decreases as $\\alpha$ decreases. In particular, this figure indicates that the test error depends polynomially on the scale of initialization $\\alpha$ . This is in line with our theory, where we also show that the test error decreases at least with the rate $\\alpha ^ { 2 1 / 1 6 }$ (see inequality (5) in Theorem 3.3). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Change of test and train error during training. In the next experiment, we set $r = 1 8 0$ and examine how the test error $\\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \\| _ { F } ^ { 2 }$ and the train error $f \\left( U _ { t } \\right)$ changes throughout training and, in particular, how this depends on the scale of initialization. To this aim, we run gradient descent with $\\mathrm { \\dot { 4 } \\cdot 1 0 ^ { 5 } }$ iterations. We see that for a small scale of initialization, $\\alpha = 1 0 ^ { - 3 }$ , which is the scenario studied in this paper, both test error and train error decrease throughout training. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We observe that in the beginning, as described our theory, both test and train error decrease rapidly. After that the decrease of both test and train error slows down significantly. Moreover, the train error converges towards zero, in contrast to the test error. One reason for the slow convergence in this phase might be that $U _ { t }$ is ill-conditioned in the sense that $\\sigma _ { r _ { \\star } }$ $( U _ { t } W _ { t } )$ is much larger than $\\Vert U _ { t } W _ { t , \\bot } \\Vert$ ⋆It is an interesting future research direction to extend our theory to this part of the training. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For large scale of initialization $\\alpha = 0 . 5$ , we observe a very different behaviour. We see that the train error converges with linear rate until machine precision is reached. However, the test error barely changes throughout the training. This scale of initialization corresponds to the lazy training regime [34], where the parameters stay close to the initialization during the training. We depict the results in Figure 5. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Number of iterations until convergence: In the last experiment, we set $\\alpha = 1 0 ^ { - 3 }$ and examine how many iterations are needed until the test error $\\| U _ { t } U _ { t } ^ { T } - X X ^ { T } \\| _ { F } ^ { 2 }$ falls below a certain threshold of $1 0 ^ { - 4 }$ for different values of $r$ obeying $5 \\leq r \\leq 3 0$ . For each choice of $r$ we run the experiment ten times and then average the number of iterations for each choice of $r$ . The results are depicted in Figure 6. We observe that increasing the number of columns $r$ from 5 to 10, i.e., a small amount of overparameterization, decreases the number of iterations needed. After that the number of iterations needed stays roughly constant. This observation is in line with Figure 3, where we have seen that overparameterization leads to fast decrease of the test error in the spectral phase (with diminishing speedup as $r$ becomes larger and larger) without affecting the other two phases. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "image",
+        "img_path": "images/708720b79c80fa2eb96e1f8e6541f2b603ed1c1fcec3c248b501d23e83e1bffe.jpg",
+        "image_caption": [
+            "Figure 6: Number of iterations required for the test error to fall below $1 0 ^ { - 4 }$ for different levels of overparameterization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6 Conclusion and Broader Impact ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "In this paper we focused on demystifying the role of initialization when training overparameterized models by showing that small random initialization followed by a few iterations of gradient descent behaves akin to popular spectral methods. We also show that this implicit spectral bias from small random initialization, which is provably more prominent for overparameterized models, also puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well. ",
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+        "text": "We think that our results give rise to a number of interesting future research directions. For example, one could extend our results to scenarios where the measurement matrices are more structured such as in matrix completion [43] or in blind deconvolution [44]. Moreover, while our main results, e.g. Theorem 3.3 do require early stopping, our simulations (e.g. Figure 5a) indicate that early stopping is not needed. It would be interesting to examine whether we can remove the early stopping requirement. It is also an interesting future avenue to examine whether the quadratic dependence of the sample complexity $m$ on $r _ { \\star }$ in our results is really needed. ",
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+        "text": "Moreover, while in this paper our main focus was on low-rank matrix reconstruction, we believe that our analysis holds more generally for a variety of contemporary overparameterized machine learning and signal estimation tasks including neural network training. This is a tantalizing future research direction. ",
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+        "text": "Despite being theoretical/foundational in nature our results have potential for broader practical impact. In particular, low rank reconstruction problems are an important component of many recommender engines and our insights may guide better algorithm and systems designs for such engines. More broadly, training overparameterized models using stochastic GD starting from small random initialization is the work-horse of modern learning ncluding deep learning and our insights may in the long term help enable more efficient/reliable training with a smaller carbon footprint and improved test accuracy. As with other technologies such insights may potentially also be used nefariously. ",
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+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "text": "M.S. is supported by the Packard Fellowship in Science and Engineering, a Sloan Research Fellowship in Mathematics, an NSF-CAREER under award #1846369, the Air Force Office of Scientific Research Young Investigator Program (AFOSR-YIP) under award #FA9550-18-1-0078, DARPA Learning with Less Labels (LwLL) and FastNICS programs, and NSF-CIF awards #1813877 and #2008443. ",
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